Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

, we obtain

Vf E .9'(X).

Now 9'(X) is dense in .9''(X). Therefore S = c e C. This proves the lemma.

Some other subgroups of G(l) will be needed in proving that the map G(l) - Sp(l) is an epimorphism. They are

i. the finite group generated by the Fourier transforms in one of the coordinates (some base of the vector space X having been fixed); ii. the group consisting of automorphisms of .9''(X) of the form

f -. e°Qf, where Q is a real quadratic form mapping X -. R; iii. the group consisting of automorphisms of .9''(X) of the form

f' -* f, where f (x) =

det T f'(Tx), T an automorphism of X.

Each of these groups has a restriction to .9'(X) that gives a group of automorphisms of .9'(X) and a restriction to .*'(X) that gives a group of unitary (that is, isometric and invertible) transformations of .i*'(X). The following definition uses these properties. Definition 1.2.

Let A be the collection of elements A each consisting of

1°) a quadratic form X Q+ X

R, whose value at (x, x') e X Q+ X is

A (x, x') = Z - + Z
(1.9)

where, if `P denotes the transpose of P,

L:X,-. X*,

P = `P:X -. X*,

Q = `Q:X -+ X*,

det L A 0; 2°) a choice of arg det L = nm(A), m(A) e Z, which allows us to define A(A) =

det L

by

arg A(A) = (rz/2)m(A).

7

1,§ 1,1

det L is calculated using coordinates in X* dual to the coordinates in X and is independent of coordinates chosen such that dx' n . A dx' = d'x. Remark.

Remark.

m(A) will be identified with the Maslov index by 2,(2.15) and

§2,8,(8.6).

To each A we associate SA, an endomorphism of Y(X) defined by A(A) I (SAf)(x) [IV,, rzi]`I2

where f' E ,9(X),

arg[i]t'2 = nl/4.

(1.10)

Clearly SA is a product of elements belonging to the groups (i), (ii), and (iii). Therefore SA is an automorphism of Y (X) that extends by continuity to a unitary automorphism of A (X) and to an automorphism of 9"(X). These three automorphisms will be denoted SA; SA e G(l). The image sA of SA in Sp(l) is characterized as follows (where Ax is the gradient of A with respect to x): (x, p) = sA(x', p') is equivalent to p = Ax(x, x'),

p' = -Ax.(x, x').

Let f' e Y (X). a(SAf')/(3x and SA(df'/dx) are calculated by differentiation of (1.10) and integration by parts; the result of these calculations gives the following relations among differential operators of Proof of (1.11).

V 8x -

Px = - SA('Lx)SA',

SA (v Ox

+ Qx} SA' = Lx;

writing (X, P) = SA(x', P'),

these relations mean

P - Px = -'Lx',

p' + Qx' = Lx

bx' E X,

p' e X

This is proposition (1.11). Definition 1.3.

We shall write Esp for the set of s e Sp(l) such that x and

x' are not independent on the 21-dimensional plane in Z(l) Q Z(l) de-

termined by the equation (x, P) = s(x', p'). Let us recall the well-known theorem that the set of sA characterized by (1.11) is Sp(l)\ESp.

Proof. Clearly sA 0 Esp. Conversely, let s e Sp(l). On the 21-dimensional plane in Z(l) Q Z(1) determined by the equation

(x, P) = s(x', p') we have, since s is symplectic,

- = - . Therefore

? d [ - ] _ - .

We assume s 0 Esp. Then x and x' are independent on the above 21dimensional plane. On this plane we define

A(x, x') = i - 1 .

(1.12)

We therefore have

dA = - , that is,

p = -As..

p = As,

x and Ax have to be independent. Hence det;k(AX X ') i4 0. Therefore s = 5A, which completes the proof. The sA clearly generate Sp(1). Thus: LEMMA 1.3.

The natural morphism G(l) - Sp(l) is an epimorphism.

By lemma 1.2, G(1) is a Lie group and

G(1)/t = Sp(l ).

(1.13)

[C is the center of G(l) because the center of Sp(l) is just the identity element.] Definition 1.4.

The metaplectic group Mp(l) is the subgroup of G(l)

9

I,§1,1-I,§1,2

consisting of those elements whose restriction to ff (X) is a unitary automorphism of ,*°(X).

We have SA e Mp(l) VA. Now the SA generate Sp(l), so the natural morphism

Mp(l) - Sp(1) is an epimorphism. By (1.13), all elements of G(l) can be written uniquely in the form cS,

where S e Mp(l),

c > 0.

Writing R+ for the multiplicative group of real numbers > 0, we obtain

G(l) = R+ x Mp(l).

(1.14)

The study of G(l) therefore reduces to that of Mp(l), which has the following properties:

Mp(l) is a group of automorphisms of S°'(X) whose restrictions to.W'(X) are unitary automorphisms. 1°) Let S' be the multiplicative group of complex numbers of modulus 1. THEOREM 1.

Then

Mp(1)/S' = Sp(l).

(1.15)

2°) Let EMp be the hypersurface of Mp(l) that projects onto Esp. Every element of Mp(l)\EMp can be written as cSA, where c e S' and SA is given by an expression of the form (1.10). 3°) The restriction of every S e Mp(l) to So(X) is an automorphism of

,°(X). Proof of 1°): (1.13) and (1.14); S' is identified with a subgroup of Mp(l). .

Proof of 2°). Let S e Mp(l)\EMp . Then the image of S in Sp(l) is some element sA, A e A; SSA' e S' by (1.15).

Proof of 3°). By 2°), S = cSA, . . SAC . Now the restrictions of c, SA, , ... , SA, to °(X) are automorphisms of , °(X ). .

2. The Subgroup Sp2(l) Of MP(l) Definition 2.1. We denote by Spz(l) the subgroup of Mp(l) that is generated by the SA.

I,§1,2

10

The purpose of this section is to prove that Sp2(l) is a covering group of Sp(l) of order 2.

In order to prove this, we calculate inverses and compositions of the elements SA .

Definition 2.2. - Given A E A, we define A* e A as follows:

A*(x, x') = -A(x', x),

1(A*) = i'A(A),

m(A*) = l - m(A). SA 1 = SA*; thus sA1 = SA+.

LEMMA 2.1.

This amounts to proving the equivalence of the following two conditions for any f and f' EY(X): Proof.

f(x) - _

(±)"2A(A)

evA(x.x') f'(x')d`x',

x

rz

i

(Iv2rz

f (x) =

) 0(A)

x

Using the expression for A given by (1.9), this is the same as the equivalence of the following two conditions:

f(x) = J Xx

f'(x') =

(jj)'detLI jev<1x'>f(x)dIx.

The equivalence is deduced from the Fourier inversion formula; the lemma follows.

To compute compositions of the SA, we will find an explicit expression for SA(e°`° ), where (p' is a second-degree polynomial. This is made possible by the following definition.

Choose linear coordinates in X such that d'x = dx' A ... A dx' and choose the dual coordinates in X*. The following Definition 2.3.

notions are independent of this choice. Let cp be a real function, twice differentiable: cp : X -+ R.

1,§1,2

11

Hess,,((p) denotes the hessian of cp, the determinant of its second derivatives.

Alternatively this is the determinant of the quadratic form X-3

Inertz((p) denotes the index of inertia of this form. It is defined") when 0. Clearly

Hess((p)

Inert(-q') = I - Inert(q), arg Hess((p) = n Inert(g) mod 2n. This formula makes possible the definition arg Hess((p) = n Inert(q).

(2.1)

Thus, for example, [Hess (9)]112 =

(2.2)

IHess((p)I112itnert(N).

If op is a real quadratic form,

rp: X a x i-- I, where R = 1R: X - X*, then Hess((p) and Inert((p) will be denoted Hess(R) and Inert(R). Hess(R) is

the determinant of the symmetric matrix R. Inert(R) is the number of negative eigenvalues of R. Clearly

Inert(R) = Inert(R-1),

[Hess(R)]1J2[Hess(-R-1)]112 = it.

(2.3)

Let 9' be a real second-degree polynomial. Let A e A be such that Hessx.(cp'(x') + A(x, x')) A 0. Denote by 9(x) the critical value of the polynomial LEMMA 2.2.

Xax't--* A (x,x') + V (x'); rp is a second-degree polynomial. We have SA(e'11)

= A(A)[Hessx.(cp' +

Remark 2.1.

A)]-1l2e"°.

(2.4)

This lemma assumes v/i > 0. Up to this point, it was

sufficient to assume v/i real and nonzero.

Proof. We know that 'It is the number of negative eigenvalues of the linear symmetric operator dx F--. dcp_

12

I,§1,2

exp [ - 2] dx =

2n.

J

L

Therefore if c e C and I arg p I< n1/2, x JexP[-1(X

Iarg f <

+ c)2l dx = VP

JJ

.

4

We then have, for any p e C,

fex[_vx - 2(x + c)zl dx =

ev(O

exp - 2u [vp + µ(x + c)] )} dx =

J

e (P

'C

where (p is the critical value of the function

x i- p'(x) - px,

where gyp'

p (x + c)2.

2v

The Fourier transform F is the automorphism of 9 defined by 1/2

(Ff')(p) =

2nli

Je<>f(x)dlxdf' E.5(X)

(2.5)

We then have, for I = 1, I arg p I < n1/2, Fe°`° =

vIvI

evv; VF,

= e"'4

11P 1/71-

Since F is a continuous automorphism of 5''(X), the preceding formula remains valid for p = -Ev, r e $; then

f 1'/E

if r > 0 iJIEI if r < 0.

In other words, when I = 1, the following result holds: Let p': X -' R be a real second-degree polynomial such that Hess p' 0; let p(p) be the critical value of the polynomial

xicp'(x) - CP,x>, we have

13

1,§1,2

Fe°`° =

[Hess p']-1ne"*.

(2.6)

Let us show that, since relation (2.6) holds for I = 1, it holds for all I >, 1. It suffices to choose the coordinates x' in X such that

(P'(x) = i (pi(x'). j=1

Now using the definitions (1.9) of A, (1.10) of SA, and (2.5) of F, we have

in the case P = Q = 0, (Snf') (x) = A(A) (Ff ') (Lx).

Then (2.6) establishes (2.4) in this case. From the definitions of A and SA, the general case is clearly equivalent to this one.

Before taking compositions of the SA,we consider compositions of the sA: LEMMA 2.3.

1°) Let A and A' E A. The condition (2.7)

$ASA' 0 E5p

is equivalent to the condition

Hess,, [A(x, x') + A'(x', x")] : 0 (the Hessian is constant).

(2.8)

2°) This condition is equivalent by lemma 2.1 to the existence of A" E A such that SASA.sA.. = e [identity element of Sp(l)].

(2.9)

A" is defined by the condition that the critical value of the polynomial x' + A(x, x') + A'(x', x") + A"(x", x) be zero. 3°) Just as (1.9) defines A by P, Q, L, let A' and A" be defined by P, Q', L' and P", Q", L". The condition (2.8) for the existence of A" is expressed as

1' + Q is invertible. A" can be defined by the formulas P" + Q' = L'(P' + Q)-"L',

P + Q = 1L(P' + Q)-1L, (2.10)

14

I,§1,2

Remark 2.2. we have

Writing A + A' + A" for A(x, x') + A'(x', x") + A"(x", x),

Inertx(A + A' + A") = A' + A") = Inert,-.(A + A' + A"). Hess x .(A + A' + A") =

Proof of I°).

(2.11)

A2(A)A2(A')

(2.12)

A2(A'*)

By (1.11), the relations

(x', p') =

(x, P) = SA(x', P'),

p")

may be written

p = Ax(x, x'), p"

p' = -As (x, x') = A'x,(x', x"),

-A'x.,(x' x").

It results from the elimination of p' and x' in these relations that (x, P) = SASA'(x", p")

The condition (2.7) that SASA' 0 Esp is then equivalent to each of the following conditions:

The elimination of p' and x' in the preceding step leaves x and x" independent. The relation

A, (x, x') + A'x-(x', x") = 0 leaves x and x" independent. For any x and x", there exists an x' satisfying this relation. Now in (1.9), det L A 0. Therefore (2.7) is equivalent to (2.8). Proof of 2°). Assumption (2.9) means that any two of the following three relations implies the third: (x, P) = SA(x', P'),

(x', P) = SA,(X", p"),

(x", P") =

P)

Then by (1.11), each of the next three relations implies the other two:

(A+A'+A")x=0,

(A+A'+A")x.,=0, (2.13)

is

I,§1,2

where

A + A' + A" = A(x, x') + A'(x', x") + A"(x", x). Now by Euler's formula, these three relations imply

A+A'+A"=0. Therefore

(A + A' + A")s, = 0, that is, (A + A')x. = 0, implies A + A' + A", = 0. Proof of 30).

We have

Hessx.(A + A' + A") = Hess(P' + Q), which gives the first statement. For the other, the three pairwise equivalent relations (2.13) can be written

(P+Q")x-`Lx' -L'x"=0, -Lx + (P' + Q)x' -`L'x" = 0,

`L"x-L'x'+(P"+Q')x"=0. (210) clearly expresses the equivalence of these three relations. Proof of Remark 2.2.

P" + Q',

By (2.10), the symmctric matrices

(P' + Q)-1,

P + Q"

can be transformed one into the other. They therefore have the same inertia. This is (2.11). By (2.10)3,

Hess(P' + Q) = (det L) (det L')/(-1)' det L". By definition 2.2, this is (2.12). Definition 2.4. 3A,

Given SA SA' SA" = e,

we define Inert(sA, 5A,, sA.-) = Inertx(A + A' + A")

We define

[see (2.11)].

(2.14)

I,§1,2

16

Inert(SA, SA,, SA,.) = Inert(sA, SA,, SA,.).

Moreover, we define the Maslov index of SA, m(SA) E Z4, by

m(SA) = m(A) mod 4.

(2.15)

§2,8 will connect this with the index that V. I. Maslov actually introduced.

Lemma 2.1 and (2.15) have these obvious consequences: Inert(sA,l, sA.', sA 1) = I - Inert(sA, SA., SA..),

m(SA 1) = 1 - m(SA),

(2.16)

m(-SA) = m(SA) + 2 mod 4.

We can at last study compositions of the SA. Consider a triple A, A', A" of elements of A such that

LEMMA 2.4.

(2.17)

SASA, SA,. = e.

Then

SASA'SA.. = ±E [E is the identity element of Mp(1)].

(2.18)

We have (2.19)

if and only if Inert(SA, SA,, SA,.) = m(SA) -

m(SA,.) mod4.

(2.20)

Remark. Condition (2.17), which is equivalent to (2.18), implies (2.20) mod 2.

Proof. Let Y E X. Formula (1.10) holds if f' is replaced by the Dirac measure with support y, given by

6'(x) = 6(x - y). We obtain (SA.(>)(x)

=

(21ri

q2

0(A )eA (s y)

from which follows, by lemmas 2.2 and 2.3,2°),

17

I,§1,2

(;)u2

iA(A)A(A'){Hessx.[A(x, x')

(SASAb'x) =

y)]}-1f2e-,A"(y.x)

+ A'(x',

Multiplying this by f'(y)d'y, where f' c- Y (X), and integrating, we get SASA' f' =

A(A(AO(j)

[Hessx,(A + A' +

which gives, by lemma 2.1 and formula (2.12),

SASA.SA = ±E. Now specify the sign. By definition 2.4,

arg[Hessx.(A + A' + A")] 1j2 = Z Inert(SA, SA., SA..) mod 2n. By definition 1.2, (2.16), and lemma 2.1, arg A(A) = 2m(SA), argA(A"*)

=

n

1

arg A(A') = 2m(SA,) = 2 [l n 2

[l - m(SA..)] mod 2n.

Therefore

arg(± 1) =

n 2

[Inert(SA, SA., SA..) - m(SA) + m(SA.1) - M(SA")] mod 271,

which proves the lemma. Recall that Sp2 (1) denotes the group generated by the SA. LEMMA 2.5.

Every element of Sp2(1) is a product of two of the SA.

Proof. By lemma 2.1, every element of Sp2(1) is a product of the SA. It then suffices to prove that given U, V, W c A, there exist B and C in A such that SUSvSW = SBSS.

(2.21)

Now, by lemmas 2.3,1°) and 2.4, for every W e A and every T a generic element of A, SWST belongs to {SA} and is generic. Therefore, for T generic,

18

SVST E {SA},

SUSVST E {SA},

ST'SW E {SA},

which gives (2.21) with Sg = SUSVST E {SA},

SC = ST1Sk E {SA}.

The restriction to Sp2(0 of the natural morphism Mp(l) - Sp(1) is clearly a natural morphism: SP2(1) - SP(1)LEMMA 2.6.

The kernel of this morphism is the subgroup

S° = {E, -E}. Therefore Sp2(1)/S° = Sp(1).

Proof. By the preceding lemma, the kernel of this morphism is the collection of the SASA.(A, A' c- A) such that sASA. = e. From this, by lemma 2.1,

Therefore, by (1.11),

A'(x, x') = A*(x, x')

Vx, x' c- X.

Consequently by definition 1.2.

A(A') = ±1 (A*), and

SA, _ ±SA.; therefore SASH- = ±E. LEMMA 2.7.

The group Sp2(1) is connected.

Proof. Given k e Z4 (additive group of integers mod 4), let Dk be the collection of SA such that

m(A) = k, or equivalently, i-kA(A) > 0. The collection of quadratic forms A satisfying A2(A) > 0 [or A2(A) < 0] is connected. Each Dk is thus a connected set in Sp2(1). Given k e Z4, let SA and SA, be such that

1,§1,2

19

-k mod 4;

m(SA) -

p' + Q has one eigenvalue equal to zero and I - 1 eigenvalues > 0. Let B and B' be elements of A near A and A' and such that

Hess,,,(B + B')

0.

Inerts,(B + B') takes the values 0 and 1. Since m is locally constant, m(Sa.l) = m(SA1).

M(SB) = m(SA),

E. By (2.20),

We define B" E A by

k + 1 in any neighborhood of the element

takes the values k and of Sp2(1). This (SASA.)-1

element thus belongs to Dk n Dk+1: pk n Dk+ 1

zA

0,

which gives the lemma.

The above lemmas prove the following theorem. Part 1 of the theorem reduces the study of Mp(l) to that of Spz(I). Its equivalent can be found in the work of D. Shale and A. Weil, but the proof we have given has es-

tablished various other results that will be indispensible to us. One of these is part 3 of the theorem. This will be used in §2,8. THEOREM 2.

1°) The elements SA of Mp(l) that are defined by (1.10) generate

a subgroup Sp2(l) of Mp(l). Sp2(l) is a covering group (see Steenrod [17], 1.6, 14.1) of the group Sp(l) of order 2. It is a group of automorphisms of E/(X) that extend to unitary automorphisms of,Y(X) and to automorphisms Of 99'(X).

2°) The formulas (2.11) and (2.14) define the inertia of every triple s, s', s" of elements of Sp(l)\Esp such that

ss's" = e [identity element of Sp(l)]. The inertia is a locally constant function (discontinuous on Esp) with values in {0, 1,

... ,

1} satisfying

Inert(s, s', s") = Inert(s", s, s') _

=I-

Inert(s"-1

s'-1, s-1).

Let Esp2 be the hypersurface of Sp2(l) that is mapped onto Esp in Sp(l) under the natural projection. The elements SA defined by (1.10) are the elements of Sp2(l)\Esp= . Let S, S', S" be a triple of such elements satisfying

I,§1,2-I,§1,3

20

SS'S" = E [identity element of Sp2(1)]. Let s, s', s" be the images of these elements under the natural projection onto Sp(l). We define

Inert(S, S', S") = Inert(s, s', s").

3°) Formula (2.15) and definition 1.2 define the Maslov index m on Sp2(1)\Y-SD,. It is a locally constant function (discontinuous on 1sP2) with values in Z4. It satisfies

m(S-') = 1 - m(S),

m(-S) = m(S) + 2 mod4,

Inert(S, S', S") = m(S) - m(S'-1) + m(S") mod 4. We shall see later that m is characterized by the last formula and the property of being locally constant. Remark 2.3. Remark 2.4.

Sp2(1) contains the three subgroups of G(l) defined in section

1 by (i) Fourier transformation, (ii) quadratic forms, and (iii) automorphisms of X.

Proof. Let S be an element of one of the three subgroups. It is easy to find A E A such that SSA = S,,., Remark 2.5.

where A' E A.

It can be shown that every S E Sp2(!) is of the form

S = S1S2S3S4,

where S3 E (i), that is, S3 is a Fourier transformation in at most I coordinates; S, and S4 E (ii), that is, they are of the form f' i-+ e"4f', where Q is a real quadratic form; and S2 E (iii), that is, S2 has the form f' i-- det T f' o T, where T is an automorphism of X. 3. Differential Operators with Polynomial Coefficients

By definition 1.1, the elements of Sp2(1) transform differential operators with polynomial coefficients into operators of the same type. Section 3 describes this transformation more explicitly. Let a+ and a- be two polynomials in 1/v, x, and p: a+(v,

x, p) _

a. (v, x)pa,

a (v, p, x) _ >paaa (v, x)

21

I,§1,3

(a a multi-index). We consider the two differential operators (v, x,

a+ a

(v,

vax

ax

) :f

,x :f

l

(v

[aa (v, .)f( )]-

vex

These two operators are identical, that is,

LEMMA 3.1.

a+

vex x )af ( )'

a, (v'

v1- a xJ1 'vax, '

a x1 ) =a ,vax

if and only if there exists a polynomial a° in 1/v, x, and p such that + a (v, x, P) _ [exp 12v

a (v, p, x)

_

aax,

aa

/] a '(v, x, p)-

[ exp 2v- -ax-,apa

1

a

a°(v, x, p).

The notation is the following: ,

=

a

ij

ax ap

a2

s

a ax, ap/ =

exp

Proof.

(xi and pi dual coordinates in X and X*);

ax api

1

°°

k/ a

1

,(

k=°k!ax,

a Y. k op/

Relation (3.3) defines a bijection a- i-- a+ such that, for all

pEX*, a+(v, x, p) = e va+ e

-

x l a )Y V ax

(v

v(P,x>a-

1a I\ (v'

x) ev

vex '

1a1

P + vaxJ aQ (v, x)

- [eXp

\

v

\Ox, ap )] a

since, by Taylor's formula, for every polynomial P: function f : X

P (p

- C,

+ v ax)f (x)

Y P.

P

P(P) (vex) f (x)

X/

(v, p, x),

C and every

I,§1,3

22

_

[exp!(. ap)] [P(P)f (x)]

The bijection a- r-+ a+ can then be defined by the relation a+

_ exp-1

(v, x, p)

[

v

a

a\

ax' ap

)]a-(v, p, x)

This is what the lemma asserts.

Let a be a differential operator that can be expressed as in (3.1) and (3.2). It is defined by the polynomial a° in (1/v, x, p) that satisfies (3.4). We say that a is the differential operator associated to the Definition 3.1.

polynomial a°.

Theorem 3.1 will describe the transform SaS-1 of a by Sc Sp2(1); Lemma 1.1 has already dealt with the case in which a° is linear in (x, p). The proof of this theorem will use the following properties.

If a and b are the operators associated to the polynomials a° and b°, then the operator

LEMMA 3.2.

c=ab is associated to the polynomial co, where

c°(v, x, p) _ }[exp 2v \ay' ap) [a°(v, x, P)b°(v, y, q)]

Proof.

2v

Cax' aqM (3.5)

Y='-

e-n If b°(v, x, p) only depends on p, then the polynomial co associated

to c = ab is

)][a4 (v, x, P)b°(P)] c°(v, x, P) _ [exp - 2v ax' ap {[exp _

{[exp

2v 1

(ax a

p + aq)]

[a+(v, x,

P)b°(q)]} 9-P

a

2v ax ap)]

[a°(v, x,

P)b°(q)}q=r

Similarly, if h°(v, x, p) only depends on x, then the polynomial associated

to c = ab is

23

I,§1,3

c°(v, x, P) =

{[ex-(- , a aY

P

)] [a°(v, x, P)b°(Y)]Lx

.

Thus if b+(x,p) = b'(x)b"(p), then the polynomial associated to c = ab is c°(v, x, P) = [exp - 2v ax

aq)] {Lexp 2v (P' a-)]

[a°(v, x, p)b'(y)] 1)y=x

-

{[exp

2v

ax' aq)

[a°(v, x,

2v

(ay aq)

+ 2v

(ay' P)]

}y_}4=p

This is (3.5) since, by (3.4),

[exp

- 2v (ax' 4)] [b

b°(Y, q)

This implies lemma 3.2, which has the following obvious consequence: LEMMA 3.3.

The operator

c = 2(ab + ba)

is associated to the polynomial

x, p) _ cosh

1

o

a

[2V(8Y'2

1

2v

o

a

ax

aq)]

[a°(v, x, P)b°(v, y, q)]lq=p y=z

If b is linear in (y, q), then

(,)]2[ao(v,x,p)bo(v,y,q)] Oy, ap -

= 0,

from which follows

cosh[

] a°b° = a°b°

therefore we have the following lemma. LEMMA 3.4.

(ab + ba).

If b is linear in (x, p), then the operator associated to a° b° is

I,§1,3

24

This lemma enables us to prove the following theorem. THEOREM 3.1.

The transform SaS-1 of a by S is the differential operator

associated to the polynomial a° o s -I [S E Sp2(l); s is the image of S in Sp(l)].

Proof. Let b be a differential operator associated to a polynomial b° that is linear or affine in (x, p); lemma 1.1 shows that theorem 3.1 holds

for b. To prove the theorem by induction on the degree of a° in (x, p), it suffices to prove that, if the theorem holds for a°, then it holds for a°b°. Since the theorem holds for a° and b°, the operators associated to the polynomials

a°b° and (a°b°) o s-1 = (a° o s-1)(b° o s-1) are, respectively, by lemma 3.4,

4(ab + ba); 2(SaS-1SbS-1 + SbS-1SaS-1) = 4S(ab + ba)S-1. The theorem thus holds for a°b°, which completes the proof.

We supplement this by a theorem about adjoint operators. Definition 3.2.

(f 19) =

Recall that.,Y(X) has a scalar product:

jf(x)d1x

bf, 9 E Af (X),

x

where g(x) is the complex conjugate of g(x). Two differential operators a and b are said to be adjoint if (af I 9) = (f I b9)

df, 9 E -*'(X).

(3.6)

Two differential operators a and b associated to two polynomials a° and b° are adjoint if and only if THEOREM 3.2.

b°(v,x, p) = a°(v,x, p) Proof. b

VvCiR, xcX, pEX*.

It is clear that (3.6) is equivalent to

(v, p, x) = a+(v, x, p),

that is to say, since v is pure imaginary, to

(3.7)

25

I,§1,3-1,§2,0

\l -2v (ax,

P/

)Jb°(v, x, P)

r

1

exp -2v (-a

)]QVx, p),

a ax, a

P

and hence to (3.7).

Theorems 3.1 and 3.2 obviously have the following corollary. COROLLARY 3.1.

If a* is the adjoint of a, then VS E Sp2(l), Sa*S-1 is the

adjoint of SaS -1. Theorem 3.2 clearly has the following corollary, which will be important later. COROLLARY 3.2.

The operator a associated to a polynomial a° is self-

adjoint if and only if the polynomial a° is real valued Vv e iR, x e X, p e X*.

§2. Maslov Indices; Indices of Inertia; Lagrangian Manifolds and Their Orientations 0. Introduction

Historical account. Following V. C. Buslaev [3], [11], §1 has defined a Maslov index mod4 on Sp2(l) by (2.15) and has connected it by (2.20) to an index of inertia that is a function of a pair of elements of Sp(l). On the other hand, V. 1. Arnold [1], [11] defined another Maslov index

on the covering space of the lagrangian grassmannian A(l) of Z(l); this index is connected to the preceding one and to a second index of inertia that is a function of a triple of points of A(l). J. M. Souriau [16] has given a variant of the definition of the Maslov index that is considered in this section.

Chapter I, §3. and chapter II use these two Maslov indices and a third index of inertia, which is a function of an element of Sp(l) and a Summary.

point of A(l). We review and modify the various definitions of these indices (Arnold's,

section 5; Maslov's, section 6; Buslaev's, section 7) so as to clarify their

properties (sections 4-8). In §3 those properties that will be used in chapter II are set forth. First of all we must recall and supplement the topological properties of Sp(l) and A(l) (theorem 3). To study these properties we follow Arnold in employing a hermitian structure on Z(l) (sections 1 and 2).

I,§2,1

26

1. Choice of Hermitian Structures on Z(1)

Let (. I ) be the scalar product defining a hermitian structure on Z(1); clearly

Im(zIz')= -Im(z'Iz) is a symplectic structure on Z(1). Now in §1,1, a symplectic structure [ , - ] was defined on Z(1). Restriction to X defines a homeomorphism between the set of hermitian structures ( I -) on Z(1) such that LEMMA I.I.

ix = X*,

Im(z I z') = [z, z'],

(1.1)

and the set of euclidean structures on X. Proof. (i) The restriction to X of a hermitian structure on Z(1) satisfying (1.1) is euclidean since

[x, X] = 0

Vx, X' E X.

Observe that, by (1.1),

(z I z') = [iz, z] + i[z, z'], and in particular

(XIX')= [ix, X]

Vx, X, C- X.

Hence

ix =

lax z E X.

2

Ox a

(ii) A given (- I ) on X defines

by (1.4), the restriction of i to X,

i,: X -> X*; the restriction of i to X i2: X* -+ X,

because i2 = -il' since i2

= -1;

hence the automorphism i of Z(1),

27

1,§2,1-1,§2,2

(1.5)

i(x, P) = (i2P, ilx); finally, by (1.2), the hermitian structure on Z(1).

The restriction to X of hermitian structures on Z(1) satisfying (1.1) is thus an injective mapping of the set of such structures into the set of euclidean structures on X. (iii) It is bijective. Indeed, the automorphism i of Z(1) defined by the given (

I

i2 = -1,

) on X, that is, by (1.5), satisfies

[iz, z'] = [iz', z],

since `i1 = i1, `i2 = i2; the function ( is clearly linear in z e C', and

I

), which (1.2) defines on Z(1),

satisfies (z', z) = (z', z), hence is sesquilinear, satisfies Ix + iyI2 = IxI2 + IyJ2, and indeed defines a hermitian structure.

Lemma 1.1 has as the following corollary.

The set of hermitian structures on Z(1) satisfying (1.1) is an open convex cone. It is therefore connected. LEMMA 1.2.

Remark 1. We choose arbitrarily one of these hermitian structures on Z(1), which we shall use to define topological notions (the Maslov indices). By the preceding lemma, these notions will not depend on this choice.

2. The Lagrangian Grassmannian A(1) of Z(1) Definition 2.1 A subspace of Z(1) is called isotropic when the restriction of [ , ] to this subspace is identically zero, that is, by (1.1), when the res-

triction of the hermitian structure on Z(1) is a euclidean structure on this subspace. Every orthonormal frame of an isotropic subspace of dimension k is thus composed of vectors orthogonal in Z(1); hence k < 1.

Definition 2.2. The isotropic subspaces of maximal dimension I are called lagrangian subspaces; the collection of lagrangian subspaces A(1) is called the lagrangian grassmannian:

XandX*EA(1).

28

I,§2,2

Let A E A(l) and let r be an orthonormal frame of A. It is a frame of Z(1): the elements of Z(1) (respectively A) are linear combinations with complex (respectively real) coefficients of the vectors that make up r. Let U(1) denote the group of unitary automorphisms u of Z(l) (that is, uu* = e, where u* = 'u, and e is the identity). By (1.1), U(1) C Sp(1).

Further let A' E A(l) and let r' be an orthonormal frame of A'. There is a unique element u in U(l) such that

r = ur' from which follows A = ua.'.

The group U(1) thus acts transitively on A(l). The same holds a fortiori for Sp(l); whence 1°) of the lemma below, where St(l) [respectively, 0(l)] denotes the stabilizer of X* in Sp(l) [respectively, U(1)], that is, the subgroup of s such that sX * = X *. Now 0(l) is clearly the orthogonal group. Lemma 2.3 characterizes St(l); part 2 shows why the stabilizer of X* interests us more than that of X. LEMMA 2.1.

1 ° We have

A(l) = Sp(l)/St(l) = U(1)/0(1).

(2.1)

2°) Let W(l) be the set of symmetric elements w in U(1), that is, the set of elements w such that 'w = w; thus w c- W(1) means w = 'w = w-'. The diagram U(l) a

u-

U'u = w E W(l)

U(1)/0(1) = A(l) a2 = UX* defines a natural homeomorphism

Then z c- a. is equivalent to z + w(A)z = 0.

I,§2,2

29

Let

z = x + iy, where x and y c X. Assume 10 sp(w(A),

where sp(w) is the spectrum of w, a 0-chain of the unit circle S'. Then

z e 2 is equivalent toy = i

e + w(A)x,

e - w(it)

(2.5)

where i(e + w(A)]/[e - w(.)] is a real symmetric matrix (that is, equal to its transpose). 3°) dim(.1 n A') is the multiplicity of 1 in sp(w(,)w-1(.1')). Remark 2.1. Part 3 is preparation for the topological definition of the Maslov index (section 5).

The proof of lemma 2.1 is based on the following lemma. Writing u E U(l) in terms of its eigenvectors and eigenvalues, the proof of lemma 2.2 is clear.

1°) Let u c- U(l). A necessary and sufficient condition for u e W(l) is that all of its eigenvectors can be chosen to be real. 2°) Every surjective mapping

LEMMA 2.2.

F : S1 - S' (S' is the unit circle in C) defines a surjective mapping W(l) a w i-+ F(w) E W(l).

The diagram (2.2) defines a mapping (2.3) since, if u and uv c- U(l) have the same image in A(l) = U(l)/O(l), then v c- 0(l), and so Proof of lemma 2.1,2°).

uv `(uv) = uv `v `u = U 'U.

By lemma 2.2,2°), given w c- W(l), there exists some u c- W(l) such that w = u2. Then w = u 'u, and so the map (2.3) is surjective. Since a. = uX *, where u c- U(l), the condition z c- ) means u-'z c- X*, or

Re(u-lz) = 0, or u-1z + u-12 = 0, or z + w2 = 0. The map (2.3) is therefore injective.

I,§2,2

30

Proof of lemma 2.1,3°).

Let w = w(,1), w' = w(A'). Then A n A' is given

by the equations

z+w2=0,

z+w'2=0;

that is, r) A':

w-'z =w'-'z,

z=

WE.

Let T be the analytic subspace of Z(l) given by the equation

T: w-'z = w'-'z. Then dim, T = k, where k is the multiplicity of 1 in sp(ww'-'). The equation

of An A' in Tjs

z+wz=0. By lemma 2.2,2°), there exists a u c- W(l) such that

-w=U2 =u-'u. Thus the equation of A n A' in T may be written

uz=iii. The isomorphism T=)

therefore maps A n A' onto the real part Rk of Ck, and so

dim, A n A' = k. The stabilizer St(l) of X * in Sp(l) has the following properties: 1°) The elements s of St(l) are characterized as follows:

LEMMA 2.3.

s(x', P) = (x, P) is equivalent to

x = slx',

P = 'Si'(P' + s2x'),

(2.6)

where sl is an arbitrary automorphism of X and s2 = 's2 is an arbitrary symmetric morphism X -+ X*. 2°) An element s of St(l) is the projection of two elements S of Sp2(l) defined by

31

I,§2,2-I,§2,3

(Sf)(W ) _ ,/d et s1 1

[e("12)Cx',s=x'>f(x')]z

(2.7)

=sI,X

Remark 2.2. We denote by St2(1) the subgroup of Sp2(1) whose projection onto Sp(1) is St(1). By Remark 2.5 in §1, St2(1) is the set of S E Sp2(1) that

act pointwise on .°(X): the value of Sf at a point x of X depends only on the behavior of fat a point x' of X (in fact on the value of f at x'). The elements of the stabilizer of X* in the group of autornorphisms of the vector space Z(1) are the mappings (x', p') i-- (x, p) Proof of 1 °).

defined by

x = six',

p = s*(p + s2x'),

where sl and s* are automorphisms of X and X* and s2 is a morphism X -a X*. These elements belong to Sp(1) when t -1 s = S1

t S2 = S2.

Proof of 2°). Formula (2.7) defines an automorphism S of Y'(X) that belongs to Sp2(1) by Remark 2.4 in §1. Clearly

x-(Sf) = S[f -s1x'],

I

ax(Sf) =

S[tsl1 (-'V ax'

+ S2x'J1 f].

Hence, for any a in .sd (§1,1),

la

a°(X,- )(Sf) vax

= Sao(s1x', ts11

la v8x

+ S2x'

f,

that is, by (2.6),

S -1 aS is associated to a° o s,

and so s is the natural image in Sp(1) of ± S E Sp2(1). 3. The Covering Groups of Sp(1) and the Covering Spaces of A(1)

The properties of these covering groups and spaces (2) follow from properties of n1 [Sp(1)] and n1 [A(1)], which are obtained by studying n1 [U(1)]. Here ik denotes the kth homotopy group, (see Steenrod [17]; we note that N. Steenrod uses the expression symplectic groups in a different sense than we do.) 2See Steenrod [17], 1.6, 14.1.

I,§2,3

32

1°) The inclusion O(l) c St(l) induces an isomorphism

LEMMA 3.1.

rzk[O(1)]

' nk[St(l)]

t1k E N.

2°) The inclusion U(l) c Sp(l) induces an isomorphism lrk[U(1)]

Vk E N.

it,k[Sp(l)]

3°) The morphism nl [U(1)]

3Y

1 --r

27ri

d(det u) EZ det u

(3.1)

is a natural isomorphism: Z.

ir,[U(1)]

Proof of 1°). The elements s of St(l) are characterized by (2.6); those for which s2 = 0 form a subgroup GL(l) of St(l). The inclusions

O(l) c GL(l) c St(l) induce natural morphisms ltk[O(1)]

itk[GL(l)]

7rk[St(l)].

The second morphism is an isomorphism, since

St(l) = GL(l) x R", where n = 1(1 + 1)/2. It has to be shown that i is an isomorphism. Now GL(l) acts transitively on the set Q+ of positive definite quadratic forms on X, and O(l) is the stabilizer of one of them. Hence

GL(l)/O(l) = Q+, where Q+ is convex.

The exactness of the homotopy sequence of this fibration (see Steenrod [17], 17.3, 17.4) proves that i is indeed an isomorphism. Proof of 2°).

U(l) c Sp(l),

The inclusions

St(l) n U(l) = 0(1) c St(l)

define a mapping (see Steenrod [17], 17.5) of the fibration

U(l)/O(l) = A(l) into Sp(l)/St(l) = A(1); its restriction to A(l) is the identity. This mapping induces a morphism of

33

I,§2,3

the homotopy sequences of these two fibrations (see Steenrod [17], 17.3, 17.11, 17.5):

nk+I[A(l)] -4 nk[O(l)] 4 nk[U(l)] p' nk[A(l)] ... no[o(l)] l i°

1 io

Ii,

i i°

i i°

7rk+i[A(l)] 4 n,[St(l)] - nk[Sp(l)] P rzk[A(l)] ... no[St(l)] This diagram, in which the lines are exact, is thus commutative. Since the mappings io are isomorphisms, it follows that the mappings it are necessarily isomorphisms. (Steenrod [17], 25.2, proves part of this by other means.) We denote by det (that is, determinant) the epimorphism Proof of 3°).

U(l) 3 u E- det u c S'

(3.2)

and by SU(l) its kernel. Since u c SU(l) when det u = 1, we have

U(l)/SU(l) = S'; (3.2) is the natural projection of U(l) onto S'. The homotopy sequence of this fibration contains the following, which is thus exact:

it1[SU(l)] -4 ni[U(l)] -°' rrl[S1] * no[SU(l)];

(3.3)

here p is induced by the morphism (3.2). Since SU(l) is connected, iro[SU(l)] is trivial. Let us compute n1[SU(l)]. SU(l) acts transitively on the sphere 521-1:Izl = 1.

The stabilizer of the vector (1, 0, ... , 0) in C' is SU(l - 1); thus SU(l)/SU(l - 1) = S21-1

The homotopy sequence of this fibration contains the following, which is thus exact: it2 [S21-1]

nl [SU(l - 1)] '' 7r1 [SU(l)]

_v

nl [Su i]

where 7x1[521-'] and n2[S21-'] are trivial for l 3 2 (see Steenrod [17], 21.2). Thus i' is an isomorphism. Now 7r, [SU(1)] is trivial, since SU(1) is trivial, and so

it1[SU(l)] is trivial.

(3.4)

I,§2,3

34

Since 1r1 [S U(1)] and no [S U(1)] are trivial in the exact sequence (3.3), p is

an isomorphism. Now 9G1[S1]3FH-

I frd- EZ

is an isomorphism. The composition of p, which is induced by (3.2), with this isomorphism is an isomorphism n1 [U(1)] -+ Z, which clearly is defined by (3.1). 1 °) The composition of the natural isomorphism n [A(1)] n1 [W(1)] [cf. (2.3)] and the morphism LEMMA 3.2.

1

nl[j'1'(l)]-3

7ri

f

ddetw)EZ

(3.5)

is a natural isomorphism (Arnold [1]): [A(1)] ^' Z. 2°) The fibration Sp(l)/St(l) = A(1) defines a monomorphism

p : Z '=' it1 [Sp(1)] - nl (A(1)] = Z,

(3.6)

which is multiplication by 2 on Z. Proof of 1 °).

The homeomorphism (2.3) allows us to define

det ). = det w e S 1;

(3.7)

the mapping

A(l)a2"detA c- S'

(3.8)

is clearly an epimorphism. By (2.2) we have

det A = det z u, if i = uX *. Hence for all u e U(1) det(uA) = det 2 u det A.

'(3.9)

The mapping (3.8) thus defines a fibration on which U(1) permutes the fibers. The fibration is A(1)/SA(l) = S 1,

I§23

35

where SA(1) denotes the variety in A(l) defined by the equation SA(l) : det w = 1.

The exactness of the homotopy sequence of this fibration proves the exactness of the sequence 71,

[SA(l)] -4 n, [A(l)] A n, [S1].

(3.10)

Since

p: n, [A(l)] - n, [S1] is induced by

det:W(l) - Si, p is an epimorphism. Let us compute n, [SA(l)]. SU(1) acts transitively on SA(1) and X * E SA(l). The stabilizer of X * in SU(1) is SO(1), the connected component of the identity element of O(l), and so we have the fibration SU(1)/SO(l) = SA(l).

The exactness of its homotopy sequence implies the exactness of

n, [SU(l)] 4 n, [SA(1)] 4 no[SO(l)], where n, [SU(1)] and no [SO(1)] are trivial, by (3.4) and the fact that SO(1) is connected. Thus n, [SA(1)] is trivial, and so pin (3.10) is an isomorphism Now p is induced by the mapping (3.8). Taking its composition with the isomorphism

7t, [S'] 3Fr-,

1

2ni

dzEZ z

we obtain an isomorphism 7r, [A(1)] - Z, defined by (3.5). In (3.6) the isomorphism Z ^- n, [Sp(l)] is the composition of the isomorphisms defined by parts 2 and 3 of lemma 3.1. The isomorphism n, [A(l)] Z is the composition of the isomorphism (3.5) and the one that induces the homeomorphism of A(l) and W(1). Proving part 2 of lemma 3.2 is thus the same as proving the following: Proof of 2°).

The fibration U(1)/0(1) = W(l) induces a morphism

1,§2,3

36

P: Z

i i[U(1)] - n [W(l)] "' Z

(3.11)

that is multiplication by 2 on Z. This fibration is defined by the mapping U(1) 9 u "_w = u `u, which satisfies det w = (det u)2. Since the isomorphisms entering into (3.11) are defined by (3.1) and (3.5), we have p:

Z3

1

Yd(det u)1--.

2ni

det u

1

27ci

fd(detu)'cZ. (det u)

This morphism p: Z - Z is evidently multiplication by 2. The two preceding lemmas have the following theorem as an immediate consequence. It is clearly independent of the hermitian structure on Z(1) used above.

a and Q are the generators of n,[Sp(l)] and n, [A(l)] whose natural images in Z are 1.

Definition 3.

1°) Sp(l) has a unique covering group, Spq(l), of order q (q = 1, 2, ... , .x) [namely: the number of points having the same projection onto Sp(l) is q]; a acts on Spq(I); a' does not act as the identity on Spq(1) unless r = 0 mod q. 2°) A(l) has a unique covering space, Aq(l), of order q; fi acts on A,(1); (lr l' does not act as the identity on Aq(I) unless r = 0 mod q. THEOREM 3.

3°) Spq(l) acts transitively on A2,(1): (aSq)) .2q = Sq(Q2A2q) = f32(SgA2q), where A2q E A2q(I),

Sq E Spq(1).

(3.12)

Example 3.1. A2(0 is the set of oriented (in the euclidean sense) lagrangian subspaces of Z(1); Sp(l) acts on A2(1).

Sp2(1) acts on A4(l). This result is essential for the theory of asymptotic expansions (chapter II). Example 3.2.

Notation 3. Denote by s the projection of Sc Spq(l) onto Sp(l); by A the projection of Aq c- NO onto A(l); by A2 the projection of A2q E A2,(1) onto A2(1); by e the identity element of Sp(l); and by E the identity element of Spq(1).

I,§2,3-I,§2,4

37

Let us choose an element X of A,,(1) projecting onto X * in A(l); XQ denotes its projection onto A9(1). 4. Indices of Inertia

Definition of the index of inertia Inert(,, A', ).") of a triple (A, A', A") of elements of A(1) pairwise transverse. We have

A0+A'=A'

A"=A" $+A=Z(1).

The conditions Z' E A',

Z E A,

Z" E

Z + Z' + Z" = 0

therefore define three isomorphisms

zE

1% A" 3 Z" a--I Z' E A'

whose product is the identity. By (4.1)

[z, z'] = [z', z"] = [z", Z];

(4.3)

this number is the value of a quadratic form at z e A (at z' E A' or at z" e A'). The isomorphisms (4.2) transform each one of these forms into the others. Thus they all have the same index of inertia, which will be denoted Inert(A, A',1").

LEMMA. The quadratic form

A3z -+ [z, z'] ER is nondegenerate (that is, has no zero eigenvalues). Proof. C E A,

Since

Take a second triple C' E A',

C" e A" such that C + C' + C" = 0.

A', and A" are lagrangian, the bilinear form

(Z, r) i- [Z, 4'] = [yr', Z"] = [Z", 4] = [C, Z'] = [Z ,

y r"]

y

= [S", Z]

is symmetric and, hence, is the polar form of the quadratic form Z H [Z, Z'].

1,§2,4

38

If this were degenerate, then there would exist z -A 0 such that

zeA,

[z, C] = 0 VC'eA,

that is,

zeA n A'. contrary to hypothesis.

Thus Inert(-, , ) has the following properties:

Inert(.,

A") = Inert(A', A", 1) = l - Inert(2,).", A').

(4.4)

Inert( , , ), which is defined when its arguments are pairwise transverse, is locally constant on its domain of definition. Inert(sA, sA', s2") = Inert(A, 2', A")

Vs E Sp(l).

(4.5)

Formulas (2.11) and (2.14) of 1,§1 defined Inert(s, s', s") for

s s' s" = e.

s, s', s" E Sp(l)\Esp,

(4.6)

The following relation exists between these two indices of inertia: THEOREM 4.

Under assumption (4.6),

Inert(s, s', s") = Inert(sX *, X *, s"-1 X *).

(4.7)

The condition s 0 Esp is equivalent to the following: sX* and X* are transverse. Remark 4.1.

Proof. We return to the notation of I,§1,2 (specifically that of lemma 2.3), setting

s = SA,

s = SA ,

Sri = SA,. ;

sA : (x', p') h-. (x, p) means p = Px - `Lx',

p' = Lx - Qx';

sA., : (x, p) i - (x", p") means p" = P"x" - `L"x,

The equations of the subspaces

A=sX*,

J' = X*,

s"-1X*

are then 1: p = Px,

2':x, = 0,

p

_

_Qx

p = L"x" - Q"x.

39

I,§2,4

Condition (4.1), Z" _ (x", p") E A",

Z = (x, p) E A,

Z + Z" E A',

may be written

z" = (-X, Q"x),

z = (x, Px),

from which follows

[z", z] = <(P + Q")x, x>. Hence, by definition (2.14) in §1,

Inert(., Z, A") = Inert(P + Q") = Inert,,(A + A' + A") = Inert(s, s', s"). The two lemmas below express the index of inertia Inert in terms of the Kronecker index KI. After we have defined the Maslov index m in terms of the Kronecker index in section 5, lemma 6.1 will deduce the relation between the indices of inertia and the Maslov index from these lemmas. LEMMA 4.1.

Sp(l) acts transitively on the set of pairs of transverse elements

of A(l).

Remark 4.2. Therefore every triple of pairwise transverse elements of A(l) has the form (sl, sX, sX *).

We recall that Inert(sA, sX, sX*) = Inert(A, X, X*). Proof.

Since Sp(l) acts transitively on A(l), it suffices to show that the stabilizer St(l) of X* acts transitively on the set of elements of A(1) transverse to X*. The element s of St(l) defined by (2.6) transforms

X:p' = 0into sX:p = `si ls2si lx, where s2 : X X* is symmetric. Now the condition that the equation p = s3x define an element of A(1) is clearly that s3 : X - X* be symmetric.

Notation 4. S1. Let

Recall that the spectrum of u e U(1), sp(u), is a 0-chain in

I,§2,4

40

(4.8)

SP(1) = sp(w),

where w is the image of) under the natural homeomorphism (2.3). Denote by (exp iB) the point in S1 with the coordinate exp iO (0 E R) and by l(exp iO) the 0-chain consisting of this point with multiplicity 1(1 e Z). We have, for example,

sp(X*) = sp(e), sp(e) = 1(1),

sp(X) = sp(-e),

sp(-e) = l(-1).

(4.9)

Let a denote a 1-chain in S1, Jul its support, a6 its boundary, and KI the Kronecker index (see Lefschetz [9] or any theory of chain intersection).

Recall that the integer KI[a1, 60] is defined when a, and a0 are a 1-chain and a 0-chain in S1 such that

I6oInIaa,I='0' is zero if I ao I n I6, I = 0,

is linear in 60 and a,, is equal to 1 if 6, is a positively oriented arc and 6o is an interior point of this arc,

satisfies KI[a1, aft] _ -KI[61, aa1]. LEMMA 4.2.

Let a and a* be two 1-chains in S1 such that

a6 = sp(ti) - sp(X*),

au* = sp(ti) - sp(X),

6 - 6* belongs to the half-circle in S1, where Im(z) > 0.

(4.10) (4.11)

Then

Inert(;., X, X*) = KI[a,(-1)] - KI[a*, (1)]. Remark 4.3.

Proof.

Lemma 5.1 will use this decomposition of Inert.

Let

Z = x + iy e 2,

Z' E X,

Z" E X*

be such that

z+z'+z"=0, where x, y e X. Clearly

(4.12)

I,§2,4

41

i = - x,

z" = - iy,

[z,, z"] = Im(z' I z") = - (x

v)

Let w be the natural image of A under (2.3). By (2.5)

y= i e+ wx e-w

(e is the identity).

Since w is unitary and symmetric, i(e + w)/(e - w) is real and symmetric. Hence Inert(1, X, X*) is the index of inertia of the quadratic form

ie + w

X=3xi-' -I

e-w

x ER,

that is, the number of positive eigenvalues of the real symmetric matrix i(e + w)/(e - w). Now the positive real axis is the image of the half-circle in S', where Im z < 0, under the homographic transformation S 3 (exp i0)

i--* i

1 + exp iO - E R.

(4.13)

1 - exp i0

We orient it positively: it forms a chain x in S'. Since (4.13) transforms the eigenvalues of w into those of i(e + w)/(e - w),

Inert(1, X, X*) = KI[y, sp(w)]. Let r be a chain in S' with boundary

ar = sp(w) - sp(ie). Then

Inert(;., X, X*) = KI[x, ar] _ -KI[r, OZ]

= KI[r, (-1)] - KI[r, (1)]. Let a and a* be 1-chains in S' such that a - r and a* - r are defined as follows:

a(a - r) = sp(ie) - sp(e), a - r belongs to the arc 0 E [0, n/2] in S'

a(Q* - r) = sp(ie) - sp(-e), a* - r belongs to the arc 0 E [n/2, n] in S'.

(exp i0) t;

I,§2,4-I,§2,5

42

Since

KI[ r - z, (- 1)] = KI[Q* - T, (1)] = 0, the preceding expression for Inert(A, X, X*) is equivalent to (4.12). Now or and rr* satisfy conditions (4.10) and (4.11) and are determined by them up to the addition of 1-cycles y and y* in S' such that y - y* belongs to the half-circle in S', where Im z > 0. Since y and y* are then homologous,

KI[y, (-1)] = KI[;,*, (01 Thus (4.12) holds for any pair (a, a*) satisfying (4.10) and (4.11).

5. The Maslov Index m on A(1) We are going to supplement Arnold's definition [1], but first we shall use the following preliminary definition.

Definition 5.1 of the index M on A4(1). Let (v,, v') be an element of A,,(l) x A,,(1) = A',(1), that is, a pair of elements of A.,(1). Let v and w(v) be the natural images of v., in A(1) and W(l) [cf. (2.3)], and let v' and w' = w(v') be the images of v.. By lemma 2.1, 3°), the condition that v and v' be transverse can be expressed as (1) 0 sp(ww'-');

sp(ww'-1), which we denote sp(v, v'), is a 0-chain in S' (see Lefschetz [9]). The mapping F n (vim, vim)

- sp(v,

V)

maps the arc r onto a 1-chain in S1 denoted sp(r). If

ar = 0, tip) - (µ., Nx), then

asp(r) = sp(, ti') - sp(µ, µ'). Suppose

,Z and 2' are transverse, u and µ' are transverse; then by lemma 2.1, 3°), (1) Ojasp(r)I.

43

1,§2,5

Thus KI[sp(r), (1)] is defined. It is an integer depending only on the homotopy class of r, that is, on OF. We denote it

p ., p.] = KI[sp(r), (1)] e Z.

M[A.

(5.2)

By Remark 1, M is independent of the choice of the hermitian structure on Z(l) used in its definition. Remark.

Properties of M. M is defined under hypothesis (5.1). M is locally constant on its domain of definition. M has the obvious additive property

+ M[ux,

v

,

ti's] =

vti> v ], (5.3)

which implies

Ar] = 0. M has the invariance property

M[S%c,SA':'; i

,p' ] =

(5.5)

dS E

If /3 is the generator of n, [A (1)] (cf. definition 3), then

fl`Ate; µx, µx] = M[A,;x;µ.,µ'] + r - r'

dr,r'EZ. (5.6)

Proof of (5.5).

Assuming hypothesis (5.1), s1. and sip' are transverse, and

the mapping

S r-4 M[S)., S;",; per,, ux] e Z is defined and locally constant. Therefore it is constant since Sp,c (1) is connected. Proof of (5.6).

Choose the arc r to be differentiable and such that

Clearly one can define 1 continuous functions

O:F=(v,x,vj- U;(vz,,v.)ER,

j = 1,

1,

such that sp(v, v')

(expi9,); O (/1'A, /x'A ) = O

A) mod 27r.

I,§2,5

44

Definition (5.2) of M gives

AlA.,2,]

2nfd6'

2n

jrd(0j)

ddet(ww'-1)

1

2ni J r det(ww - 1)

f d(det w)

1

2iri

r

det w

1r

d(det w')

2ni J r -d-et w'

_

-r

r

by (3.5) and definition 3. Hence (5.6) follows by the additive property (5.3) of M. In order to recover assumption (4.11) of lemma 4.2, we need the following definitions.

Definition 5.2

X., is the point of A,,,(1) that projects onto X in A(1) and

that may be joined to X* by an arc y of Ax (l) whose spectrum sp(y) belongs to the half-circle in S', where lm(z) > 0. Definition 5.3.

The Maslov index is the function m given by

m(;,, 4) = M(2,, 4; X *, X.). This allows us to formulate lemma 4.2 as follows. LEMMA 5.1.

Inert(2,

2',

For any triple 2")

= m(A.,

,)

,

A of elements of A,,,(I),

m(2., 1 ) + m(2,, 2 ,,

).

(5.7)

Proof. By (5.6), the right-hand side of (5.7) depends only on (2, A', 2"). By Remark 4.2 and the invariance property (5.5) it suffices to establish the following special case of (5.7):

Inert(2, X, X*) = m(;.,, X.) - m(1.,, X*) + m(X,,, X*).

(5.8)

Definition 5.3 of m and the additive property (5.3) of M give

X* , X.]

m(1.,,, X.) = m(2,

,

X*) - m(X., X*) = M(1.z, X*;

X*].

Definition 5.1 of these two values of M makes use of two arcs r and r* of A'(1) such that

iar = (2,w, Xx) - (X*, XJ,

OF* = (A,, X*) - (X', X*).

45

1,§2,5

We choose these arcs as cartesian products,

r*=y* x x*,,,

f=y x x,,,,,

where y and y* are then arcs of AW(1) such that

X,*

ny=gym

oy* = L -X".

We have

a(y-y*)=XX Definition 5.2 of X,,allows us to choose y and y* such that sp(y - y*) belongs to the half-circle in S', where Im(z) ? 0. Denote a = sp(y),

a* = sP(y*)

Since the homeomorphism

w(X) _ -e,

defined by (2.3) has the values

w(X*) = e,

definition 5.1 of M gives M[A

,

X*, Xj = KI[a, (-1)],

M[A, X**; X., X*] = KI[a*, (1)] The formula (5.8) to be proved is thus identical to formula (4.12), which

holds because or - a* = sp(y - y*) satisfies condition (4.11). LEMMA 5.2. We have

m(L, X.) + m(7.'', n.") = 1. Proof.

Substitute the expression (5.7) for Inert into (4.4).

The following lemma supplements lemma 5.1. LEMMA 5.3. Every function n:Oq, A) i-+ n(2q, Aq)

and A' transverse, with values in an abelian group, locally constant on its domain of definition, such that, for pairwise transverse ., A', A", defined for Aq, Aq e Aq(1),

(5.9)

1,§2,5

46

).q) + n()

,

)) = 0,

(5.10)

is identically zero. Proof. By (5.10), n is constant in a neighborhood of any pair (:tq, 2q), transverse or not. Therefore n is constant on Aq (l), which is connected. By (5.10), its value is 0.

We have proved the following theorem. THEOREM 5.

1°) The Maslov index (definitions 5.1-5.3) is the only function

M ( ; ., m:(A ),2x)F-'m().

)EZ

defined on pairs of transverse elements of A. (1) that is locally constant on its

domain and makes the following decomposition of the index of inertia possible:

Inert(A, )',.2.") = m(A,,, 2x) - m(2,°, Ate) + m(A', A

(5.7)

).

2°) Taking into account definition 5.2, this function has the following properties:

m(da A') + m(2', A,°) = 1;

(5.9)

m(s), SA') = m(Am, Ax)

VS E Sp. (1);

m(l'2,,, /I''A') = m(J.., %;°) + r - r'

m(X*,, Xj = 0,

Vr, r' E Z;

Xm) = 1.

(5.11) (5.12) (5.13)

Proof of 1 °). Lemmas 5.1 and 5.3.

Proof of (5.9).

Lemma 5.2.

Proof of (5.11) and (5.12). Proof of (5.13).

Definition 5.3 of m, (5.5), and (5.6).

Definition 5.3 of m and (5.4); (5.9).

Formula (5.7) clearly implies the following: If A, A', are four pairwise transverse elements of A(l), then Remark 5.1.

2"'

Inert(A, 2', 2") - Inert(., A', 2) + Inert(., %", ti") - Inert(,', A", A"') = 0. Remark 5.2.

M[2 x.,

A

We call any transverse pair (p., µ,) E AI(1) such that

; p, , µz] = m(1.

,

A')

VA, A' E Ax(1)

47

I,§2,5-1,§2,6

a basis. For example, (X *, X_) is a basis. By the additive property (5.3) of M, the condition that (pm, i') be a basis can be expressed as

m(pm, t4) = 0. 6. The Jump of the Maslov Index m(),,,, , d' ) at a Point (2, A'), Where

dim 2n2'= 1 Maslov [11] defined his index, up to an additive constant, by the expression of its jump across the hypersurface YA2 in A'(1), which is the set of pairs (L , ).) of nontransverse elements of A.(1). Theorem 6 will make the expression for this jump explicit.

First of all let us establish some properties of U(1). By y we denote a differentiable arc of U(1): 7:1- 1, 119 0 F- U (O) E U (I),

ue = du 00.

Let exp(i/(0)) be a simple eigenvalue of u(O) and let z(O) be a corresponding eigenvector. Then LEMMA 6.1.

IIZ(e)I

% = (u_1(o)uZ(o)Iz(o)).

Recall that if U is a Lie group with generic element u, infinitesimal transformations Xk, and Maurer-Cartan forms CO, then Remark 6.1.

u-1 du (hence, in particular, u-1ue) may be written

u-1 du = Iwk(u, du)Xk; k

see E. Cartan [4]. Remark 6.2.

Since U(1) is the unitary group, (1/i)u-1(O)u9 is an arbitrary

I x l self-adjoint matrix characterizing the vector ug tangent to U(1) at u(0).

Proof. Since exp(i>!i(O)) is a simple eigenvalue, k(0) is a differentiable function of 0, and the vector z(0), which is defined up to a scalar factor, may be chosen to be a differentiable function of 6. Then differentiating the relation

u(0)z(0) = z(0)exp(ii(0))

I,§2,6

48

and multiplying on the left by (1/i)u-'(0), we obtain

lu

luez

+ l zo = -u- ' zo exp(io) + z(6)Oe t

i

Taking the scalar product with z eliminates ze and gives (6.1) because (u-ize exp(ii) I z) = (Z e I exp(- i/i) uz) _ (z e l z) since u is unitary.

Denote by Eu the set of u e U(1) such that (1) e sp(u); recall that (1) denotes the point of S' c C with coordinate 1. The following lemma has the sole purpose of interpreting the assumption of lemma 6.3 geometrically.

If u(o) E Eu, (1) is a simple value of sp u(o), and z(o) is a corresponding eigenvector, then the condition that uo define a direction tangent to Eu at u(o) takes the form LEMMA 6.2.

(u-'ouozozo)) 1

= 0.

(6.2)

Proof. u(o) is a regular point of E,,,. By lemma 6.1, every direction tangent to Eu at u(o) satisfies (6.2). The hyperplane of directions satisfying (6.2) thus contains the tangent plane to Eu at u(o); but Eu is a hypersurface in U(l).

If the arc y in U(1) intersects E. only at the point u(o), if the eigenvalue I of u(o) is simple, and if the corresponding eigenvector z(o) LEMMA 6.3.

satisfies

(U'O)UZOZO))

0

(i.e., if y is not tangent to Eu), then

KI[sp y, (1)] = sign( U-'(o)uoz(o) I I z(o) Proof.

.

Let exp(iir(O)) be the eigenvalue of u(O) near 1. Evidently

KI[spy,(1)] = sign/e(o) if Le(o) j4 0, so (6.3) follows from (6.1).

(6.3)

49

I,§2,6

Notation. EA3

EA2 denotes the set of nontransverse (1t, X):

AZ(1).

F denotes a differentiable arc in A2(1): [-1, 1[3 0 i-- (,(6), A'(B)) e A2(1).

w(9) and w'(B) denote the natural images of ,(O) and )'(O) E A(1) in W(1): see (2.3). LEMMA 6.4.

If the arc F in A2(1) intersects EA2 only at the point (L(o),

A'(o)) and if

dim ),(o) n ).'(o) = 1,

z(o) E ).(o) n ti (o),

(w'-'(o)z(o)lz(o)) 9 I

nw''-' (o)z(o) z(o)

then

KI[sp T', (1)] = sign

WOW-1(o)z(o) I l

z(o))

f

_ (ww'-'o)zo)Izo))}. 1

(6.4)

Proof. The image of the arc r E A2(1) in U(1) is the arc y : [-1, 11 -3 B

u(O) = w(O)w' 1(B).

By definition,

KI[spF, (1)] = KI[spy, (1)]. By lemma 2.1,3°), intersects Ev only at the point A = o, and 1 is a simple value of sp u(o). By lemma 2.1,2°), z(o) + w(o)z(o) = o,

z(o) + w'(o)z(o) = o,

so z(o) - u(o)z(o).

Now u-1ue

= u-1wew-lu - wew'-1;

I,§2,6

50

thus, because u is unitary, (u-'(o)uez(o)I z(o)) = (wew-'(o)z(o)j z(o)) - (wBw'-'(o)z(o)Iz(o))

Hence, by lemma 6.3,

KI[spy,(1)] = sign{(WW

'(o)z(o) I z(o))

- (ww'1(o)z(o) Iz(o)l,

and (6.4) follows. LEMMA 6.5.

Let

0 Hz(0) be differentiable mappings of [-1, 11 into A(l) and Z(l) such that Z(O) E 2(B).

Then Iwew-1zIz)

( Proof.

= [ze,z(0)]

By lemma 2.1,2°),

z+wa=o, so

zo + wee + wee = o. Hence (ze I z) -

(wow-1 z I z)

+ (wee I z) = 0,

where (wze I z) = (Ze I

w-1z)

_ -(291 Z) _ -(Ze II Z).

From (1.1), (6.5) follows.

The left-hand side of (6.4) can be expressed in terms of the Maslov index by (5.2) and definition 5.3; the right-hand side of (6.4) can be expressed in terms of the symplectic form [ , ] by lemma 6.5. Hence the following theorem, which will allow us to establish theorem 3.2 of §3.

51

I,§2,6-I,§2,7

THEOREM 6.

Let

[-1, 1]30(Z,z')eZ2(l) be two differentiable mappings such that

zEA,

z'E)

for each 0;

z = z' : 0 for 0 = 0;

:i and 2' are transverse for 0 : 0;

the function

[-1, 1] -3 0H[z,z']ER vanishes only at 0 = 0 and vanishes only to first order. Then there exists a constant c e Z such that C

for [z, z'] < 0

1 +cfor[z,z']>0. 7. The Maslov Index on Spm(1); the Mixed Inertia Let

S, S', S" E Sp.(l)\Esp" be such that

SS'S" = E (the identity element); let s, s', s" be their projections onto Sp(l). Formula (4.7) of theorem 4.1, which relates the definitions of the inertia on A(l) and Sp(l), and formula (5.7) of theorem 5, which relates the inertia on A(l) to the Maslov index, yield, by the invariance property of the Maslov index (5.11),

Inert(s, s', s") = m(SX*, X*) - m(S'-1X*, X*) + m(S"X*, X*). We rewrite this formula as (7.3), by virtue of the following definition:

Definition 7.1 of the Maslov index on Sp,(l). If S e SpJ)\Esp. we define

m(S) = m(SX*, X*).

(7.2)

This definition makes sense by Remark 4.1: S Esp. is equivalent to the condition that SX* and X* be transverse. Let us supplement this formula with a lemma analogous to lemma 5.3:

1,§2,7

52

LEMMA 7.

Every function

n:Si-+n(S) defined for S e Spq(l)\Espg,

with values ip an abelian group, locally constant on its domain of definition,

such that

n(S) - n(S'-1) + n(S") = 0 when SS'S" = E, is identically zero.

This lemma, combined with theorem 5 and (3.12) gives the following theorem. THEOREM 7.1.

1°) The Maslov index defined by (7.2) is the only function

Z

m:

that is locally constant on its domain and that makes the following decomposition of the index of inertia possible: under hypothesis (7.1),

Inert(s, s', s") = m(S) - m(S'-1) + m(S").

(7.3)

2°) This function has the following properties:

M(S) + m(S- `) = 1,

(7.4)

m(a'S) = m(S) + 2r,

(7.5)

where a is the generator of 7r, [Sp(l)] (definition 3). Definition 7.2 of the mixed inertia. Let

s e Sp(l)\Esp; A, 2' a A(l), transverse to X* and such that 2 = s2'.

(7.6)

The mixed inertia is defined by the formula

Inert(s, 1, A') = Inert(sX*, X*, A) = Inert(X*, s-1X*, 1'),

(7.7)

the last two terms being equal because of the invariance (4.5) of the inertia. The following theorem is evident. THEOREM 7.2.

Under assumption (7.6),

I,§2,7-I,§2,8

53

Inert(s-', A', A) = I - Inert(s, A, i.'), Inert(s, A, A') = m(S) - m(A.z X*00 ) + m(A', X*o)

(7.8)

(7.9)

if A,, = SA',, and if s is the projection of S E Sp,,(1).

Now we express the mixed inertia in terms of the inertia of a quadratic form, as we did for Inert(s, s', s") (§l,definition 2.4) and Inert(., A', A") (§2,4). This result will be used in section 10. Under assumption (7.6), we have s = sA(§1,1) and, by (2.5), the equation of A' is p' = cp;,,(x'), where cp' is a quadratic form on X. Then

THEOREM 7.3.

Inert(s, A, A') = Inert(A(o, ) + (p'( )). Proof.

(7.10)

By (7.7) and the definition of Inert(., A', A") (section 4),

Inert(s, A, A') = Inert(X*, s-'X*, 1.')

is the inertia of the quadratic form x' i-- [z, z']

for z = (x, p), i = (x', p'),

(X + X', p + p')

S(X', P) E X*,

(X, P) E X*,

Relations (7.11) may be rewritten

x=o,

p' =-A(o,x'),

p+p'=cp.,,(x+x'),

whence

x = o,

p = Ax.(o, x') +

This implies

[z, z'] = = 2A(o, x') + 2N'(x'), and therefore (7.10).

8. Maslov Indices on A,(/) and SpJ1) Let Aq, A' E Aq(1) and S E Spq(1) be the projections of A,, A, E A,,(1) and

S., E Sp,,(1). By (5.12) and (7.5) (where a and /i are the generators of n' [Sp(l)] and n, [A(1)] ), the relations m(Aq, Xq) = m(tir,

mod q,

m(S) = m(S,,,) mod2q,

(8.1)

I,§2,8

54

define functions m. They are again called Maslov indices. From theorems 5, 7.1, and 7.2 and lemmas 5.3 and 7, they evidently have the following properties. THEOREM 8.

1°) The Maslov index defined by (8.1)1 is the only function

defined on transverse pairs of elements of Aq(l) that is locally constant on its domain and that makes the following decomposition of the index of inertia possible: Inert(.lq, 'q, n,9) =

.19) -

.1q) + m(),9, )9) mod q.

(8.2)

2°) The Maslov index defined by (8.1), is the only function m: Spq(l)\EsPq H Z2q

that is locally constant on its domain and that makes the following decomposition of the index of inertia possible:

Inert(s, s', s") = m(S) - m(Sr-1) + m(S") mod 2q.

(8.3)

3°) These functions have the following properties: m(.,q, :t9) + m(ti9, ).q) = I mod q,

m(S) + m(S 1) = I mod 2q,

Inert(s, A, ).') = m(S) - m(A2q, XZq) + m().Zq, XZq) mod 2q

(8.4) (8.5)

if S E Spq, ),Zq = S.'Iq. We recall that Spq(l) acts on A2q(l).

By §1,theorem 2,3°), assuming q = 2, theorem 8,2°) identifies:

the Maslov index that formula (2.15) of §1 defines on Sp2(l) mod 4, the Maslov index that (7.2) and (8.1) define on Spq(l) mod 2q. Definition 8. Let A E A (§1,definition 1.2). Let us give a definition of SA E Spq(l)\EsPq that coincides with that of §1 for q = 2. Recall that m(sA) is defined mod 2. By formula (2.15) of §1, m(sA) = m(A)

mod 2. Then by (7.5), there exists a unique element of Spq\EsPq, denoted S.4. such that

its projection onto Sp(l) is sA ;

m(SA) = m(A) mod 2q.

(8.6)

55

I,§2,8-I,§2,9

The mapping A-3

is clearly surjective and, if q = oc, bijective. If q -A co, the condition

S,, = Se. is equivalent to the following:

A(x, x') = A'(x, x')

Vx, x' e X,

m(A) = m(A') mod 2q.

9. Lagrangian Manifolds An isotropic manifold in Z(l) is a manifold on which d = 0; i.e., >dp; A dx1 = 0; i.e., d[z, dz] = 0. i

(9.1)

Its tangent plane is isotropic, and hence it has dimension < 1. A lagrangian manifold V is an isotropic manifold such that

dimV=1. Let V be the universal covering space of V (see Steenrod [17], 14.7). Evidently (9.1) means that there exist functions cp : V

R,

1k: l

- R,

defined up to additive constants, such that dcp = ,

VI(x, p) = (0(x, p) - i,

dtji = z[z, dz]

cp is the phase of V and i(i its lagrangian phase. The I-plane A(z) tangent to V at z = (x, p) e V is obviously lagrangian.

The apparent contour Ev of V is the set of z E V such that %(z) is not transverse to X*. On V\E,, and on its tangent l-planes, x may serve as a local coordinate. By (9.2), the equations of V and its tangent I-planes are then V: p = (p.(x),

.l(z):dpi = i cps,x. dxk. k=1

Each element of Sp(l), S: Z(l) -3 Z' f-+ sz' = z E Z(I),

(9.3)

1,§2,9-1,§2,10

56

maps every lagrangian manifold V' in Z(1) into a lagrangian manifold V in Z(l), with lagrangian phase

O(z) = i/i'(z') + const.

for z = sz'.

If z = (x, p) and z' = (x', p'), the phases o and cp' of V and V' are then related by (p (x) -

i < p, x> = cp'(x') - 1 < p', x'> + const.

(9.4)

If s = sA 0 Esp(,), then p = Ax, p' _ -Ax, by §1,(1.11), and so

tp(x) = cp'(x') + A(x, x') + const, (9.5)

p = (px = Ax, LEMMA 9.

p' = (px, = -Ax..

Let SA e Sp(l)\Esp; let z' _ (x', p') e V'\E,,, be such that

sAZ' = z = (x, p) e V\E,,, where V = sAV'. Evidently these relations define a diffeomorphism x' -+ x of the local coordinates of V and V'. Its jacobian takes the form d'x _ Hessx.[A(x, x') + cp'(x')]

d'x'

A2(A)

(9.6)

(In the calculation of Hess,,, x and x' are viewed as independent.) Proof.

By (9.5), the diffeomorphism x' f-- x satisfies

cp', (x') + As (x, x') = 0. Then (9.6) follows, because, from §1,1,

A2(A) = det(-Ax:x,,).

10. q-Orientation (q = 1 , 2, 3, ... , co) The notions defined in this section enable us to supply a complement to formula (9.6). This is theorem 10, which will be crucial in the sequel. By a q-orientation of A E A(l) we mean a choice of a it2q E A2q(l) with natural projection A. A q-oriented lagrangian manifold, denoted Vq, is a lagrangian manifold V together with a continuous mapping V -3 Z H A2q(Z) E A2q(1)

57

I,§2,10

that, when composed with the natural mapping A2q(1) - A(1),

gives the mapping V -3 z f--' A(z) e A(1), where ).(z) is the tangent 1-plane to V at z.

Each element of Spq(1) maps a q-oriented lagrangian manifold Vq into another. A q-orientation of V is characterized by the values taken by the locally constant function V\Ey

zf

m(Xzq, 2q) E Z2,-

By (5.12), a change in this q-orientation is equivalent to the addition of a constant E Z2q to the function m.

The statement of theorem 10 will be simplified by the following definition. Definition 10. The argument of d'x at a point of Vq, where the tangent 1-plane is A2q e A2q(l), is defined by the formula arg d`x = rcm(XZq, ).Zq) mod 2q7[.

(10.1)

For example, by (5.13),

argd'x = 0 on X2y.

(10.2)

Let SA E Spq(1)\Espq

(definition 8),

and let Vq be a q-oriented lagrangian manifold in Z(1); denote Vq = SA Vq .

Let ) 2q and 2q be the tangent planes to V9 and Vq at z' and z = sAZ'. In

formula (8.5), taking S = SA and s = SA, we have, by (8.6) and §1, definition 1.2,

m(S) = m(A) =

2

n

arg(t (A)).

By theorem 7.3 and equation (9.3) for ).'we obtain that Inert(s, inertia of the symmetric matrix

is the

I,§2,10-I,§3,1

58

z

x,; x [A(x, x') + T'(x')] By §1,definition 2.3 of arg Hess, this is

1 arg Hessx, [A(-x, x') + p'(x')]. n

Thus, by (8.4), formula (8.5) may be written m(Xzq, 2q) - m(XZq, A'2q) _

argHessx.[A + lp]

- 1 arg A2 (A) mod 2q, whence, by definition 10, we have the following theorem. THEOREM 10.

If Vq and Vy are two q-oriented lagrangian manifolds such

that Vq = SAV4, where SA e Spq(l)\Espq,

then not only are the two terms of (9.6) equal, but so are their arguments mod 2gir.

It is the special case q = 2 of this theorem that will be used (see §3, corollary 3).

§3. Symplectic Spaces 0. Introduction

Symplectic geometry makes it possible to state the preceding results in the following form, which will be used in chapter II. Z(l) has been provided with a symplectic structure, and moreover with a particular frame consisting of a pair (X, X*) of transverse lagrangian I-planes. It is important to state conclusions that are independent of this choice of a particular frame. 1. Symplectic Space Z A symplectic space Z consists of R21 together with a symplectic form, that is, a form

59

I,§3,1

x Rz1a(z,z')"[z,z']ER that is bilinear, alternating, real valued, and nondegenerate. We then have

[z, z'] = - [z', z]; [z, z'] = 0 for a given z and all z' R21 only if z = 0. Z(l) is provided with the symplectic structure defined in §1,1; the isomorphism of the symplectic structures of Z and Z(l) is obvious and has the following consequences.

The subspaces of Z on which [ , ] vanishes identically are called isotropic ; their dimension is <, 1. The isotropic subspaces of dimension 1 are called lagrangian subspaces.

The collection of lagrangian subspaces A(Z) is called the lagrangian grassmannian of Z; A(Z) is homeomorphic to A(l) and therefore has a unique covering space Aq(Z) of order q c- {1, 2,

..., oo}.

The projection of Aq E Aq(Z) onto A(Z) is denoted A. The following definition (see §2,4) makes sense on Z. Let A, A', A" e A(Z) be pairwise transverse. Inert(A, A', A") is the index of inertia of the nondegenerate quadratic form Definition 1.1.

z H [z, z'] = [Z', Z,] = [z", z],

(1.1)

where z E A,

Z C -A"

Z" E

z + Z' + Z" = 0.

Clearly its values belong to {0, 1,

... , 1), and

Inert(A, A', A") = Inert(.', A", A) = 1 - Inert(A, A", A').

(1.2)

Let Y.^9 denote the set in A9(Z) consisting of pairs of nontransverse elements of Aq(Z). By theorem 8 and (8.1), we have the following theorem. THEOREM 1.

For each q c {1, 2,

... , oo}, there is a unique function

m : A9 (Z)\1^, -, Zq,

called the Maslou index, that is locally constant on its domain and that satisfies Inert(A, A', A") = m(A.q, A) - m(Aq, A') + m(AQ, Aq) mod q.

It has the following properties:

(1.3)

I,§3,1-I,§3,2

60

Aq) = I modq,

(1.4)

m(Aq, Aq) = m(A, Ate,) mod q.

(1.5)

Theorem 6 of §2 applies to the jump of this Maslov index across Ego. Definition 1.2. - A lagrangian manifold V in Z is a manifold of dimension I on which

d[z, dz] = 0;

(1.6)

its tangent 1-plane is lagrangian. In other words, on the universal covering space V there exists a function

0:

R such that d = [z, dz],

(1.7)

i

called the lagrangian phase, which is defined up to an additive constant. 2. The Frames of Z A framet3t of Z is an isomorphism

R:Z-+Z(1) respecting [ , ]. If R and R' are two such frames, then RR'-t is called the change of frame: RR'-t E Sp(l).

(2.2)

Clearly, if A, A', A" E A(Z), then RA, R1.', RE' E A(1) and

Inert(A, A', A') = Inert(RA, RA', RA").

(2.3)

If V is a lagrangian manifold in Z, then RV is a lagrangian manifold in Z(1) with phase tpR given by

(pR(z) = O(z) + '< p, x>, where Rz = (x, p) E X ®X

Z(1).

(2.4)

The apparent contour of V relative to the frame R is the set ER of points in V at which the tangent l-plane is not transverse to R X *. On V\ER, x may serve as a local coordinate. By lemma 9 of §2 we have the following theorem. 'The use of the letter R to denote a frame comes from the initial of the French word repere. [Translator's note]

1,§3,2-I,§3,3

THEOREM 2.

61

Let x and x' be the local coordinates defined on V\(ER U ER,)

by two frames R and R' such that RR'-' a Sp(l)\Esp. Then there exists A (§1,2) such that SA = RR'-'; let rp'(x') = cpR,(z) be the phase of R'V. Then the local diffeomorphism X 3 x' r+ x e X has the jacobian d`x

_ Hessx.[A(x, x') + cp'(x')]

d'x'

A2(A)

(In the calculation of Hess,., x and x' are viewed as independent.)

The frames we have just defined do not allow us to fix the q-orientations if q > 1. Remark 2.

3. The q-Frames of Z The q-frames of Z do allow this: each q-frame (q E [1, 2,

... , oe}) consists

of

i. an isomorphism jf:Z -+ Z(l) respecting ii. a homeomorphism hR: A2q(Z) -+ A2,(1) whose natural image is the homeomorphism A(Z) - A(1) induced by jR.

If R and R' are two such frames, then the change of frame RR'-' consists of 1. S = .1RIR'1 E SP(1),

ii. H = hRhR., a homeomorphism of A2,(1) whose natural image is the homeomorphism of A2(l) induced by s (cf. §2, example 3.1). To define H knowing s it suffices to give HX Zq E AZq(l) with the projection sXz E A2().

There are q elements of Spq(l) with image s e Sp(l). By §2,(3.12), they map X *4 into the q elements of A2q(l) with image sX*2 e A2(1). The unique element S E Sp,(1) with images E Sp(l) and such that SX Zq =

HX 2q thus induces the homeomorphism H of A2q(l). It characterizes RR'-', which we denote S : RR'-1 E SPq(l)

(3.1)

R denotes either jR or hR; we write

(x,p)EX O+ X* = Z(1), R : A2q(Z) 3 %.2q t--+ R22q E AZq(l).

I,§3,3

62

Evidently m(A2q, )'2q) = m(RA2q, RA'2q) mod 2q

VA 1,, A'2q E A2q(Z).

(3.2)

R maps a q-oriented lagrangian manifold V in Z into another in Z(1). If A2q E A2q(Z) is the tangent plane to V at z, we define

m,e(z) = m(R-'Xzqi A2q)E Z2q.

(3.3)

If x is the local coordinate of V\ER (q-oriented) defined by the q-frame R, we define

argd'x = nmR(z) mod 2gir. Example 3.

(3.4)

On R -1X24, arg d`x = 0 mod 2gir by §2,(10.2).

Theorem 10 of §2 evidently allows us to supplement theorem 2 as follows. THEOREM 3.1.

In the statement of theorem 2, suppose that V = Vq is

q-oriented and that R and R' are q -frames. Then RR' -' = SA E SP q(l)

(3.5)

and the arguments of the two sides of (2.5) are equal mod 2g7r.

Let us recall that §1 gave the definitions 1.2 and 2.3 of arg A(A) and arg Hess. By §2,(8.6),

m(A) = m(SA) mod 2q.

The following special case of this theorem will be used in part (iii) of the proof of theorems 4.1-4.3 of II,§1.

If q = 2, the half-measure [d'x]'/2 on the lagrangian

COROLLARY 3.

2-oriented manifold V is defined by

arg[d`x]1/2 =

Jr

mR(z) mod 2n.

We have

[dIx]1/2 = ,A) {Hess..[A(x, x') + 9'(x/)]11/2[d'x']1/2.

(3.7)

In a neighborhood of a point of ER, where dim). n R-1X* = 1, mR(z) has the following expression, which follows from §2, theorem 6.

63

I,§3,3

THEOREM 3.2.

Let ).(z) be the tangent 1-plane to the lagrangian manifold

Vatz;zEERmeans

dim(.nR-'X*)> 0. We stay in a neighborhood of a point of ER at which

dim(ti n R-'X*) = 1; then for z e ER, the projection of R)i(z) onto X parallel to X* is a hyperplane: i

icjdxj=0.

j=1

j

1°) There exists a regular measure w on V such that for z E ER, for all and all k

dx' A

A dxj-' A dpK A dxj+' A

. A dx' = cj cKrJ.

(3.9)

2°) If V = V9 is q-oriented, there exists a constant c E Z2, such that

mR(z) = c mod 2q for d'x/m < 0, (3.10)

mR(z) = 1 + c mod 2q for d'x/ra > 0, provided that d'xl'm vanishes to the first order where it vanishes (namely on ER).

Remark. A regular measure to on V is a differential form of maximal degree 1 on V with everywhere nonvanishing coefficient.

Proof of 1°). The cj entering into (3.8) are not all zero. Say c, # 0; then, on V at z, dxZ A .. A dx' * 0; 3k such that dpk A dxz A . . A dx'

0.

(3.11)

Since Ejdpj A dxj = 0 on V, we have, for all k,

dp1 A dx' A dxZ A dxk A dx' +dpkAdxkAdx2A . where the cap suppresses the term it covers. By (3.8)

dx' = -Ckdxk mod(dx2, ... , dzk, ... , dx'). c1

(3.12)

64

I,§3,3

Denoting

w= dp1 Adx2 A...Adx', 1

we may write (3.12) as C I Ck m= d pk A dx 2

A ... A dx',

(3.13)

which is (3.9) for j = 1. By (3.11), (3.13) implies

m i4 0on Vatz. Using the expression

dx' = --- dx' c;

mod (dx2, ... , dx', ... , dx')

in the right-hand side of (3.13), we obtain (3.9) for j > 1 and c; 0 0. If j > I and c; = 0, then the two sides of (3.9) are obviously zero. Proof of 2°). Suppose cl 0 0 at a point of ER. By (3.9), for z e V near this point, we can choose the coordinates (pl, x z, ... , x') of Rz in Z(1) as coordinates on V. Then on V, x1 and p are functions of (pl, x2, ... , x'). The equation of ER is

axl(P1, x2, ... , x') R

0.

aPl

Let t"'(z) be the vector in A(z) having the coordinates

(dpi = 1,dx2

=...=dx'=0).

Then in Z(1), RS'(z) has the coordinates

(dxi

ax1(P1,x2, ...,x') =

aPl z

dx2 = 0, ...,dx' = 0,

)

dp; _ aP,

Let C(z) be the vector on R - 1 X * such that

RC(z) = dx = 0, dp = aP(Pl, x2,

aPl

x')

65

I,§3,3-I,Conclusion

1'(z) is ayyfunctioyyn of z c V. Then

b(z) = b'(z) for Z E Y-R ;

ax I 'x '(z)] = [RC(z), RC'(z)] = dpl _ P,

(3.14)

because, in view of (3.9), the obvious relation

dx' A ... A dx' = Ox'(P),x2, ...,x')dpl

A

dx2 A...Adx`onV

Opi

means Ox' apl

d'x

Now by §2, theorem 6, for z r= V\ER,

mR(z) = m(R 1X

,

A,,(z)) =

C'(z)] < 0 for 1 + c for [Oz), Nz)] > 0,

c

c a constant c- Z. Considering (3.14), (3.10) follows.

4. q-Symplectic Geometries

If R is a q-frame, clearly the group

R-' Spq(l)R = Spq(Z) acts on Z and its lagrangian manifolds while preserving their q-orientations; this group is independent of R by (3.1). F. Klein defined a geometry by specifying a manifold and a Lie group acting on that manifold. Each of the groups Spq(Z), where q c (1, 2, ... , cc }, thus defines a geometry on Z which it is convenient to call the q-symplectic geometry.

Conclusion

This chapter in §1 defined a unitary representation of the group Sp2(!) in the Hilbert space K(X), where dim X = 1. As a result of §2, in §3 the study of this group was subsituted by that of the isomorphic group Sp2(Z) that defines the 2-symplectic geometry of Z, where dim Z = 21. Earlier §2

66

I,Conclusion

and §3 detailed the properties of the index of inertia and the Maslov index, already introduced in §1. The interest of these various properties is to make possible the definition and study of a new structure, lagrangian analysis, which is the subject of the next chapter.

I I Lagrangian Functions; Lagrangian Differential Operators Introduction Summary.

In Chapter I we studied only differential operators with

polynomial coefficients and functions defined on all of R' = X. The aim of chapter II is to extend those results. In §2, we consider a symplectic space Z of dimension 21 provided with a 2-symplectic geometry. We define

lagrangian functions and lagrangian distributions, on which Sp2(Z) acts locally;

lagrangian operators, more general than differential operators, which are transformed by Sp2(Z) like differential operators with polynomial coefficients (see I,§l,theorem 3.1).

Each lagrangian function U is defined on a lagrangian manifold V in Z; in each 2-frame R, U has an expression UR, a function with formal values defined on V\ER. In each frame R, a lagrangian operator has an expression aR

=

+ V, X,1Ia-lJ1

aR

V ax

that is a formal differential operator of order < oc, acting locally on the UR.

A change of frame

S = RR"' e Sp2(l) (see 1,§3,3)

is a local operator that transforms UR, into UR =

aR' into aR = SQR'S-1,

and hence transforms aR, UR' into aR UR = S(aR. UR.).

It follows that lagrangian operators act on lagrangian functions and on lagrangian distributions.

All the expressions aR of a single lagrangian operator a are readily obtained from a single formal function a° defined on Z. In geometry, a change of frame leaves invariant (or changes linearly)

the value of a scalar (or vector) function at a point; but an essential

68

II,Introduction-11,§1, 1

characteristic of lagrangian analysis is the following: at each point z of V, the group Sp2(l) of changes of frame acts on the germs of expressions UR of a lagrangian function U and not on the values at z of these expressions. U. = SUR, may have singularities on ER, but has none on ER.\ER, n ER, whereas the singularities of UR, are on ER.. Thus the singularities of the expressions UR of U are not singularities of U; they may be described as apparent singularities. Their nature can be made explicit by making use of the Maslov index.

Historical account. V. P. Maslov [10] elucidated this essential characteristic of lagrangian analysis, even though he only studied the projections of lagrangian functions onto X without defining either lagrangian oper-

ators or lagrangian functions explicitly. He only used a subgroup of Sp 2(1), depending on a choice of coordinates of X : the subgroup generated

by Fourier transforms in a single coordinate.

§1. Formal Analysis 0. Summary In section 1 we define and study asymptotic equivalence classes of functions.

In section 2 we define formal functions; formal functions defined on X with compact support are asymptotic equivalence classes. Then we may integrate them (section 3) and transform them by Sp2(l) and by differential operators, whose definition can be generalized (sections 4 and 6). We deduce (sections 4 and 6) that Sp2(l) and formal differential operators act locally on formal functions defined on lagrangian manifolds V or their covering spaces V; these functions may be compactly supported or not. In section 5 we study their scalar product. Formal functions on V, which are no longer asymptotic equivalence classes, enable us to define lagrangian functions in §2. 1. The Algebra ' (X) of Asymptotic Equivalence Classes The algebra -4(X) Let I be the purely imaginary half-line

I = ill, 301EC. I(X) denotes the algebra of mappings

f:I x X-3 (v,x)F-- f(v,x)EC

69

II,§1,1

all of whose x-derivatives are continuous in (v, x) and satisfy 81

x9

Sup

8x J

(v,x)EI X X

dq,rEN`;

f(v, x) < oc

Sp2(l) and the differential operators with polynomial coefficients in (1/v, x),

a=

a+

(, x, -1-alOx/I=a_v, 1

v

vOX

V

x

act on V (X ).

If 1 = dim X = 0, 24(X) is denoted :#?: thus.1 is the algebra of bounded

continuous mappings I - C.

The algebra (X) Let N E R. Let -IN(X) be the set of f e .4(X) such that the mapping (v, x) - vNf(v, x)

still belongs to ,4(X). Clearly 24N(X) is an ideal in 2(X) that shrinks as is an ideal in 24(X). We define N grows; hence -4,)(X) = the algebra 16(X) as follows:

W(X) = 24(X)i2x(X)

(1.2)

l'(X) is an algebra over C; its elements are called asymptotic equivalence

classes. The condition that the elements f and f of .(X) belong to the same class is

f - f'E2N(X)

dNeR+.

Clearly, "eN(X) = -qN(X)114.(X)

is an ideal in 16(X) that shrinks as N grows:

f w,(X) = to}.

NeR,

Let f and f e :4(X) with classes f and f' e''(X ). To express the equivalent relations

f - f e 24N(X),

f-

' e'6,(X), where N 5 cc,

(1.5)

II,§1,1

70

we shall write one of the following:

f=f

modvN,

f = f' modf'E f modvN.

(1.6)

If I = dim X = 0, '(X) is denoted W. Clearly '(X) is a subalgebra of the algebra of functions X

W.

Thus an element f of '(X) has a restriction to any subset of X and a support

Supp(f)

X.

Remark 1.

Let F be an entire holomorphic function

(1.7)

F:Ci - C such that F(O) = 0;

for every fi c fi, the F(fl , ... , f,) are in the same class, denoted j). Thus d. f eW(X)(j = 1, ..., J), F(f,,...,.Tj) Ew(X). Let F be a Holder function F(.r,,...,

F : C' -p C such that F(O) = 0. We similarly define F(.Ti , ... , .fi) E W

bfi E W,

j = 1, ... , J.

For example, if f e ', then j f1, Re f, and Im f are defined. f c W is real if Imf = 0; then f+ (the positive part) and f_ are defined. f = 0 is written 0 1 and defines a partial ordering on Re W, the set of real elements of 1W: let f and E Re 16; f > 0 and g > 0 imply f + j > 0

0; f 0 (that is, -.r > 0) excludes f > 0 (that is, 0). However, f' may satisfy neither f < 0 nor f > 0.

and f g > 0; 12

0
Let F be a Holder (and increasing) function

F : R - R such that F(0) = 0;

df E Re W we have F(f) E Re q' (the order relation on Re q' being preserved).

In particular we have the Schwarz inequality

Irilz]gi llz r

z]l/z J

dri, 9i e 1K.

II,§1,1

71

The differential operators with polynomial coefficients in (1/v, x) given by

l al _( l a a=a+rlv,x,vax/-a \ vex'x are clearly endomorphisms of -4(X), -4N(X ), 'K(X ), and 'N(X ). They act

locally: the restriction of of to an open set S in X only depends on the restriction off to this open set: Supp(a.T) c Supp(.T) As in §1,3, the differential operator

1 al= aa = a+(v, x, --) vax

(v, 1- a

vax

,

x)

is associated to the polynomial a° in (1/v, x, p) given by

a°(v, x, P) = lexp

= [exp

- 2v Cax' 6)] a+ (v, x, P) 8

1

2v

a

Cax' Op

] a- (v, p, x).

Integration is defined on '(X): if f c .J.A(X ), the function v i-- f f(v, x)d:x x

belongs to S. Its asymptotic equivalence class, which depends only on the class f of f, is denoted J(v, x) d'x c 16; 1.

f is called the asymptotic integral.

The scalar product (I) is a sesquilinear form on '(X) x W(X) with values in W defined by

(T, 9) =

ff(v,x)g(v,x)dlx, x

where f e f c'(X), g c 4 c'(X);

(1.9)

II,§1,1

72

g(v, x) is the complex conjugate of g(v, x); v = -v. Thus

(J,f) _ (f, 9) The seminorm

II

is the function W(X) - 16 with values >, 0 defined by

IIfll2

llf l + llg LEMMA 1.1.

I(f,g)l -<

,

g11 in 16.

Sp2(1) forms a group of automorphisms of c(X) that are

unitary, that is, preserve seminorms and scalar products. Proof.

Let SA E Sp2(1)\Esp2. Its definition in I,§1 by the integral (1.10)

shows that SA transforms the asymptotic equivalence class f' into the class SA,' defined by the asymptotic integral evA(x.x'f(x,)d'x'.

(27riJL2A(A)

(SAf')(x) = /

fX

This formula makes sense because multiplication by e"A, where vA is purely imaginary, clearly preserves asymptotic equivalence. The SA act on '(X), composition being the same as when they act on Y(X ). Now they generate the group Sp2(1); thus this group acts on W(X). Since Sp2(l) is unitary on ,'(X), it is unitary on W(X). Theorems 3.1 and 3.2 of i,§1 and their corollaries give the following. LEMMA 1.2.

The transform

SaS-'

of the differential operator a by

S e Sp2(l) is the differential operator associated to a° o s-1; s denotes the image of S in Sp(1); s acts on the space Z(1) of vectors (x, p).

LEMMA 1.3. A necessary and sufficient condition for the differential operators a and b to be adjoint is that

b°(v,x,p)=a (( P In particular, a is self-adjoint when a°(v, x, p) has real values for V E I,

(X, P) E Z(1).

S transforms adjoint operators into adjoint operators.

II,§1,2

73

2. Formal Numbers; Formal Functions

Theorem 2.2 connects the following definitions with the previous ones. A formal number is a formal series

u = u(v) = Y- E 1',e°`0', Vr

(2.1)

jE rEN

where J is a finite set, aj, e C, qj e R, and cpj -A cpk if j -A k. The expression (2.1) for a formal number is unique, by definition.

The set of formal numbers is a commutative algebra F for which the addition and multiplication rules are obvious. We give the topology defined by the following neighborhood system .NV(N, e) of the origin:

NE N, eeR+, uE,N'(N,e)means Ylajrj <e

dr4N.

jEJ

A formal function on X is a mapping u : X -+ the form

.

that can be put into

a-'

u = u(v) =

(2.2)

V

jEJ rEN

where J is a finite set, ajr: X -+ C, cpj: X - R, and ajr and cpj are infinitely differentiable. The mapping of u may be put in the form (2.2) in many ways at a point where two of the cpj are equal in a neighborhood of that point. The value of u(v) at x is denoted u(v , x )

=

E E aJr(rx) evNj(x) V

jEJ

.

(2.3)

Moreover, u(v) has a support

Suppu(v) c USuppaj,r.

(2.4)

j.r

The set .F(X) of formal functions on X is an algebra over .f, which is commutative. The set of formal functions on X with compact support is a subalgebra .FO(X) of .F (X). The cpj are called phases, the aj = ErEN(xjr/vr) amplitudes.

II,§1,2

74

The set F° [respectively,

of elements of F [respectively,

.F(X )] with vanishing phase forms a subalgebra of.

[respectively, . (X )].

Let Z denote a symplectic space with a 2-symplectic geometry, V a lagrangian manifold in Z (I,§3,1 and 4), and V the universal covering of V. A formal function UR on V consists of

i. a 2-frame R of Z ii. a mapping UR = UR (v) =

r- e"'PR : V rEN V r

where 4pR is the phase of R V (I,§3,2),

ar : V - C is infinitely differentiable. Clearly,

Supp UR = U Supp ar

(2.6)

r

Given R and V, the set of these formal functions UR is a vector space over 3F°, denoted F (V R); the set of those of its functions with compact support is a subspace 3F°(V, R). We give F(17, R) the topology defined by the following neighborhood system /V(K, p, r, e) of the origin:

K is a compact set in V; p,reN,eE1(i+; UR e .A'^(K, p, r, e) means that the derivatives of the as (s < r) of orders < p have modulus < e on K.

ER denotes the apparent contour of V for the frame R (I,§3,2). ER denotes the apparent contour of V, that is, the set of points of V that project onto ER. We shall deduce the properties of -flV\ER, R) from those of .f°(V\ER, R). The latter will be deduced from the properties of .F°(X) using the morphism resulting from the composition of the two morphisms that will be defined in theorems 2.1 and 2.2, respectively.

If z e Z and Rz = (x, p), then we write x = RXz. The composition of the natural projection V - V and the restriction of RX to V

Notation.

is denoted 1

: V -p X.

75

II,§1,2

There exists a natural morphism

THEOREM 2.1.

]ZR:

o(V\ER, R) - .9Fo(X)

(2.7)

called the projection. It is defined as follows. Let UR E .F o(V\ER, R); u = FIR UR is given by

u(v, x) _ X UR(v, z).

(2.8)

ZERX'X

IIR is a monomorphism if any period of cPR is non vanishing.

Remark 2.

Proof. Formula (2.8) makes sense because RXIx n Supp UR is a finite set, since Supp UR is compact in V\ER and I is a local homeomorphism V\ER . X. There exists a natural monomorphism of the algebra .F,(X) into the algebra W (X) that allows the convention

THEOREM 2.2.

.Fo(X) c 'e(X). It is defined as follows. Let u

=

jEJ

a

e"`lj E J`o(X );

(2.10)

17

put

T'-1 a

uN

E Y-','ev4:I xX -C.

jEJ r=O

ti.

(2.11)

1°) There exist functions f c P4(X) such that f - uN = 0 mod 1/vN VN. 2°) All these functions belong to the same asymptotic equivalence class f e (X); the mapping u i-- f defines a natural morphism E--' r, (X). 3°) This morphism is a monomorphism.

Proof of 1°).

u(V, X) = I

It suffices to consider the case in which (2.10) reduces to a'(x) e°O(X). Vr

(2.12)

For all g e M (X) vanishing outside a compact set of X we use the norm s

IgI, = Sup (1 a } g , where v E I, X E X, +sj < r. V, z. r \ V 8x

(2.13)

II,§1,2

76

We choose an increasing sequence of numbers 0<µo<µl<...

such that

2'I xre"' I r

< µr,

lim u, = oG,

(2.14)

and we choose a sequence of continuous functions go, E, , ...: I -+ R+ such that 0 < E,(v) < 1, Er(v) = 0

for IvI < µr,

Er(v) = 1

for 2µr < I i ' ! .

(2.15)

Now we define

f

(v , x

)_

e, (v)

=o

-( ) eV(p(x) .

( 2. 16)

This series converges since, from (2.14)2 and (2.15)2, the number of nonzero terms is finite on any bounded interval in the domain of definition of v. In accordance with (2.11), we define 1

ar(X)evrp(x)

UN(V, X) r=0

(2.17)

V

By (2.15)3, for 2µN < IV 1,

If(V) - uN+l(V)I N < '=L Y_

N+1

Er(V) I

V

I,

lore I r

Now (2.14)1, (2.15)1, and (2.15)2 imply

E,(v)2'Ia,e"'I, < IvI. Hence, for 2µN < I VI,

If(V) - UN+1(V)IN < E

r=N+1

because 1

2rVIr

2IVIN 2 - (1/IVI)

IN I

I vI ; thus

If - UNIN = 0 mod N.

(2.18)

77

11,§ 1,2

Let us prove a more general statement :

If-uNIM=0modv

VM, N e N.

(2.19)

If M < N, (2.19) follows from (2.18) and the fact that M. If N < M, (2.19) follows from the relations

If - U11

increases with

< If - UMI M + J UM - UNIM,

I f - uMI M = 0 mod

vM,

and hence mod

M-1a

I

I UM - UNI M = 0 mod

-

vN,

because UM - UN = Y-

, e°v. V

V IV

Since the supports of f and of the uN belong to Supp u = U Supp ar, which is compact, (2.19) implies

x°(f - UN)IM = 0 mod

VM,N,geN;

vN

hence

f e R(X),

f-UN=0modI V

Proof of 2°). If f and f' c- .V(X) and satisfy

f - uN = f' - UN = 0 mod

I

VN,

then

f - f' = 0 mod N V

VN.

Proof of 3°). Assume

U - I = 0; then

UN(V1 x) = 0 mod N V

VN e N, Vx e X.

II,§1,2

78

It has to be proved that

u(v, x) = 0

Vx E X.

It suffices to consider the following case: u is a formal number. Assume that in expression (2.1)

aJ, = 0 for r < N

dj E J;

then we have 1 uN+1(V)jQJ _ Vy ae"i = 0 mod VN+ 1'

whence lim I aJNe'lli = 0, jEJ

which implies

aJN=0

Vj.

Theorem 2.2 follows. Asymptotic integration, elements of Sp2(l), and differential operators with polynomial coefficients are morphisms COROLLARY 2.

0(X) - T,

fFo(X) - `6(X),

.F0(X) -. .FO(X).

The end of this section describes the properties of these morphisms. We use the following theorem in §2, 3 to study the norm of lagrangian functions. THEOREM 2.3.

u=

1°) Let

x E 'F0 rcN Vr

he a formal number with vanishing phase. The condition that it be real,

uERey°, is a; is real, that is, i-'a, is real `dr. The condition u > 0 is

II,§1,2

0<

79

as

v'

that is, i`as > 0,

,

where s is the first r such that a.: 0. Thus every u E Re F ° satisfies either u > 0 or u <, 0. Hence > defines an order relation on Re go. 2°) The condition u'12 E Re .° is equivalent to the following:

u > 0;

u c .f °; Notation.

the integer s (defined above) is even.

u'12 may be chosen in two ways; we take u'/2 > 0.

Proof. Let U = Z,EN(a,/V') E .F°; let s be the first r such that a, 0. If a,v-' is real valued for every r, the proof of theorem 2.2 constructs a real-valued function f c u, so that u c Re go. It follows that

Re u =

Im u = E Im a ,

Re '

rEN

rEN

V

Thus u c Re .y ° if and only if a,v-' is real for every r.

Let us study uSince as

2

mod

u = v2s

1

V2s+1

the assumption u112 c Re s is even,

° implies:

as/i' > 0, u e Re y ° (see remark 1).

For the converse, it evidently suffices to prove the following:

u1j2 e Re.y ° when a° > 0 and u e ReJF°. This is clear because the condition 2

.

a

V.

is equivalent to the condition f3 = a° and to certain conditions giving (i°/i, (for each r = 1, 2, ....) as a real linear function Of 0tr and Mr-1

(0 < t < r). Now u'12 E Re .F° implies u > 0 (see remark 1). Hence, if u e Re F

then ;v` > 0 implies u > 0. Thus u c Re _F' satisfies either u > 0 or

u<0.

80

II,§1,3

3. Integration of Elements of .po(X)

The essential properties of formal functions will be deduced from the following theorem, which supplements some classical results (the method of stationary phase). Its proof requires the use of lemmas 3.4 and 3.5 in place of the lemma of Marston Morse [12], by which a function with a nondegenerate critical point is transformed locally into a quadratic form by a change of coordinates. We briefly recall the other lemmas needed for the proof of this theorem. By corollary 2, asymptotic integration is a morphism

:.F0(X) - W. sx

Clearly,

u(v, x) d`x e 9

1, when the phases of'u are all constant. Theorem 3 shows that

vy2 ju(v , x) d`x c- .F in

the important case considered in corollary 3. Let u E .FO(X).

THEOREM 3.

1°) The value of f xu(v, x)d`x depends only on the restriction of u to an arbitrarily small neighborhood of the set of critical points of the phases of u; this value is 0 if this set is empty. 20) Let

u(v) =

x'

e" ° e FO(X),

rrN v

where cp has a single nondegenerate critical point xc on Supp u. Let cpc = cp(xc) be the critical value of p; arg i112 = nl/4 and v Hessc cp be the value of ,/Hess at xc (I,§1,definition 2.3). Then

f u (v, x) d`x = Jx where

nil11Z /I (I2VI

(Hessc(P)112

V1

rEN

r

(3.2)

II,§1,3

81

I0((p a) = ao(xc),

II r

I,(cp, a) =

(3.3)0

2(r-s)

S=O ;=0

1*r S+;

(r - s + j)!j !

81

ax) F0'(x)xs(x)1 ))x=x (3.3),

4), dD*, and O have the following definitions.

Notation. The second-order Taylor expansion of cp at xc is written

(p(x)=qc+ 1(x-xc)+O(x),

(3.4)

where O vanishes to third order at xc and b is a quadratic form X -+ R; b* is its "dual" form, that is, (D* (p) is the critical value of the function x H (D(x) +
(3.5)

In other words, if M is the value of the matrix ((piX,) at xc, then

*(p) _ -i,

b(x) = z<Mx, x>,

where M = 'M: X - X*.

(3.6)

The following is evident. COROLLARY 3.

Let u e .F0(X). If all the critical points of all the phases

of u are nondegenerate on Supp u, then vq2 f

u(v,x)d'xe9;

x

the phases of 0 2 f x u d'x are the critical values of the phases of u on Supp U.

It clearly suffices to prove theorem 3 when u is replaced by a function f given by

f(v, x) = a(x)e'1*),

(3.7)

where a e !2(X) (that is, x is infinitely differentiable and has compact support). The theorem then results from the following lemmas, the first of which proves part 1 of the theorem. LEMMA 3.1. We denote by xc the critical points of (p belonging to Supp f If they are all nondegenerate, then

a(0)e°1'(")d'x mod N sx

v

II,§1,3

82

is a linear function of the values of the derivatives of a of order <2N at the points xc. Proof.

This amounts to proving the following:

If a vanishes to order 2N at the points xc, then f f d`x = 0 mod N. v X

(3.8)N

Now (3.8)o is obvious. Suppose (3.8)N holds. If a vanishes to order 2(N + 1) at the points xc, then there exist fi e 9 such that

8icpxi,

vanishes to order 2N + 1 at the points xc.

Hence by (3.8)N,

f,

-

ae"°d'x = - f v

i Oxi

X

e"'d`x = 0 mod

1

which proves (3.8)N+,

The next lemma is proved similarly.

Let a e 9(X) (Schwartz space; see I,§1,1). Let (b be a nondegenerate quadratic form X - R. Then LEMMA 3.2.

a(x)e"m's)d'x mod N v

1.

is a linear function of the values of the derivatives of a of order <2N at 0. More explicit is the following. LEMMA 3.3.

Let a e .9'(X). Let 4) be a nondegenerate quadratic form

X - R. Then 1,

o (x)e"m(x)d'x (2mi)h/2 IVI

)

[Hess(D]-1/2I rev' rtN

(D*r

I

(

x

modvl .

x(x) =0

Proof. We know (see I,§1, proof of lemma 2.2) that e-'``1 ' dx

f-,

= 1p

1 '2 for larg pj < 7t/2.

II,§1,3

83

Hence, by differentiating with respect to µ,

= 2 µ(2µ)'l/z (20f

W

r° x 2i e-uX'/2dx

r!

-l/zµ rexp?µd / `7` z

2

xz.

L

z=0

Moreover, xzr+1

a-uX=/2

dx = 0

`dr E N.

Thus for any I x I diagonal complex matrix M, with nonzero eigenvalues whose arguments e I - n/2, n/2I, and for any polynomial P : X - C, P(x)e-m`(X)dix

J.

= (2ir)'/z[Hess cQ -112[eY',(a/3X)P(x)]x=o,

(3.9)

where (D,(x) =

i<Mx, x>,

'P (P) = i< P, MC 1P);

arg Hess (Dc is the sum of the arguments of the eigenvalues of ct explicitly, we write

.

More

where v e ill, x[ and b+ and D are two quadratic forms X -+ R, independent of v; (p+ is positive definite; we assume D is nondegenerate. The assumption that the matrix M, is diagonal is superfluous because a change of coordinates reduces two such forms to

+(x) = >x,

fi(x) _ > cix; i

i

,

where ci

0.

Let e c -9(X) be such that s(x) = 1 for x near 0. Lemma 3.2 and (3.9) give \1/2

C

2v

J(x)P(x)e v(x) &.(x)dix

= CHess

-

\ - uz (D+

[e' (era:)P(x)]s=o mod `1 ;

(3.10)

V

argHess((D - (1/v)(D.+) is the sum of the arguments of the eigenvalues

I1,§1,3

84

of b - (1/v)1+, which belong to 10, its. The argument of -v = Ivl/i is -n/2. The right-hand side is a function of 1/v, which is holomorphic for 1/v = 0. Hence, mod 1/v', it is an element of F whose limit, when 1 + vanishes, is

[Hess

mod

1

v

from the Taylor series of this function; arg Hess d) is given by I,§1, definition 2.3. Thus, mod 1/v°°, the left-hand side of (3.10) is an element of F; by lemma 3.2, its limit, when 1 + vanishes, is

f,

e(x)P(x)e"O(x)dCx.

Thus e(x)P(x)evo(x)dtx

=

(21ri

1/2

[Hess(D]-in[ecu")m'(a/Ox)P(x)]x=o modvI ;

(3.11)

IVI

lemma 3.3 follows by approximating a at the origin by its partial Taylor series P and applying lemma 3.2.

Lemma 3.3 proves part 2 of theorem 3 in the particular case where the phase cp is a quadratic form 1. The general case is deduced from this special case by applying the following lemma to the remainder of a partial Taylor series. The proof of the lemma is analogous to that of lemma 3.1. LEMMA 3.4.

0e 10, 11,

Let

xeX.

Let a: (0, X)HOC(o,x)eC

be infinitely differentiable with compact support, vanishing to order 2N when x vanishes. Let cp : (0, x) i-- cp (0, x) c- R

be infinitely differentiable and such that on Supp a, for every

0,

85

II,§1,3

(p: x r-+ 9(0, x) has only one critical point, which occurs at the origin and is nondegenerate. Then

JdOJc(Ox)evcP(0x)d1x = 0 mod

I

(3.12)

V IV,

x

o

From this lemma we deduce the following. Let [p : X - R be infinitely differentiable with only one critical point, which occurs at the origin and is nondegenerate, and critical value zero: LEMMA 3.5.

(px(0) = 0,

(p (0) = 0,

Hess (p(0)

0.

Let

cp(x) = I(x) + O(x) be its second-order Taylor expansion at the origin: 4) is a quadratic form; O vanishes to third order at the origin. Let a:X - C be infinitely differentiable, with support that is compact. Then

Y

Jx

`O'(x)a(x)evO(x)d`x mod

j=0 j! x

(3.13) v

This series converges in .`f since, by lemma 3.3, vi

f Oj(x)a(x)e''m(x) d'x = 0 mod v-('/2)-[(i+l)/2] x

where [ Proof.

.

] denotes the integer part of .

-

that is, the greatest integer

By (3.12) and Taylor's formula,

e= 2N-1 vlOJ J! v

e

vm

(1 -

+v 2N

J=1

0

e)2N-1

(2N - 1)!

O2Ne

0.

(3.14)

Lemma 3.4 gives v2N f xX

f1(1

-

0) 2N-1

o

0 mod N

(3.15)

v

since O2N vanishes to order 6N at the origin. When N tends to infinity, (3.14) and (3.15) imply (3.13).

II,§1,3-II,§1,4

86

Proof of theorem 3,2°). We may replace u by a function f given by (3.7); qp has a single critical point that is nondegenerate. We may assume that this critical point is the origin and the critical value is zero. Then by (3.13) and lemma 3.3, a(x)e,9(Y) drx_ 27ri

V2

(Hess (D)-112 1

,1 r !j ! vr-i

IvI

a

f

I

[®'(x)a(x)]jmod 1

\ax/

x=0

,

v

(3.16)

where r, j e N. Denote the exponent of v by q = r - j. Since O(x) = 0 mod Ixt3, the condition { }x=o : 0 implies 3j 2r, that is j < 2q. Let -

O*q+j[Oj(x)a(x)1JIlj Iq(o; a) _ 2 x=0 i=0 (q + j)i j z

1

i

(')

in particular,

Io((p; a) = 00). Now (3.16) can be written

ess)

= fX

gIq((p; a),

12

(FV

qeN

whence part 2 of the theorem follows. 4. Transformation of Formal Functions by Elements of Sp2(l)

Each S E Sp2(l) induces an automorphism S: ''(X) -+ W(X) (see section 1) whose restriction to

.FO(X) c '(X) (theorem 2.2) is a monomorphism:

S :.F0(X) - W(X). Theorems 4.1, 4.2, and 4.3 develop the properties of this monomorphism.

Let Z be a 2-symplectic space, V a lagrangian manifold in Z, R' a 2 -frame of Z, and THEOREM 4.1.

87

II,§1,4

S E Sp2(l);

R = SR' is then a 2 -frame of Z (I,§3,3). 1°) There exists an isomorphism, induced by S and denoted S,

u ER,, R') , o(V\ER u ER,, R),

(4.2)

characterized by the two following properties:

i. It is local. That is, if UR. E. '0(V\ER u ER', R'),

UR = SUR, E'F0(V\ER U ER', R),

then UR(v, 1), the value of UR at z e I, is a linear function of the values of UR. and its derivatives at the same point f. ii. The following diagram is commutative: y0(V\ER u ER,, R') - 'Fo(V\ER U Z`R., R) nRl

(4.2)

1-14

Fo( X)

S

) W(X )

.

(4 . 1)

It thus defines a morphism

S o IIR, = rIR o S:.FO(V\ER u ER,, R') - '0(X).

(4.3)

2°) The composition law of the morphisms (4.1) and (4.2) is that of Sp2(l). Remark 4.

The restriction of (4.1),

S:IIR. O(V\ER u ER,, R') - IIRAO(V\ER U ER,, R) - .FO(X),

(4.4)

is thus an isomorphism. While (4.2) is local, (4.4) is not local, but only pointwise: If U' E IIR,.,O(V\ER U

R'),

u = Su',

then the value u(v, x) at x e X is a linear function of the values of u' and its derivatives at the points of the finite set RXRX'x n Suppu'. The local character of the morphism (4.2) is made explicit as follows. THEOREM 4.2.

11x,e°(0" E'0(V\ER U ER., R').

UR. = rEN

Then

Let

V'

88

II,§1,4

where R = SR',

UR = SUR. E .Fo(V\LR U ER,, R),

is given at i by U

UR(V,

Z)

x3r+1/2 1

E (dtX = rEN

Vr

JR,, (' ; a) e

vroA(=)

(4 . 6)

where

x'=Rzz,

x=RXZ,

(4 . 7)

[dtx] 112 is defined in I,§3,corollary 3.

'IR.r is a linear function of the derivatives of the as (s <, r) of order < 2(r - s)

on V Its coefficients are infinitely differentiable functions of z E V\LR.; these functions depend on V, R', and S. (4.8)

JR.OIZ, x) = 00(2)

Formula (4.6) retains a meaning for all UR. E y (V \ER, , R') and i E V\ER U ER' ; therefore the following theorem results from the preceding one. THEOREM 4.3.

Formula (4.6) defines an isomorphism induced by S and

denoted S:

S:.F(V\ER U ER., R') , .F(V\ER u ER., R).

(4.9)

The composition law of these isomorphisms is that of the group Sp2(l).

Proof of the preceding theorems. that the restriction

I : Supp UR. --+ X of RX : V

Let UR. E Ao(V\ER V ER., R') be such

X

is a diffeomorphism, with inverse denoted AX 1 The case S 0 Esp,. Then S = S,1 (I,§1,1). Let OR. be expressed by (4.5); U' = rIR. Ua.

is given at x' by

u'(v, x') = E lxr(x')eVP'(x'), r rEN l

where

(4.10)

II,§1,4

89

a;(x') = a,(RX 1 x'),

cp'(x') = cpR,(RX 1 x') for x' e Supp u',

a; (x') = 0 for x' 0 Supp u' = RX Supp UR..

Since u' e po(X) c '(X), we have u = Su' e '(X). By I,§1,(1.10), we have the following expression for u at x :

_ u(v' x) -

()2 IVI U1 f 2niA(A)

X rEN Vr

/

ar(x

)e"1n(x,X)+w'(x)]dlx'.

(4.11)

We apply theorem 3 to this asymptotic integral. i. For each x e X let us find the critical points of the phase X -3 x' -- A(x, x') + 1p' (x') e R

(4.12)

in Supp u'; that is, the x' e Supp u' such that

As.+9',,=0.

(4.13)

Let us write

i = AX 1x',

R'z = (x', p'); that is, p' = q . e X*.

(4.14)

Relation (4.13) becomes

p'+Ax,=0. By I,§1,(1.11), this means that there exists p e X* such that (x, P) = sA(x', P')

-

Thus (4.13) is equivalent to

x=

RxR'-1(x', p')

= RXR' 1(x', p').

By (4.14) this says that (4.13) is equivalent to

x=fix. Therefore, on Supp u' the critical points of the phase (4.12) are the points

x' = I' i, where 1 e RX 1x n Supp UR..

(4.15)

This set is finite since Supp UR. is a compact set in V\ER and RX is a local diffeomorphism in a neighborhood of this compact set. ii. Let us find the critical values of the phase (4.12). At a critical point x' defined by (4.15), let

II,§1,4

90

Iii = (x, P),

R

= (X" p');

from the preceding formulas,

p' = -Ax..

p= Ax,

Hence, since A is a quadratic form in (x, x'),

A(x, x') = z - i , that is, in view of I,§3,(2.4), A(x, x') = cPR(z) - APR (z), where cpR,(z) = cp'(x').

Therefore at the critical point x' defined by (4.15), the value of the phase (4.12) is (PR(4

iii. The square root of the Hessian of this phase (4.12) is given by I,§3, (3.7): J1/2.

{Hess..[A(x, x') + (p'(X)]}l,Z =

(4.16)

While calculating Hess.-, x and x' are viewed as independent. Afterwards

we give x and x' the values x = RXi and x' = RXi so that x and x' are local coordinates of the same point i e V\tR u R. Taking these values for x and x' in the right-hand side, it follows from (4.16) that Hess.. [A (x, x') + p'(x')] : 0. Consequently theorem 3 can be applied to the calculation of (4.11). iv. Completion of the proof. This application of theorem 3 to (4.11) gives U(V, X) _ Y_

(4.17)

UR(V, z),

zERx1X

where: UR has the same support as OR. ;

defining x = RXi and x' = RXi, the value of OR at Z is UR(v, i) = A(A)[Hess..(A + gy0')]_1/2 Y 1 I,(A + (p'; x') e"*-('). ti'

Now on one hand, (4.17) means

U = nRUR,

(4.18)

II,§1,4-II,§1,5

91

but on the other hand, by (4.16), (4.18) is equivalent to (4.6). Indeed, by (3.3) Ir(A + (p'; a') is the value of a linear function of the derivatives of

the as(s < r) of order <2(r - s); the coefficients of this function are rational functions of the derivatives of cp'; the common denominator of these rational functions is [Hessx.(A + (p')]3i, whose value is given by (4.16).

Thus theorems 4.1 and 4.2 hold for SA E Sp2(1)\Esp2 The case where S E Esp2. By I,§1,lemma 2.5, given S e Esp, there exist SA and SA' E Sp2(1)\Esp2 such that S = SA, SA.

Since the morphisms (4.1) and (4.2) are composed like the elements of Sp2(l) that induce them, the compositions of the morphisms induced by SA and SA, depend only on S = SA.SA; these are then morphisms induced by S. Given UR. such that Supp UR c V\±R' U ESR', we can choose SA

sufficiently close to the identity so that tSAR, n Supp UR, = 0, whence follow theorems 4.1 and 4.2, which imply theorem 4.3. 5. Norm and Scalar Product of Formal Functions with Compact Support

Let V and V' be two lagrangian manifolds in Z, V and V' their universal covering spaces, R a 2-frame of Z, c°R and wR the phases of R V and R V,

0 and 0' the lagrangian phases of V and V. Let UR = Y- laR reYWR E fo(V\iR, R), reN V'

is aR sev'6 E Col V'\i' , R),

UR = SEN

with projections U = IIRUR,

u' = IIRU'R.

Definition 5.1. Under assumption (5.1) the scalar product of UR and UR is the asymptotic class (UR I UR) = (u I u') E

(5.2)

where (u I u') is defined by the asymptotic integral (1.9). Thus (URI UR) and (UR UR) are complex conjugates and the Schwarz inequality applies:

92

II,§1,5

(UR I UR)112(UR I UR)1f2.

(UR I UR)I

The seminorm of UR is (UR UR)112 = iI u the triangle inequality.

0; this seminorm satisfies

The value of (UR I UR) depends only on the behavior of UR and UR at the pairs of points of V and V' projecting onto the same THEOREM 5.I.

point of V n V'. This value is 0 if V n V' = 0. 2°) If V = V', which implies cpR = (pR, then

(UR Ua)

= Y ( .1)5 .,s

a .°F,

aR..(z)aR.s(z

(5.4)

zEV ;.Z

V

where:

x=Rxz,d'x>0; i and i are the points of Supp UR and Supp OR' projecting onto z. In (5.4), fi(i) - O(z) is one of the periods of 0: cY = 2

f [z, dz], where y is a cycle in V. r

Thus the phases of (UR I UU) are these periods c., of 0.

3°) If V and V' are transverse, then (5.5)

vy2(OR I UR) a

If V and V are transverse and intersect at a single point z that is the projection of a single point i of Supp UR and a single point i of Supp UU, then 1/2

2

li

(UR I UR)

Hess

'

3.-112P,

aR, aR

a°[0()-OV')]'

where

(p and (p' are defined near Rxz by

W(x) = (PR(RXIx n V), V(x) = (6(RX1x n Hess((p - (p) = {Hessx[(p(x) - (p'(x)]}x°R:z ;

II,§1,5

93

P, is a sesquilinear function of the values of the derivatives of OCR, . and YR,s (s < r) o f order < , 2(r - s) at 2 and z' ; the coefficients of this function depend on the behavior of V and V' at z; (5.7) PO(aR, aR) = a R,o(Z)aR.o(fl 4°) [Invariance under Sp2(1)]. Let S E Sp2(1). Under the assumption

U' c- FO(V'\ R V t'SR, R),

OR E .FO(V`tR U ±SR, R),

which is stronger than (5.1), we have (5.8)

(UR I UR) _ (SUR I SUR).

Proof of 1°), 2°), and 3°).

u and u' can be written in the form

u(v, x) = Y vv(v, x)e""AX) where vi(v, x) _ Y_ I, v;.,(); rEN '

jEJ

where vk(v, x) _

u'(v, x) = Y dk(v,

lr

vk,r(X);

rEN V

kEK

J and K are finite sets. By part 1 of theorem 3,

(u I u') _ f u(v, x)u'(v, x) d'x x

depends only on the behavior of the pairs vie"wJ, vke"c1 at the critical points of cps - wk, that is, at the points X E X such that ca,, = cpk,.

In other words, (OR I UR) depends only on the behavior of OR and UU at pairs of points (z, z') E V x V' such that

Al = Iii'

= (x, c'p

) = (x, 9"J

These pairs are the pairs of points of V x V' projecting onto a single point z of Vn V. 1°) and 2°) follow, as does 3°), by (3.2) and (3.3).

Proof of 4°). Lemma 1.1 and part 1 of theorem 4.1.

The invariance of (UR UR) under Sp2(1) stated in part 4 of theorem 5.1 raises the following problem: to give invariant expressions of the right-hand sides of (5.4) and (5.6). Theorems 5.2 and 5.3 will give such expressions mod(1/v).

94

II,§ 1, 5

Notation. define

Let , and n' be regular positive measures on V and V'; we

argil = argil' = 0. We assume_V and V' are both given 2-orientations: If z e V, x = Rxi, then d'(Rxz) = d'x '1

is a function V - R

n

vanishing on iR whose argument is given by I, §3, corollary 3: arg d'(Rxz) = rcmR(z) mod 4ic.

We define Qo : V - C by [d1(1x2)]h/2

f3 (2) =

aR,o(2), where 2 e

From formulas (4.6) and (4.8) of theorem 4.2, Qo is invariant when we replace R by SR without changing V ; $o is called the lagrangian amplitude of UR. This amplitude depends on the choice Definition 5.2.

of the measure n and the 2-orientation of V. Substituting the expression (5.9) for aR,o and ax the following.

o

into (5.4), we obtain

Suppose V = V', which implies ii = ii"; let Qo and Qo denote the lagrangian amplitudes of UR and U. Then

THEOREM 5.2.

(URIUR)= r

Qo(Z)Qo(fle"[V'(=)-PVC0]n

J ZEV Z.Z

mod 1,

(5.10)

v

where i and 2' are the points of Supp UR and Supp UU projecting onto z.

Notation. (Continued) Let 0 and 0' be the lagrangian phases of V and V'[1, §3,(1.7)]. We make the same assumptions as in part 3 of theorem

5.1. Let il, and , E A,(Z) be tangent to the 2-oriented manifolds V and V' at i and 2', respectively; let A and A' be their natural images in A(Z). Let 10 (and lo) be the measure on A (and %') that is translation invariant and equal to P1 (and rl') at i (and verse (as in part 3 of theorem 5.1); thus

V and V' are assumed trans-

II,§1,5

95

Z = A Q+ A';

110 A no is a measure on Z.

(5.11)

By (5.1) and part 1 of theorem 5.1, the two sides of (5.6) are zero unless we assume

z' E V'\R,

z E V\ER,

(5.12)

that is,

RA and RI' are transverse to X*. Substituting the expression (5.9) for aR.o and aR.o into (5.6) we obtain, under assumption (5.12), Cvl J112 (UR I UR)

[Hess( - Wr)] mod 1

112

[ d`

]112 r

U1

xz)

]1o2

xz ) (5.13)

.

V

Lemma 5 transforms this formula into the following.

If V and V' are each given a 2-orientation, are transverse, and intersect only at a point z that is the projection of a single point i of Supp UR and of a single point f of Supp UR, then THEOREM 5.3.

L VI

q Od

(UR I UR)

l2

112

mod

(5.14) V

where m is the Maslov index, defined mod 4 (I,§3,3) and rlo A 11o and d 21z are the measures on Z defined, respectively, by (5.11) and (5.15).

1°) The symplectic structure of Z defines a measure on Z that is invariant under translations and under Sp(Z):

LEMMA 5.

d21z

where

(dx A dp)1 = (-1)1(1 -1112d1x A d'p,

(5.15)

II,§1,5

96

dx n dp = > dxj n dpj ;

Rz = (x, p),

j=1

{xj} and { pj} are dual coordinates on X and X *. 2°) Under assumption (5.12), with the notation (5.9),

d'(Rxz) d'(Rxz )

Hess((p

d21z

(5.16)

no Ano

arg Hess(rp - (p')

d'(Axz) d'(Rz )I

L C

= nm(1.4i A4) mod 4n.

(5.17)

n'

n

The value of dp n dx on a pair of vectors z, z' is [z, z j

Proof of 1°).

Proof of (5.16). Since RA and RA' are lagrangian and transverse to X by (5.12), their equations have the form

RA:p = O(x),

R.Z':p = (D ,(x),

(5.18)

where 0 and 0' are two quadratic forms X -+ R. Since A and A' are transverse,

- (D')

Hess(

0.

By (5.11), each z E Z decomposes uniquely as the sum of a vector in A and a vector in A':

Rz = (x + x', is(x) + 'D

(x')) E Z(1),

where x and x' e X are unique. It follows by (5.15) that (-1)1(1-1Y2d'[x

d2'Z =

_

+ x'] A d'[''x(x) +' '.(x')]

(-1)1(1-1W2d'[x + x'] A d'[d'x(x)

(- l)t(t+1)/2 Hess((D

-

a)s(x)]

- cD')d'x A d'x'

_ (-1)tu+1U2 Hess((D - tb') d'(Rx()d'(Rx(') no A no, no

where

no

e A, ('e A'; hence (5.16) follows.

Proof of (5.17).

By definition [I,§3,corollary 3, (3.2) and (3.3)]

argd(RxZ) = nmR(z) = nm(X4, RA4) mod 41r;

I

II,§1,5-II,§1,6

97

thus

nm(A4, ,a) - arg d`(xz) + arg d`(Rxz ) n

n

= nm(14,,a) -

nm(X*a, RA4) + m(X a*, Rya)

= n Inert(X *, RA', R)) mod 4n

by I,§2,(8.2). Then, by the definition of arg Hess [I,§1,(2.1)], to prove (5.17) it suffices to show that

Inert(q) - tb') = Inert(X*, R)', R)). Proof of (5.19). RA:p = (D .,(x),

(5.19)

Let us write down the equations of R..., RA', and X*:

R)':p' _ Dx(x'),

X*:x" = 0.

The definition of Inert (I,§2,4) makes use of z' E RA',

z E R1.,

z" E X* such that z + z' + z" = 0.

Define z = (x, P),

z' = (x', P),

z , = W" P");

x"=0,

p+p'+p"=0,

hence

x+x'=0,

[z",z]=(P",x)=-(P,x>-(P',x>

_ - (cb (x), x> + (b' (x), x> = 2 b'(x) - 2 b(x). Consequently,

Inert(R), R)', X*) = Inert(O' - 0). (5.19) follows by I,§2,(4.4) and I,§1,definition 2.3, of the inertia of a form.

6. Formal Differential Operators Definition 6.1.

Let 0 be an open set in Z. Let

denote the set of

formal functions with vanishing phase

a° = 1 1r ar rEN v

(6.1)

98

II,§1,6

such that the a, : 0

C

are infinitely differentiable. We give the vector space y °(0) the topology defined by the following system of neighborhoods .N'(K, p, r, e) of the origin:

p, r e N,

K is compact in 0;

Ee

a° E A' (K, p, r, E) means that on K the derivatives of x5 (s < r) of order p have modulus <E. Let us say that a° is a polynomial if (v, z) i-- a°(v, z)

is a polynomial in 1/v and in the 21 coordinates of z. By the Weierstrass theorem the set of polynomials in.F °(Z) is dense in. °(Q). y °(S2(1)) is defined similarly by replacing 0 by a domain S2(l) of Z(1) _

XQ+X Every 1-frame R of Z (hence, a fortiori, every 2-frame R) induces an isomorphism .97'(Q) 3 a " aR e

°(SZ(1)), where Q(I) = RSZ,

defined by the formula a°(v, z) = aR(v, Rz)

Vv, z.

The formulas (3.4) I,§1, a,+, (v, x, P)

r

1

= CeXp 2v

08x' 8

8p

] a° R(v,.x, p),

(6.3)

aR (v, P, X) =L Cexp 2v - - `\8x a 8p a

I] ae(v, x, P)

define two automorphisms of y °(Z(l)) that are inverse to each other: o + aRHaR,

o aR"aR.

Definition 6.2 of formal differential operators rests on the following lemma.

II,§t,6

99

Let

LEMMA 6.1.

u = ae"°e.F(X),where a = E lrar. reN V

At a critical point x of the phase cp we have h

u(v, x) = 0 mod vlhl(z .

(v ax

Proof. It suffices to consider the case a = a°. Then at a critical point of q , h

a) u = vlh Phe"W (h is a multi-index), v where PP is a polynomial in the derivatives of a of order e 10, IhII and in the derivatives of vcp of order e 12, h] (since qpx = 0). Each monomial

in Pti is a product of derivatives whose orders sum to I hI. Hence the power of v in such a monomial is <, Ih1/2. The lemma follows. Definition 6.2.

Let

(y, x) = qq(x) - qq(y) - <9,,(y), x - y>

(6.5)

denote the remainder of the first-order Taylor series of the phase cP of (6.4) at the point y. To an element a° of .F°(SZ) and to a frame R, let us associate the local endomorphism aR of .F(X) and .F°(X), called a formal differential operator, defined by the two equivalent formulas (ax u) (v, y) Ja'lhPhR

hl1 l h

( v \

(V, X, P) l

aR

1 a h [IhIc2_ \v

2Ph

(v ,

l )h [a(v,

x)e°o(Y,xl] lx=Y

e"w(Yl

P=Wr(y)

p, x)a(v, x)e

vo(y.x )

J

"W(Y)

x=v P

e

(6.6)

W,(y)

By lemma 6.1, each of these two formulas makes sense when (y, (py) e SZ(l): we say that aR is defined on S1(l) = R.

Proof of the equivalence of these two formulas. Leibniz's formula gives (compare I,§l,proof of lemma 3.1)

100

Y

II,§1,6

h!

a,

p, x)(

p)h

(V

v(v,

x)]

ax)h[aIhla_(v

1

= f j!k! =:

k

a1zj+klaR(v, p, x) 1 a (ap)'+k(v axy v ax

uv (, x )

1 alk'aR (v, x,, p) (1 akll(V, X) v OX) (ap) k!

because

a

1

j!

a

[exp v ax ap

(ap)'(v ax)'

]

aR (v,= aRp,(v, x,x.p).

Justification of definition 6.2. Let us show that it is equivalent to the definition in section 1 when a° is a polynomial, that is, when aR is a classical

differential operator with polynomial coefficients in (1/v, x). In this case (6.6) means (aRU)(v, X)

=

{a+

v]}Y

v' x, P +

V

x

aX) `U(v' x)e

.

IP=(O,

Now aR

(V, X, P +

x)e-v(P.x-Y>] V

ax) `u(v,

=

e-tiaR

(V, X,

ax)U(V, X). V

Then, taking a+ (v, x, av/ax) = a- (v, av/ax, x) in the sense of section 1, we have (aRU)(v, x) = aR (V, X,

V

ax) U(v' x) = aR (v'

V ax'

X) U(V9 X).

Notation. Denote (aRU)(v, x) = aR (V, x,

v

-V, ax) u(v, x) = aR (

V aX,

X) U(V, X).

Let aR denote the unique local Definition 6.3. Let V c Q, a° E R) such that endomorphism of aRfIR = IIRaR.

(6.7)

aR, being local, extends to an endomorphism of.F(V\ER, R), which has the following properties.

11,§1,6

101

THEOREM 6. 1°) The formal differential operator aR is local (in the sense of part 1 of theorem 4.1); hence

Supp(aRUR) c Supp UR n Supp(a°). 2°) The mapping

`-V°(SZ) X .-(V\tR, R) 3 (aR, UR) r--+ aRUR E y (V\ER, R) is continuous. 3°) Each S E Sp2(1) transforms aR into an operator

aSR = SURS-

1

defined as follows: aSR(SUR) = S(aRUR) for all UR E

U 'SR, R).

aR and its transform aSR are associated to the same function a°, which satisfies

a°(v, z) = aR(v, Rz) = asR(v, SRz)

Vv, x.

(6.9)

4°) The formal differential operators aR and bR are adjoint, that is, (aRUR, UR) = (UR, bRUR)

'UR, UR,

if and only if b°(v, z) = aa(vz)z)

Vv, z.

5°) If aR and bR are two differential operators associated to elements a° and b° of 3 °(Z), then their composition cR = aR o bR is associated to the element c° of.F°(Z) defined by the formula

{[exp

c°(v, z)

=

- 2v (az' a?

)J [a°(v, z)b°(v, z')] }Z=: ;

(6.10)

there

Cazl

a

'N

x' tip'/-\ - \ex

p

(6.11)

where Rz = (x, p), Rz' = (x', p'), is clearly independent of R. Remark 6. co = a° o b° + (1/2v)(a°, b°) mod(1/v2), (,) being the Poisson bracket; see II,§3,(3.14).

II,§1,6-II,§1,7

102

Proof of 1°). The definition of aR. Proof of 2°). Use the definition of aR and the definitions of the topologies of .F°(Q) (section 6) and of R) (section 2). Proof of 3°.), 4°), and 5°).

Apply I,§1 (theorem 3.1, theorem 3.2, and

lemma 3.2) to the special case in which a° is a polynomial. This special case implies the general case, since (6.8) is continuous and the polynomials are everywhere dense in.&°(SZ).

Corollaries 3.1 (adjoint of an operator) and 3.2 (self-adjoint operator) of I,§1 clearly apply to formal differential operators. 7. Formal Distributions

In sections 1 and 2, it is not possible to replace .9(X) by .99'(X): the proof of theorem 2.2 breaks down because So'(X), unlike Y(X), does not have a countable fundamental system of neighborhoods of 0. The algebra

Definition 7.1.

R) is the set of functions

f:V\ER-.C that are infinitely differentiable and compactly supported. x = RXZ is used as a local coordinate on V\ER.

Given a compact set K of V\ER and a positive number b, for every 1-index r, we let

B(IC, {b,}) =

{f e

R)

Supp f c IC,

< b,f.

The subsets of these B(K, {b,}) are by definition the bounded sets of -9 U \tR, R). Definition 7.2.

f' :

The vector space

R) is the set of linear mappings

(V\R, R) - C

that are bounded on each bounded set of -9.

These f' are distributions; they have supports and derivatives of all orders with respect to x = RXZ ; these derivatives are distributions. We make the convention -9 c -9'; -9' is a vector space over the algebra -9.

II,§1,7

103

The value of f' E -9' at f e -9 is denoted

f'(z)f(f)d'x, where x = AXi. flP

Then

1a V ax

is defined on .9' by the requirement that it be self-adjoint:

f

[1f'()]JdI x =

IIV

f (Z)

ax f (z

ff,

d'x.

Given a bounded set B in -9(V\ER, R), we let

I FIB = Sup f7 f'(zf(Z)dix feB

< 00.

The topology of 9'(V\ER, R) is defined by the following fundamental system of neighborhoods of 0, each depending on a bounded set B of -9 and on a number e > 0: Al" (B, e) = { f' I If'IB , e}.

The f' with compact support form a subspace 2 of -9' that clearly is dense in -9'. It can be proved (as in L. Schwartz [13], chapter VI, §4, theorem IV, theorem XI and its comment) that -9 is dense in !'o, and thus R) is dense in Definition 7.3.

R).

A formal distribution UR on V consists of

i. a 2-frame R of Z, ii. a formal series

-

«r

a "),tR

reN V

where (PR is the phase of R V and a, e

Supp UR = U Supp a" c V.

R).

II,§1,7-II,§2,0

104

The set of formal distributions UR on V forms a vector space R). Its topology is defined by the following fundamental system of neighborhoods of 0, each depending on a bounded set B of R) and on two numbers c > 0, N > 0:

1''(B,s,N) _ {URIIa,IB < efor r < N}. Clearly .

-o(V\ER, R) is dense in

.

'(V\ER, R).

By (4.6), the isomorphism (4.9) defined by S = RR'-' E Sp2(l) extends to an isomorphism

S:.f '(V\ER V ER., R') - .y'(V\ER U ER., R) that is continuous and local. Similarly, by (6.6), the formal differential operator aR extends to an endomorphism

aR:R) -

R)

that is continuous and local and satisfies theorem 6. The scalar product of UR and UR, a formal function and distribution

on the same V\ER, is defined by formula (5.4) of theorem 5.1 when Supp UR n Supp UR is compact; it is invariant under Sp2(l) (part 4 of theorem 5.1) and satisfies theorem 5.2.

§2. Lagrangian Analysis 0. Summary

Synthesizing the properties of formal functions and formal differential operators from §1, we define and study lagrangian operators, lagrangian functions and distributions, and their scalar products in §2. These three notions make up the structure called lagrangian analysis. In section 1, we define lagrangian operators and study their inverses. In section 2, we define lagrangian functions on the covering space V of a lagrangian manifold V; their scalar product (' ) is defined when their supports are compact. This makes it possible to define lagrangian functions on V in section 3; their scalar product is defined when the intersection of their support is I

II,§2,0-II,§2,1

105

compact; section 3 requires the datum of a number v° e iJO, 001; it uses the "restriction" of v to the value v° in the expressions UR of a formal function U. This process of defining lagrangian functions on V has a theoretical justification, which is the possibility of defining their scalar product (I.), and a pragmatic justification, namely, the process of many "approximate calculations." An example is quantum physics, where 2ni/h plays the role of our variable v e ill, 001, h being Planck's constant; here h is taken to be infinitely small without comparing the orders of magnitude of the numerical values taken by the terms coming into play when h is given its physical value; h is given this physical value after completing the calculations in which h is assumed to be infinitely small. Another historical example is celestial mechanics; H. Poincare has elucidated how such a

process is an approximate calculation capable of predicting celestial phenomena with extreme precision using divergent series in which ultimately only the first terms have to be retained. V. P. Maslov wants to obtain "asymptotics" by the same process, that is, by making approximate calculations. In chapter III, we shall apply this process to cases in which it is certainly not an approximate calculation, and we shall recover numerical results that are in agreement with experimental results. Therefore, to make this process coherent is "a problem that poses itself and not a problem that one poses" in the sense that H. Poincare meant it. 1. Lagrangian Operators Let Z be a symplectic space and Q an open set in Z. Definition 1.1. a R= aR( v, x'

Let a° e ,y °(Q) (II,§I,defnition 6.1). Let

)

1 8 1_ _(v, 1 8 v ax J -

aR

V

X

x

be the formal differential operator associated to a° and to the 1-frame R (II,§l,definition 6.2). The lagrangian operator associated to a° is the collection a = {aR } of these formal differential operators aR (a° given; R arbitrary); aR is called the expression of a in the frame R. Theorem 2.3 will justify this terminology. We say that a is defined on Q. We define

Supp(a) = Supp(a°)

II,§2,

106

If

a(v,z) = ceC

Vv, Z'

then a is denoted

a=c. Formula (2.9) will justify the following definition. Definition 1.2.

Two lagrangian operators a and b associated to a° and

b° e F°(S2) are adjoint when b°(v, z) = a°(v, z)

Vv, z;

equivalently, by II,§1,theorem 6,4°), a and b are adjoint when aR and bR are adjoint for every R.

Notation.

The adjoint of a is denoted a*.

Formula (1.2) and the resulting formula (2.2) will justify the following definition. Definition 1.3.

The composition a° o b° of a° and b° e f °(S2) is given by

(a° o b°)(v, z) _ {(exp

where

- 2v[a_, 1

1

1

)I a°(v, z) b°(v,

Oz-

'

is defined by II,§1,(6.11).

° , [aZ 3z'

(1.1)

The composition a o b of the lagrangian operators a and b is the lagrangian operator associated to a° o b°. By part 5 of II,§1,theorem 6, (1.2)

(a o b)R = aR o bR.

Since composition of the aR is associative, composition of the a° and composition of the a are associative. It is easy to verify this directly by observing that (1.1) implies (a° o b° o c°)(v, z)

exp -

l

a

a

l

2v az' aZ' - 2v

a°(v, z)-b°(v,

a

a a a 1 LaZ' az" - 2v az ' 0z"

z")}z.z-=.

.

(1.3)

107

II,§2,1

This formula easily extends to the composition of any number of elements of Clearly

Supp(a o b) c Supp(a) n Supp(b),

(1.4)

(a° o b°)(v, z) = a°(v, z) b°(v, z) mod 1,

(1.5)

V

the right-hand side being the product in F ° of the values of a° and b°: (a° o a°)(v, z) = Fcosh 2v rLa L

z

l a°(v,

z). a°(v,

J

z')

(1.6) =-_

The set of lagrangian operators defined on S2 is thus a noncommutative algebra F °(S2) over the algebra 9' of formal numbers with vanishing phase; 3 °(S2) has an identity element. (Inverse of a lagrangian operator) The operator a has an inverse in the algebra of lagrangian operators defined on S2 if and only if THEOREM 1.1.

a°(v, z) : 0 mod

l

Vz E S2.

(1.7)

V

Recall that, by definition, a°(v, z) is a formal series: a°(v, z) = Y_

1,

,EN t'

a,(z)

(1.8)

Condition (1.7) means

a°(z) j4 0

Vz E S2.

Proof.

(1.7a)

The equivalent conditions (1.7) and (1.7a) are necessary by (1.5). Conversely, if these conditions hold, a right inverse a' of a is an operator associated to an element

a'(v, ') = Y v,a'(-) ,EN

of

°(S2) satisfying

{(exp that is.

d - 2vf ez-,c oz' )a°(v, z)a'0(v, z')}Z=Z. = 1, J 1 1

108

II,§2,1

ao(z)ao(z) = 1,/

((1.10)0

a0(z)ar(z) = Cr(z),

(1.10)r

where cr(z) is a linear combination of the Oz

)'a,(z)]

z

such that 0 < Isl = Isl = r - t - t', t' < r. These conditions determine a'. Therefore a has a right inverse and, similarly, a left inverse; these are identical since the composition of elements of F °(Q) is associative. The theorem follows. The definition of scalar products (sections 2 and 3) will use the following. Definition 1.4. A partition of unity is a collection (aj} of lagrangian operators defined on Z such that Uj Supp a3 = Z is a locally finite covering

of Z and

a, = 1 (identity operator).

(1.11)

This partition is said to be finer than an open covering Uk S2k = X of X when every Supp a3 belongs to at least one of the Stk.

(Partition of unity) 1°) There exist partitions of unity finer than a given open covering of X. 2°) These can be chosen so that the operators a3 are self-adjoint. 3°) More precisely, these partitions of unity can be chosen such that each a, has the form THEOREM 1.2.

a, = b* o b;,

Supp b; = Supp a3,

where b3 is a lagrangian operator and b# is its adjoint. Proof of 1 °) and 2°).

It suffices to prove the theorem when the operators

a; are replaced by C'-functions a;°:Z -+ R that are independent of v. It is thus a classical result.

Proof of 3°). Given a covering Uk S2k = X, we choose Cr-functions b°: Z -+ R such that

Y [b°(z)]2 = 1,

i

Suppb° c S2k for some k.

109

II,§2,1-II,§2,2

It follows by (1.6) that I

b° o bV = Y

2

where (3,:Z -+ R, flo = 1.

reNv

Let us try to find

c= o

I

Vzr Y.,

where Yr : Z - R,

yo = 1,

reN

such that

c°oc° = Yb°ob°, that is, cosh

1 2v

C

Yv az aZa, J ""

r v

zr

Pr'

This condition defines Yr as a function of Pr and of the

[()5

[J)s Y(Z)J

Y,

(z)]

such that

0 < I s l - Is l = 2(r - t - t'),

0 < t < r;

thus co exists, is unique, and has an inverse by theorem 1.1. Now

bi=b*,

c=c*,

Ybobi=coc; i

therefore the

ai = (bioc-1)*o(bioc-1) constitute a partition of unity finer than the given covering Uk S2k = X. 2. Lagrangian Functions on V

Let Z be a symplectic space provided with a 2-symplectic geometry. Let V be a lagrangian manifold in Z and let V be its universal covering space. Theorem 4.1 of §1 justifies the following definition. Definition 2.1.

A lagrangian.function 0 on V is defined by the datum of

110

II,§2,2

a formal function UR on V\ER for each 2-frame R, these formal functions satisfying

= RR'-'V,, on V\ER U ER.

UR

VR, R'.

UR is called the expression of U in the frame R. The support of U, Supp U, is the subset of V defined by the condition

Supp UR = (V\ER) n Supp U

VR.

This definition makes sense since S = RR'-' is local.

The set of lagrangian functions on V and the subset of those with compact support are vector spaces over F; they are denoted F(P) and .FF(V). Their dimension is infinite, as is proven by the following theorem. (Existence) Every UR. E F0(V\ ER., R') is the expression in R' of a lagrangian function U on V with compact support. THEOREM 2.1.

Proof.

Let R be any 2-frame. Let

S=

RR'-' E Sp2(l),

UR = SUR. ;

UR is a formal function defined on V\ER U ER, which is identically zero in a neighborhood of ER.. We extend its definition by making the following convention:

UR = 0 on ER,\R n ER.. Then OR becomes a formal function defined on V \ER ; U = { UR } is clearly a lagrangian function and Supp U = Supp UR.. II,§l,theorem 4.2 proves the following. (Structure) Let iG be the lagrangian phase of V, rl a positive regular measure on V, and x = 1 L E X, where i e V. We make the convention [rl] "2 > 0. We give V a 2-orientation; recall that [d'x] is defined by I,§3,corollary 3 using the Maslov index. THEOREM 2.2.

1112

We have URIV,

[' rEN

l3r+I/2 -d

/

1

(

yr YRr( )E

(2.1)

111

II,§2,2

on V\iR, where the YRr are infinitely differentiable functions V -+ C.

flRO is independent of R, is denoted f3o, and is called the lagrangian amplitude. Remark 2.1.

In (2.1), the exponent 3r + i cannot be decreased: see

III,§3,theorem 4.2, where the polynomial f, has degree 3r and the rational function g, poles of order 3r. II,§1,theorem 6 justifies the following definition. Let n be an open neighborhood of V. Let a be a lagrangian operator on n and U a lagrangian function on V. Let R and R' be 2-frames Definition 2.2.

and S = RR'-' E Sp2(l). Then aRUR = S(aR' UR,);

thus the formal functions aRUR form a lagrangian function, which we shall denote a V.

In other words THEOREM 2.3.

The algebra of lagrangian operators a, b defined on SZ

V

acts on the vector space f (V) [and .FO(V)] of lagrangian functions U [with compact support] defined on V; a(bU) = (a o b)U ;

(2.2)

Supp(aU) = Supp U n fl-' Supp a,

(2.3)

ft being the projection of V onto V c Z. Definition 2.3 of the scalar product (I ). Let U E .FO(V) and Let us first assume that the following condition holds:

There exists a 2-frame R such that

Supp U n ER = 0,

2.4)

Supp U' n Ex = 0'

This condition means

UR E 9'0(P \i,, R),

UR E

R).

We then define

(U U') _ (UR UR) E r6 for each R satisfying (2.4).

2.5)

II,§2,2

112

The right-hand side is an asymptotic class by II,§1,definition 5.1. It is independent of R by part 4 of II,§1,theorem 5.1.

Let U and U' no longer satisfy (2.4). Let Y ai = 1,

> aj. = 1

(2.6)

I

be two partitions of unity (definition 1.4) sufficiently fine (theorem 1.2) so that for every choice of (j, j') there correspond 2-frames Ri,i, satisfying the condition

Supp d, n ER,., = 0;

= 0,

Supp ai n ER;.;

thus (ai U I a;. U') is defined Vj, j'.

We define

(UI U') _ Y_ (aaUIaj.(Y)E`'. i.i'

The right-hand side is independent of the choice of partitions of unity (2.6); indeed, if

Ibk=1, k

>bk'=1 k'

is a second choice satisfying (2.7), then Y_ (aiUl a,, C') i1 j,

(ai OR,,

R;.r) =

(ai o bk UR;.r ai, o b,',, UR;.r)

i.f, k, k'

_ Y (bk U l bkXP), k,k'

since Ri,i. can be replaced by Rk,k' . THEOREM 2.4.

(Scalar product) 1°) The scalar product (- .) is a function I

of pairs U and U' of lagrangian functions on V and V' with compact support. It is sesquilinear over F ° and takes values in ' (asymptotic equivalence classes).

(UJU') and (U'IU) are complex conjugate and depend only on the

113

II,§2,2

behavior of U and U' at the pairs of points of V and V' projecting onto the same point of V n V.

If VnV'=0,then (UIU')=0. (aUI U') = (UJa* U') if the lagrangian operators a and a* are adjoint. (2.9)

2°) If V = V', which implies Vi = 0i', then

(UI U') E',

(2.10)

the phases of (U I U') being the periods of 0 (that is, the values of i f 7[z, dz] on the cycles y of V); Y(t')]g

flo(z)flo(z

(U I U') =

mod

(2.11) V

ZEJV z,zY

where the notation is that of theorem 2.2 and z and z are the points of

Supp U and Supp U' projecting onto z in V. We have

0 '< (U I U) E'.

(2.12)

The seminorm of U, U U)112 E

0

satisfies the triangle and Schwarz inequalities:

(Ui

(U U')I

U)112.(U'

U,)112 in W.

3°) If V and V are transverse (where V and V are given 2-orientations), then v1/2(U U') E

';

(2.13)

1riJt/2 (U I U') C2lvl

= I :Evn v,

no A no

d21z

1/2

Y

0

(z

mod 1

(2.14)

V

where

no A no and d

21z

are the measures on Z defined at the point z c- V n V'

by II,§1,(5.11) and II,§1,(5.15) respectively,

1 and 2' are the points of Supp U and Supp U' projecting onto z,

II,§2,2

114

m is the Maslov index defined mod 4 (I,§3,theorem 1),

A4 and , are the 2-oriented lagrangian planes tangent to f/ and fl' at i and i', respectively.

Proof of 1°). Use definition 2.3, part 1 of II,§1,theorem 5.1, part 4 of §1,theorem 6, and definition 1.2 of the adjoint of a lagrangian operator. Proof of (2.10).

Use §l,definition 2.3 and §1,(5.4).

Proof of (2,11). Use §1,theorem 5.2 and the fact that aU mod(1/v) is multiplication of U by the function ao. Proof of (2.12).

We use a partition of unity [part 3 of theorem 1.2]

Yb*obi = 1 sufficiently fine so that for every j there exists a frame Ri satisfying

Supp bi n IR, = 0. We have

(UI U) = >(bJUR I biUR.) > 0 (§1,definition 5.1).

Proof of the Schwarz inequality. By §l,definition 5.1, we have the triangle and Schwarz inequalities in R [§1,(5.3)] and in W (§I,remark 1); hence,

(UIU') _

(bi UR, I bi UR;)

(bi UR; bi QR ) (bi UR, I bi UR, )1'

2

(b, UR, I bi UR,

)1/2

)1112

[(bJUR,IbJURJ)] 11

[

(b, UR, I b, OR 1

J

U,)

The triangle inequality follows from the Schwarz inequality.

Proof of 3°).

Use §l,theorem 5.3.

115

II,§2,2-II,§2,3

Remark 2.2.

If V = V', (U U') is defined by the right-hand side of §1,

(5.4) under the assumption.

Supp U' n ER = 0-

Supp U n ER = 0,

If this assumption is not satisfied, the right-hand side is a divergent integral in view of theorem 2.2 (structure). Thus definition 2.3 of (- I ) gives a meaning to this divergent integral. 3. Lagrangian Functions on V

We are interested in functions on V rather than functions on V, that is, multiforms on V. The definition of lagrangian functions on V requires the datum of a number vo ; it will be convenient (see, for instance, Maslov's quantization, II,§3,6) to choose vo E i]0, oo[. The first homotopy group n, [ V ] acts on V: V is the quotient of V by

n1[V]. Clearly it,/[ V] leaves ER invariant /VR;

(YZ) = OM + cy!

(PR(YZ) = (PR(Z) + C

where c., = 1 J [z, dz]

dy e n 1 [ V];

that is,

q0 /-, = 0 - Cy!

(PR0Y-1

=(PR-CY.

Definition 3.1 of the group n, [V] of automorphisms of

OR = Y aRrr evv- E f (V\i,! R),

Let

Y E n, [ V].

(3.1)

V

Define the transform of OR by y not as UR

o 7-1 = e-vcy

y

aRr

Y

R)

V

but as yIIR = e(°-V,)C, UR o Y-1 = e-vocr Y aR.r 0 Y rEN

Clearly

evWR E

R).

(3.2)

II,§2,3

116

Y' (Y UR) = (Y"01R,

so n1 [V ] is a group of automorphisms of .y (V\ER, R). It clearly follows from definition (3.2) and the definition of the operator S E Spz(1) in II,§1 (lemma 1.1, part 1 of theorem 4.1, and theorem 4.3) that S and y commute: If

UR. E .f(V\±R. U :R, R') and R = SR', then

(3.3)

Y(SUR') = S(YUR,).

If UR are the expressions of a lagrangian function U on V, then yUR are the expressions of a lagrangian function on V; denote it by yU. Then (Y OR = yUR and n1[V] is a group of automorphisms of .F(). Clearly Vy e 7r, [ V], V O e

(V ), Va a lagrangian operator,

Supp(yU) = y Supp U;

(3.4)

y and a commute, that is, (3.5)

Y(aU) = a(yU).

A lagrangian function on V is a lagrangian function U on V having the following three properties, which are clearly pairwise equivalent and thus independent of the choice of the 2-frame R : Definition 3.2.

i. U is invariant under n1 [V], that is,

yU = U

Vy E n1 [V] (see definition 3.1);

ii. a "°`YaR,, o Y-1 is independent of y c- n1 [V] Vr E N;

iii. the function

UR =

URe(vo-v)WRI rEN

V

called the restriction of UR to vo, takes the same value at all points of V having the same projection onto V Therefore we may write: UR : V \ER - .y °.

The set of lagrangian functions on V[with compact support in V] is a Hence it is the set of points in V projecting onto a closed subset of V,

117

II,§2,3

which will be called the support of U in V and denoted Supp U. Then

Supp UR =Supp U\ER n Supp U,

t1R.

The set of lagrangian functions on V [with compact support in V] is a vector space over the algebra .F', It is denoted .y (V) [.y°(V )]. Clearly, by (3.5) the following theorem holds.

The algebra of lagrangian operators a, b defined on 0 acts on the space .y (V) of lagrangian functions on V; THEOREM 3.1.

V

Supp(aU) c Suppa n Supp U.

(a o b)U = a(bU),

The scalar product of lagrangian functions on V will be defined using the following lemma and definitions. LEMMA 3.1.

There exists a natural epimorphism

.y°(v)3CJ-+U

Y YUE.y°(V).

(3.7)

yER1[V]

Proof. Definition (3.7) makes sense: it defines U by a finite sum, since for every 2 e V, there are only finitely many y such that yz e Supp U. Clearly U is invariant under n1[V] and has compact support. Since y commutes with partitions of unity (definition 1.4, theorem 1.2), to prove (3.7) is an epimorphism it suffices to prove the following: Every U E FO (V) with support in a simply connected subset w of V is the image under (3.7) of an element U of .y°(17). This is proved as follows. The connected components of the subset of V projecting onto w are the transforms of any one of these components (b by the elements y of n1 [ V ]. Let U be the lagrangian function on V that vanishes outside th and equals U in th; then yU vanishes outside y(b and equals U = yU in y(b, hence

U = Y yU. Let us express the notion of restriction to v° more explicitly. Definition 3.3.

e°9 - e'ow

Let 5 be the ideal of F generated by the elements cp e R.

Each element of .5 may be written uniquely:

118

II,§2,3

I aJ(e`'PJ - ev0 j),

J finite,

ai E F °,

gyp; E R.

JEJ

.F is the direct sum of f and F °. Let us call the projection of F onto .F ° the restriction of F to vo: .FE) U = E c e"ip' F- 3 u ° = E a i e"O (P; E -IF' = -F/J. In particular, if U E flV), then the restriction of UR(v, 2) E to vo is the value UR (v, z) of the restriction UR of UR to vo at the point z, the projection of z onto V. Definition 3.4.

Any additive subgroup M of ' having the properties

.F c M c 16, M = 47-(complex conjugate) is called a submodule of 16. Multiplication in the algebra W makes .elf a module over the algebra .S. In particular,

ERimply ue"°E.11.

Let .' be the submodule of M whose elements are uJ(e°°i - ev00,),

J finite, u, E .i,

(p; E R.

JEJ

Let us call the quotient

..lf -+ -# / .N = ..lf °

the restriction of Al to vo; tf° is a module over the algebra F°. Since .N' = .N', conjugation in .,lf ° is induced by complex conjugation in M. Example 3.1.

If ..l' = .IF, then J(° _ -IFo

Example 3.2.

Let (1/IvI)s9,

...

denote the images of 9,

...

under the

multiplication 16 Z) J-+

1IsfE16,

sER+.

V

If .4Y = (1/Ivls).F, then tf° = (1/Ivl3),a-'°. Definition 3.5 of the scalar product ( I ). Let V and V' be two lagrangian manifolds in Z. Let.,# be a submodule of ' (definition 3.4) such that

II,§2,3

119

V U E .f°(V),

(UI U') E .,H,

U' E .F°(V')

(3.8)

For any U E .F°(V), U' E -FO (V'), there exist U E .`°(V), U' e .f° (V') such

that

U = E TO, YEidV]

U' =YEn1[V'I E y' U'

according to lemma 3.1. By definition, the scalar product of U and U' is (U I U') = (U I U')°, the restriction of (U I Cl') E .,H to v0.

(3.9)

Thus the scalar product has values in Mo. This definition makes sense by the following lemma. LEMMA 3.2.

Proof.

(UI U')° depends only on U and U'.

Let

Eb*obj=1 J

be a partition of unity (definition 1.4); we have

(UI U') _

(b;CI Ib;U');

it then suffices to prove that (bU I b;U')° depends only on

E yb;U = bjU and E y'b;U' = b;U'. YEn,[Vl

YeAdv ]

By theorem 1.2, it suffices to prove the lemma when the following two conditions hold: i. There exists a 2-frame R such that UR E 10'0(1 \'R, R), UR E

R).

ii. There exist two simply connected open sets, w in V and w' in V', that are projected injectively into X under RX and contain the supports of U

and U:

SuppUcwcV,

SuppU'cci cV.

The connected components of the subset of V projecting onto w are the open sets yw in V, where y e nl[V] (see the proof of lemma 3.1). Given x e RX Supp U, let

II,§2,3

120

z be the unique point of co such that Rxz = x, i be the unique point of to projecting onto z in V. By definition 2.3, §1,(1.9), and definition 5.1,

(fl UR)(V, X) - MR UR)(V, x)d'x.

(U I U') =

(3.10)

fX

By

definition (§1,theorem 2.1), if x E Supp(HRUR) = Rx Supp U, then

(nR UR) (v, x) = E UR(V, Y

' Z),

YE1,IV]

that is, by (3.2)

(nRUR)(v, x) = E e('0-Vk' YUR(v, Z)' YE1 11 VI

We substitute this expression for 11R UR and the analogous expression for

11RUR into (3.10). Using definition 3.4 of .N' and (3.7), we obtain the formula (U I U') =

UR(v, Z) - UR(v, Z )d'x

mod Al

(3.11)

J

under the assumptions (i) and (ii).

This formula (3.11) proves lemma 3.2 and is of use in proving the following theorem, part 1 of which is clear.

(Scalar product) 1°) Let V and V' be two lagrangian manifolds in Z. The scalar product (- I -) is a function of pairs U and U' of lagrangian functions on V and V' that is sesquilinear over F°.

THEOREM 3.2.

(U U') is defined when Supp U n Supp U' is compact. (U I U') e tl°, a module over .F' depending on V and V'. (U U') depends only on the behavior of U and U' in a neighborhood of Supp U n Supp U'.

(U I U') = 0 when Supp U n Supp U' = 0. (U I U') and (U' l U) are conjugate. (aU I U') = (U I a* U') when a and a* are adjoint lagrangian operators.

2°) Suppose V = V. Then (U I U') E .F° (algebra of formal numbers with vanishing phase):

0 <, (U I U), equality implying U = 0.

(3.12)

121

II,§2,3

The norm of U,

0I IUII = (UIU)112 E.°,

(3.13)

satisfies the triangle and Schwarz inequalities:

11U + U'II , IIUII

+ 11 U'11,

I(UIU')I,

IIUII'IIU'Ilin.Fo;

if the equality holds, then U and U' are proportional: either U = 0 or there exists a c .F ° such that U' = a.U. Let Ue(v, z) = a,,0(2)e°-O-() mod

-

(3.14)

V

11z

mod 1

(3.15)

V'

where z denotes any one of the points of V projecting onto z in V; (3.15) assumes V is given a 2-orientation that defines the Maslov index MR(z) and arg d'x (1,§3,corollary 3); the lagrangian amplitude fo is independent of the choice of R, but depends on the choice of >l, a regular measure on V such

that arg rl = 0. Then, using analogous notation for U' and denoting Rxz by x,

(3.16)

xR,o(z)xe,°(z)d'x = flo(z)#0' (z)'1

is a regular differential form on V, which is independent of R, and

(U I U') = J aR,o(z)xR,o(z)d`x

v = J 0(Z)flo(z), mod-.

(3.17)

v

3°) Suppose V and V are transverse. Then v'12 (UIU')E.F

I)112(UIU) 1J

E [Hess(cp zE Vn V'

^

in (3.18),

"'OCR, 0(z)xeo(z)e"°[wR(=) wK(=)] mod-

(3.18)

V

12

nodziZ zCVnV'

- T')]

o

mod-; V

(3.19)

II,§2,3

122

p and cp' are defined near Rz by (P(x) = (PR(1 x'X n P),

(P'(x) _ 9R(RXlx n P')>

Hess(cp - cp) _ {Hesss[gp(x) - (P'(x)]}x in (3.19),

= R1Z>

-

>G is the lagrangian phase of V,

ti4 e A4(Z) is the 2-oriented lagrangian plane tangent to P at 2, m is the Maslov index defined mod 4 (I,§3,theorem 1), 'lo A '1o and d 21z are the measures on Z defined by (5.11) and (5.15) (§1).

Proof of (3.16).

Use §1,(5.9).

Using a sufficiently fine partition of unity Ej b* o b, = 1,

Proof of (3.17).

it suffices to prove (3.17) under the assumptions (i) and (ii) used in the proof of lemma 3.2, which imply (3.11). Since V = V', formula (3.11) can be written (U I U') = j UR(v, z)URU (v, z)d'x, x

(3.20)

and hence proves (3.17). Proof of (3.12). Using the same partition of unity, it suffices to prove (3.12) when (3.20) holds. Then (3.12) is evident, as well as the following

more precise statement: If U # 0, then

(UI U) = '.Y

OE'

co

,whereseN,

=2s V

Proof of (3.13).

0[2, > 0-

Use the preceding formula and part 2 of §1,theorem 2.3.

Proof of the Schwarz inequality. Suppose U and U' are not proportional.

Then there exists a e .°" and s a N such that

U=aU'+

IS v

U",

where the lagrangian amplitudes $' and /3 of U' and U" are not proportional. A classical calculation gives U12.II U'l12

- I(UI

U')12

= IVI2s[I

U'II2.

U,, III - I(U' I U")12].

123

I1,§2,3-II,§2,5

It then suffices to prove that the right-hand side is > 0 in 3F°. In other words, it suffices to prove that U and U' satisfy the strict Schwarz inequality when their lagrangian amplitudes fl° and j30 ' are not proportional. Thus, from (3.17), we have U112.1 U' I

2

- I(U 101 2 =

I!l°(Z)12q. fy I a(Z)12n Jv f°(Z)fia(z)n

Hence, from the classical Schwarz inequality, 11U112.1

U'112 - I(UIu')12 = po mod1,where 0 < paER.

Thus, by II,§1,theorem 2.3, 1(U I U')1 < 11 U

.

I

U111.

Proof of the triangle inequality.

Use the Schwarz inequality.

Proof of 3°). Using a partition of unity, we may assume that (U I U') is expressed by (3.11). Then (3.18) follows from part 3 of theorem 5.1 (§1) and (3.19) of theorem 5.3 (§1).

Without assumption (i) (made in the proof of lemma 3.2), the integral in the right-hand side of (3.20) diverges (see theorem 2.2 on the structure of lagrangian functions). Thus definition 3.5 of (1) gives a Remark 3.

meaning to this divergent integral.

4. The Group Sp2(Z)

The group Sp2(Z) of automorphisms of the 2-symplectic geometry of Z (see I,§3,4) clearly transforms lagrangian operators into lagrangian operators. The images of lagrangian functions under this group are lagrangian functions. The group leaves the scalar product invariant. 5. Lagrangian Distributions

Lagrangian distributions are defined by replacing functions by distributions (§1,7) in definitions 2.1, 3.1, and 3.2. Theorems 2.1, 2.2 (the !'R, being distributions), 2.3, and 3.1 apply to distributions just as to functions. Definition 2.3 and theorem 2.4,2°) apply to the scalar product (U 1 U')

II,§2,5-II,§3,1

124

of U and U', a lagrangian function and a distribution both defined on a lagrangian manifold V when Supp U n Supp U' is compact. Definition 3.5 and theorem 3.2,2°) apply to the scalar product (U I U') of U and U', a lagrangian function and a distribution both defined on a lagrangian manifold V, when Supp U n Supp' U' is compact.

§3. Homogeneous Lagrangian Systems in One Unknown 0. Summary

The existence of an inverse a-' of a lagrangian operator a under the assumption of theorem 1.1 of §2 shows that it is the homogeneous lagrangian systems in one (§3) or several (§4) unknowns whose study is nontrivial. §3 and §4 begin this study, which will be concluded in chapters III and IV with the special cases of the Schrodinger equation and the Dirac equation, the latter of which is a system in four unknowns. 1. Lagrangian Manifolds on Which Lagrangian Solutions of a U = 0 Are Defined

Let a be the lagrangian operator associated to a formal function a° : S2

.FO (S2 an open set in Z)

whose value at z is

a°(v, z) = E 1 ar(z). ,ENV

THEOREM 1.

The support of a solution U of the equation aU = 0 is a

lagrangian manifold V on which ao = 0.

Consider a point in V at which ao i4 0. Replace Q by a neighborhood of this point on which ao 0. Then a-' exists on S2 and aU = 0 implies U = 0 on S2. Proof.

We shall assume that ao is real valued and that the subset of S2 where ao = 0 is a regular hypersurface W given by the equation W : H(z) = 0,

HZ

0.

V is therefore a submanifold of W such that

dim V = l;

d[z, dz] = 0 on V.

II,§3,1-II,§3,2

125

V is studied by applying E. Cartan's theory of differential forms to the pfaffian form [z, dz].

2. Review of E. Cartan's Theory of Pfaffian Forms") We first recall some well-known results that are set forth, in particular, in the treatise of Y. Choquet-Bruhat [6]. Let co be a differential form defined on Z that in section 2 will be any infinitely differentiable manifold of finite dimension. E. Cartan expresses co locally using a minimal number of functions

The number n of these functions is called the rank of w. These functions are the first integrals (independent and maximal in number) of a completely integrable system of pfaffian equations called the characteristic system of co. E. Cartan explicitly describes this system. In other words, this characteristic system is equivalent to

dit=...=dff=0; there exists a differential form za such that w(z, dz) = m(f (z), df (z)), where

f = (fl, ... , f,)

Let co be a pfaffian form defined on Z (that is, a differential form of degree 1 in the differentials). Let i, denote the interior product [6] with a tangent vector of Z; the characteristic system of dw may be written icdw(z, dz) = 0

V.

(2.1)

The rank of dco is even; denote it by 2n. Let

f=(ft,...,f2.) be 2n independent first integrals of (2.1) locally. Then there exists a differential form m' such that dw(z, dz) = m'(f (z), df (z)):

evidently dm' = 0. Then m' = dm locally. In other words, locally there exist a pfaffian form m of 2n variables and a function fo such that ' See Cartan [5].

11,§3,2

126

w(z, dz) = dfo(z) + rm(f (z), df(z)), where f = (fl, ... , f2.)-

(2.2)

The rank of w is evidently 2n or 2n + 1 depending on whether fo is a composition with f = (f1, . . , fen) or not. In the first case it is possible to choose w such that fo = 0. .

Let us state some less familiar facts with which E. Cartan [5] (chapter XII, "Les equations qui admettent un invariant integral relatif") has supplemented this result; we recall their proofs. LEMMA 2.

Let w be a pfaffian form; let 2n be the rank of dw. Locally there . : . , f2,, such that

exist 2n + 1 numerical functions fo,

w(z, dz) = dfo(z) + I f2j-1(z)df2j(z);

(2.3)

j=1

fl, -

-

-

,

f2, are 2n independent first integrals of the characteristic system

of dw.

f1, . . . , fen is not an arbitrary sequence of first integrals of this characteristic system: see remark 2.2. Remark 2.1.

Let f2 be a first integral of the characteristic system of do). The restriction of dw to the hypersurfaces f2 = const. has even rank <2n, so its rank is 52(n - 1). Let f4 be a first integral of the characteristic system of this restricted form. Continuing in this fashion, the restriction of dw to the manifolds Proof.

f2 = const.,

f2; = const.

.. ,

has rank < 2(n - j) and therefore vanishes for some value J of j such that J <, n. In other words,

dw = U on the manifolds f2 = const.,

..

,

f2, = const.

Then there exists a function fo such that w = dfo mod(df2,

.

,

df2,);

that is, J

w = dfo + I f2i-1 df2;. J=1

But do) has rank 2n, so J = n and f1, ... , fen are 2n independent first integrals of the characteristic system of dw.

II,§3,2

127

E. Cartan has supplemented this lemma by using the notion of the Poisson bracket as follows. Let co be a pfaffian form that has been put in the form (2.3). Let g and g' be two first integrals of the characteristic system of dw. Each is then a composition: Definition 2.

z"g(z) = G[f1(z), ...,f2.(z)],

z"g'(z) = G'[f1(z), ...,f2. (z)]

Then there exists a composition of the same type, called the Poisson bracket of g and g' and denoted (g, g'), such that

n(dw)n-1 A dg n dg' = (g, g')(dw)".

(2.4)

This bracket

(g, g') _ -(g" g), which is bilinear and independent of the choice of f = { f1, be expressed as follows, using one such choice: 8G

(g, g') (Z) _ 1

3G'

af2;-1 Oft;

8G' aG

- 42j-1 8f2; I=I(z)

... ,

may

(2.5)

Hence every triple g, g', g" of such first integrals satisfied the Jacobi identity: (g, (g', g")) + (g', (g", g)) + (g", (g, g')) = 0.

g and g' are said to be in involution when (g, g') = 0.

By (2.4), the pairs fJ,fk (1 < j < k) in the expression (2.3) for w are in involution with the exception of the pairs f2j_1, f2j; these

Remark 2.2.

satisfy

(f2;-1, f2;) - 1. Let g1, . . ., g9 be independent first integrals of the characteristic system of dw. The restriction of w to a manifold Remark 2.3.

91 = const.,

... ,

gq = const.

reduces the rank of dw by an even number that E. Cartan [5], chapter XII, section 124, has made explicit: it is 2 (q - r) if all the Poisson brackets (ga, gk) (j, k E 11, ... , q}) are compositions with g 1, ... , gq and if the

II,§3,2

128

exterior form

Ak

Y- (9j. j, k

in the variables

, ... ,Q has rank 2r on this manifold.

In particular (q = n, r = 0), we obtain the following theorem, which will enable us to establish theorem 3.1. These theorems shed some light on the beginning of the study of the Schrodinger equation. Let w be a pfaffian form; let 2n be the rank of d(o. 1°) Locally, the characteristic system of d(o has n-tuples { f2, f4, f2ri} of independent first integrals that are pairwise in involution.

THEOREM 2.

2°)) Given one {of these n-tuples, a quadrature locally gives n{+ (1 functions f0, f1, f3, , f2'. , f2n-1 such that to is expressed by (2.3); f1, f2, f3,

are independent first integrals of the characteristic system of d(O. Proof of 1°).

Lemma 2 and remark 2.2.

Proof of 2°).

By assumption,

df2 A df4 A ... A df2

0,

(dw)..-1

A df2j A df2k = 0

yj, k.

In other words, there exist differential forms Oj such that (dw)n

1 = Y Oj A df2 A ... df2j ... A df2,. j=1

(^ suppresses the term that it covers). This relation expresses the existence of pfaffian forms Oj such that

dw =

Oj n df2j, j=1

or equivalently, the condition

dw = 0 mod(df2, .

.

,

df2,.),

that is, the existence of a function fo such that

to = dfo for f2 = const.,

... ,

f2n = const.,

or the existence of n + I functions fo, f1, f3, f2,.-1 satisfying (2.3).

II,§3,3

129

3. Lagrangian Manifolds in the Symplectic Space Z and in Its Hypersurfaces Let

w(z, dz) = z[z, dz],

(3.1)

that is, in a frame R w(z, dz) = 2' - ' 2 = - zd , where Rz = (x, p);

(3.2)

dw evidently has rank 21; its characteristic system is dz = 0. Lagrangian manifolds V in Z are manifolds of dimension 1 on which do) = 0. These are manifolds given by local equations (3.3)

P = (PR,s(x)

outside the apparent contour ER of V, where (PR is the phase of V in R; coR

is an arbitrary function when V is arbitrary. Given x e V, it is possible to choose R such that x ER and use the local equations (3.3) in a neighborhood of x. But if we want to use a single frame, we must supplement the preceding result as follows (x' and pp will denote dual coordinates of x and p). The lagrangian manifolds in Z that project under RX : z r+ x to a manifold in X of dimension k S 1 given by equations

LEMMA 3.1.

x'=F;(x',.-..,xk),

k+1

j<1,

(3.4)

are the manifolds in V given by local equations (3.4) and A =

a(PR(xl, ... , xk) ax`

!

;

OF.

j=k+1 ax

1

i < k,

(3. 5)

where coR is the phase of V in R and x'.... , xk, Pk+11 ... , p, are local coordinates on V; cp, is an arbitrary function of x', ... , xk when V is arbitrary. If k = 1, the equations of V reduce to (3.3).

If k = 0, Visa plane: x = const.,

p arbitrary,

cpR = const.

II,§3,3

130

Proof + 1 d< p, x> = < p, dx >

Pi +

2

+

I \\\

1

F; P;J dx'. 1=k+I ax

Then dw = 0 on V if and only if there exists, on V, a function WR of (x', ... , xk) satisfying (3.5). The condition dim V = ! is that , pI) be independent on V. Obviously cpR is the phase (x`, xk, pk+I, of V.

Given a function f : Z R and a frame R, fR denotes the function Z(!) R such that fR(x, p) = f (z) for Rz = (x, p); that is, Notation.

fROR = f. Put JR,xk

afl,

fR,x = (fR.x', ...,fR,x,)

W is a hypersurface in Z given by the equation

Hz # 0.

W: H(z) = 0,

(3.6)

ww andfw are the restrictions of w and f to W. (E. Cartan) dww has rank 2! - 2; its characteristic system is

LEMMA 3.2.

Hamilton's system

__

dx' HR,Pi(x, P)

dpk

on W

dj, k e {1, ..., !},

(3.7)

HR,x°(x, P)

where the vector (HR,P, - HR.x) is evidently tangent to W.

Proof.

The condition

(d(ow)'-' A dfw = 0 on W

(3.8)

may be written

(dw)'-' A df n dH = O on (dw)'

W.

Then by (2.4) and (2.5), where we choose f2J_1 = p; and f2J = x' in order to identify (2.3) and (3.2), condition (3.8) may be written (fR,P,HR,xj - HR.P;fR.x;) = 0 for H. = 0. J=1

(3.9)

II,§3,3

131

(dcow)'_1 # 0. But (dcow)' = 0; thus the It does not hold identically; thus 2(I - 1). Hence the equivalent relations (3.8) and (3.9) rank of d( ow mean that fw is a first integral of the characteristic system of do)w. Hence this system is Hamilton's system (3.7).

As local coordinates on W we shall use 21 - 2 independent first integrals of Hamilton's system (3.7) and a function t such that (3.7) may be written Remark 3.

dp = -HR. x(x, p)dt.

dx = HR,p(x, p)dt,

(3.10)

Definition 3.1.

Given a function g: Z - R, define g_ E Z at the point z by

dg = [dz, 9Z]

(3.11)

in other words, (3.12)

R9z = (9R,P, -9R,x) Hamilton's system (3.10) may then be written dz = H_ dt.

(3.13)

By (2.5), where we choose f2J_1 = p3 and f2J = x' in order to identify (2.3) and (3.2), the Poisson bracket of g and g' is (3.14)

(9, g') = <9R.x, 9R,,> - <9R,x, 9x,,> = [9z, 9=].

The tangent vector K = H. of W is called the characteristic vector. The curves in W tangent to this vector at each point of W, that is, the solutions of Hamilton's system (3.7), are called characteristic curves.

The following classical theorem explicitly describes the lagrangian manifolds in W, that is, the resolution of the nonlinear first-order equation HR(x, (Px) = 0,

in which the unknown is a function (p. THEOREM 3.1.

1°)

Hamilton's system (3.7) has (I - 1)-tuples

of independent first integrals that are pairwise in involution, that is,

(dw)'-Z A dh; A dh,, = 0 on W

Vj, k e 11,

.

. ., I

- 1;

.

(3.15)

II,§3,3

132

Given one of these (1 - 1)-tuples, a quadrature locally defines l functions

go,91, ,91-1 such that 1-1

w=dgo+

(3.16)

g3dh3 on W. =1

g1, ... y1_1, h1, ... , hi-1 constitute 2(l - 1) independent first integrals of Hamilton's system (3.7). If t is defined by (3.10), then

t, h1, ...,h1-1,91, ...,9,-1

(3.17)

form local coordinates on W; go is a function of these coordinates. 2°) Lagrangian manifolds V in W are manifolds in Z that locally

i. are fibered by the characteristic curves

t arbitrary, h 1 = const.,

... ,

h1

g, = const.,

... ,

g1 _ 1 = const.,

_ 1 = const.,

ii. have the lagrangian manifolds in the space Z(1 - 1) with the coordinates

x' = (h1, ... , h,-1),

(3.18)

P = (91, ... , 91-1)

as a base for this fibration. 3°) If the local coordinates (3.17) are used on W, we have the following local equations of a lagrangian manifold V in W, after suitably permuting the

indices ( 1 , ... , 1 - 1), choosing k e (0, ... , l - 1} and choosing 1 - k real numerical functions of k variables Fo, F,.,,, ... , F,_1:

k+ 1 <,j,
h;=F;(h1,...,hk), 3Fo(h1, ... , hk)

V:

91 =

1-1

OF.

;_k+1 ahi

ah1

g;,

1

.

k;

(3.19)

t, h 1, ... , hk, 9k + 1 , , g, -1 are local coordinates on V. The lagrangian phase of V is given by .

V/(t, h1, ..

,

-

hk, 9k+1 .

.

.

, 91-1)

=9o(t,h,,...,hk,9k+1,...,g1-1)+Fo(h1,...,hk).

(3.20)

II,§3,3

133

If k = I - 1, the system (3.19) means


Oh;

(3.21)

If k = 0, it means It = const., Proof of 1°).

g arbitrary,

F0 = const.

(3.22)

Use theorem 2 and lemma 3.2.

Proof of 2°). By (3.16), the condition that dw = 0 on V is equivalent to the following: the mapping

V3zH(h1i...,h;-1;g1,---,g,-,)EZ(1-1)

(3.23)

sends V onto a manifold B in Z(l - 1) satisfying Edh; A dg; = 0. Then

B has dimension <1 - 1. A necessary and sufficient condition for dim V = I is that

i. dimB = I - 1, that is, B be lagrangian; ii. (3.23) map a characteristic curve of V onto each point of B. Proof of 3°). Lemma 3.1, where Z, x, p,

V,

dcpR = on V,

Z(1 - 1), x', p' [see (3.18)],

B,

dFo = Y_ g;dh; on B,

are replaced by :-1

j=1

so that (3.16) gives

w=dgo+dFoon V.Now w=day. Section 4 will use the following definition.

Given a lagrangian manifold V in W, let us call a regular measure q on V that is invariant under the characteristic vector K of W an invariant measure on V ; in other words, an invariant measure on V is any differential form q defined on V that is homogeneous of degree l in the differentials and is annihilated by the Lie derivative Y. in the direction Definition 3.2.

of K:

Y.q = 0.

(3.24)

II,§3,3

134

Since, by definition,

YKg = i drl + d(i,,q) and drl = 0, (3.24) means

d(i,,j) = 0.

3.25)

1°) Using the local coordinates x = Rxz on V\ER, the con-

THEOREM 3.2.

dition that rl(z, dz) = XR(x)d'x,

where xR : V\ER - R,

(3.26)

be an invariant measure on V may be written

fa

/ ax, XR(x)HR.P(x, coR,x)

= 0.

(3.27)

By (3.10), this means that along the characteristics generating V 0HR,PJ(x, (PR,.)

dxR dt

+

j

(3.28)

XR = 0

ax,

more explicitly,

dXRR +

j

P)

L =1

+

j H.,,,,, (x, P)1PR,.,x'

j=1k=1

XR = 0-

(3.29)

P=WR>

2°) Using the local coordinates (3.17) on V, the condition that

rl = r(t, h, g)dt n d'-'h n d'-lg, where z: V - R,

(3.30)

be an invariant measure on V may be expressed in these words: the function i is independent of t. Proof of 1°).

(3.31)

In the specified coordinates, the components of K are

dx = HR,p/lx, (PR.x),

(3.27) reexpresses (3.25).

Proof of 2°).

In the specified coordinates, the components of K are

II,§3,3-II,§3,4

135

(dt, dh, dg) = (1, 0, 0); (3.31) reexpresses (3.25).

4. Calculation of aU The definition of a lagrangian operator, that is, formula (6.6) of §1, can be expressed more explicitly as follows.

Let a be a lagrangian operator associated to a formal

Notation 4. function

1

a°(v, -) = I

r.N yr

a°(-)

defined on S2 c Z. Let W be a hypersurface of Q given by the equation

W:H(z)=0,

HZ:0.

Let V be a lagrangian manifold in W. Let q be its phase in a frame R. Let

be an infinitely differentiable function; x = RXz will serve as a local coordinate on V\ER . Let us exclude the following case: the operator a vanishes on W; namely, by (6.6): a°(v, x, p) and all its derivatives vanish on W. Let rl = XRd'x be a positive invariant measure on V (section 3). 10) We have

THEOREM 4.

aR

v'

X,

[«(x)evw.(s)]

v fix

=

etiNR(x)Lr

(X,

)x),

(4.1)

where Lr is a differential operator of order < r depending on a, V, and R; Lo(x) = ao(x, APR,J.

2°) If a° = H, then

L° = 0,

L, (X,Oxa = Xa2dt(«X-112),

where d/dt is the derivative along the characteristic curves of W [see (3.10)],

that is, the Lie derivative s,.

II,§3,4

136

3°) More generally, suppose that ao vanishes to nth order on W.

1
1 <m
-

Lo=L,=...=Lm_,=0,

L.(X'ax)a(X)- ,RZM(-,dt

) (aXReZ)ifm

oc.

The differential operator M depends on a and W but is independent of V and R. It is not identically zero on W. Its order is <m, with equality holding only if m = n. In general, m = 1. Hence, if n > 1, then L, is generally multiplication by a nonzero function.

4°) If a° = H and if HR is polynomial in p of degree s with principal part H('), then

L,=0forr>s, 1/a

exp2

l

L(x, P)

(4.s)

\3x' L)1 R

HR (x, p).

Thus if s = 2, formulas (4.3) and (4.5) explicitly describe a.

Proof of 1°).

Formula (6.6) of §1 is made explicit as follows. By formula

(6.3) of §1,

aR (v, x, 1

1 a(X)evw.(x)

_

v 3x f

1 eYw.(x)l, (v, x, rEN v

\

1 a(x),

(4.6)

aXJ

where lr, a formal function of v, is the differential operator of order r: lr (VI x, r k

ax I 1

2-111

k! r.Jo.....J' I !J° !... Jk!

° aRx1p1+Jo+...+: (v, x,

(a Jo 9R.x'

Ox (4.7)

the sum

EH,J°,...,j, extends over the collection of (k + 2)-tuples (I, J°, ... , Jk) of 1-indices satisfying the condition

II,§3,4

137

2
III+IJoI+... +IJkI - k=r,

(4.8)

this condition clearly implies

III + IJoI + k < r,

2 < IJkI < r + 1 fork 3 1.

Formulas (4.6) and (4.7) clearly imply (4.2) and (4.1), where L, is independent of v, while lr depends on v.

(4.2) gives Lo = 0. We have L, = Ir. For r = 1, condition

Proof of 2°). (4.8) means

or

k = 0, k = 0,

or

k=1,

either

III = 0,

I J° I = 1;

III=1, III=0,

IJoI=0; IJoI=0,

J1I=2.

Thus d

L1

c'a

X, fix) a

1

+ 2 j H,,,., (X, p)

HR.P(X, coR,x)

+jIi k-1

j=1

I

HR,p p,

(x, P)(PR,x'x' t

;

JP°(DR,, (PA'.

this relation is equivalent to (4.3)2 by (3.10) and (3.29).

Let k e N; there exists a function °b°: S2 -+ C such that

Proof of 3°).

a°o = °b° H"°

,

where

no = n and °b° is not identically zero on W if n

oc,

no = k + 1 and °b° = 0 on W if n = oc. Since

a° o b° = a° b° mod I[[see §2, (1.5)],

there exists a lagrangian operator c such that aR

(v' X' V OX

= VCR (V, X, V OX

o6R

(v'

X' v

ex)

o [Hit

(v' X'

Ox V

Ano

II,§3,4

138

If c° does not vanish on W, let 'b° = co, n, = 0. If co vanishes on W, we apply the same reasoning to c as has been applied to a. Proceeding by recursion, we obtain a decomposition of the De Paris type [7]: h

_

CV, X,

aR

j

V (X

1^I

X,

Vj'bR

mod

X) O [HR V

CV, x, V

CVO

(x)]

1 Vk+ 1

where

hk, nj> 0 nj=k+ Iwhen

0onW.

By part 1 of the theorem, n HR

[a(x)eP,(z)]

v. x, v

ax)

= 0 mod

(4.9)

i.

Let m(k) =

inf

(J + n j

jE{o,...,k}

if m(k) > k, we then have aR (v, x

1 (1

0 mod

V OX J

Vk+1'

But we assumed that a does not vanish identically on W. Therefore we can choose k such that

m(k) 5 k. We define m = m(k); let J be the collection of j such that

j + nj = m. By 1°), jbR (v, x, (1/v)a/ax) is mod(1/v) multiplication by the function V. Then, by (4.9), the De Paris decomposition gives aR Cv X, 1 S) [a(x)evv-(Y)] V ax -0

jY vj ib0(Z)[HR (V. X, v (x

m1 [2(x)e°'"(s)] mod

+i

139

II,§3,4-II,§3,5

where

1 <m<, n, J is finite and nonempty, 0 e J if and only if m = n, jb° does not vanish on W. Therefore, by (4.3)2 the relations (4.4) hold with LmC'

- YjEJib°(z)XR'I\

Jm ,(aXRlf2

L. is not zero; its order is <m; its order is m if and only if m = n. In general, c° does not vanish on W, and therefore n, = 0, m(k) = 1,

m=1. Proof of 4°). a° = H, so 1, = L,. By (4.7), where a° = H, we can supple+ IJkI < s; that is, ment condition (4.8) by the relation III + IJOI +

k+r<s.

If r > s, it cannot be satisfied ; thus L, = 0. If r = s, it requires k = 0; (4.8) becomes III+IJOI=s

and (4.7) becomes 2-ICI

L3(X, p) _

I!

HR'z'pl (X, P),

the sum extending over the collection of 1-indices I such that III <- S.

In other words,

L(X, p) _

\ (i-,

)H(Xp),

P/

where it is possible to replace E j=O by XT .0, which gives (4.5).

5. Resolution of the Lagrangian Equation a U = 0

By theorem 1, solutions of this equation are defined on lagrangian submanifolds V of the manifold given by the equation ao = 0.

II,§3,5

140

Notation 5.

Assume that the equation

ao=0 defines a hypersurface in Z. Let W be one of the parts of this hypersurface where it is regular. Let UR =

1

rtN

xR re

°WA

be the expression in a frame R of a solution defined on a lagrangian manifold V in W. Notation 4 is kept. By theorem 4, the condition that aU = 0 on V\ER may be written (/z,-

Ml

\

r

[XR 1l2(X)aR.r(X) +

XR 1J2

s=1

j'm+s (X'

\

aR.r-s(X) = 0

Vr E N;

(5.1)r

the first of these equations may be written dt)fJo(z) = 0, where Qo(z) =

M(z'

XR1;2(X)aR,o(X)

(5.1)0

is the lagrangian amplitude of U, which is independent of R. Theorem 2.2 of §2 supplements equations (5.1) as follows: XR 3r- 112OCR

r

is infinitely differentiable on V,

(5.2)

even at the points of ER; recall that XR1 = 0 on ER.

If ao vanishes to nth order on W, where n > 1, then, in general, the operator M is multiplication by a nonzero function M : W -+ C. Thus the equations (5.1) imply that the support of U is a Remark 5.1.

lagrangian submanifold of the manifold in Z determined by the equations

H(z) = M(z) = 0. The study of this case requires a generalization of sections 3 and 4: it is necessary to construct lagrangian submanifolds V of a given manifold W in Z when the codimension of W is > 1; it is necessary to describe explicitly aU when U is defined on such a lagrangian manifold V. We shall not study this case and exclude it.

II,§3,5

141

Let us show how conditions (5.1) and (5.2) enable us to solve the equation

aU = 0 using a single frame R.

Let K be an arc of a characteristic generating V. Assume that the principal coefficient of M does not vanish on K. 1°) Let z' c- K. Let U' be a solution of the lagrangian equation aU' = 0 defined in a neighborhood of z' in V. Then there exists a neighborhood of K in V on which the equation aU = 0 has a unique solution U such that U = U' in a neighborhood of z'. 2°) Let R be a frame such that ER is transverse to K and XR' does not vanish to infinite order on K n ER. Then UR is the expression in R of a LEMMA 5.

lagrangian solution U of the equation aU = 0 in a neighborhood of K if and only if UR satisfies (5.1) and (5.2).

Notation used by the proof. VK is a neighborhood of K in V of the form

VK=B x I, where B is the ball Ibi

1 in R`-', I is the segment ItI < const. in R; the segments b x I are the characteristics; 0 x I is the given characteristic K, z' _ (0, 0); U; Ij = I is a finite covering of I, V = B x Ij, j an integer.

Proof of 1°). We make a choice of VK and Vj having the following properties :

U' is defined on V0;

to each V; there is associated a frame R; such that V; n ER = 0;

I, is the segment j - I < t < j + 1. If U has been defined on Vo u V, u . . . u V _ 1 (j > 0), then using (5.1) in the frame Ri makes it possible to extend successively the definitions of OCR,, 0, ... , aR.,r, ... , UR, U to Y; these extensions are unique.

Proof of 2°). The expression UR of a lagrangian function U, defined in a neighborhood of K, satisfies (5.1) and (5.2). It remains to prove the converse.

Let R be a frame, V,, a neighborhood of K, and

II,§3,5

142

UR =

; IaR,,e

'wR

rcN V

a formal function defined on VK\ER n VK satisfying (5.1) and (5.2).

It has to be proved that, in a neighborhood of K, UR is the expression of a lagrangiani function. Let z e K. By 1°), there exists a unique lagrangian function U defined in a neighborhood of K whose expression in R is UR in some neighborhood of z'. It has to be proved that in some neighborhood of K, its expression in R is still UR.

Since this expression, like UR, satisfies (5.1) and (5.2), it suffices to prove the following uniqueness theorem: If the formal function (5.3) vanishes in a neighborhood of a point of K, then it vanishes in a neighborhood of K. We choose VK and V having the following properties:

UR=OonV0; Ij is the segment j - 1 < t

j;

VK n ER = UjB,, where Bj = VV n Vj+1 = B x j. Assume we have proved that

UR=O onVouV1u...u(VJ\BJ),

j>0.

By (5.2), xR 1naR,, has an infinite number of derivatives on B, that all vanish; then (5.1) gives successively aR.o = 0, aR,1 = 0, ... , aR.r = 0, ... , UR = 0 on VV+1\Bj+1

This lemma shows that the differential operators M and L, have very special properties. The following is a consequence of part 2 of this lemma. Let W be a hypersurface in Z on which do = 0: assume that the operator a is not zero and that the principal coefficient of M does not vanish. Let V be a lagrangian manifold in W. Let R be a frame. Let p = LRd`x be an invariant positive measure on V; assume that XR 1 does not THEOREM 5.

vanish to infinite order on ER and that ER is transverse to the characteristics of W generating V. Then 1

rEN V

vWR

143

II,§3,5-II,§3,6

is the expression in R of a lagrangian solution U defined on the universal covering space V of V if and only if UR satisfies (5.1) and (5.2). Remark 5.2.

Lemma 5 and theorem 5 evidently apply to solutions U

of the equation I

aU = 0 mod

sG1V,

V

defined mod(1/v`) on V.

6. Solutions of the Lagrangian Equation aU = 0 mod(1/v2) with Positive Lagrangian Amplitude: Maslov's Quantization Definition 6.1.

We call an infinitely differentiable function

H: S2 - R (S2 an open subset of Z)

such that H.

0 on the hypersurface W given by the equation

W: H(z) = 0 a hamiltonian.

In classical mechanics, Hamilton's system (3.7) governs the movement of particles and, more generally, that of holonomic mechanics. Let a be the lagrangian operator associated to a hamiltonian H; then it is self-adjoint (§1,6). Solving the equation

aU=0modIV2 amounts to finding lagrangian manifolds in W having an invariant measure that we assume is chosen > 0. Indeed, by (4.3), equation (5.1)0 is written dfl0

dt

=0

and means that the lagrangian amplitude of U is constant on each of the characteristics generating V. In other words, floe is invariant. We require it to be > 0. In other words, we require that the lagrangian amplitude Qo of U satisfy the condition Io > Jo-

(6.1)

II,§3,6

144

(Recall that this is the case in physics, where the amplitude should change more slowly than e°' and not oscillate around the value 0.)

Definition 3.2 (§2) of lagrangian functions on V requires the datum of a purely imaginary number vo that will be denoted by

vo=,

h>0,

(6.2)

in chapter III because in quantum physics 2nh is chosen to be Planck's constant. Since 1/2

xR,o = flo

dx

,

where f3o 3 0,

the condition that U be lagrangian on V mod(1/v) amounts to )1,12

d`x

being defined (that is, uniform) on V. From the definitions of (d'x)112 and ,111/2 (I,§3,corollary 3 and II,§2,theorem 2.2), this condition is the following. Definition 6.2. A lagrangian manifold V satisfies Maslov's quantum condition when the function 2ni

APR

- 4mR,

where vo = i/h,

is defined mod 1 on V; this condition is independent of the choice of the frame R. Remark 6. If V is oriented, in the euclidean sense, MR is defined mod 2 on V. Then Maslov's quantum condition assigns to each period 1 f ,,[z, dz]

of 0, that is, to each period nh mod 2nh.

< p, dx> of r°R, one of the two values 0 or

If V has a 2-orientation, mR will be defined on V mod 4 and this condition requires that cpR and t be defined on V mod 2nh. This will not be the case in the applications that are given in chapter III. We have just proved the following theorem. 1 °) Let a be the operator associated to a hamiltonian H. A lagrangian solution U of the equation THEOREM 6.

I1,§3,6-II,§3,7

145

aU=0mod z V2 with lagrangian amplitude >-O can be defined mod(1/v) on a manifold V if and only if V simultaneously has the following three properties: i. V is a lagrangian manifold in the hypersurface W: H(z) = 0; ii. V has an invariant positive measure (invariant under the characteristic vector of H); iii. V satisfies Maslbv's quantum condition. 2°) The datum of V having these three properties defines U up to a constant

factor if and only if the invariant measure on V is unique, that is, if every function that is infinitely differentiable on V and constant on each of the characteristics generating V is constant on V. This theorem will be applied in III,§2. 7. Solution of Some Lagrangian Systems in One Unknown Theorems 7.1 and 7.2 supplement theorem 6. They are applied in 111,§1 and III,§3, respectively. THEOREM 7.1.

Let a(') (j = 1, . , l) be lagrangian operators associated to

l hamiltonians

H('): fl -. R (S2 an open subset of Z) that are independent and pairwise in involution; that is,

d1l"' A .. A dH(')

0, (dw)1-2 A dH(') A dH(k) = 0

Vj, k,

(7.1)

where (o = j[z, dz]. 1°) The system

a(')U=...=ac1'U=0

mod12

V

has a lagrangian solution U defined mod(1/v) on the connected manifold V if and only if V simultaneously satisfies the following two properties: i. V is a connected component of the manifold in Z given by the equations

V: H111(z) _ ... = H(1)(z) = 0, which imply that V is lagrangian.

(7.3)

II,§3,7

146

ii. V satisfies Maslov's quantum condition (definition 6.2). Then the lagrangian amplitude of U is constant when d21z

n

A dH(l

dHcl) A

(see remark 7)

(7.4)

is chosen as the measure on V. Thus, when V has been chosen, then U is defined mod(1/v) up to a constant numerical factor. 2°) The Lie derivatives £f 1 in the direction of the characteristic vectors i of the H111 commute. If V is compact, then V is a torus whose translation group is generated by the infinitesimal transformations IKj.

Recall that d21z is defined by §1,(5.15); (7.4) means that ry is the restriction to V of any form 1z on Z such that Remark 7.

dHc11 A ... A dH11) A ?1z = d21z;

rl is clearly independent of the choice of ?1z; rl is invariant under each K.

Proof of 1°). From theorem 1, the support of U belongs to one on the connected components V of the manifold given by the equations (7.3). Conversely, let V be one on these components. The rank of do) is 21

because its characteristic system is dz = 0. By theorem 2 (E. Cartan), locally there exist functions go,

. . .

, g, such that

1

= dgo + Y- gj dH c>> ; J=1

then the restriction a, of w to V satisfies dcoy = 0.

Now dim V = l; thus V is a lagrangian manifold.

Moreover this result could be deduced from (3.22) of theorem 3.1, 3°). Let R be a frame of Z; let (x, p) = Rz; the condition that H(') and H(k) be in involution is expressed all(j) OH (k)

ax'

that is,

OH (k)

3H1;i

ap/-Cax'-ap/-0'

II,§3,7

147

Y,,; H(k) = 0,

(7.5)

where $4, is the Lie derivative in the characteristic direction of H('): Kj = 8H(j) , 8p

-

OH(j) Ox

From (7.5) follows

5 K, d H(') = 0;

moreover the definition of the Lie derivative gives

d21z=0, hence, by the definition (7.4) of rj,

YKl7 =Q

VJ.

Then in view of theorem 4,2°), the condition that a lagrangian function U defined on V satisfy

a(i)U = .. = a(()U = 0

mod 12

is that its lagrangian amplitude /J satisfy YK'YO = 0

Vi;

since the H(') are independent, that is, since they satisfy (7.1)1, this condition can be expressed as /Jo is constant on V.

Then, by section 6, the condition that U be lagrangian on V mod(1/v) is equivalent to Maslov's quantum condition. Proof of 2°).

By §2,(1.1), the commutator of a(') and a(k),

ja(i), a(k)I = a(1) o a(k) - a(k) o a(1)

is the operator associated to the formal function

- 2 {sinh

2v

a] [-Oz-, az'

H(j)(z)H(k)(Z')j -Z

since .H(') and H(k) are in involution, this formal function vanishes

II,§3,7

148

mod(1/v2). It is an odd function of v, so it vanishes mod(1/v3), from which follows Jaci), a(k)]

= 0 mod 3

.

v

From theorem

4,2°),

aR) (aeYWrt) = V evWR yR2

xi (ayR 1/2) mod V2

da.

so that aci) a(k) ae°wR Re R]( )

= V12 eYwR yRlie

I -1/2) mod 1 . y /a, . XR V3

thus, by (7.6),

[yxh zJ = 0 Then, if V is compact, the infinitesimal transformations

, Yx'

generate an abelian group of homeomorphisms of V, whence 2°). We supplement theorem 7.1 as follows.

Let a(') (j = 1, , 1) be lagrangian operators that commute with each other and are equal mod(1/v2) to operators associated to l independent hamiltonians H(') [that is, satisfying (7.1)1]; these hamiltonians are THEOREM 7.2.

then in involution [that is, satisfy (7.1)2]. Let us study the problem of defining

mod(1/vr+'), on a connected manifold V, a lagrangian solution U of the system

a(1)U =

= a(')U = 0 mod

1 2,

r , 1.

(7.7)r

Assume that conditions i) and ii) of theorem 7.1 are satisfied; indeed they are necessary. Then this problem has a solution U if and only if, in addition, the following two conditions are satisfied: iii. A solution of (7.7)r_ 1 has to exist on V (we assume it is explicitly known).

iv. A function f -+ C that is defined by integration of a closed pfaffian form on V\ER, and by the condition that it have polar singularities on ER, has to

be a function V - C. (Knowledge of this function explicitly solves the problem.)

II,§3,7

149

When (i), (ii), (iii), and (iv) are satisfied, then the solution U of (7.7), is defined on V up to a factor that is a formal number with vanishing phase. Preliminary to the proof. Let R be a frame of Z; let V satisfy (i). We restate theorem4,1°),2°) as follows. Let /3: V\ER -+ C; then a(i)+(v x, 1 R

O

(x) 3(x)e'`°",

(Xx2

I

VOX/

;C'n/2(x)ev9-(s) rY

Mr (x' OX) /3(x),

V'

(7.8)

where Mi is a differential operator of order <, r, depending on a(i) and V:

Mo=0,M; The assumption that the operators a(i) commute obviously may be stated as follows: r

Mrk-s+ll

X lMjs+l,

=0

Vj, k, r.

(7.9)

S=O

Proof. Let U be a lagrangian function defined on V with lagrangian amplitude /30 = 1. Let UR(v, x) =

vR/2ev(R(x)

l

1x

(7.10)

rENV

be its expression in the frame R, where fio = 1, fr: V\ER - C. Suppose that U is given mod(1/v') and satisfies (7.7)x_1: #0, ..., Nr-i are given and by (7.8) satisfy s

Y Mj /3S_, = 0 for j = 1, ... , 1, s = 1, ... , r.

(7.11)

t=1

The condition that U satisfy (7.7)r is r+ 1

MisQr+1-s = 0

(7.12)

dj ;

s=1

this condition is a system of I equations that defines M1 Qr

Vj;

i.e., Y,, /jx

by means of / 0,

Vj;

i.e., dflr,

... , /3r_1. Since the Mil = SP, i commute, the condition

of local integrability of this system, that is, the condition that the ex-

II,§3,7

150

pression for d/3, given by this system be a closed pfaffian form, is expressed by r+1

r+1

M i o Y Msflr+1-s - M i o MS J3,+1-s = 0 s=2

s=2

or, replacing s by s + 1, by

Y- [Mil, M;+lll'r-s + Y- lMy+1, Mil/3r-s s=1

S=1 r

(7.13)

+ E (MS+1 ° Mil ` Ms+1 ° Mi)i3r-s = 0. S=1

Now, with t and s replaced by t + 1 and r - s + 1, (7.11) implies r

k+j // M+1 ° Ml'r-s + Ms+lk ° Mr+1Yr-s-1 J

s=1

s.1

- 0,

where Es.r signifies Es-1 Ei-i ; that is, I,, extends over the set of integer pairs (s, t) such that 1 ' < s,

1 ' < t,

s +t < r.

In other words, (7.11) implies r

s=1

r

MS+1 ° Mi#.-s +

s-1

s-2 r-1

MS r+1

0 Mt+1flr-s = 0,

and similarly r J

r

Mk

Ms+1 ° M1 /3, s + S=1

s-1

J Mk Mr+1 ° Ms

s=2 t=1

r+1 /3,

s = 0,

so that the condition (7.13) of local integrability may be written r

s

Y-

Y-

[r

tMt +1,

k

Ms-r+lli3r-s = 0.

s=1 1=0

Thus it is satisfied by the assumption of commutivity (7.9). Then the system (7.12) defines a function

(ar P \±l - C up to the addition of a function that is locally constant on V\ER. By this choice of /3, in (7.10), UR locally becomes the expression in R of a lagrangian solution of (7.7), on V\ER.

II,§3,7-II,§3,Conc1usion

151

Then, by lemma 5,1°), which holds for systems mod(1/v'+Z), there exists a lagrangian solution of (7.7), on V whose expression in R is UR, up to the addition to /3, of some function that is constant on each component of V\ER. By theorem 5, the addition of this function to P. makes xR 3r1, infinitely differentiable on V; /3, is then defined up to the addition of a function that is constant on V, that is, a multiple of fo.

By assumption (ii), Maslov's condition is satisfied; by section 6, U is lagrangian on V mod(1/vr+1) if and only if fi, is a function: V --+ C. The theorem follows.

8. Lagrangian Distributions That Are Solutions of a Homogeneous Lagrangian System

Theorem 1 and lemma 5,1°) apply to solutions that are lagrangian distributions; condition (5.2) should then be stated as follows: xR3'-1J2aR,r is

a distribution on V.

Without trying to extend either lemma 5,2°) or theorem 5 to distributions, we merely remark that theorems 7.1 and 7.2 apply to solutions that are lagrangian distributions. Hence the following theorem holds. (Regularity) Under the assumption of theorem 7.2, every lagrangian distribution U that is a solution of the system (7.7), is a lagrangian THEOREM 8.

function mod(I/v'+1) Conclusion

V. P. Maslov [10] called the "solutions defined mod 1/v" studied in sections 6 and 7 "asymptotics". But there is no reason for them to be equal mod(1/v) to a lagrangian solution U of the equation aU = 0 (see theorem 7.2 and III, §3), and there is no reason for the expression UR of one such solution U to be the asymptotic expansion of a solution of the differential operator or pseudodifferential operator that aR can formally define. This is shown by the examples

considered in chapter III. The most evident feature of this lagrangian analysis, which was motivated by the study of V. P. Maslov's treatise [10], is that it is a new structure.

It is formal. Therefore, in physics, this analysis could be reasonably applied only to the nonobservable quantities of quantum theory.

II,§4,1

152

§4. Homogeneous Lagrangian Systems in Several Unknowns Let us generalize the simplest results of §3: theorems 4, 5, 6, and 7.1. 1. Calculation of Em _1 am U. 11

We shall extend a result of [8], section 8, and elucidate its proof; by doing this we shall generalize theorem 4 of §3. Notation 1. Let a be a p x p matrix whose elements a" (m, n = 1, ... , p) are lagrangian operators associated to formal functions with vanishing phase; they are elements of a matrix

a° = I v, a,°, where a,° : S2 -+ C"'. reN

Let W be a hypersurface in f) given by the equation

W: H(z) = 0, where H.

0 on W.

Let V be a lagrangian manifold in W; let WR be its phase in a frame R. Let

c(x, p) = a° (z) for (x, p) = Rz.

b(x, p) = ao(z)

(1.1)

Let a = {al, ..., a,,} be a vector whose components are infinitely differentiable functions am. r \Y'R - C;

x will serve as a local coordinate on V\ER, Let q = XRd`x be a positive invariant measure on V (§3,3). THEOREM 1.

aR (v' x'

10) We have [a(x)eYwR(x)]

V

ax

=

v evwR(x)L, reN

{ x, `\

Ox

)x)

,

(1.2)

where L, is a p x p matrix whose elements are differential operators of order
(1.3)

2°) Suppose detb = H. There exist two nonzero p-vectors f and g that are functions of z e Z such that on W

153

II,§4,1

bf = 0,

1bg = 0;

i.e.:

0, M=1

> gnb" = 0;

(1.4)

n=l

choose these (which is possible) so that on W, fmg" is the minor of b' in the matrix b; that is,

dH = Y fmg" dbn mod H.

(1.5)

m,n

Let

g"un for any vector u = (u1, .. . uµ).

(g, u> n=1

Evidently, by (1.3) and (1.4),

(g(x,'PR.X), Lo(x)a(x)> = 0.

This formula is supplemented by the following, where y is an arbitrary function V\ER -- C, and where d1dt is the derivative along the characteristic curves of W, that is, the Lie derivative PK : (x, (PR,X),

XR2

L1(x,

fl

d (Y CR 112)

[1'(x)f(x, (PR,.)] (1.7)

AX, R,x)Y(X).

Here J(x, p) is defined on W by the formula

J=z

[(g" bm)fm + bn (g", f,.) + 9"(bn, fm)] + m,n

in which

, m.n

gncnfm

(1.8)

denotes the Poisson bracket, defined by §3, (3.14).

J has the following properties, all obvious except for the first: Remark 1.1.

J only depends on b and c and the restrictions fw and gw

offandgtoW. Remark 1.2.

Multiplying f by h: Z -+ C\{0} multiplies gw by h-' and

adds d(log h)/dt to J. Remark 1.3.

If the matrix b is symmetric and the matrix c is antisymmetric,

then J = 0, since it is possible to choose f = g.

II,§4,1

154

Suppose the matrices b and is are self-adjoint. In particular, this is the case when the matrix a is self-adjoint, that is, when Remark 1.4.

a' = (an,,)*

dm, n.

Then the values of J are purely imaginary, since it is possible to choose 9 = f

Proof of the theorem.

Part 1 follows from §3, theorem 4. To prove part 2,

let

a = aR,

b' =

(P = (PR(X), b nz

b n ( x, 1Pz),

=

a6 m

na

x , p) ,

P=N. a9n(x, P)

9

Op

= w:

so that abm

ax`

I

n

b x, + E b P; (px,x;,

,

1

-'9; - 9Xt + L 9P,(Pxix" j=1

J=1

(1.9)

where x' and p; are the components of x and p (1, j = 1, ... , 1). By the definition of a+ in §1, (6.3)-(6.6), an m (V,

x,

e"'bn (x, Pz)am + eV

V ax

bnP,P;(Px'x' +

+ (2 mod

l

1 V

hence, by (1.9) and the definition (1.2) of L 1,

\g, L 1(X' ax) a) J,m,n

Qn( I

9nbnpl aXm + m.n

.

npj +

V

Thus, by choosing am = If,,, it follows from (1.5) that

b P' as 1 bn ;Pl + cn I or. 1J

11,§4, 1

155

<9,Lj(yf)>

=YHp+[2I ;n9nbp'-xm Y1 2im,n9nasi(bv;fm)+

+

E9ncnfnl ;

hence by (1.5) and the definition of d/dt on W,

<9, LI(yf)> = do + 2 E aa;xp;y + Jy,

(1.10)

l

where 1

J

E 9nbnp

2

;

Y axjbnp fm +

2

j, m, n

j, m, n

9nCnfm M' n

Now, by §3, (3.28), a

Y-

ax,

Hpj

dx1e.

1

_

XR

dt

hence (1.10) is equivalent to (1.7). Moreover, by (1.4),

axj

(Yg-b-)

I = 0;

a

hence (1.11) can be written _ bn m

9n

afm

ab

ag

axj

axj

axj

fm

J=

1

2j

Y-

mn

nm 9 Cn fm-

+ m,n

fm m p;

np;

9v,

By (1.9), this is -bm

fm

J=1

Y-

2j,mn

,nx'

f

mp

because

n

9n

m

nx' m

n p;

9xn '

+

nm

9 Cn fm, m,n

(1.12)

II,§4,1-II,§4,2

156

n

9 bnpi m

Jmpr

n

(Px'x' = 0,

gPl

r"PI

bnp

9;;

since the above determinant is an antisymmetric function of (i, j). Clearly, (1.12) is equivalent to (1.8).

Proof of remark 1.1.

J=

(1.8) can be written ('

9n(bf, Jm)] + E 9ncm fm -

[(9n,

2

11

m,n

m,n

Now, by (1.4) there exist regular functions F. such that

Ybnlm = HFn. m

Hence, on W, where H = 0,

J=

d9n

fm)J +

+ m,n

9ncmfm; m,n

thus J only depends on b, c, f, and the restriction of g to W. 2. Resolution of the Lagrangian System aU = 0 in Which the Zeros of det a°° Are Simple Zeros Section 5 of §3 is easily extended in this case.

Notation 2.

Notation 1 is kept. By theorem 1,1°), solutions of the system

t a' U. = 0

(2.1)

M=1

are defined on lagrangian manifolds V in the hypersurface W given by the equation W : H = 0, where H = det b (by assumption, H. Let UR =

1 V

reN

OfR,,e

N

0 on W).

(2.2)

157

II,§4,2

be the expression, in a frame R, of a solution

U=(U1,...,U") of the system (2.1), which will be written

aU = 0; the U. are lagrangian functions defined on V c W; the a,,r are functions V\ER -+ C". In view of theorem 1, the condition aU = 0 on V\ER may be written Lo(x)aR,r(x) + s= 1

L, \x' Oz)

Vr e N.

s(x) = 0

xR,

(2.3),

By (1.6), it implies 9(x, PR,x), Y- Ls S=1

x, )R.Sx))

= 0.

(2.4),

Let M 0 be one of the matrices V - C"x" such that the relation Lo(x)a(x) = a'(x), where = 0 and a(x), Y(x) c- 0. is equivalent to the existence of y : V\ER -* C such that a(x) + Y(x)f(x, (PR.x) = M0(x)a'(x).

Then, under condition (2.4),, equation (2.3)r may be written aR,r(x) + YR,r(x)f(x, (PR,.) + I Mo(x)L., (X,

X)aR,r-s(x) = 0,

(2.5),

s= 1

and, by (1.7), equation (2.4)r+1 may be written d

dt

(YR,rXR 112) + JYR,rXR

112

a

r

+

9(x, (PR,x),

E Ns+

X,

U Ox)aR,r-s(x

)

1J2 XR-

= 0,

3=1

where N3+1 Cx, ax) a(x)

L 1 Cx,

)[M0(x)L(x.

- Ls+1 x,

-

cx

a(x).

)(x)]

(2.6)r

II,§4,2

158

Theorem 2.2 in §2 supplements equations (2.5), and (2.6), as follows (r e N): aR.,XR

3P-1/2 and thus YR,,XR 3r-"Z are infinitely differentiable on f/, (2.7),

even at the points of tR; recall that XR' = 0 on ER.

Lemma 5 of-§3 is easily extended. It shows that the conditions (2.5),, (2.6) and (2.7), make it possible to solve the system aU = 0 using a single frame R. More precisely (compare §3, theorems 5 and 6), the following theorem holds. THEOREM 2.1.

Let V be a lagrangian manifold in the hypersurface W

defined by (2.2); let V be its universal covering space. Let R be a frame. Let n = xRd'x be a positive invariant measure on V; assume that xR' does not vanish to infinite order on ER ; assume that ER is transverse to the characteristics of W that generate V. Then

,e'9 x UR = Y 1IR Vr rEN

is the expression in R of a lagrangian solution U = (Ul, ... , U"), defined

on V [or on V], of the system aU = 0 if and only if, for every r E N, the vectors OCR, : V\tR -+C" and the functions YR,, : V\ER -. C satisfy the conditions: (2.5),,

(2.6),,

Remark 2.

YR,e"ow.: V\'R _' C]

(2.7), [and

This theorem applies to solutions of the system

aU = 0

model,.

They are defined mod(1/vs+') up to the addition of

where

ae'-O-: V -. C" [or V -+ C"]. Evidently we have the following theorem. THEOREM 2.2.

[Reduction mod(1/v2) of a system to an equation]

We

keep the assumption of theorem 1, 2°), which defines H and J. The existence

of a solution on V c W [or on V] of the lagrangian system

aU = 0 mod- , that is, 17

an Um = 0 mod

-

,

(2.1)2

II,§4,2-II.§4,3

159

is equivalent to the existence of a solution on V [or V] of the lagrangian equation

a' U' = 0 mod

1

2,

where a' is the lagrangian operator associated to the formal function

To each solution U of the system (2.1)2 there corresponds a solution U'

of the equation (2.8) such that

U = U'f mod 1. V

In chapter IV, a reduction theorem analogous to the preceding one will be used in the special case of Dirac's equation. 3. A Special Lagrangian System a U = 0 in Which the Zeros of det ao Are Multiple Zeros The following extension of theorem 7.1 of §3 is used in chapter IV. (Theorem 7.2 of §3 admits an analogous extension.) THEOREM 3.

Let a(k) (k = 1,

.

. .

,

1) be µ x µ matrices whose elements

are lagrangian operators. Assume ac" is associated mod(1/v2) to the matrix

H(k)E +

J(k)

1V

where E is the µ x µ identity matrix, H(k) : 0 -. C, and j(k) : 0 - C"' is a p x µ matrix. Let V be the manifold given by the equations

V: H")(z) = ... = Hcn(z) = 0.

(3.1)

Assume that the a(k) commute mod(1/v3) and that

dH"l) A

.

A dHm"

0 in a neighborhood of V.

Then the following hold. 1°) The hamiltonians HW are pairwise in involution: V is a lagrangian manifold; the measure on V

II,§4,3

160

d 21z

n

dH'1' A

A dH"'

V

is invariant Vk under the characteristic vector x'kl of H'kl, which is tangent to V. If V is compact, then V is a torus whose translation group is generated by the infinitesimal transformations

2°) Let U = (U1, ... , UU) be a vector whose components U. are lagrangian functions; U satisfies the lagrangian system

a'k) U = 0 mod ldk V2 if and only if U is defined on V and the lagrangian amplitudes /3 = (/31,

... ,

of its components satisfy the system of first-order partial dif-

ferential equations (H(k), Q)

+ J(k)$ = 0, where

Q) = ./ 4k) /3

and is the Poisson bracket (3.14) of §3. This system is equivalent to a completely integrable system d/3 = w/3,

that is, d/i = Y-wn /3m, m

where the elements w' of the p x p matrix co are pfaffian forms defined on V. In addition to (3.3), /3 has to satisfy the "quantum condition"

d`xl

1/z

/3e"°w': VEER - C°

Remark 3.

The condition that (3.3) be completely integrable may be

stated as follows:

dw=wna,

(3.6)

where CO A w is the matrix whose elements are the Y wh A (am, . Hence it is h=1

satisfied; it is equivalent to

(H(`', J(k') - (H'k' J''') + j(i)j(k) - J'k) J(n = 0

Vi, k,

where by assumption (H'`', H(k') = 0, and by definition

(H')J(kl) _ Y H(') j(k) _ j=1

P,

j=1

J(k) H'', x' pj

II,§4,3

161

in an arbitrary frame R; (3.6) or (3.7) is equivalent to the condition that the a(') commute mod(1/v3).

... , p) be lagrangian functions defined on V with lagrangian amplitudes fl,,,; let U = (U1i ... , and f3 =

Proof of 1°). (/3,,

Let Um(m = 1,

. , (i ). By theorem 4 of §3,

aR+(k) CV,

x, v

8x) U

v

x

2evw. L(k)

a mod vz

,

(3.8)

(XI

where L(k) is a u x p matrix whose elements are first order differential operators; the principal part of L(k) is L,,mE, where 2,&, is the Lie derivative

in the direction of the characteristic vector x(k) of H(k) and E is the u x it identity matrix. By assumption, the a(k) commute mod(1/v3). Thus the L(k)(x, a/ax) commute. Thus their principal parts commute. Therefore the )c (k) are tangent to the manifold V given by equation (3.1). [See the proof of theorem 7.1, 1°) of §3).]

Proof of 2°). L(k)

By (3.8), the system (3.2) is equivalent to the system

(x, ax) Q = 0

Vk;

(3.9)

since the 2 commute, it is possible to find local coordinates t 1, ... , t, on V such that (3.10) Ctk

By theorem 4 of section 3, L(k)

(X, 8x)

at k

+

J(k)

(3.11)

Thus system (3.9), in which the unknown function 13: V -+ C" has to satisfy (3.5), is equivalent to the system

d/3 =w/3, on V, where )

Y J(k) dtk k=1

(3.12)

162

II,§4,3

is a it x u matrix whose elements are pfaffian forms. Since the a)) commute, the Lik) defined by (3.8) commute, which by (3.11) is expressed OJ(k'

OJ(i)

Oti - Otk

+ j(i)j(k)

- J(k)Jci) = 0

(3.13)

or, by (3.12), d(w = 0.) A co. This is the condition of complete integrability of the system (3.4). By (3.10), this condition is also expressed by (3.7).

III

Schrodinger and Klein-Gordon Equations for One-Electron Atoms in a Magnetic Field

Introduction

Summary.

The most interesting problems in the theory of linear and

homogeneous partial differential equations are the eigenvalue problems. Their essential feature is that they have solutions only exceptionally. Examples of lagrangian problems having the same feature are given in this chapter. These problems assume 1 = 3,

Z(3) = X Q+ X*,

X = X* = E3 (euclidean space).

They concern the lagrangian operator a associated to some convenient hamiltonian: this hamiltonian gives rise to a Hamilton system admitting two first integrals defined on Z(3), namely, the length L and") one of the components M of the vector x A p in V. H may be the hamiltonian(2) of the nonrelativistic or relativistic electron under the simultaneous influence of the electric field of a stationary atomic nucleus and a constant magnetic field (Zeeman effect). Then H depends

on a parameter: the energy level E of the electron. The operator a is the Schrodinger or (in the relativistic case) the Klein-Gordon operator. The energy levels for which our lagrangian problems have a solution coincide with those defined by the problems that are classically studied regarding these operators. The advantage of the lagrangian point of view is its simplicity. By applying theorem 7.1 of II, §3, these energy levels are obtained in §1 by a quadrature. The latter is easily calculated in the Schrodinger and Klein-Gordon cases using the method of residues. In §1, we determine solutions defined mod(1/v) on a compact lagrangian manifold of the lagrangian system: 1

a U = (aL2 - const.) U = (aM - const.) U = 0 mod 1,

(1)

V11

where aL= and am are the lagrangian operators associated to the first integrals L2 and M. Here theorem 7.1 is applied. Three integers introduced by the Maslov quantization,

1, m, n such that I m < I < n, 'The author uses A - to denote the vector product in V. [Translator's note] 'Or the hamiltonian of the harmonic operator: see §3, remark 4.3.

III,Introduction

164

characterize the equations having solutions (thus the energy levels) and also the solutions. These are Schrodinger's three quantum numbers. The lagrangian manifolds on which these solutions are defined are 3-dimensional tori T(l, m, n) [see theorem 7.1, 2°)], given by the equations T(l, m, n): H(x, p)

L2(x, p) - const. = M - const. = 0.

These constants have the same values as in (1) and depend on (1, m, n). In §2, we determine solutions of the lagrangian equation

a U = 0 mod 12

(2)

that are defined mod(]/v) on a compact lagrangian manifold V and that have lagrangian amplitude > 0. This amounts to a formal problem analogous to the boundary-value problem whose study constitutes the classical study of the Schrodinger equation; the condition concerning behavior at infinity in the classical problem is now replaced by the condition that V is compact. Always, the condition of existence is the same: it is characterized by a triple of quantum integers; but the solution corresponding to such a tripel is not necessarily unique. In §3, we determine solutions defined on a compact lagrangian manifold of the lagrangian system

aU = (aL2 - const.)U = (aM - const.)U = 0,

(3)

where the constants are formal numbers that are real mod(]/v2) and H is the hamiltonian of the relativistic or nonrelativistic electron; then a, aLs, and aM commute. Here theorem 7.2 is applied. Solutions again are characterized by a triple of quantum integers (1, m, n). Solutions of problem (1) are, mod(]/v), those of problem (3). In §4, we recall the problem classically posed regarding the Schrodinger and Klein-Gordon equations: to find a function

u:E3-C, that is square integrable-as is its gradient-and that satisfies the partial differential equation

au = 0.

(4)

In §4, we recall the resolution of this problem in order to show how it

165

III,Introduction

differs essentially from that of the preceding problems. We observe that all these problems define the same energy levels, but we do not explain why this happens.

The difficulties encountered in §2 and the length of the calculations used in §3 and §4 contrast with the simplicity of §1; §1 justifies the following conclusion.

Applying the Maslov quantization (I1,§3,6 and 7) to an atom with one electron placed in a constant magnetic field gives the observable quantities the same values as does wave mechanics. However, the Maslov quantization is directly related to corpuscular mechanics, hence to the old quantum theory. Nevertheless it does not have the shortcomings of the old theory. It has a logical justification (chapters I and II); it does not require the determination of the nonquantized trajectories of the electron, CONCLUSION

but only the knowledge of the classical first integrals L and M of Hamilton's system, which defines these trajectories.

The probabilistic interpretation of this quantization is the following: In the state defined by a choice of a triple of quantum integers (1, m, n), the point (x, p), representing both the position x and the momentum p of the electron, belongs to a 3-dimensional torus T(l, m, n) in the 6-dimensional

space Z(3) = E3 O+ E3; the probability that (x, p) belongs to a subset of T(l, m, n) is defined by the invariant measure q on this torus T(1, m, n) [see §1,(3.16)].

Remark.

Let

(1, m, n) 0 (1', m', n')

be two distinct triples of quantum integers. They define (§1) tori whose intersection is empty: T(l, m, n) n T(1', m', n') = 0.

Let U and U' be two lagrangian functions defined on T(1, m, n) and T(1', m', n'), respectively. Then their scalar product is [II,§2,theorem 3.2,3°)]

(UI U') = 0. Historical note.

V. P. Maslov did not give this application of his quanti-

zation. He only studied the case of quantum numbers tending toward infinity, that is, the "correspondence principle" in quantum mechanics.

III,§1,1

166

§1. A Hamiltonian H to Which Theorem 7.1 (Chapter II, §3) Applies Easily; the Energy Levels of a One-Electron Atom, with the Zeeman Effect Theorem 7.1 of II,§3 assumes that I functions on 0 c Z(1) are given that are pairwise in involution. In chapter III, we choose I = 3 and choose a classical triple of such functions. 1. Four Functions Whose Pairs Are All in Involution on E3 m E3 Except for One

Let X = X* = E3 denote 3-dimensional euclidean space. Let us apply chapter II to

1=3,

Z(3)=E3rE3,

thus making a choice of a frame Ro of Z : theorem 5 of II, §3 spares us the use of another frame. But in E3 we use both a fixed orthonormal frame (I,, 12, 13) and a moving orthonormal frame. Let

xeX=E3,

PEX*=E3;

let (x,, x2, x3) and (pl, P2, p3) be the coordinates of x and pin (1,, I2113): 3

P = Y_ Pj1j' J=1

Let us define five functions of (x, p):

R(x) = Ixl, P(p) = I pI, Q(x, p) = , L(x, p) = Ix A PI, M(x, p) = x1p2 - x2 p, (third component of x A p); they are connected by the obvious relations

L2 + Q2 = P2R2,

IMI _< L, 0 _< P,

0 <, R.

(1.2)

Then the vector 13 has a priviledged role: it will be, for example, the direction of the magnetic field producing the Zeeman effect. In E 3 Q E 3, the characteristic system of d< p, dx) is

dx = dp = 0. Every function E3 m E3 -+ R is a first integral of this system. Thus the

167

III,§ 1,1

Poisson bracket ( , -)(definition 2 of II,§3) of two such functions is defined. By formula (3.14), of II,§3,

(L, M) = (L, Q) = (L, R) = (M, Q) = (M, R) = 0,

(Q, R) = R. (1.3)

From theorem 2 (E. Cartan) of II,§3, a quadrature locally defines four real numerical functions f o, f 1, f 3, f 5 on E 3 (D E 3 such that

= dfo + f1dL + f3dM + f5dR.

(1.4)

Let us now define the functions ff explicitly. Let (J1i J2, J3) be the orthonormal moving frame, defined for L

0, such that

x n p=LJ3,

x=RJ1i which implies

P 0 0,

p = QR-1J1 +LR-'J2,

R

0.

Let w1, w2, and w3 be the infinitesimal components of the displacement

of that frame relative to itself (G. Darboux-E. Cartan); these are the pfaffian forms

wl = <J3, dJ2> = -<J2, dJ3>,

w2 = <J1, dJ3>,

w3 = <J2,dJ1> such that

dJ1 = w3J2 dJ3 = w2J1

- w2J3, - w1J2.

dJ2 = ())1J3 - w3J1,

Recall that exterior differentiation of these relations gives the equations

dw1 = 0)3 A w2,

d()2 = w1 A w3,

dw3 = w2 A w1

(1.8)

(which are the structural equations of the orthogonal group). By (1.7), differentiating (1.5)1 gives

dx = (dR)J1 + Rw3J2 - Rw2J3,

(1.9)

from which follows

< p, dx> = QR-1 dR + Lw3

(1.10)

by (1.6).

In order to transform this formula into a formula of the form (1.4),

III,§ 1,1

168

let us introduce the Euler angles CD, `P, and O; these are the parameters ±13, that of the frame (J1, J2, J3) that are defined as follows for J3

is, IMl < L: 0 is the angle between 13 and J3 ; 0 < 0 < it; a rotation by CD around 13 transforms (11,12, 13) into (I1, 12, 13) such that

12sinO = 13 A J3; a rotation by 0 around 12 transforms (Ii,12, I3) into (I' l, I2, J3); a rotation by `P around J3 transforms (Il, I2, J3) into (J1, J2, J3). It is obvious that

M = L cos 0,

(1.11)

CD and `P are defined mod 2n. We have the classical formulas:

J1 = (cosCDcos`Pcos0 - sin (D sin `P)I1 + (sin CD COST cos 0 + cos CD sin T)12 - cos `P sin 013 ,

J2 = (- cosCsin`PcosO - sin (D cos')11

(1.12)

+ (- sin Csin `PcosO +cos(D cos'F)12 + sin `P sinO13, J3 = cos(1) sin 011 + sin CDsin 012 + cos013;

wl = - cos`PsinOdC + sin `PdO, cot = sin `P sin 0 dC + cos `P dO,

(1.13)

cos = cos 0 dC + d'. By (1.11) and (1.13)3, the explicit expression we sought for formula (1.10) is

p, dx> = Q

R + Ld`P + MdC.

(1.14)

This fundamental formula is of the form (1.4), in agreement with E. Cartan's theorem, which is our guide. Complementary formulas.

By (1.7), differentiating (1.6) gives

dp = [d(QR-') - LR-1w3]J1 + [d(LR-1) + QR-'0)3]J7. + [LR-1co1 - QR-'w2]J3.

(1.15)

169

1II,§ 1,1

By (1.9), d3x = dx1 A dx2 A dx3 is given by

d3x = R2(dR) A a)2 A (A3;

(1.16)

hence, by (1.15),

d3x A d3p =

LR R

or, replacing the o )j of (1.11),

A dQ A dL A (01 A (02 A (J)3, their expressions (1.13) and eliminating O by means

d 3x A d 3 p= dL A dM A dQ A

R A dD A d`P.

(1.17)

Let 516 be an open subset of Z(3) = E3 Q E3 on which

IMI
L

O on S16 ; thus by (1.2)1, P

O, R

0.

Let V be a lagrangian manifold in 06 (dim V = 3). Let us replace each of the preceding functions and differential forms by its LEMMA 1.

restriction to V. 1°) The apparent contour ER. of V is the surface in V where

ER°:dR A w2 A w3 = 0 on V.

(1.18)

2°) In a neighborhood of a point of ER°, there exists a differential form m on V of degree 3, that is nowhere zero, such that on ER.

dQ A cot A 0)3/m > 0,

dL A dR A cot/rw >1 0,

dRAco3Awl/w>, 0,

(1.19)

where the three left-hand-side functions do not vanish simultaneously. There exists a constant c such that, in a neighborhood of this point of V, the Maslov index MR. has the value

when dR A 2 A (D3/rv < 0 1 + c when dR A cot A (J)3/0 > 0.

`c MR°

jl

(1.20)

III,§1,1-III,§ 1,2

170

Proof of 1°).

Use expression (1.16) for d'x.

Proof of 2°). Let us calculate the jump of mR. across ERo by applying theorem 3.2 of I,§3. By (1.9), (1.15), and (1.18), the components d;x and

dap of dx and dp in the frame (J1, J2, J3) satisfy on V, at the points of 2:R0

dtp A d2x A d3x = R2[d(QR-') - LR-'w3] A w2 A w3

=RdQAw2Aw3, d1x A d2p A d3x = -RdR A [d(LR-') + QR-'w3] A w2

=dLAdRAw2, d1x A d2x A dap = (dR) A 0-)3 A [Lw1 - Qw2]

=LdRAw3Awl. The existence of w satisfying (1.19) and (1.20) follows by this theorem (theorem 3.2, of I,§3).

2. Choice of a Hamiltonian H Let

be an infinitely differentiable function in involution with L and M, that is, by (1.14), a composition with the functions L, M, Q, and R:

H(x, p) = H[L(x, p), M(x, p), Q(x, p), R(x)].

(2.1)

H[ ] is an infinitely differentiable function defined on an open subset 04 of the space R4 with the coordinates L, M, Q, R. Assume that, on 524

0 < R,

I MI < L,

(HQ, HR)

(0, 0) when H = 0.

(2.2)

HQ denotes the function aH[L, M, Q, R]/8Q. By (1.3), the three functions H, L, and M of (x, p) are pairwise in involution, so that the results of II,§3 can be applied explicitly to the lagrangian operator a associated to H. From theorem 2 (E. Cartan) of II,§3, a quadrature locally defines four real numerical functions on 526, 90,91+92,93,

171

111,§ 1,2

such that < p, dx> = dgo + g1 dL + 92 dM + g3 dH.

(2.3)

We shall use this formula when H = 0; it is made explicit by lemma 2, assuming

H = 0. According to (1.14), the quadrature is the definition of the function 0 by (2.8).

Let W be the hypersurface in 06 given by the equation

Notation.

W : H(x, p) = 0

(Compare II,§3). Let (Lo, M0) be a pair of real numbers such that IM01 < Lo; let V[L0, M0] denote any connected component of the subset of W given by the equations

L(x, p) = Lo,

V [LO, MO] : H(x, p) = 0,

M(x, p) = Mo.

(2.4)

W is the union of the V [La, M0]. By theorem 7.1, of II,§3, V [LO, M0] is a lagrangian manifold. It is the topological product of

the 2-dimensional torus with the coordinates D and T mod 27r, a connected curve I [Lo , M0] in the open half-plane with the coordinates

Q, R > 0. The equation of this curve is (2.5)

T [L0, MO]: H[L0, MO, Q, R] = 0.

By (2.2)3, this curve does not have any singular points. Let us define a real numerical function t of [L, M, Q, R] on W up to the addition of a function of (L, M) by the condition

dt = RHQ[L,IM,

RHR[L,dM,

Q, R]

Q, R]

on F [L, M];

(2.6)

when the curve T[L, M] is a closed curve, then t is defined modc[L, M], where c [L,

[L'

M] _

- fr[L,M] RHQ

_

dQ

.

-fr[L,M] RHR'

(2.7)

III,§1,2

172

t is monotone on F; (L, M, t, (D, `P) forms a system of local coordinates on W.

Let us define another real numerical function S2 on W up to the addition of a function of (L, M) by

A2 = Q

R on F[L, M];

(2.8)

when the curve F[L, M] is a closed curve, then

S2

is

defined

mod2nN[L, M] where

N[L , M] =

dR >0 1 f r[LM] R 1

Q

(2 . 9)

.

S2 is not monotone on F. Let its differential be denoted by

dc2[L, M, t] = Q R + 1 [L, M, t] dL + µ[L, M, t] dM.

(2.10)

The properties of the functions ) and µ thus defined will be specified in §2. Now we can give the explicit expression of the restriction of (2.3) to W. LEMMA 2.

1°) The restriction of the pfaffian form

w= to Wis

cow=d(S2+L`P+MD)-O +`P)dL-(µ+(D)dM.

(2.11)

2°) The characteristic system of dww is Hamilton's system: dx

dp

HH(x, P)

H,,(x, p) '

H(x, p) = 0.

(2.12)

Its first integrals are compositions with the functions

L, M,2+`P,µ+ b;

(2.13)

L and M are in involution; 2 + `P and p + (D are also in involution. 3°) On the solution curves of this system, that is, the characteristic curves of W, we have

_ --

dx

dt = H,(x,

p)

dp

HH(x, p)

_

dR RHQ[L, M, R, Q]

_ dP _

d(D

HL[' ]

HM[' ]

173

III,§1,2

dQ ,dL=dM=0.

(2.14)

RHR[-] 4°) On W d 3x A d 3

dH

_ dL n dM n dt n d(D n d`)'.

PI

(2.15)

w

Proof of 1°). Use the expression (1.14) for in Z and the expression (2.10) for Q dR/R on W. Proof of 2°). Lemma 3.2 of II,§3 (E. Cartan) proves that (2.12) is the characteristic system of dww and that dww has rank 4. Now, by (2.11),

dww, =dL n d(.1 +`Y)+dM n d(µ+(D); thus the four functions (2.13) are first integrals of this system (see E. Cartan's definition of a characteristic system, II,§3,2). The pair (L, M) and (2 + `F, p + 1) are in involution by remark 2.2 of II,§3. Proof of 3°). The same lemma (E. Cartan) and the expression (1.14) for prove that the characteristic system of dw,, is dR

d`P

RHQ[L,M,Q,R]

HL[

_

dD

dQ

HM[']

RHR[']

_

dL 0

_

dM 0 '

H = 0;

thus this system is equivalent to (2.12). Obviously dx

_

HP(x, p)

<x, dx> <x, HP(x, p)>

_

R dR <x, LP>HL + <x, MP>HM + <x, QP>HQ

Since x A (p + sx) is independent of the real variable s,

d(x A p) = 0 for dx = 0,

dp parallel to x;

therefore,

<x, LP> = <x, MP> = 0. Moreover,

<x, Q> = R2. The preceding five relations imply

III,§1,2-III,§1,3

174

dx Hp(x, p)

_

dR RHQ[L, M, Q, R]

_

d'Y

_ d) _ _

HL

dQ

RHR'

HM

dL = dM = 0

and thus (2.14), by (2.6) and (2.12).

Proof of 4°).

Formula (1.17) can be written

d 3x n d3 p= dL n dM n dH n dR n d(l) n d'Y, RHQ

where, for H = 0, dR RHQ

= dt mod(dL, dM)

by (2.14); hence (2.15) holds, the following meaning being given to its left member: d3x A dap

dH

1W

denotes the restriction ww to W of any form w of degree 5 such that dH A m = d 3 x A d 3 p ; ww is clearly independent of the choice of w.

3. The Quantized Tori T(1, m, n) Characterizing Solutions, Defined mod(1/v) on Compact Manifolds, of the Lagrangian System 2) aU=(aL=-LoU=(aM-Mo)U=O mod v

1

.1) (3.1)

a, aL2, and ay denote the lagrangian operators associated, respectively, to the hamiltonians H, L2, M,

which are in involution; Lo and MO are two real constants such that IM0I < Lo. From theorem 7.1 of II,§3, solutions of this system are lagrangianfunctions U with constant lagrangian amplitude, defined mod(1%v) on those

manifolds V[L0, M0] defined by (2.4), whether compact or not, that satisfy Maslov's quantum condition (II,§3, definition 6.2). V[L0, M0] is chosen to be connected. The measure nv on V, which is invariant under the characteristic vectors of H, L2 - Lo, and M - M0, and which serves to define the lagrangian amplitude, is

III,§1,3

175

nv = dt A d' A dW

(3.2)

from (2.15) and formula (7.4) of II,§3.

Recall the statement of Maslov's quantum condition: The function 1

Zh

cOR° - 1 MRo

(whereh =

i

is real

o

has to be uniform on V mod 1. By (2.11), where L = Lo and M = M0, the phase cpR, of V [L0, M0] (defined in I,§2,9 and I,§3,1) is

(pR0 = 0 + LOT + MO(D.

(3.4)

Let us calculate the Maslov index of V [L0, M0]. LEMMA 3.

1°) The apparent contour ER0 of V[L0, M0] is the union

E, u E2 of two surfaces in V[L0, M0] given by the equations E2 :HQ[L, M, Q, R] = 0.

E, :'Y = 0 mod 7r;

2°) The Maslov index m mR° _

of V[L0, M0] is

[i-p] -

[V [L0, M0] 77

up to the addition of an integer constant; [ Proof.

(3.5)

R

] denotes the integer part.

Apply lemma 1. By (1.11) and (1.13),

0)1 = - cos'I' sin O dW, L00)3 = L0 d'I' + M0 da)

0w2 = sin W sin O dcb,

on V[L0i M0]; hence, by (2.6),

dR A w2 A 0)3 = -RHQsin'I'sin0dt A d'I' A dD,

(3.6)

dQ A 032 A W3 = RHR sin W sin O dt A d'I' A d(D,

(3.7)

dR A w3 A 0), = -RHQcos'I'sinOdt A d'! A dcb,

(3.8)

where R sin O : 0 by (2.2), and (2.2)2 . By lemma 1,1°), ER° is given by the equation HQ sinW = 0, from which follows part 1 of lemma 3.

III,§1,3

176

In a neighborhood of a point of E1\E1 U E2, HQ cos `P # 0; in lemma 1,2°), we can choose

w = dR n w3 n w1;

dR n W2 n w3/w = tan `P,

whence by this lemma mRo =

[i: `Pl + const. n

J

By (2.2)3, in a neighborhood of a point of E2\E1 U E2' HR sin `P in lemma 1,2°), we can choose

0;

dR A 0)2 A m3/m = - HQ/HR,

w = dQ A wz A 0)3; where

MR,, = const. - C1 arctan L

(3.10)

].

HR

The global expression for mRo follows from the two local expressions (3.9) and (3.10).

By (3.4) and (3.5), Maslov's quantum condition may be formulated as follows: The function 1

h 2n

4n

arctan

HQ HR

+ Lo _ h

1

`P

Mo do

2

27c

it 2n

is defined mod 1 on V[L0, Mo].

If V[L0, Mo] is compact, that is, if the curve F[L0, Mo] is a closed curve, by (2.9) this condition is that

IN[Lo, M] + 1, 0 2

h

LO - 1, 2

1M

h°

be three integers. Denote them by

n-1,

1,

m

in order to recover the classical notation of quantum physics. Since N > 0 and IMOI < Lo, we have

Iml<1
177

III,§1,3

Lo-2=1,

1MO=m

be two integers such that Iml _< 1. Thus the conclusion of this section is the following theorem. The connected manifolds on which the solutions of the lagrangian system (3.1) are defined are THEOREM 3.

1°) The compact manifolds V [L,, M0] defined by (2.4) such that there exist three integers 1,

in,

n

satisfying the conditions

Imll
(3.11)

Mo = hm,

Lo + N[LO, M0] = hn;

(3.12)

thus V[L0, M0] is a torus, which will be denoted by T(1, m, n); the coordinates at a point of this torus are t mod c [L 0, M0] defined by (2.6)-(2.7),

`P mod 27t,

b mod 2n. (3.13)

2°) The noncompact manifolds Y [L,, M0] defined by (2.4) such that there exist two integers 1,

m

satisfying the conditions (3.14)

Imi < 1,

Lo=h(l+z),

Mo=hm;

(3.15)

thus V[L0, M0] is a product of a 2-dimensional torus with the coordinates `P mod 2n, D mod 2n, a line, half-line, or segment with coordinate t. We provide V [Lo, M0], whether compact or noncompact, with the follow-

ing measure, which is invariant under the characteristic vectors of H,

L2 - L0,and M - Mo: tlv = dt n d(D A d`Y;

(3.16)

178

111,§1,3

then the solutions of (3.1) defined on V[L°, M°] are lagrangian functions with constant lagrangian amplitude. From now on, we shall limit ourselves to the study of compact lagrangian manifolds (for example, that of the electron belonging to an atom). Remark 3.1.

Remark 3.2.

The characteristic vectors of LZ - Lo and M - M° are

KL2_L0:dL=dM=dt=0,

d'h=2L°,

d(D = 0,

(3.17)

1M_MU:dL = dM = dt = 0,

d'P = 0,

d(D = 1,

(3.18)

respectively. It is immediately verified that these vectors leave q, invariant. Proof of (3.17). It follows from (1.2),, then (1.5), and (1.6), and finally (1.5), and (1.12) that KL2_LQ is the vector tangent to V[L°, M°] such that

dx = 2LLP = 2(R2p - Qx) = 2LRJZ =

2LOx(L,

M, t, T, (D) OT

This proves (3.17).

Proof of (3.18).

It follows from (1.1), then (1.5), and (1.12), that Km-m, is

the vector tangent to V[L°, M°] such that

dx =

(-x2, x1, 0) =

8x(L, M, t,'P, (D) O(D

Remark 3.3. In agreement with theorem 7.1,2°) of II,§3, when the manifold V[L°i M°] is compact, hence a torus, the characteristic vector of H

[see (2.14)],

x:dL=dM=0,

dt = 1,

dP=HL,

d'P=HM,

(3.19)

and those of L2 - Lo and M - M° generate a group of translations of V[L°, M°], namely, the translations in the following coordinates [see (2.7) and 111,§2,(1.5) and 111,§2,(2.3)]:

t modc[L°, M°],

'P + A[L0, M0, t] -

NL[L°, M t mod 2n, c[L0, M0]

:p + p - NM t mod 2ir. c

179

111,§ 1,4

4. Examples: The Schrodinger And Klein-Gordon Operators

Let us choose H (x, p)

=-

[ P,

- K[R' M

(4 1)

2

.

where K: R+ Q+ R --+ R is a given function. In other words, H [ L , M, Q, R ] =

2

[L2 +

Q2

- K[R, M]]

.

(4 . 2)

1 2

Theorem 3 can be applied immediately.

The condition that (2.4) define at least one compact hypersurface V [LO, M0], which is a torus, is that the function

L2eR be positive between two consecutive zeros R1 and R2 (0 < R1 < R2). Then (2.9) defines R3

N[L,, M0] = Remark 4.1.

1

R

rz

/K[R M0] - Lo dR

0.

R

if K is an affine function of M, then, by (4.1),

P(._)H(xP) 7x'

= 0,

and consequently [II,§1,definition 6.2 and II,§1,(6.3)], the expression in Ro of the operator associated to H is a

2vzA

2R2K[R'v(xlax z

- x2ax )J, 1

where A = 1:3J=' a2iax;,

the multiplication by any function of R commutes with the operator

1(XI a - Xz

v

Cxz

Example 4.

a

ax,

We call the operator associated to the hamiltonian

180

III,§1,4

H(x , p)=2

[P2 +A(M)-2B M-+C M)1

(4. 6)

Rz

R

where A, B, and C:R -+ R are some given functions, the SchrodingerKlein-Gordon operator. Let AO = A(Mo),

Co = C(Mo).

Bo = B(Mo),

The condition that there exist at least one compact manifold V[LO, Mo] defined by (2.4) may be expressed as follows:

Ao>0,

L2O+Co>0,

Bo>-,/A0V/L+C

(4.7)

When it is satisfied, V [Lo, Mo] is unique, and by (4.3),

N [Lo, Mo] = 2n

f~ - AOR, + 2Bo R - Lo - Co R ,

where the integral is calculated along a cycle of C enclosing the cut JR1 i R21; taking residues at R = oo and R = 0 gives

N[Lo, Mo] =

BO

- v[L:o + Co > 0.

v' Ao

The statement of theorem 3 becomes the following.

Let a be the Schrodinger-Klein-Gordon operator, that is, the operator associated to the hamiltonian (4.6). Let aLl and am be the operators associated to the hamiltonians L 2 and M. Then the lagrangian THEOREM 4.1.

system

aU = (aL: - Lo) U = (am - MO)U = 0 mod

a V

has solutions defined on a compact manifold if and only if there exists a triple of integers (l, m, n) such that

Lo=h(l+Z), Iml

l < n,

Mo=hm,

(I + )z +

(4.10) Coh-z

> 0,

AO =Boh-z[n-l-I+

(l+1)2+Coh-z]

z

(4 . 11)

If this condition is satisfied, this compact manifold is unique: it is the torus T(l, m, n) given by the equations

III,§1,4

181

T(l, m, n): H(x, p) = L(x, p) - Lo = M(x, p) - Mo = 0.

(4.12)

We provide this torus with the invariant measure (3.16); then the solutions of (4.9) defined mod(l/v) on T(l, m, n) are the lagrangianfunctions with constant lagrangian amplitude.

Notation. (Related to physics and used only at the end of this section)

In E3, we choose an electric potential sago : E3 -+ R and a magnetic potential vector ('41, d2, a3): E3 - E3 satisfying the physical law

i=i axi

= 0;

(4.13)

they are time independent. The trajectories in E3 of the relativistic or nonrelativistic electron of

mass µ and electric charge -F < 0 are the solutions of Hamilton's system, defined by the hamiltonian given by

sii(z)12-E

H(x,p) = 1

2µi=1

CPi

+

- Fsdo(x)

c

(4.14)

J

in the nonrelativistic case, and by sii(z)12

H(x, p) = 2µ iY-i [pj +

- 2µc2 [E +

2 + iµc2

(4.15)

in the relativistic case. Here c is the speed of light, E is a constant, which is the energy of the electron whose position x and momentum p satisfies H(x, p) = 0 (energy including the rest mass µc2 in the relativistic case). By (4.113)

a'x,

ap) H(x, p) = 0.

Then by// I1,§1,(6.3) and II,§l,definition 6.2, in the nonrelativistic case the operator associated to H is the Schrodinger operator

a= 2µ

v2 0 + ce Ed'(x) v ax j + i

c2

E i

(x)1

-E

- Fdo(x),

(4.16)

182

III,§1,4

and in the relativistic case it is the Klein-Gordon operator

a=

+ 2z

2Lyz

I.i 1

+ E2 I d2]

0

-

2-t- [E + Ed0]2 +

µ2z (4.17)

Let us choose these hamiltonians and operators as follows: d0 is the electric potential of a nucleus with atomic number Z, placed at the origin; the magnetic field is constant, parallel to the third coordinate axis, and of intensity .* . In other words,

-Zrxz,

SIo(W ) = R ,

z(x) = Z. x1,

S13 = 0. (4.18)

Let us neglect the . Z terms, as is done in physics; the hamiltonians (4.14) and (4.15) of the electron become hamiltonians of the form (4.6)

H(x, p) = Z

[P2

µ

+ c .M - 2µE -

2µR2

Zl

(4.19)

in the nonrelativistic case, and

Hx ( ' P)

2µ '

I

P2+Ec M+ µz c 2

Ez_2e2ZE-e4Zz cz

c2R2

c2R

(4.20)

in the relativistic case. The Schrodinger operator becomes

a=2µ[v A+E

(xla3Z-xzOX

µz -2µE-2RZ(4.21)

the Klein-Gordon operator becomes

a=

1

1

2µ vz

+ ec xl aX2

xz

8x1

+ µzcz - Ez cz

2e2ZE e4Zz - czR - czR zl. (4.22)

Relation (4.11)3 defines E as a function of (l, m, n); in the course of this calculation, two constants, which are classical in physics, appear. z

he

-

137,

the dimensionless fine-structure constant,

183

III7§1,4

p=

,

2u

the Bohr magneton (j3.)(° has the dimensions of energy),

as well as the function given by

F(n, k) =

1

.

2

aZ

1 +

(4.23)

2

The statement of theorem 4.1 becomes the following. Let a be the Schrodinger operator (4.21) or the Klein-Gordon operator (4.22), H being (4.19) or (4.20). Let aL= and aM be the operators associated to L2 and M. For certain values of the constants E, Lo, and M0, the lagrangian system THEOREM 4.2.

aU = (aL2 - Lo )U = (am - Mo)U = 0 mod z

(4.9)

has solutions defined mod(1/v) on compact manifolds. These values of E, Lo, and MO are expressed as functions of a triple of integers (1, m, n) such that

ml < I < n and, in the Klein-Gordon case, l + z 3 aZ; these expressions for E, Lo, and MO are -ucZ

E_

azZz

2n2 +

j3.*'m in the Schrodinger case,

E2 = uc2(uc2 + 2f3.°m)F2(n, l + Z) in the Klein-Gordon case,

Lo = h(l + Z),

Ma = hm.

(4.24)

(4.25) (4.26)

These solutions of (4.9) are defined on the tori (4.12). Let us provide these

tori with the invariant measure (3.16); then these solutions are lagrangian functions, defined on these tori, with constant lagrangian amplitude. Remark 4.2.

±E

Physicists, neglecting j32 and j3a2, simplify (4.25) as follows:

uc2F(n, I + Z) + f.#°m.

The minus sign concerns antimatter (u < 0). The energy levels (4.24) and (4.25) are those given by the study of the partial differential equation Remark 4.3.

184

I I I,§ 1,4-III,§2,0

au = 0, whose unknown u:E3\{0} -+ C and its gradient have to be square integrable (III,§4).

§2. The Lagrangian Equation aU = 0 mod(1/vs) (a Associated to H, U Having Lagrangian Amplitude > 0 Defined on a Compact V) 0. Introduction

In §1, we studied problem (1) (introduction to chapter III), that is, a system of three lagrangian equations. In §2, we study the first of these three equations and, more precisely, problem (2) (introduction to chapter III). Each solution of problem (1) is a solution of problem (2), by theorem 3 of §1. In the exceptional case of a hamiltonian H independent of M (for Summary.

example, Schrodinger and Klein-Gordon without a magnetic field), a rotation in E3 acting simultaneously on x and p obviously transforms solutions of problem (1) into solutions of problem (2) that are no longer solutions of problem (1). Without considering this exceptional case, theorem 1 of §2 constructs solutions of problem (2) that are not solutions of problem (1): a solution of problem (2) defined on a torus T(l, m, n) (theorem 3 of §1) is not necessarily unique up to a constant multiplicative factor.

On the other hand, theorem 2 shows that even if H depends on some parameters, these tori T(l, m, n) are, in general, the only compact lagrangian

manifolds V on which solutions of problem (2) are defined. Moreover, theorem 3.1 specifies that in the Schrodinger and Klein-Gordon cases problems (1) and (2) define the same energy levels: the classical levels. Remark. Theorem 1 of §2 does not have any physical significance: it takes

into account the condition that some numbers, which in the Schrodinger and Klein-Gordon cases are measurements of physical quantities, take rational values. Let us call a problem that when posed for the SchrodingerKlein-Gordon equation has the essential features of the classical problem (4) (introduction to chapter III) a well-posed problem. Problem (1) is wellposed by theorem 4.2 of §1. From the preceding remark, the main result of CONCLUSION.

§2 is that problem (2) is not well-posed.

III,§2,1

185

1. Solutions of the Equation aU = 0 mod(1/v2) with Lagrangian Amplitude -> 0 Defined on the Tori V [L0 , Mo)

In section 2 of §1 we chose H and defined the manifolds

W:H=0,

V[L0,Mo]:H=L-Lo=M-Mo=0.

Any compact lagrangian manifold V in W is a union of characteristics K of H (whose paramter t varies from - cc to + oo); V is thus a union of compact closures K of such characteristics. Properties of a characteristic K of H with compact closure K. By (2.13) of §1, such a characteristic K stays on a torus V[L0, Mo] and is given by the equations

K:`Y-To+).[L0,M0,t]=(D -(o+p[L0,M0,t]=0 (0o, `1'o constants)

on this torus. Recall that the coordinates of a point of V[L0, M0], (`1', 4), t), are defined

(mod 2n, mod 2n, mod co), where co = c[L0, Mo].

More explicitly, let R3 be given the coordinates (`Y, D, t) and let Z3 be the additive group of triples of integers (S, q, () acting on R3 as follows: Z3 -3 ((1, q, C): (P, 4), t) i -+ (P + 2nd, 1) + 2n7, t + c0S);

the quotient of R3 by this group Z3 is V[L0, Mo]: there exists a natural mapping

R3 -' R3/Z3 = V[Lo, M0].

(1.2)

Given a function F: (L,

M,

Obviously

A,R=A,Q=0.

t

M, t].

III,§2,1

186

By the definitions (2.8) of 0 and (2.9) of N in §1,

O0[L, M, t] = 2nN[L, M]; then, by (1.3) and §1,definition (2.10) of

..

and

2ndN[L, M] = AA [L, M, t] dL + A,u[L, M, t] dM; that is, A,,l = 2nNL,

0,u = 2nNM.

Let

NL,, = NL[LO, MO],

NMo = NM[L,, MO]

We can now describe K. LEMMA 1.1.

NL° _

1°) Assume NL° and NM° are rational:

-L1 N,'

Ml,

NMo =

where (L,, M, N1) E Z3,

N,

G.C.D.(L,, M1, N,) = 1

(1.6)

(that is, L,, M,, and N, are integers with greatest common divisor 1). Then K = K is a closed curve given by the equations (1.1). More precisely, the equations (1.1) define an open curve R (that is, a curve homeomorphic to R) in R3. By (1.5) and (1.6), the subgroup 2 of Z3 generated by (L1, M,, N,) leaves ft invariant; we have

K=K=R/Z.

(1.7)

2°) Assume that NL° and N,M° are connected by a unique affine relation L, NL,, + M, NMO = N,

(1.8)

where (L,, M,, N,) E Z3, G.C.D.(L,, M,, N,) = 1. Then K is the 2-dimensional torus T 2 defined in V [L., M0] by the equation

L,{`P - To + 2[L0, M0, t]} + M,{0 - b0 + u[L0, M0, t]} = 0. (1.9)

More precisely, equation (1.9) defines a surface R2 in R3 homeomorphic to R2. By (1.5) and (1.8), the subgroup 22 of Z3 given by the equation

Z:LjI +M,q+N1 = 0

(1.10)

III,§2,1

187

acts on 1l2. We have

T2 = It2/22.

(1.11)

In order to describe the generators of Z2, let M2 = G.C.D.(L1, N1),

L2 = G.C.D.(M,, N1), N2 = G.C.D.(L,, M1).

(1.12)

M2 and N2 divide L1i G.C.D.(M2, N2) = 1 by (1.8)3; then there exists an integer L3 and similarly integers M3 and N3 such that L1 = L3 M2 N2,

Ni = L2 M2 N3.

M1 = L2 M3 N2 ,

(1.13)

Z2 is generated by its three elements (0, M2N3, -M3N2),

(-L2N3, 0, L3N2),

(L2M3, -L3M2, 0), (1.14)

which evidently are connected by the relation L3(0, MZN3, -M3N2) +

M3(-L2N3, 0, L3N2)

+ N3(L2M3, -L3M2, 0) = 0.

(1.15)

3°) Suppose there is no affine relation with integer coefficients connecting NL° and NMO. Then K is the torus V[L0, Mo].

Proof. The subset of R3 whose natural image in V[L0, Mo] is K is defined by the condition

C ` I ' - To +.[LO, MO, t]b - 0o + u[Lo, MO, t] 2n

2rc

JJ

E G,

where G is the image of Z 3 in the additive group R 2 under the morphism

Z33(

b)~'(S + NLOS,fl + NM00ER2.

Then K is the natural image in V[L0, Mo] of the closed subset of R3 defined by the condition C`I'

- To +2 [Lo, M0, t] (D - .bo + 2[Lo, Mo,

E

t] where G is the closure of G; G is a closed subgroup of R2. Three cases occur: (1°) G is discrete, (2°) dim 6 = 1, (3°) 6 = R2.

III,§2,1

188

1°) G is discrete, that is, C = G. This is the case if and only if NL° and NM° are rational (use the rapidity of convergence to an irrational number

of its rational approximations given by its continued fraction development), that is, if and only if (1.6) holds. Then K = K; the elements of the

subgroup Z of Z3 leaving invariant the curve R c R3 given by the equations (1.1) are the

N1 = L1C,

rl, l;) in Z3 such that

N,tl = M1 C-

By (1.6)4, N1 divides C. Thus Z is generated by (L1, M1, N1) E Z3. 2°) dim G = 1. Then G is the set of (0, t) e R2 satisfying a condition of the form

L10 + M1t E Z, where L1, M1 E R;

(1.17)

by the definition of G, G is the subgroup (1.17) of R2 if and only if (1.8) is satisfied and G is not discrete, that is, by (1°), if and only if NL° and NM° are connected by a unique affine relation with integer coefficients, which is (1.8).

Assuming this hypothesis, by definition (1.16) of k and definition (1.17) of G, K is the image under (1.2) of the manifold R2 in R3 given by equation

(1.9); obviously (1.10) defines the subgroup Z2 of Z3 that leaves A2 invariant; and (1.10) implies by (1.8)3 and (1.12) that

L2, M2, and N2 divide , ri, and t;, respectively.

(1.18)

By (1.13), the three elements (1.14) are contained in Z2. On one hand, (1.14)1 generates the subgroup of Z2 given by the equation

=0, because G.C.D.(M2N3, M3N2) = 1 by (1.12)1, (1.13)2, and (1.13)3. Thus

G.C.D.(N3, M3) = 1, which implies that the values taken by in the subgroup of 22 generated by the elements (1.14)2 and (1.14)3 are all the multiples of L2. Thus, by (1.18), the three elements (1.14) of Z2 generate

2.

3°) R2. By (1°) and (2°), G = R2 if and only if NL° and NM° are not connected by any affine relation with integer coefficients. If G = R2, then K is the image of R3 under (1.2); thus K = V[L0, M0].

Invariant measures on V[L0, M0]. Recall that V[L0, M0] has a measure > 0 that is invariant under the characteristic vector K of H [(3.2) of §1] :

189

III,§2,1

tlv = dt A dO A dW. Every measure on V [LO, M0] that is invariant under K is the product of

tlv and a function V[L0, Mo] -+ R that is invariant under K, that is, constant on the closures k of the characteristics K of H staying on V[Lo, Mo]. Then the following lemma is an obvious consequence of lemma 1.1. 1°) Suppose that NL° and NM,, are the rational numbers (1.6). Then the characteristics of H staying on V[L0, Mo] are the closed curves given by the equations LEMMA 1.2.

K(c1, c2): `P + %[Lo, Mo, t] = c1,

b + p[Lo, Mo, t] = c2;

el and c2 are constants defined mod 2n

M2

N,

27c

= L2N3,

2n

L2

Nl

=

2n

M2 N3,

respectively, where

L2 = G.C.D.(M1, N1), N3 = N1/L2M2 E Z.

M2 = G.C.D.(L1, N1),

The measures on V[L0, Mo] that are invariant under the characteristic vector K of H are given by

F(`f' + )[Lo, Mo, t], cD + p[Lo, Mo, t])r)v,

(1.19)

where

is an arbitrary function with periods 27r(M2/N1) and 27t(L2/N,) in the first and second arguments, respectively. 2°) Suppose that NL, and NM,, are connected by the unique affine relation

(1.8). Then the closures K of the characteristics K of H staying on V[L0, Mo] are the tori given by the equations

T2(co): L1 {`P + ).[Lo, Mo, t] + M1 {db + p[Lo, Mo, t]} = co,

(1.20)

where co is a constant defined mod 27t. The measures on V[L0, Mo] that are invariant under the characteristic vector K of H are given by

F[L1(`P + A) + M1(b + u)]'lv,

(1.21)

where F[ ] is an arbitrary function with period 2n. 3°) Suppose that there is no affine relation with integer coefficients connecting NL,, and NM,. Then the closure K of each characteristic K of H

190

111,§2,1-11 1,§2,2

staying on V[L0, Mo] is V[LO, Mo]. Every measure on V[L0, Mo] that is invariant under the characteristic vector x of H is given by (1.22)

const. g v .

The following theorem is an easy consequence. THEOREM 1.

1°) The tori V[L0, Mo] on which a lagrangian solution U of

the equation 12

aU = 0 mod

(a is the operator associated to H)

V

with lagrangian amplitude >,0 may be defined are defined by the condition (3.11), (3.12) of §1: there exist three integers 1,rn,n such that

Iml

1
Lo = h(l + i),

Mo = hm,

Lo + N[LO, Mo] = hn.

2°) A necessary and sufficient condition for the solution U defined on such a torus to be unique up to a constant factor is that the derivatives of N, NL[LO, Mo],

NM[LO, Mo],

are not connected by any affine relation with integer coefficients.

Proof. By theorem 6 of II,§3, the condition that there exist such a solution on V[LQ, Mo] is Maslov's quantum condition. The condition that it be unique is the condition that the invariant measure nv on V[L0, Mo] be unique (up to a constant factor). In section 3 of §1, Maslov's quantum condition is formulated as (3.11)-(3.12). The condition that the invariant measure be unique is given in lemma 1.2. 2. Compact Lagrangian Manifolds V, Other Than the Tori V [Lo , Mo], on Which Solutions of the Equation a U = 0 mod( I/v2) with Lagrangian Amplitude >,0 Exist

We shall show that such manifolds V exist only exceptionally. The calculation of their Maslov index (lemmas 2.3 and 2.4) uses the following properties.

III,§2,2

191

Other properties of the characteristics K of H with compact closures. Differentiating the definition (2.10) (§1) of A and p gives

R dQ n dR + dJ n dL + dp n dM = 0, where L, M, Q, R are functions of (L, M, t) satisfying H[L, M, Q, R] = 0; then

HLdL+HMdM+HQdQ+HRdR=0, whence, by elimination of dQ:

[j_dR+d1]AdL+[!jy-dR+dp]AdM=o; RQ

thus there exist three real numerical functions p, a, and r of (L, M, t), defined when HQ * 0, such that

dA _ -

RddR + p dL + a dM, RHQ L

,u = -

dR + a dL + T dM.

RHQ By the expression (1.5) for A. and O,u and definition (2.6) of tin §1, these relations imply A,p = 21tNL1,

A,a = 21tNLM,

A,[L, M, t] _ -HL[L, M, Q, R],

O,t = 21tNMz,

(2.2)

p, _ -HM.

(2.3)

Let us describe explicitly the singular part of p, a, t, and PT - a2 when HQ = 0; by (2.1),

p = RLHL RHQ

T =

RMHM RHQ

+ L,

a = RMHL + M = RLHM + AL, RHQ

RHQ

+ AM;

thus

pt - a2 = RH [RLHLPM - RMHLPL - RLHMI.M + Q

+ i.L/M - MAL-

III,§2,2

192

Since H[L, M, Q, R] = 0,

HRRL+HL +HQQL=HRRM+HM+HQQM=O. When HR :j6 0, these relations make it possible to eliminate RL and RM from the preceding expressions for p, a, T, and pr - 62; by assumption (2.2) of §1,

HR # 0 when HQ = 0; hence: LEMMA 2.1.

The functions

2

P

+

HL

PT - a2 +

Q

RHQHR,

1

HLHM

+ RHQHR,

2

HM + RHQHR'

_

RHQ HR [Hi 12M - HLHM(uL + AM) + H2 AL]

are bounded in a neighborhood of the points where HQ = 0. Properties of compact lagrangian manifolds V in W.

V is generated by

characteristics of H with compact closure, which thus stay on tori V[L0, Mo]; then the functions

L, M, N = N [L, M],

c = c [L, M]

are defined on V. Let V2 be the open subset of V where dL A dM 96 0 if it is not empty. When dL A dM = 0 on V, let Vt be the open subset of V where (dL, dM) : 0 if it is not empty t3) Then Vt and V2 are lagrangian manifolds, not

necessarily compact, that are generated by characteristics K of H with compact closures k; they contain these closures. When V2 and Vt do not exist, then V is one of the tori V[L0, Mo]. By the following lemma, Vt and V2 only exist if the graph N : (L, M) i--+ N [L, M]

contains a rectilinear segment with rational direction.

Thus the consequence of theorem 2 stated in the introduction (§2,0) follows from this lemma. 'This notation means that dL and d.41 are not simultaneously zero. [Translator's note]

III,§2,2

193

LEMMA 2.2.

1 °) On V2, N is an aff ne function of (L, M). More precisely,

L1dL + M1dM + N1dN = 0, where (L1, M1, N1) E Z3,

(2.6)

G.C.D.(L1, M1, N1) = 1.

V2 is defined in W by the datum of a function F of two variables and by the equations

`P + I[L, M, t] + FL[L, M] _'V + µ[L, M, t] + Fr,[L, M] = 0.

(2.7)

More precisely, in the space R5 with the coordinates (L, M, `P, 'V, t), equations (2.7) define a 3-dimensional manifold P2. By (1.5), where

NM = -M,/N,,

NL = -L1/N1,

the elements q, c) of the subgroup Z of Z3 generated by (L1, M, , N,) E Z3 act on V2 as follows: (5, n, C): (L, M, T, (D, t) i-- (L, M, `P + 2irS,'V + 2nq, t + c [L, M] S ).

(2.8)

We have

V2 = V2/Z

(2.9)

V2 has a measure that is invariant under the characteristic vector of H, given by

qv = dL A dM A dt. 2°) On V1, L, M, and N are affine functions of a single variable s. More precisely,

dL -dM=dN=ds L1

M1

(2.10)

N1

where (L1, M1)

0,

(L1, M1, N1) E Z3,

G.C.D.(L1, M1, N1) = 1.

V, is defined in W by the datum of three functions of s-the affine functions

L and M satisfying (2.10) and F-and by the equations

L = L(s),

M = M(s),

L1{`P + %[L, M, t]} + M1{dV + u[L, M, t]} + F,(s) = 0.

(2.11)

More precisely, equations (2.11) define a 3-dimensional manifold V1 in R5

III,§2,2

194

on which the subgroup V2 of Z3 given by equation (1.10) acts according to (2.8). Recall that this subgroup is generated by its three elements (I.14). We have

V,=V,/Z2.

(2.12)

V, has a measure that is invariant under the characteristic vector of H, given by

qv = (M, d'1' - L, d(D) A ds A dt. 3°) The phases of V, and VZ in the frame Ro (section 1 of §1) are given by

(pR0 =Q+L''+MD+F. Remark 2.1.

(2.13)

(2.8) and (2.10) in §1 define

12 up to the addition of a function F of (L, M) , and µ up to the addition of its derivatives FL and FM.

Given V, or V,, it is possible to choose 0 such that

F=0 in (2.7), (2.11), and (2.13).

In order to quantize V, we shall use only the following consequence of (2.13) and the definitions (2.7) of c[L, M] and (2.9) of N in §1: choosing F to be zero, the function

NR° - 2nNc[Lt M]

- LT - MO

(2.14)

is defined on V, and on V2. Preliminary to the proof. II,§3 applies with

1=3,

h,=L,

go=S2+LT+MO,

Formula (2.11) of §1 shows that theorem 3.1 of

h2=M,

91 = -''-A,

9z= -(D -u,

the phase cpR, replacing the lagrangian phase. Hence any lagrangian manifold V in W is given locally by equations of one of the four following types:

III,§2,2

195

T +i +FL[L,M] =0+,u +F,,t=0;

(2.15)1

M = f(L),

(2.15)2

`V + A + fL(L)(D + u) + FL(L) = 0;

the result of the permutation of (L, T), (M, 0) in (2.15)2; L = const.,

M = const.;

(2.15)3 (2.15)4

F and f are functions of one or two variables. The phase (PR. of V is expressed by (2.13), where F = 0 in the fourth case. Proof of 1°).

Locally, V2 is given by equations of the form (2.15)1. Hence

V2 is generated by characteristics K staying on disjoint tori V[L0, Mo]. Their closures K are disjoint and are contained in V2 ; dim V2 = 3, so that

dim K = 1. Then by lemma 1.1 the values of NL and NM on V are rational numbers. Thus the functions NL and NM are constant on Vand satisfy (1.6), which expresses (2.6), whence 1°) and (2.13).

Proof of 2°). Locally, V1 is given by equations of the form (2.15)2 or (2.15)3, that is, equations of the form

M = M(s), L = L(s), L5(s)('Y + A) + MS(s)(G + u) + F,(s) = 0,

(2.16)

where (Lc, MS) -0 0. For each value of s, let T(s) be the manifold in W given by equations (2.16): dim T(s) = 2. Since V1 is generated by characteristics given by the equations

L = const.,

M = const.,

'Y + A = const.,

0 + µ = const.

and contains their closures, V1 contains the closures T(s) of the T(s). Let us determine the T(s). By expression (1.5) for i,A and D,µ and since L.,NL + M.,NM = Ns, T(s) is the image in W of the set of ('Y, 0, t) E R3 such that L,

'Y + A[L, M, t] + MS 0 + p [L, M, t] 2n

+ 1 Fs E G(s), 27[

271

where G(s) is the image of Z 3 in the additive group R under the morphism

Msrl + N5

.

196

III,§2,2

Then T(s) is the image in W of the set of ('F, (D, t) e R3 such that LS'I'

+ ..[L, M, t] +

Ms(D + µ[L, M, t]

+ IFS

2n

2n

n

e G(s),

where G(s) is the closure of G(s) and thus a closed subgroup of R. G(s) = R, and thus T(s) is the 3-dimensional torus V[L(s), M(s)] unless

G(s) is discrete, that is, unless there exists (L M,, N,) E Z3 such that

-

L' L,

MS

_ N.

M,

N,

G.C.D.(L M,, N,) = 1.

Since dim V, = 3, this must be the case for every s. The functions MS/LS and NS/LS have rational values; thus they are constants. The integers

L,, M, , and N, are independent of s; and s can be chosen so as to arrive at (2.10); thus 2°) and (2.13) hold. Quantization of V.

Let us impose Maslov's quantum condition on V. In order to express this condition, let us calculate the Maslov index MR,, of V2 and of V, using lemma 1 of §1. We make F = 0 in lemma 2.2 (see remark 2.1). LEMMA 2.3. 1 °) The functions p, a, r are defined on V2. Let R be the one-point compactification of R. Then the function F : V2 -. $ given by

F(L , M , t) =

pL2 + 2aLM + TM'

L(L2-M2)(pi-a2) E $

(2 . 17 )

is defined on V2\E", E" being the subset of V2, where

LM

M2

(2 . 18 )

L2

2°) If V2\E" is connected, then

1'I' + 1 arctan F(L, M, t) mR° = 17r n

the integer part of )

(2.19)

on its universal covering space. 3°) The function M&

-

n

- 2 c[L, M]

is defined (that is, uniform) on V2.

(2 . 20)

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197

Assume a function

Remark 2.2.

N: (L, M) i-4 N[L, M] satisfying (2.6)

is given. Choose H satisfying (2.9) of §1. If this choice is generic, then dim E" = 1, Proof of 1°).

V2\E" is connected.

Recall that (L, M, t) are local coordinates on V2. By (2.2)

and (2.6),

1 °) follows.

Proof of 2°). The apparent contour of V2. Formulas (1.11) and (1.13) in §1 give the relation L sin E)

W2 A

G)3

=Lsin`Ydb Ad`Y+L2 -M2(MdL-LdM)A(Ld`Y+Md(b) (2.21)

on W. Differentiating (2.7), where F = 0, gives

d'Y + pdL+adM =dD+adL+tdM =0 moddR on V2, from definition (2.1) of p, a, and T. By (2.6) of §1,

dR = RHQdt mod(dL,dM), hence L

R sin 0

dR A w2 A w3 = G(L, M, t,'Y) dL A dM A dt

(2.22)

where G denotes the function

G = -HQ[L(pt - a')sin`Y + pL2 L2aLM2 zM2cos'Yl, which is regular on V2 by lemma 2.1. Then by lemma 1 of §1, the apparent contour ER°of V2 is

ER°=E'UE",

III,§2,2

198

where E" is defined by (2.18) and E' is the surface in V2\E" given by the equation

E': tan 'P + F(L, M, t) = 0,

(2.23)

where F is defined. by (2.17). Calculation of mRa.

Formulas (1.11) and (1.13) in §1 give

L 0CO3 A wl sin

= Lcos`Pd(D A d'P - Ly-

2(MdL- LdM) A (Ld'P + Md(b) (2.24)

on W, from which, by the same calculations used to deduce (2.22) from (2.21),

L

k -sin 0

dR A (03 A wl = Gy,(L, M, t, T) dL A dM A dt.

(2.25)

Relations (2.22), (2.25) and lemma 1 of §1 give

mR = const. for G/Gy, < 0,

mR = 1 + const. for G/Gs, > 0 in a neighborhood of a point of E'. This result obviously holds for any function G vanishing to first order on E', for example, the function

G = 1'P + 1 arctan F(L, M, t) mod 1. n

n

Thus, on V2\E", m,0 is expressed locally by (2.19), up to an additive constant; 2°) follows.

Proof of 3°). First suppose that H and S2 are generic (remark 2.2). Then V2\E" is connected and

LHL + MHM : O for HQ = 0, which implies by lemma 2.1 that the function

f = pL 2 + 2QLM + TM 2: V2\E" - k

199

III,§2,2

is defined. By (2.19), the function MR.

-

1

n

`l, - 1 arctan F is defined on V2\E";

(2.26)

n

by the definition of F, where I M I < L from assumption (2.2) of §1, we have

F/f < 0 in a neighborhood of the points where F = 0; F = 0 is equivalent to f = 0, so that arctan F + arctan f is defined on V2\E";

(2.27)

we see that

arctan f + arctan f = const.;

(2.28)

by lemma 2.1, 1/f = 0 is equivalent to HQ = 0 and HQ H f <0 R

in a neighborhood of the points where HQ = 0; thus arctan

+ arctan H4 is defined on V2\E";

(2.29)

HR

f

now, from the orientation of I'[(2.9) of §1], arct a n

HR + 27zc[Lt M] is d efine d on ['[L, M]

.

(2 . 30)

The formulas (2.26)-(2.30) prove 3°) for H generic; 3°) follows. LEMMA 2.4.

1°) Define a constant No and a function r on Vl by the

relations

L1M-M1L=No,

T + +Mlr=dD+µ-Llr=0.

(2.31) (2.32)

The functions (r, s, t) are coordinates of the points of V1. A function F : V1 - $ is defined by the formula

200

III,§2,2

F(r, s, t)

_

N2

(2.33)

L(L2 - M2)(pL1 + 2aL,M, + TM;)

when V, is generic. 2°) If V, is generic (No # 0; HL2L1 + 2HLML,M, + MM2M2 HQ = 0), then MR° =

[!w

+ arctan F(r, s, t)] ([ ] is the integer part).

0 for

(2.34)

In

3°) The function (2.20) is defined on V,. Proof of 1°). Formula (2.11)2, where F = 0 (remark 2.1), justifies the definition (2.32) of r. By (2.2) and (2.10),

A,(pL2 + 2oL,M, + TM2) = 2n[L1NL2 + 2L,M,NLM + M2 NM2] 2N

= 2n ds2 = 0. Proof of 2°).

The apparent contour of V1.

Differentiating (2.32) gives

d'P+(pL1 +aM,)ds+M,dr=dI +(aL, +TM,)ds-L,dr = 0 mod dR by (2.10) and definition (2.1) of p, a, T. Whence by (2.21) L

Rsin0

dR A w2 A cu3 = G(s, t, 'P)ds A dr n dt,

(2.35)

where 2

G(s,t,'F) = HQ[L(pL2 + 2aL,M, + TM1)sin'P + L2 N0M2cos'If RHQLLdt

mod(dL, dM), that is, modds. By 1°) and lemma 2.1, since dR = the function G: V, - $ is defined and regular on V, Thus, by lemma I of §1, the apparent contour ER° of V, is given by the equation

ER,,: tan 'P + F(r, s, t) = 0. The same calculation used to deduce (2.35) from (2.21) enables us to deduce the formula Calculation of mR°. L

R sin 0

dR A (a3 A 0), = Gyi(s, t,'P)ds A dr n dt

III,§2,2

201

from (2.24). Applying lemma 1 of §1 as is done in lemma 2.4, it follows that mR° can be expressed by (2.34). Proof of 3°).

mR°

1'f, n

Suppose that V1 is generic. By (2.34),

- 1 arctan F(r, s, t) is defined on V1.

(2.36)

it

From (2.33), where I M I < L by assumption (2.2) of §1, and from lemma 2.1,

F = 0 is equivalent to H. = 0, and FHR < 0 HQ

in a neighborhood of the points of V1 where H. = 0; thus

arctan F + arctan HQ is defined on V1.

(2.37)

R

Formulas (2.30), (2.36), and (2.37) prove that the function (2.20) is defined on generic V1, hence on every V1. 1°) VZ satisfies Maslov's quantum condition if and only if, on Vz, the functions L, M, and N are connected by a relation

LEMMA Z.S.

L1CL-2+M1M+Nl(N+21

=f+N0,

(2.38)

where

L1, MI, N1, No E Z,

N1 : 0,

G.C.D.(L1, M1, N1) = 1.

(2.39)

2°) V1 satisfies Maslov's quantum condition if and only if, on V1, the functions L, M, and N are connected by the three relations (L1, M1, N1) A

\

L - 2, M, N +

= h(Lo, Mo, No),

(2.40)

62I

where A denotes the vector product in E3 and

L1,M1,N1,Lo,MO,NoEZ, G.C.D.(L1i M1, N1) = 1,

Li + MI :g 0, (2.41)

LoLI + MOM1 + NON1 = 0.

Thus only two of the three relations (2.40) are independent.

III,§2,2

202

Preliminary to the proof. A manifold V satisfies Maslov's quantum condition (II,§2, definition 6.2) if and only if the function

21 (PR - 4 mR is defined mod 1 on V. 7rh

Then by (2.14) and (2.20), V, or V2 satisfies this condition if and only if the function L h

1 `F M0 (N j )-t +2) 2n+ h2n+(h +2 c[L,M]

(2.42)

is defined mod 1 on V, or V2.

Proof of 1°). By lemma 2.2,1°) the function (2.42) is defined on V2i V2 = V2/2, where Z is generated by (L1, M1, N1)

and acts on V2 according to (2.8). Thus Maslov's quantum condition is

(

+2)L,

+M, +

+')N, EZ; \

whence 11°).

Proof of 2°). By lemma 2.2,2°) the function (2.42) is defined on V, V, = P, /22, where 22 is generated by

(0, M2N3, -M3N2),

(-L2N3, 0, L3N2),

(L2M3, -L3M2, 0),

L2, ... , N3 being defined by (1.12) and (1.13), and Z2 acts on V, according to (2.8). Thus Maslov's quantum condition is M

M2 N3 -

(iNNi

+ 2) M3 N2 E Z,

-(L +2)L2N3+(N+2)L3N2GZ,

(

+ )L2M3

-

L3M2 E z.

By (1.13), this quantum condition is equivalent to the condition (2.40)(2.41) supplemented by the following one: Lo, M0, and No are multiples of L2, M2, and N2, respectively. Now

203

III,§2,2

L2 = G.C.D.(M,, N,), ... ; thus this last condition is a result of (2.41); 2°) follows. THEOREM 2.

aU = 0 mod

Assume that the lagrangian equation 1

(a the operator associated to H)

V

has a solution U that has a lagrangian amplitude > 0 and is defined mod (1/v)

on some compact lagrangian manifold V other than the tori V[L0, Mo] studied in theorem 1; then the graph of the function

N: (L, M) -- N[L, M]

[see III,§1,(2.9)]

contains a rectilinear segment given by a pliickerian equation

(LI, MI, N,) A (L

- h, M, N + Z I = h(Lo, M0, NO)

(2.40)

such that

Li + M1

L1, M1, N1, Lo, Mo, No E Z,

36

0,

(2.41)

G.C.D.(L1, Ml, N1) = 1, COROLLARY.

LOLL + MOM, + NON1 = 0.

The condition

NoEZ is necessary and sufficient for there to exist such a segment in a planar domain belonging to the graph of N and satisfying an equation

L;(L-2) +M,M+Nf(N+hl=hNo,

(2.43)

where

E,, Mi, Ni E Z,

L' 1

G.C.D.(L'I, M,, Ni) = 1, Remark 2.3. section 3).

+ 1 0 0, 2

(2.44)

No a R.

This corollary makes it easier to apply the theorem (see

Proof of the theorem. By II,§3,theorem 6, V must satisfy Maslov's quantum condition. Since V is not one of the tori V[L0, Mo], V must contain

III,§2,2-III,§2,3

204

a lagrangian manifold V2 or V1 satisfying this quantum condition. Then by lemma 2.5 the graph of the function N contains either a rectilinear segment with the equations (2.40)-(2.41) or

a planar domain with the- equation (2.38)-(2.39) and hence such a segment.

Proof of the corollary. The condition No E Z evidently emplies that such a segment exists. Conversely, assume that in the space R3 with coordinates (L, M, N), the plane with the equations (2.43)-(2.44) contains a line with

the equations (2.40)-(2.41). This assumption is expressed by the four relations

L1L'1 + M1Mi + N1Ni = 0, N0 =

MiNo

- NiMo = Nj'L0 L1

- LiNo =

LIMO

M,

- MiL0 N1

(two of the last three result from the other two). These relations imply NaEZsince G.C.D.(L1,M1,N1)= 1. 3. Example: The Schrodinger-Klein-Gordon Operator Let us choose a to be the operator associated to the hamiltonian H defined by (4.6) of §1, where A is assumed to be an affine function of M, B and C are

assumed to be constant, and B > 0. In §1, we studied the system 1

aU = (aL: - L')U = (am= - M0)U = 0 mod Z v

and recovered the classical energy levels. In studying the single equation

aU = 0 mod Z V

we shall find again the same condition for existence, thus the same classical energy levels. THEOREM 3.1.

aU = 0 modv1

The lagrangian equation (3.1)

205

III,§2,3

has a solution U with lagrangian amplitude >,O defined mod(1/v) on a compact manifold V if and only if there exists a triple of integers (1, m, n) satisfying condition (4.11) of §l,theorem 4.1.

Under this condition, neither the unicity of V nor the unicity of U up to a constant factor is assured. Remark 3.

Under condition (4.11) of §1, the existence of such a solution of (3.1) is assured by theorem I and also by theorem 4.1 of §1. By theorem 1, there exists such a solution of (3.1) defined on a torus V[L0, MO] only if this condition is assumed. By theorem 2 and formula (4.8) of §1, which gives the value of the function N as Proof.

N[L, M] .11/1 M

-

+ C,

the existence of such a solution on a manifold V other than a torus V[L0, Mo] requires that the graph of N contains a line segment. Thus it requires that one of the three following cases occur. First case: A(M) = Aa is independent of M; C = 0. Then by (3.2) the graph of N is the plane with the equation

L+N=

B ,r^_o

Theorem 2 and its corollary require the existence of an integer n such that B

hn.

Ao

Then condition (4.11) of §1,theorem 4.1 is satisfied since it is independent of I for C = 0 and independent of m for A(M) independent of M. Second case: C = 0; A depends on M.

By (3.2), the only lines contained in the graph of N are given by the equations

M = Mo, N + L =

Ao, where MO = const.,

Their pluckerian equations are then

Ao = A(M0).

(3.3)

III,§2,3

206

(1,0,-1) A

(L-2,M,N+21=(MO,-

A ,1l'10).

Theorem 2 requires the existence of integers m and n such that

$ f= hn.

MO = Aim,

(3.4)

Ao

From (3.3) and (3.4) it follows that

n > I m I because N > 0 and L > I MI by assumption [see (2.2) of §1]. Thus condition (4.11) of §1 is satisfied. Third case: A = Ao is independent of M; C 0 0.

By (3.2), the only lines contained in the graph of N are given by the equations

L=Lo,

N=

where Lo = const.,

Ao

-v'Lo+C, Lo + C > 0.

Their pliickerian equations are

/

(0,1,0)A(L-2,M,N+2 =I = `f o \ //

"[LO

C+2,0,2-Lol.

Theorem 2 requires the existence of integers 1 and n such that B

+Lo-\/Lo+C=hn,Lo=h(l+

VIAo

0 < 1 because 0 < Lo; I < n because N > 0 by assumption. Thus condition (4.11) of §1 is satisfied. Choose a to be the Klein-Gordon operator (4.22) of §1. Suppose the magnetic, field .$ 0 0. Then the tori T(l, m, n) defined by (4.12) of §1 are the only compact manifolds on which there exists a lagrangian solution U of the equation THEOREM 3.2.

a U = 0 mod i with lagrangian amplitude >,0.

207

III,§2,3-III,§3,1

Proof. The proof of theorem 3.1 shows that the necessary condition

stated in theorem 2 can not be satisfied when a is the Klein-Gordon operator (C 0 0) and ' 0 (A depends on M). Conclusion

We do not pursue this difficult study of the equation aU = 0 mod(1/v2). In particular, we do not describe the lagrangian manifolds other than the tori T(l, m, n) on which there exist solutions of the Klein-Gordon equation when .f = 0 or solutions of the Schrodinger equation.

§3. The Lagrangian System

aU = (am - const.) U = (aL - const.) U = 0 When a Is the Schrodinger-Klein-Gordon Operator 0. Introduction

In §3, we study the lagrangian system that was solved mod(l/v2) in §1. In section 1, we determine the condition under which theorem 7.2 of II, §3 applies.

In sections 2, 3, and 4, we apply this theorem under assumptions that become more and more strict. These assumptions finally amount to the assumption that a is the Schrodinger-Klein-Gordon operator. Existence theorem 4.1 is finally obtained. Remark 0.

A Voros orally pointed out that these properties of the

Schrodinger and Klein-Gordon equations extend to the case where the electric potential is any positive-valued function of the variable R, if the energy level E is not constrained to be a real number, and if it can be taken to be any formal number with vanishing phase. 1. Commutivity of the Operators a, aL2, and am Associated to the Hamiltonians H (§1, Section 2), L2, and M (§1, Section 1) We want to determine when theorem 7.2 of II,§3 applies to these operators, that is, when they commute. 10) am and a (thus, in particular, am and aL=) commute. 2°) aL, and a commute if and only if

LEMMA 1.

HM2Q = HM2R = 0

YL, M, Q, R.

(1.1)

III,§3,1

208

Proof. Let a and a' be the lagrangian operators associated to two hamiltonians H and H'. By formula (1.1) of II,§2, their commutator

a 0 a' - a'oa is associated to the formal function given by

-2 r

1

1

(2r + 1)! (2V) 2r+1

)]2,4 H(x, p)H'(x', P')

XxpX'ap

P =P

Suppose H and H' are in involution, which is the case for H(§1,2), L2, and M(§1,1) by (1.3) of §1. Then the first term of (1.2) is zero. If H' is a polynomial of degree 2 in (x', p'), then all of the other terms evidently are zero;

thus a and a' commute and part 1 of the lemma follows. Suppose that H' is a polynomial that is homogeneous of degree 4 in (x', p'). Then by (1.2), a o a' - a' o a is associated to H"/v3 where H" is the hamiltonian given by H"(x, p)

24

(ax /-)'

H(x, p)H'(x', p')

/-)]

x

Two applications of Taylor's formula show that the term of H'(x' + y, p' + q) that is homogeneous of degree 1 in (x', p') and degree 3 in (y, q) is

6 [(q, aP / + \Y, ax

)] 3H (x', P') _

[('aq) +

(X ay)] H'(y, q);

thus, if H' is homogeneous of degree 2 in each of its two variables, then

H"(x, p) =

4

[(p' HP (ap' ax)>

- \x' Hz, (ap' ax)!] H(x, p).

In particular, if H' = L2, that is, H'(x', p') = Ix' A p'I2, then H"(x, p)

1

/

a

=2PA --+XA P

a

a

Ox ' a P

A

a

\

ax/J

H(x, P).

In this formula, for each j the pair of operators

and Op n

r p n ca + x A z) P

;

z)

c

commutes.

(1.3)

209

III,§3,1

The operator

Cpna +xn x P is an infinitesimal rotation acting on x and p; thus it annihilates L, Q, and R. Obviously,. Cp

A a + X A Xl M(X, P) = x3P - P3x.

f

P

Suppose H is a composition with L, M, Q, and R [§1,(2.1)]. Then (1.3) becomes

H "(x, p) = 2 (a A X, HMX3P - HMP3X )

\P

Let

X F = P1

F,,

X2

X3

P2

P3

FX2

FT,

YF =

,

X1

X2

X3

P1

P2

P3

Fp2

Fp3

Fp,

for any function F of (x, p). The preceding expression for H" may be written H"(x, p) =

12 aax9/HM 3

1

- 2a

0 H"t.

P3

Now, the linear differential operators I and Y evidently annihilate P2, Q, and R 2, hence L2, by (1.2) of §1; moreover,

-M = -l-L2

1aL2

9M

219X3

2 eP3

Thus, letting !2F denote the functional determinant

1F

aL2 of

aL2 of

ax3 OP3

aP3

ax3,

the expression for H" becomes

H"(x, p) = 4'!HM2

210

III,§3,1-III,§3,2

Now, the linear differential operator .9 evidently annihilates L and M; moreover,

z9Q=P2x3-R2P3 and R -9R = Qxj - R2P3x3; thus the preceding expression for H" becomes i z 2 H"(x, p) = i (P2HM2Q + R HM2R) x3 - R HM2RP3x3 - 2R HM2QP3 (1.4)

By §1.1, p3/x3 is independent of (L, M, Q, R). Thus the condition

H"(x, p) = 0

Vx, p

is equivalent to (1.1), which proves part 2 of the lemma. 2. Case of an Operator a Commuting with aL2 and am

Suppose (1.1) holds. Lemma 1 proves that theorem 7.2 of II,§3 applies to the lagrangian system

aU = (aL2 - cL)U = (am - cM)U = 0 mod 1 2, v

r

where cL and cm are two formal numbers with vanishing phase such that CL

-Lo=cM-M°=Omod v2.

(2.2)

The lagrangian operators aL2 and am have the following expressions ail and am' in R° by I,§1,3 and the formula 1

O0x,

eXp2v

a

2

ap)] L

1

(x, p) =

C1 +

2v

(') a

a

1

+ 8v2

a

a

ap)2]

ax

.(p2R2 - Q2)

=PZRZ - QZ -

2Q- 2v2 3 V

namely aL2(v' x' vex

v (A0

2,J

(2.3)

211

III,§3,2

where (4) 2

a =R2A->xx-2zx. axe j, k aXi OXk k

i

is the spherical laplacian (2.24), which acts on the restrictions of functions

to spheres R = const.; =

+

aM (V,

x, v8X )

V

xl aX2 -

x2aX

which is an infinitesimal rotation. Let us require that the unknown U be defined mod(1/vr+') on a compact lagrangian manifold V. By theorem 3 of §1, i. V is necessarily one of the tori

V = V[L0, M0] = T(l, m, n); H = L2 - L2 = M - M0 = 0, which part 1) of this theorem defines. ii. The lagrangian amplitude /30 of U is necessarily constant. If (30 = 0, (2.1), reduces to (2.1)r_1: hence we impose the condition

J30 = const.: 0.

(2.5)

The problem defined by the system (2.1),, condition (2.5), and the condition that V be compact will be called problem (2.1),. Notation. The invariant measure is given by (3.16) of theorem 3 of §1; d3x by (1.16) and (3.6) of §1; arg d3x = nmRO by definition [(3.4) of I,§3]; MR. is given by (3.5) of §1; arg rl, = 0 by definition; hence the value of the

function x, defined and used in II,§3, and the value of its argument are, respectively,

x=

?IV

= [R3HQ[LOi M0, Q, R] sin 'Psin ®]-', where O = const.;

dr

arg x = - arg HQ - arg sin T, where arg H. = -n[(1/n) arctan(HQ/HR)], arg sin 'P = n['P/n]. (Recall that [ ] is the integer part.) The apparent contour of V is 'As usual: 0 = Ej(a/ax;)Z.

III,§3,2

212

ER°: HQ sin `I' = 0; thus X: V\ERO - R.

(2.7)

Let U be a lagrangian function on V with lagrangian amplitude

f30 = const.: 0. Its expression in Ro will be denoted UR0 (v) = fZ-#(v)e"R-, where fl(v) _ Y

#R

.

2.8)

Vs

By the structure theorem 2.2 and definition 3.2 of lagrangian functions in II,§2, the function X-3,135: V

C

is regular even on ERO

Let Dt,, DL, and DM be operators such that LaU]RO = V A e"-PR, DHl3, V

L(ae - Lo) U]R" =

V X eMR. DLQ, V

DH, DL, and DM commute since a, a,!, and am commute. Let us use the local coordinates (R, `P, (D) on V\ERO. LEMMA 2.1.

DM =

1°) With this choice of coordinates,

8

(2.10)

a(D.

2°) If U is a solution of the equation

(am - cM)U = 0 mod r1 2, v

where cm is a formal number with vanishing phase such that cm = MO modhv,

(2.11)

213

III,§3,2

then 1

cm = M. mod

r+21

$ depends, mod(1/v'+'), only on the coordinates (R, `I').

Proof of 1°). Let us calculate DM by using theorem 4 of II,§3: in this theorem, the hamiltonian M - Mo is substituted for H. By (3.18) of §1, the characteristics of this hamiltonian are given by the equations

dt=dT=0;that is,dR=dP=0. The parameter of these characteristics is b, which is substituted for tin this theorem. We obtain (2.10).

For r = 0, 2°) is obvious. Proceeding by induction on r, we can assume 2°) is true when r is replaced by r - 1. Then Proof of 2°).

cM=Mo+Mr+1

M,+1

C.

By (2.9) and (2.10), (2.11) is equivalent to

M,+i f o ,where J3o = const.: 0. Now, fi, is a function of (D having period 27r; thus

M,+1 =0,

00&

=0,

from which 2°) follows.

Notation. From now on, we assume that (i depends only on the variables (R, `I'). By (2.10), DH and DR° commute with a/a(D and thus act on functions

of(R,`I'). LEMMA 2.2.

DL $

= 2L o

1°) In the local coordinates (R, `P, (D), d-r

d`If

ra _ F1 1

aT

where T is the v ariable

T = cot F

v

J

_s X 4v

(2 . 12)

III,§3,2

214

and F is the operator with polynomial coefficients given by

FQ = F, /i +

OT

[F2

F, and F2 being the polynomials in T given by

5M2x2 Z LO +M0 8Lo(Lo Mo2)

F1 (z)

F-2 (T = (MO T2+ LO)(T2 +1). )

2Lo(L2O

-

M02)

2°) Let U be a lagrangian function that is defined on the torus Y [LO, M0],

has a lagrangian amplitude Qo j6 0, and is a solution of the system

(aL2 - cL) U = (am - MO) U = 0 mod,

(2.13)

where CL is a formal number, with vanishing phase, such that

cL = L p mod V2'

Let E be the set of points on the curve F[LO, M0] [(2.5) of §1], where

E:HQ=0. Then

cL = Lo - 4v2 mod v,+2

(2.14)

and fl(v, R, `F) = g(v, R)f(v, T) mod

vr+j,

(2.15)

g being some formal function with vanishing phase defined on

I'[L0, M0]\E and f being defined as follows. 3°) There exists a unique formal function f defined on R, of the form f (v, T) _

sf, (T), where r e V

such that

(2.16)

215

III,§3,2

df = 1 Ff' dT

(2.17)

V

fo = 1, fs is a real polynomial in z, of degree 3s, having the parity of s, (2.18) and

= 1 + vF2(IdT

- fd }

(2.19)

[I is the complex conjugate off ; (2.19) expresses f2s using f1,

1 f2s-11 .

Any solution of (2.17) mod(l/v') is, mod(1/v'), the product of f and a formal number with vanishing phase. Remark 2. Let U' be the formal function of x that is homogeneous of degree 0, is defined for

M0R < L0'/xi + x2, and is given by f(v ) ev(LOT+Mom)

Uvx=

('

\/sinq

)

(2.20) '

U' satisfies

(Xi-a V

2

- x2 as )U

vI2

0U' = R2 (Lo + 4v2) U'.

1

(2.21)

Proof of 1 °). Let us calculate DL using theorem 4 of II,§3 : in this theorem the hamiltonian L2 - Lo is substituted for H. By (3.17) of §1, the characteristics of this hamiltonian are given by the equations

dt = dD = 0; that is, dR = dQ> = 0. The parameter of these characteristics is `Y/2Lo, which is substituted for t in this theorem. The right-hand side of formula (4.5)2 in this theorem has to be replaced by

1/a a\l

exp2 C

2

r

ax' ap/ J L (x' P) = Ll +

1 /a a\ 2

x' ap

1/a a\2l + 8 ax ap/ J

(P2R2-Q2)=P2R2-Q2-2Q-32

216

III,§3,2

This theorem gives DLL = 2Lo a# + 1 [x_1/2A0x1/2)

- 2 fl],

(2.22)

where A° is defined by (2.4) in terms of the coordinates (x1, x2, x3). By (1.5) and (1.12) of §1, we have R

a OR

=

a

; thus R2 a22 J=1 'axJ 8R x

=

02

2: XJxk

Jk

8xJaxk

(2.23)

using the coordinates (R, T, (D) in the left-hand sides and the coordinates (x1i x2, x3) in the right-hand sides. Then definition (2.4) of 0° is formulated as

0

°

R2A-R2

az

2R-a.

Mk'

(2.24)

oR

Now, formulas (1.5) and (1.12) of §1 give

R = - sin O cos `P, where cos O =

MO

by §1, (1.11);

0

hence, on V, KRX, '1'

J = 0,

R20'P = cot

R2

(1

\T

`

,

W

)= /

1

+

Cot 2 O

`P '

- cot201 sin2'P

The expression for R 2 t acting on functions of (R, `P) follows. Substituted into (2.24), it gives Cot

0° =

C1 +

2

0)

2

T a.j + cot `P1 - sm2 P)

(2.25)

on these functions; hence, from (2.6), X-112Ao(NX112) = ,/sin `P Do (1V

I sT.

-2L°dTFfl

+ a13,

by a trivial calculation. Thus (2.12) follows from (2.22).

(2.26)

III,§3,2

217

Proof of 2°). By lemma 2.1, /3 locally depends only on the variables (R, `P). By this lemma and (2.12), fi° depends only on R; by (2.5), )3o is not identically zero.

For r = 0, 2°) is evident. Proceeding by induction on r, we can assume that P°, ... , Pr_, are polynomials in r whose coefficients are functions of R and that 2°) is true when r is replaced by r - 1. Then CL

2L

5

L°2

v0Lr+1 modv,l 2,

4v2 +

where L,+1 E C;

thus equation (2.13)1 may be written

DC

Vc# +

4i,2

- 2Lr+1r+1 1 33 = 0 mode,+2 o

(2.27)

By the induction hypothesis, this equation holds mod(1/v'+ 1); thus, by (2.12), it may be written duir = (Fflr_ ) dT + L,+ 1 fo dT for R = const.

Then, since F is an operator with polynomial coefficients, fl, is the sum of a polynomial in T = cot W and the function Lr+ 11 P, where f o # 0. But 13r is a function of IF with period 21r. Hence Lr+ 1 = 0,

Fir

is a polynomial in T;

thus (2.27) may be written df3 =

(FfJ) A mod v,l for R = const. 1

If 3°) isv true, (2.15) follows.

Proof of 3°). written

fo = const.,

The condition that (2.16) satisfy (2.17) mod(1/v') may be df, dT

= Ff,-1 for s = 1, ... , r - 1.

Thus f, is a polynomial of degree 3s in r containing an arbitrary constant of integration. Then f is well defined up to multiplication by a formal number with vanishing phase. Let us choose the constants of integration to be real. Then the f, are real.

III,§3,2

218

There exists a unique choice of the constants of integration of the f 2,- 1 such that the f, have the parity of r. Proof of (2.19). Let f be the complex conjugate off. Since v is purely imaginary and F is real, (2.17) implies dT

v

dT d

FI,

(f) = v (IFf - fFf

that is, by the definition of F,

d(ff)= dT

v

dT

(F2L) dT

-f

1[WdT

AdCF2df dT

v

F2Cf4I

dT

- fdlll. dT JJ

thus

if -

1

F22

v

T-

f df dT

c

=

,

where co = 1,

SseN ENV

that is, since the f, are real, for all s e N c2s+1 = 0, 2s-1

2s

Z

d

2F2 Y (-1)sf,'dTf2s-1-s' + C2, ,'=o

s =o

If s > 0, this formula expresses f2s using f1, ... , f2s-1 and c2s, which is cancelled by an appropriate choice of the constant of integration of f2,. Proof of (2.21)2.

DJ

By (2.12) and (2.17),

4vf'

that is, by the definition (2.9) of DL,

[a2(vx,

- L+

1(f fe^)

0

or, bythe expression (2.3) for at= and the expression (2,6) for x, since Do acts

on the restrictions of functions to spheres R = const. and since cpR can be expressed by (3.4) of §1, where Q only depends on R, CV2

Do - Lo -

4v2} U'(v, x) = 0

219

III,§3,2

where U' is defined by (2.20); U' is homogeneous of degree 0 in x; (2.21)2 follows from this by the definition (2.24) of A0. Proof of (2.21)1. From the definition (2.16) off and the expression (2.10) for DM it follows that

DMf = 0, or, by the definition (2.8)-(2.9) of DM,

(a+ - Mo)(f

.fe"w"")

= 0.

In the beginning of section 2, we obtained

-1( xl x2

+ am

V

-

X2

-8x° J / 1

Thus a;, annihilates functions of the single variable R. Then by the expression (2.6) for X and the expression (3.4) of §1 for (PR., the preceding relation is equivalent to (2.21)1.

There exists an operator D, acting on formal functions g

LEMMA 2.3.

defined on ['[LO, Mo]\E, such that DH[f(v, `P)g(v, R)] _ .f (v, `V)Dg(v, R).

(2.28)

Locally,

D=

D.' R, d M .N y` 1

,

(2.29)

where DS(R, d/dR) is a differential operator defined on ['[L0, M0]\E and Do = RHQ

Proof.

Since DH commutes with DL, which is expressed by (2.12),

d'I' Ca = DH

(2.30)

dR

V

Fl DH [ f (v,'I')g(v, R)]

[dz

d`I' O-T

- 1 F .f (v, `')g(v, R)0. v

Thus by lemma 2.2,3°), DH[ fg] is multiplication of f by a formal function, which is defined on ['[L0, M0]\E and denoted Dg.

220

111,§3, 2

Theorem 4 of II,§3 proves that D has the form DH = sGN vS

DH,., (R,

T, 8R , 8`I')

where DH S is a differential operator. It follows that D has the form (2.29). Since the characteristics of H satisfy (2.14) of §1, the same theorem proves

that

D1(fg)dt = d(fg) mod I for dR = RHQdt, dP = HLdt. But

f=1mod!; V

thus

DNg = RHQ dR mode . This relation is equivalent to (2.30).

10) Problem (2.1)r, defined at the beginning of section 2, is solvable if and only if THEOREM 2.

CL

=Lo -4v2,

cm =Mo.

2°) Problem (2.1), is equivalent to problem (2.31)r, which is stated as follows: To define a formal function mod(1/vr+1) on F[Lo, Mo]\E, g(v) _

1

sEN V

gs, where go = 1, gs: F[Lo, Mo]\E -+ C,

such that

Dg = 0

mod(1/vr+'),

(2.31)r

(HQ)3rg is regular, mod(1/vr+1), on F[Lo, Mo]. Any formal function satisfying condition (2.31)r is, mod(1/vr+l), the product of g and a formal number with vanishing phase. The condition that g be a solution to (2.31)r evidently is equivalent to the following :

III,§3,2-III,§3,3

221

g is a solution of problem (2.3

RHQdg,/dR + E;=1 D,g,-, = 0 on r[L0, M0]\E,

(2.32),

(HQ)3rg, is regular on r[Lo, M0].

On r[L0, M0]\E, define the formal function e"n

U"(v) = /R3HQ, g where S2 is defined by (2.8) of §1. Define U' by remark 2 and lemma 2.2. Then

U'(v)U"(v) is the expression URo of a solution U of problem (2.1)r. Every solution of the system (2.1), defined on V[LO, M0] is, mod(1/vr+l), the product of that solution and a formal number. 3°) Suppose that g is a solution of problem (2.31)r_1. Then there exists a

function gr: r[L0, MO] \i -* C satisfying (2.32) where r is replaced by its universal covering space 1'". The function g, is defined up to an additive constant; moreover, gr is defined on F[L0, Mo]\E if and only if the equivalent problems (2.1), and (2.31), are solvable.

Proof of 1°).

Use lemmas 2.1,2°) and 2.2,2°).

Proof of 2°).

Use (2.6), (2.8), lemmas 2.1,2°), 2.2,2°), and 2.3, and theorems 5 and 7.2 of II,§3.

Proof of 3°).

Use the above theorems.

3. A Special Case

In order to supplement theorem 2, choose

H[L, M, Q, R] = 2 {P2

=

R2

K[R, M] 5

2RZ {L2 + Q2 - K [R, M] }

as in §1,4, and choose K to be an affine function of M. Then condition (1.1) is satisfied. By (4.5) of §1, the expression in Ro of the operator associated to

His a

2v2A - 2RZKLR,v(x18x\\

2

x2i?x)j.

1

(3.2)

III,§3,3

222

Moreover,

Lo + Q2 - K[R, Mo] = 0 on the curve I,[L0, Mo].

(3.3)

E is the set of points on this curve where Q = 0. LEMMA 3.

D

R

We have

[dR

V

G]

where G acts on functions g:11[Lo, Mo]\E -+ C and is given by

Gg=G19+dR[G2dR] G1 and G2 being functions defined on I'[L0, Mo]\E given by Gl(R, Q) = GQR),

G2(R, Q) _

-2 Q ,

(3.6)

where 32R(KR)2 + g(K - L2)(KR + RKR2).

G3(R) =

Proof. The operator D may be described as follows with the aid of II,§3, theorem 4: in formula (4.5) of this theorem, s = 2 and H(2)(X'

2 L eap

ox ' ap

P) =

[

2

=

2 P2;

(3.7)

the equations (2.14) of §1 giving the characteristics imply that

dR =

dt,

d'1' = L0 dt;

R then, by the definition (2.9) of D11, when /3 only depends on the coordinates (R, Yf) of V, this theorem gives DH/3dt = d/3 + -1 X-1r2A(fX112)dt

I1I,§3,3

223

for dt, dR, and d`' satisfying (3.8); in other words, /3 Q

__

/ _LO

2vX '120(QX'n

OR R + 8YJ RZ +

DHR

By (2.6)

O = const.;

x = [QR sin IF sin O]

therefore, by the definition (2.24) of Do, which acts on the restrictions of functions to spheres R = const., X

112 0(flX112)

sin

= RZ

A. is-in -IF

R

R(8RZ

+

OR)(/QR).

Replacing Do by its expression (2.26) and substituting the result into (3.9) yields DH

02 QR (8R2 2v

l - QoQ R OR +

LzdT(8 +R d'Y

8r

8 2

Q

+R

1 )1 2 vF+8vR

Choose

)(v, R,'1') = f(v,'Y)g(v, R), where f is the formal function of 'Y defined by lemma 2.2 and g is a formal function defined on I'[L0, Mo]\E. In accordance with lemma 2.3, we obtain

DH(fg) = fDg with Dg _ _ Q

9

R IdR +

2v IQ (RZdRZ + 2R dR/ (

QR) + 8v QR1.

By a trivial calculation, we deduce the expression (3.4) for D, with G expressed by (3.5) and G1 and G2 given by G1

_1d RdQ 4 dR [QZ

1R dQz

dR + 8 Q3 (dR)

GZ

__1R 2Q

224

III,§3,3

Now by the equation (3.3) for F[Lo, Mo],

Q2 = K[R, Mo] - Lo hence the expression (3.6) for G, follows.

The preceding lemma permits a statement of the properties of problem (2.3) to which problem (2.1), has been reduced by theorem 2.

A function g: F[L0, Mo]\E - C is said to be even or odd

Definition 3. when

9(Q, R) = ±g(-Q, R)

V(± Q, R) E I'[Lo, Mo]\E.

(Complement to theorem 2) Assume (3.1). 1°) If problem (2.31), has a solution g such that go = 1, then it has a

THEOREM 3.1.

unique solution such that

gs is real and has the parity of s (s < r),

go = 1, and

99 = 1 -

I

2v

do) mod dRdRvr+,. -g

R (9 dg

1

(3.10)

Q

Formula (3.10) means that, for 2 < 2s < r, 2s R 2s-1 d

sE

(-1)5925 s95 =

Q

S'o(-1)sg,'dR92s-1-5;

(3.11)

thus it expresses 92, using g1, ... , 92,-1

2°) If problem (2.31)2,_, has a solution, then problem (2.31)2, has a solution.

Remark 3. The formal function U" defined on F[Lo, Mo]\E by theorem 2,2°) is evidently given by

U"(v) = g

QR

e °o .

( 3 . 12)

It satisfies

0 U"(v R) = R Z { K [R Mo] - Lo - 4v2 U (v, R), where A is the laplacian, specifically (d 2 /dR 2) + (2/R)(d/dR).

(3.13)

III,§3,3

225

Proof of 1°). Assume that 1°) has been proved when r - 1 is substituted for r and that problem (2.3)r is solvable. Then, by theorem 2,2°) and lemma 3, gr is defined up to a constant of integration by the conditions dg,

Q3rgr is regular on F[L0, M0].

= Ggr_1 on F[L0, Mo]\E,

(3.14)r

Now, G changes parity. If r is odd, then a convenient choice of the constant of integration makes gr real and odd. If r = 2s is even, then 92, is even and can be chosen to be real. We have dg

_

dR

modvrl1,

Gg=G19+dR[G2dR]

1

By a trivial calculation analogous to the proof of (2.19), we deduce (3.10),

up to the addition of a formal number E,(c,/v`), which is canceled by conveniently choosing the constants of integration of the g2,-

Proof of 2°). Assume that r = 2s is even and that problem (2.31)2s_1 is solvable. By theorem 2,3°), problem (3.14)2s has a solution g2s when t is replaced by its universal covering space t. The proof of (3.11) remains valid; (3.11) proves that 92s is defined on F[L0, Mo]\E; thus 92., is a solution of problem (3.14)2..

Proof of remark 3. By the expression (3.2) for a and theorem 2,2°), {v2A

RZK[R'v

z

xzax

)

][U(v,x)U (vex)]

0,

1

where U' is homogeneous in x of degree 0 and U" only depends on R. This implies

A(U'. U") = U'- AU" + U" - AU' ; hence (3.13) follows by (2.12)2.

Let [R 1, R21 c R c C be the set of values taken by R on F[L0, Mo]\E. Let o. be a simply connected neighborhood of the real Notation.

closed segment [R 1, R21 in the complex plane C; R E C. THEOREM 3.2.

Assume that K is holomorphic in co. Then Q, defined by

(3.3), is holomorphic in (t)\[R1, R21.

III,§3,3-III,§3,4

226

1°) If problem (2.31)25 has a solution g, then, for all r _< 2s,

Q3i94, R) = (-Q)3'g.(-Q, R) is the value of a junction of R that is holomorphic in w. In particular, 92, is a function of R that is meromorphic in w, with poles R1 and R2. 2°) Problem (2.31)25+1 is solvable [and thus problem (2.31)25+2 also] if and only if the primitive of (Gg2i)dR is defined (that is, uniform) in w\[R1, R2].

Suppose 1°) is true. [1°) is evident for s = 0]. Then, by (3.14), the function g25+1: I'[L0, Mo]\t - C is obviously the restriction of the primitive of (Gg2s)dR to the edges of the cut [R1, R2] in C, which is Proof.

defined on the universal covering space of w\[R1, R2]. Obviously, 925+1

is defined on I'[L0, Mo]\E if and only if this primitive is defined on w\[R1, R2) Assume that this condition is satisfied. With a convenient choice of the constant of integration, this primitive 925+1 is an odd meromorphic function of Q in a neighborhood of R1 and R2. Like Q, it takes opposite values on the edges of the cut [RI, R2]. For R near R1 or R2i Q is near 0 and g25 is an even function of Q having a pole of order 6s at Q = 0. From (3.3), (3.5), (3.6), and (3.14),

ddQ 1 = dQ G925 = 2-93-n 925 la - 4dQ[ RKR Q ddQ2s

,

where KR

0.

Thus g25+1 is locally an odd function of Q having a pole of order 3(2s + 1)

at Q = 0. Consequently Q3c2s+ng25+1 is holomorphic in w.

g25+2 is defined by (3.11). Then Q3(2,+2 '92,+2 is holomorphic in a). Consequently 1°) holds when s is replaced by s + 1. 4. The SchrSdinger-Klein-Gordon Case

In order to establish, for any r, the solvability of problem (2.1) which we have reduced to problem (2.31) an appropriate assumption is clearly necessary. Assume that K is a second-degree polynomial:

K[R,M] = -R2A(M) + 2RB(M) - C(M).

227

III,§3,4

In other words, the expression of a in R0 is the Schrodinger-Klein-Gordon operator (§1,4) and A, B, and C are affine functions of M. Then in theorem 3.2, co = C. In the definition (3.5) of the operator G,

G3 is a polynomial in R of degree 3 and G1 and G2 are functions of R holomorphic in C\[R 1, R21 and at infinity, where G1 vanishes to second

order. If g is holomorphic in C\[R1, R2] and at infinity, then so is the

primitive of (Gg)dR. Then all the g, exist and are holomorphic in C\[R1i R21 and at infinity. Since g, is holomorphic at infinity and since Q 3rg, is holomorphic in C, Q 3'g, is a polynomial in R of degree 3r. Thus we

have proven the following two theorems. (Existence and uniqueness) Assume that the expression of a in R0 is the Schrodinger-Klein-Gordon operator (example 4 of §1). 1°) The condition that the lagrangian system THEOREM 4.1.

aU = (aL2 - CL)U = (am - cM)U = 0,

(4.1)

where cL and cm are two formal numbers such that

= v2, o-cM-M0=0mod 2

CL

1

have a solution defined on a compact lagrangian manifold V is the following:

i. CL = Lo + (1/4v2), CM = MO ;

ii. V is one of the lagrangian tori V [L0, M0] = T(l, m, n) defined by theorem 4.1 of §1.

2°) There exists a lagrangian solution U of (4.1) defined on such a torus V having lagrangian amplitude

#0=1. Any lagrangian solution of (4.1) defined on V is the product of U and a formal number with vanishing phase. Remark 4.1. Vx : R 1

The projection of V on X is

x < Rz>

MoJxJ < Lo

xzl +

xzz,

where R1 and R2 are the two roots of the equation

AOR2-2B0R+C0+Lo=0

III,§3,4

228

(A0, Bo, and Co are the values of A, B, and Cat Mo.) (Structure) There exists a unique solution U of (4.1) defined on such a torus V having the following structure: Its expression THEOREM 4.2.

UR,, in the frame Ro (§1,1) has the form (4.3)

U& M = U'(v) U"(v).

Using the local coordinates x e V, on V, U'(v) is a formal function of x that is homogeneous of degree 0 and satisfies

V

I x1

a z

- xz as

1 U'(v , X )

= M U'( v, x ) , O

i (4.4)

V2

AU

Rz l L0 + 40 J

(v' x)

U'(v, x).

U"(v) is a formal function of R satisfying x)

V

=1

{K[R,

M0] - Lo _ 4vz} U"(v, x).

4.5)

U' and U" are defined by the formulas

U' v x (

'

fv' t)ev(r.ovV+Mom)

)

,sin`'

'

(4.6)

U"(v, x) = g(v, Q, R)evo, QR

where T = cot `P and Q is the function of R defined by (2.8) of §1; arg sin 'Y

and arg Q have jumps of +n at the points 'P = 0 mod 7r on R and Q = 0 on ['[L0, M0], which are oriented in the directions dP > 0 and Q dR > 0. Moreover,

f(v) _

?r rEN V

g(v) =rfNI .rgr

fr

are formal functions with vanishing phase defined on R and on I'[L0, M0]\E, respectively, by the following set of properties:

df = 1 Ff, dT

v

dg

dR

=

1

v

Gg,

(4.7)

III,§3,4

229

the differential operators F and G being given by Gg

Ff = F1J + dT[Fzd-r

G1g

+ dR

[G2]

where F1 and F2 are the even polynomials in T of degree 2 and 4 defined by lemma 2.2,1°); QSG1 and QG2 are the polynomials in R of degree 3 and 1

defined by (3.6); fo = 1, go = 1; the functions f, and g, are real and have the parity of r;

Jf =

1

+

1

Fz

v

f d f - f df ) C

dT

dT

99 =

1

+

1

v

G2

(

dg

dg 9

dR -

9

dR (4.9)

These formulas (4.9) give the even functions f2, and 92, in terms of fl, , f2,-1 and g1, .. , 925-1. The odd functions fzs+1 and 92,+1 are defined by the quadratures df2 +l = (Ffzs)dT,

dg25+1 = (Gg2,)dR.

f, is a polynomial in T of degree 3r. Q3rg, is a polynomial in R of degree 3r. Remark 4.2.

Thus g25+1 - G2dg2 /dR

is

the odd function

on

F[L0, M0]\E that is the primitive of the differential form G1g2,dR, namely, of a form II'(R)Q-65-sdR, where IT is a polynomial of degree II(R)Q-6,-3 6s + 3. By the preceding theorem, this primitive shall be where fI is a polynomial of degree 6s + 3. Let us give a more direct proof of this essential fact. LEMMA 4.

d fl(R) dR Qzs+,

For each s e N, the differentiation

- fI'(R) Qzs+3

(4.10)

(Q2 a polynomial in R of degree 2, discriminant Q 0) defines an automorphism fI " IT of the vector space of polynomials of degree 2s + 1.

Let Fl be a polynomial of degree 2s + 1; fI(R)Q-2s-1 is holomorphic at infinity; thus its derivative has a double zero at infinity. Proof.

Consequently (4.10) defines a polynomial fI' of degree 2s + 1. Hence the mapping

II " rI'

230

III,§3,4-III,§4,O

is an endomorphism of a finite-dimensional vector space. Now it

is

evidently a monomorphism; thus it is an isomorphism. Conclusion

This §3 is concerned with finding, for dim X = 3, a lagrangian system, with one lagrangian unknown, having a solution defined on a compact lagrangian manifold unique up to a multiplicative factor. In this §3, we have found a system of this type: the system used in wave mechanics for studying atoms (or ions) with a unique and spinless electron.

Replacing the Schrodinger-Klein-Gordon hamiltonian by the harmonic oscillator hamiltonian for E3, Remark 4.3.

H(x,p)2 [P2+AR2-2B], u

that is, choosing

K[R, M] = -AR4 + 2BR2 where A and B are affine functions of M, one obtains another lagrangian system (2.1) belonging to the same type. This can be shown by computations similar to those performed above, where R2 plays the role that R played above. The energy levels are still those of wave mechanics. Remark 4.4.

Of course, the study of the harmonic oscillator in R is

simpler; in E3, for A and B independent of M, it gives new lagrangian

solutions for the operator associated to the harmonic oscillator, but again the same energy levels.

§4. The Schrodinger-Klein-Gordon Equation 0. Introduction

The following classical boundary-value problem will be called problem (0.1): To find nonzero square-integrable functions

that have square-integrable gradients and that are solutions of the differential equation

I1I,§4,0-I1I,§4,1

231

au=0,

(0.1)

where a is the differential operator associated to the hamiltonian

H[L, M, Q, R] = 21 P2 - R2 K [R, M]}

(0.2)

,

K being an affine function of M [compare (3.1) and (3.2) of §3]. This operator is defined by substituting vo = i/h for v in the lagrangian operator associated to H [(4.5) of §1]. Hence it is a

2V2

-

2RZKrR,

o(x1ax2

- x2az1)]

In section 2, we assume (compare §3,4)

K[R,M] = -R2A(M) + 2RB(M) - C(M),

(0.4)

where A, B, and C are affine functions of M. Then a is the SchrodingerKlein-Gordon operator. In §4, we briefly recall the solution of the classical problem (0.1) in order to observe two facts:

i. There are formal analogies between its solution and that of the lagrangian problem (2.1) of §3, which is to solve

aU = (aL= - CL)U = (aM - c;,i)U = 0 on compact V. ii. The condition for the existence of a solution to this lagrangian problem [or to this problem mod(1/v2), namely, to (3.1) of §1] is the same as that for the classical boundary-value problem (0.1) in the Schrodinger-KleinGordon case, that is, under assumption (0.4). This assumption is essential. Remark 0.

For a suitable choice of A, B, and C, the Schrodinger-

Klein-Gordon equation is the Schrodinger or Klein-Gordon equation (4.21) or (4.22) in § 1, where the

.

2 terms have been omitted. It is customary

to treat the terms of these equations that are linear in .' by "perturbation" theory. We do not do this, but we rigorously solve problem (0.1) in order to show that assertion (ii) is rigorous. 1. Study of Problem (0.1) without Assumption (0.4) Review of the properties of spherical harmonics ui,m.

The set of poly-

nomials in x c- E' that are harmonic and homogeneous of degree 1 is a

232

III,§4,1

vector space over C of dimension 21 + 1. A basis consists of the polynomials I and m integers,

R'ui,m,

lml '< 1,

defined by the following system, where ul.m is homogeneous of degree 0 and thus satisfies R2 Au' = A0u' [see (2.4) and (2.24) of §3]:

Aui.m + R21(l + 1)u',, = 0,

I x1

- x2ax )u',m

axa

\

= imui.m.

i

z

(1.1)

Let S2 be the unit sphere in E3 and let a be the usual measure on S2. Then o J'S2

2

, Ul,m

OX

f

a = 1(1 + 1)

IU.ml2a

S2

and for all (1,, ml)

(112, m2)

S/2

J

x Ul2'm2/ O = fS2 U13,m1Ul2,"2Q = 0.

a

, > is the (sesquilinear) scalar product. The restrictions of the ul,m to S2 form a complete system of functions on S2 : every square-integrable function u : E3 -+ C with square-integrable gradient has a unique expansion

where <

u

ul,mul, m, I'm

where the u;',,, are functions only of the variable R > 0, such that u12d3x = Y

114",mlafo

R2luml2dR < o0

l,mJ. S2

JET

and a

L

ax

a,

2

u

d3x =

J +E

I'm

l.m

2

OX

Ul,m

2

a. J

ml2a

I

I

U! 2

Jo

+ ac

u,%ml2dR

R2

2

d

dR

Ul,m

dR
Resolution of problem (0.1). By (0.3) and (1.1), the condition that u be a solution of problem (0.1) is formulated as

233

II1,§4,1

dz

(d 2 + R dR)u` m(R) + Rz {

K[R, hm] - 1(1 + 1) u';,m(R) = 0 (1.2)

for all 1, m. By writing Ru;,m = v, we may evidently state that result as follows.

Problem (0.1) is equivalent to problem (1.3), which is to find integers 1, m and nonzero functions v:10, + oo I - R such that THEOREM 1.1.

Iml

<, 1,

z

dRZ + Ry

hzK[R,hm] - 1(1 + 1)}v = 0

and

+00 (I+' z)V2dR<

()2dR < oo.

oo,

fo

a

dR

Thus problem (0.1) is solvable if and only if the following problem is solvable: To find two integers 1, m and a nonzero squareRemark 1.1.

integrable function u: E3 -+ C with square-integrable gradient such that

lml
au = 0,

Dou + 1(1 + 1)u = 0,

(x, Y - x2 _ ) u = imu. z

(1.5)

i

Recall that system (1.5) plays an essential role in physics.

Equations (1.1) and (1.2) and the system (1.5) are obtained formally by replacing

Remark 1.2.

v,

U',

U"

U

by

uium, u in equations (2.21) and (3.13) and in the system (2.1) of §3, where account is taken of theorem 3 of §1 and theorem 2 of §3, which require

234

11 1,§4,1-111,§4,

cL

Lo=h(l+2Mo=hm,

LO

-4vz,

that is, by (2.3) of §3,

aL:-CL

v2Ap-h2[l+2

2vh][1+2+2vh]

= -h2[Ao + l(l + 1)] for v = i/h,

( + Q-CM= v X1 1

d)

8 2

-X2-

- hm.

1

Assume that K is a holomorphic function of R at the origin. Let

Co = -K[0, hm],

y = .%(l +

21)2

+ Coh-2.

(1.6)

Apply Fuchs's theorem, that is, the theory of regular points of analytic ordinary differential equations: see, for instance, [19], section 10.3. Fuchs's theorem constructs two independent solutions of equation (1.3).

If 27 is not an integer, the respective quotients of these solutions with R±Y+I/2 are holomorphic at the origin. If y is purely imaginary, then every nonzero solution of (1.3) makes the integrals (1.4) diverge at the origin. The same is true if y = 0 (see Fuchs). Assume y > 0. The solution v of (1.3) that is a product of Ry4 2 and a holomorphic function at the origin makes the integrals (1.4) converge at the origin, while the other solutions of (1.3) make them diverge (see Fuchs). Thus, since problem (1.3) is equivalent to problem (0.1), the following theorem holds. THEOREM 1.2.

If K is holomorphic at the origin, then problem (0.1) is

solvable if and only if the following problem is solvable: To find two integers l and m such that

Iml
y>0,

and such that the nonzero solution v of (1.3), whose quotient with Rl+1r2 is holomorphic at the origin, satisfies

ii

v2dR < oo,

f

()2dR

< oc.

(1.7)

2. The Schrodinger-Klein-Gordon Case

In this case, that is, under assumption (0.4), the preceding theorem can be supplemented. Let

III,§4,2

235

MO = hm,

a=

Bo = B(M0),

AO = A(M0),

Co = C(M0),

= Boh-z.

Aoh-1,

Equation (1.3) becomes the confluent hypergeometric equation: d 2V

dR2 +

[

2

+

yZ R1/41z R

0,

where a2, fl, and y e R, y > 0. If a2 > 0, then choose a > 0. vR-Y-112 is an entire Let v be the nonzero solution of (2.2) such that function of R ; it is defined up to a multiplicative constant (Fuchs). LEMMA 2.

The convergence of the integrals (1.7) is equivalent to the

following condition:

+ 2 - }' is a positive integer.

(2.3)

aProof. Express v by v(R) = RY+1/2e-aR E CsR',

cs E C.

(2.4)

seN

Substituting into (2.2), we obtain (Fuchs) the recurrence formula defining the cs in terms of co :

s(s + 2y)c5 = 2[a(s + y - z)

- fl]c_1

Condition (2.3) holds if and only if E c5Rs is a polynomial.

It implies a2 > 0; thus a > 0 and thus the integrals (1.7) converge. Now let us prove that (1.7) does not hold when (2.3) is not satisfied. Case when a > 0: If EScSR' is not a polynomial, then, for an appropriate choice of the sign of co, cs > 0 for s near + oc. Let E E 10, a[. From (2.5) we have

scs > 2ec5_ 1 for s near + oe ;

thus ESc,R' > c'e2,1 for R near + oc, where c' = const. > 0, and the integral (1.7)1 diverges.

III,§4,2

236

Case when az < 0: The classical integral expression for v gives an asymptotic expression for v which is also classical: for R near + oc, v(R) = ceaRR-Pla + ce-aRRP1a + .

.

where c and care constants and a is purely imaginary. Hence the integral (1.7)1 diverges. (See Whittaker and Watson, [19], chapter XVI, "The Confluent Hypergeometric Function: Asymptotic Expansion.") Case when a

= 0, f < 0:

v(R) = Ry' 1/2

By (2.5),

c,R5, where cs > 0

Vs;

sEN

thus the integral (1.7)1 diverges.

Case when a = 0, $ > 0: R-r-'/zV satisfies a differential equation with linear coefficients. Laplace's expression for its solution gives

v(R) _v k JT I t-2y-'e`2 R(-t-')dt where T is the boundary of a half-strip in C containing the cut I - oo, 0[.

By replacing T by a path on which Re(t - t-') -< 0 (=0 only for t = i) and assuming that R is near + oo, we obtain the asymptotic value

v(R) =

R+ ce-2i,/2%1R] + ...

Thus integral (1.7)1 diverges.

Theorem 1.2 and lemma 2, where a and $ are defined by (2.1) and y by (1.6), prove the following.

When a is the Schrodinger-Klein-Gordon operator, then, for each of the following problems, namely, THEOREM 2.

1°) the classical problem (0.1) that we have just reviewed, 2°) the lagrangian problem (2.1) in §3,

aU = (aL: - cL)U = (am - cM)U = 0 on compact V, 3°) the same problem mod(1 jv2), that is, problem (3.1) in 41,

the condition that there exist a solution is the same: the existence of a triple of integers (l, m, n) satisfying condition (4.11) of §1, theorem 4. Remark 2.

A comparison of theorem 1.2 and theorem 3.1, 1°) of §1

III,§4,2-1I I,§4,Conclusion

237

proves that the preceding theorem does not hold for every operator a associated to a hamiltonian H of the form (0.2). Conclusion

Although the classical boundary-value problems and the lagrangian problems are completely independent, they define the same energy levels for the Schrodinger and Klein-Gordon equations. As a matter of fact the experimental values of the energy levels agree with those of the Dirac equation, which is studied in the following chapter.

I

Dirac Equation with the Zeeman Effect

Introduction In §1, we solve, mod(1/v2), a homogeneous lagrangian problem in several

unknowns. That is one of the simplest problems to which theorem 3 of II,§4 can be applied. The resolution of this problem introduces the quadruple of quantum numbers that arises in the study of the Dirac equation. In §2, we use lagrangian analysis to reduce the Dirac equation mod(1/v2)

to the simpler system solved in §1. This reduction is analogous to the reduction theorem 2.2 of II,§4. Thus we prove that the lagrangian solutions

of the Dirac equation (one-electron atoms in a magnetic field) defined mod(l/v2) on compact manifolds have energy levels that are exactly those for which the classical solution of this equation exists (taking into account the Zeeman effect and even the Paschen-Back effect).

§1. A Lagrangian Problem in Two Unknowns In this §1, we give an application of II,§4,theorem 3. 1. Choice of Operators Commuting mod(1/v3) As in III,§1,1, we let

X = X* = E3, R(x) = Ix

x,pcE3,

P(p) _ Cpl,

Q(x, p) = ,

L(x, p) = Ix A Ph

so that

L2 + Q2 = P2R2.

(1.2)

Let (M1, M2, M3 = M) be the components of x A p. By III,§l,(1.3), the following formulas define three functions H"1, H« and H13j that are pairwise in involution (see [1I,§l,2): H")(x, p) = H[L(x, p), M(x, p), Q(x, p), R(x)], (1.3) H13 (x, P) = L2 (x, p), P) = M(x, P) j121, To use theorem 3 of I1,§4, we must choose three it x it matrices and J(3) that are functions of (x, p) c E3 O+ W. These must be such that the matrices of lagrangian operators associated to the three matrices

H'k)E +

J'k)

(E the p x p identity matrix)

(1.4)

IV,§ 1,1

239

commute mod (1 /0). Since the H(k) are in involution, it follows by remark 3 of II,§4 that this commutivity condition is equivalent to the condition that for all i, k,

(H(h J(k)) - (H(k) J( )) + J(n J(k)

- j(k) j(i) = 0,

(1.5)j k

where (, ) is the Poisson bracket. We choose p = 2. Then the values of the J(k) are matrices acting on vectors in the space C2, which we provide with a hermitian structure. We choose the iJ(k" to be self-adjoint. (The vectors in C2 are called spinors.)

Notation.

Let ao = 1 be the 2 x 2 identity matrix. Every 2 x 2 self-

adjoint matrix has the form x0c0 + or, where xo is a real number and a is a self-adjoint matrix with zero trace:

x1 - ix2

x3

Q = Q[x1, X2, x3]

x1 + 1x2

-x3

= x1Q1 + x262 + x363,

(1.6)

where (x1, x2, x3) e E3 and

Q1 =

(01

10),

62 =

(0

i

1

00,

Q3 = (0

--

)

are the Pauli matrices, which satisfy the relations

Qk = 1,

Cork = -ak(ri,

Q1Q2Q3 = i.

(1.7)

It follows that

Q2[x] _

N2.

(1.8)

Remark 1.1. Recall the consequences of (1.8): Let u2 be a measurepreserving automorphism of C2, that is, u2 E SU(2); u2Q[x]u2 1 is self-

adjoint and has zero trace; thus it is of the form a[y]. Let y = ux. Then

Q[ux] = u2Q[x]uj By (1.8), u preserves Ixl2. More precisely, u is a rotation in E3, that is, u E SO(3), whence we obtain a morphism SU(2) 3 a2 H u e SO(3),

IV,§1,1-IV,§1,2

240

which proves that SU(2) is the covering group of SO(3) of order 2. It is the universal covering group of SO(3) (see Steenrod [17], p. 115). LEMMA 1.

Assume

J(1j = if a3 + iga[x A p], is a datum, where

f(x, p) = f [L(x, p), M(x, p), Q(x, p), R(x)],

g = g[L, M, Q, R].

Then (1.5) is satisfied by choosing J(2) = 0,

P) = - 2Q3.

Remark 1.2.

Relations (1.5) evidently remain valid when arbitrary com-

(1.10)

plex numbers (that is, their products with the identity matrix denoted by co = 1) are added to the H)`) and P'`). (H12!, J(1)) = 0 because, by III,§1,1, L is in involution with Q, R, and M = M3, and similarly with M1 and M2. Proof of (1.5)1.2.

Proof of (1.5)1,3. (H)3), J(1))

Since M is in involution with L, Q, and R,

= ig(M, a[x A P]) = iga[-M2, M1, 0]

(1.11)

because an immediate and classical calculation gives

(M. M1) = -M2,

(M, M2) = M1.

Moreover, (1.7) implies

a3a[x A p] - o[x A p]a3 = 2icr[-M2, M1, 0]; whence, by the definitions (1.9) and (1.10) of Jj1) and J(3), J(3)J(l)

_ j(l)j(3) _ -iga[-M2, Ml, 0];

(1.5)1,3 results from (1.11) and (1.12). The proof of (1.5)2,3 is evident.

2. Resolution of a Lagrangian Problem in Two Unknowns

Notation. Let

af.s'

aL2,

am

(1.12)

IV,§1,2

241

be the 2 x 2 matrices of lagrangian operators associated to the 2 x 2 matrices

H[L, M, Q, R] + v f [L, M, Q, R]a3 +'g[L, M, Q, R]a[x A P], L2,

M.

(2.1)

Let us study the solutions U = (U1, U2) of the system

at,gU = (aLl - Lo =

(a,M

2i Lol') J

V

U

- Mo - 'm'+ 2v a3) U = 0 mod

z,

V

where Ul and U2 are two lagrangian functions defined on a compact lagrangian manifold V, l' and m' are real numbers near 0 whose squares are negligible, and Lo and Mo are two real numbers. (The interest of this system is shown by lemma 1 and remark 1.2.) As in 111,§ 1, let V [Lo, Mo] denote the lagrangian manifold in E3 Q E3 given by the equations

V [Lo, Mo] : H [Lo, Mo, Q, R] = L(x, P) - Lo = M(x, P) - Mo = 0. (2.3)

When this manifold is compact, it is a torus whose points have the coordinates

t modc[L0,M0],

'P mod 27r,

(D mod 27r.

F[Lo, Mo], t, and c[Lo, Mo] are defined in III,§1 by (2.5), (2.6), and (2.7), respectively. Assume that on V [Lo, Mo]

(f - ZHN)c[L, M] and Lgc[L, M] are near zero and have negligible squares. For V[L0, Mo], and hence I'[L0, M0], compact, let .fo =7
J r[L,,M0]

I

Jr[L0,M0]

.f[Lo,Mo,Q,R]dt,

g[Lo, Mo, Q, R] dt,

ML

fo + 2 o.fogo + go > 0, 0

(2.4)

IV,§1,2

242

where IMOI < L0. We shall see that f0, g0i and c have negligible squares. THEOREM 2. The compact manifolds V in E3 Q+ E3 on which solutions U of the system (2.2) are defined are the tori V [L0, M0] such that

Lo=h(l+i-I'), h(l +

i) +

N [h(1

M0=h(m-rn'),

+ 4), hm] = fin ± hs[h(1 + ' ), hm],

(2.5)

where 1, m - i, and n are three integers satisfying the inequalities Iml

l+z,

1
(2.6)

V and m' are numbers near 0 that are arbitrary unless m = I + '-z. Then they must be chosen so that

IM01 < L0.

U will be denoted by U+ or U_ depending on whether (2.5)3 holds with the choice + or -. The lagrangian amplitude Remark 2.1.

A, f2) of U = U+ will be denoted by /3± . It satisfies the relations /3±(t + c[L0, M0], `E, (D) = c±i #+(t,'p, (D), lf+(t, `Y + Zit, (D) = c±2 f3±(t, IF, (D),

(2.8)

fl±(t,'I','D + 2n) = c±3 f3± (t, Y', (D) The

are complex numbers with modulus 1 given by

Cfi = e 2rzi(l'NL[Lu.Mo]+m'N.d[Lo,Mo]+e[Lo,Mo])

C+2 =

e2nil '

c,

±3 = -e 2nim '

2.9)

Proof. By lemma I and remark 1.2, theorem 3 of l1,fi4 can be applied. Then the V on which a solution U of the system (2.2) is defined are the covering spaces of the manifolds V given by the equations (2.3). The compact V are the tori V [L0, M0]. Let us determine when U is defined on V [L0, M0], that is, when q (j)'2e0ofl: V [L0, M0] -' C2,

(2.10)

where R0 is the frame we are using,

arg rl = 0,

arg d 3x = mR,,

and mRo and rpR are the Maslov index and the phase of V[L0, M0] in R0.

IV,§1,2

243

By (3.5) and (3.4) of IIL§1,

- I arctan HR1, where [ ] = integer part,

1In

mRo =

YJ

(2.11)

JJ

PRO=S2+LO'P+M0D, where 0 is defined in III,§1 by (2.8).

We now finish applying theorem 3 of II,§4. By the remarks 3.2 and 3.3 of III,§1, the characteristic vectors K, KL2, and KM of the hamiltonians H, L Z, and M are

d(D = Hy;

KL':dt=0,

d'P = HL, d`P =2L0,

KM:dt=0,

d'!=0,

dl = 1.

K:dt = 1,

dD=0;

Then by this theorem the system (2.2) is equivalent to the condition 8

at PT

; Q + ifQ0 + igQ[x n P]I3 = 0, a0 +H +H ' aT Mad - il'/3 = 0,

-cq)

im'/3 +

2

r3 f l = 0.

In other words, it is equivalent to the pfaffian system

df1 + i{ fa3dt + gx[x A p]dt - l'(dT - HLdt) - (m' - za3)(d(D - HM dt)}Q = 0,

(2.12)

which is guaranteed to be completely integrable by this theorem. By (1.5) of III,§1,

x A p = LOJ3; thus by (1.12)3 and (1.11) of III,§1,

(r[X A p] = L [Q, cos 4) sine + Q2 sin 0 sine + Q3 cos O], where

cos o =

MO

Lo

Then, by (1.7),

(2.13)

244

I V,§ 1,2

Q[x A p] = L001(cos 0 + ir3 sin (D) sin 0 + L0Q3 cos O = Loule'm°3sin O + L0Q3 cos O = Loe im°,/2(o1 sin 0 + Q3 cos O)eio";12

Moreover, the functions of [L, M, t], 7., and µ, which are defined by (2.10) of III,§1, satisfy

,=

- HL,

µt = - H

(2.14)

by (2.3) of 111,§2. Let

/3 = e-i°3m;2+ir(1F+x)+iM(m+µ)y

(2.15)

Then the system (2.12) may be written

dy + i[(f - ZHM)Q3 + Log(Q1 sin 0 + r3 cos O)]ydt = 0.

(2.16)

It has now become evident that this system is completely integrable: here y is independent of 0 and 'Y and is only a function of t.

Condition (2.10), which is that U be defined on V[L0, M0], is then formulated as

( ex )1j2

evoL2+voLoY`+voMom-io3m/2+ii'(`Y+x)+im'(mtµ). . V

C2.

11

Now S2, A, and p only depend on t and, by (1.4) and (1.5) of III,§2, increase by

AtS2 = 2tN[Lo, M0],

A,.1 = 2rzNL[Lo, M0],

Ap = 2nNM[Lo, Mo]

(2.17)

when t increases by c[L0, M0N is defined by (2.9) of 111,§ 1.

Let vo = i/h (h > 0); then, by (2.11), condition (2.10) that U be lagrangian on V may be formulated as follows, since e-"3 = -1: There exists e e R such that

y(t + c[L0, M0]) = e2" y(t),

(2.18)

N[L0,M0]+1'NL+m'NM+Z+e=Lo+l'-Z

= M0+m'-z=0mod 1. (2.19)

IV,§1,2

245

The search for solutions y of (2.16) satisfying (2.18) is a classical problem:

(f - 4Hm)r3 + Log(Q1 sin 0 + o3 cos O) is a self-adjoint matrix with zero trace; it is a periodic function of t; the period is c[L0, Mo]; then the 2 x 2 matrix u(t) defined by

du + i[(f - IHM)Q3 + Log(Q1 sin 0 + o3 cos O)]u dt = 0, u(0) = 1,

(2.20)

is a unitary matrix with determinant 1, that is, u c- SU(2); it satisfies

u(t + c[L0, Mo]) = u(t)u(c[L0, MO]); the general solution of (2.16) is

y(t) = u(t)d, where 6 E C2;

y satisfies (2.18) if and only if 6 is one of the eigenvectors of the matrix u(c[L0, Mo]) E SU(2). Let e±2aie[Lo.Mo]

(0 -< e[Lo, Mo]

i)

(2.21)

be the eigenvalues of u(c[L0, M0]). Then, in (2.18),

e = +e[L0, Mo]. Thus, there exists a lagrangian solution U of (2.2) defined on the torus V[L0, Mo] if and only if there exist three integers 1, m - Z, and n such that

ILo+1'+1N[Lo, MO]+1'NL+m'NM=n+e[Lo,MO], Lo + I' - 2 = 1,

Mo + m' = m, where IMo < Lo.

(2.22)

This assertion does not assume that 1', m', (f - HM)c[L, M], and Lgc[L, M] have negligible squares on V[L0, Mo]. z U and /l will be denoted by U± and #± depending on whether (2.22) holds with ±e[L0, Mo]. By (2.15), (2.17), and (2.18), where e = +e[L0, Mo], fl± satisfies (2.8), where the c±k have the values (2.9). Assume that the squares of 1' and m' and the squares of the derivatives of he are negligible. Then (2.22) reduces to (2.5). The integers 1, m - Z, and n must satisfy condition (2.5)3, which is independent of l' and m'. Since 1 and n are integers, since a is near 0, and since N > 0, (2.5)3 implies

IV,§1,2

246

I < n. Since 1' and m' are near 0, (2.5)1 and (2.5)2 imply Iml < 1 + Z, which completes the proof of (2.6).

It remains to prove that (2.4) is an approximate expression for e[Lo, Mo]. Since we have assumed that, on V[Lo, M0],

(f - 1HM)c[L, M] and Lgc[L, M] are near zero and have negligible squares, the solution of (2.20), modulo these squares, is

u = 1 - i [ r (f + Z pt) dt a3 + L0

g dt (a, sin 0 + a3 cos O) fo

0

by (2.14)2. In other words, choosing the right-hand side to be in SU(2), like u,

u = e-i[fo(J+j,.

2)dta3+L0S0gdt(a,sin0+a,cos®)

Thus, by (2.17), definition (2.21) of e[Lo, Mo], and definition (2.4) of fo and go, the eigenvalues +2rce[Lo, Mo] are approximately the eigenvalues of the self-adjoint matrix with zero trace given by for3 + go(al sin 0 + a3 cos 0). Now, by (1.7) and (2.13), its square is

(fo + go cos 0)2 + go sine 0 = f o + 2 Mo fogo + go ; 0

hence the approximate expression (2.4)2 for e[Lo, Mo] follows.

The-quantum numbers

Remark 2.2. 1,

m,

n,

±1

become those used in the classical solutions of the Dirac equation by imposing a condition stricter than (2.6) less strict than the condition that, on account of (2.7), would result from

the choice 1' = m' = 0, namely,

lmI
1
This condition is as follows.

(2.6)*

IV,§1,2

247

Definition 2. In III,§1, we solved a system in one unknown, analogous to the system (2.2) when 1' and m' are chosen to be zero. In III,§1, the lagrangian amplitude of the unknown is necessarily constant. It is in conformity with the nature of the amplitude a and of the phase v0q

in physics: a must vary more slowly than e"OQR. Let us make the present situation as similar as possible: consider a quadruple 1,

m,

n,

+1

satisfying the condition (2.5)3 for the existence of a solution of (2.2) that is lagrangian on V[L0, M0]; this condition is independent of (l', m'); require that at (1', m') the function

(l',m')i_4Ic±1-112+Ict2-12+ c13+12ER takes on values close to its minimum; in other words, by (2.9) and since l'2 and m'2 are negligible in comparison with I' and m', at (l', m') the function (I', m') i-- l'2 + m 2 + [l'NL + m'N,f + E]2 takes on values close to its minimum; if such a choice, namely, (1', m') near +

2 (NL, NM), 2 1+NL+NM

satisfies the condition (2.7), then and only then will we say that the quadruple (l, m, n, + 1) is admissible.

In §2 (Dirac equation), the following case arises.

(NL, NM) is near (-1, 0); thus (1', m') is near (+ e/2, 0). By (2.7), the admissible quadruples (1, m, n, ± 1) are those that satisfy condition (2.5)3 and the condition Example 2.

0
Iml±2

(the signs correspond). It is customary in quantum mechanics to let

.j+1 and to use quadruples of quantum numbers

(1, m, n, j = I + 2).

(2.23)

248

IV,§1,2-IV,§2,1

They are admissible if they satisfy (2.5)3 and

0'< 1
(2.24)

Im( _< j.

Then for each admissible choice of (1', m') there correspond solutions U of (2.2) that are equal up to a proportionality factor. This factor is a constant e C.

§2. The Dirac Equation 0. Summary

The Dirac equation is a system in 4 unknowns. Theorems 2.1 and 2.2 of II,§4 do not apply in this case because the zeros of det ao are double zeros.

Theorem 1, whose statement resembles that of theorem 2.2 of II,§4, reduces, mod(1/v2), this system in 4 unknowns to a self-adjoint system in 2 unknowns. By suppressing terms that are negligible in view of the order of magnitude of the magnetic field, this system is transformed in section 2 into the reduced Dirac equation, which is a system of the form solved in §1. In section 3, we observe that the energy levels defined by this reduced Dirac equation in lagrangian analysis are those defined by the classical resolution of the Dirac equation, even when the magnetic field is strong enough to produce the Paschen-Back effect. But the probability of the presence of an electron obtained in section 4 differs from that obtained in wave mechanics; it is connected with the first quantum theory. 1. Reduction of the Dirac Equation in Lagrangian Analysis

Suppose we are given two infinitely differentiable mappings and a constant: A : E3 \{0} - R + ;

B : E3 -. E3 ;

C E R+.

Let a' and a" be the two 2 x 2 matrices of lagrangian operators whose expressions in Z(3) = E3 Q+ E3 are a

A(x)+a[vex+B(x)l,

a"=A(x)-a

rax+B(x)1'

249

IV,§2,1

where a- is the 2 x 2 matrix defined by (1.6) of §1; a' and a" are self-adjoint. They are associated to the self-adjoint matrices

A(x) + a[p + B(x)] and A(x) - a[p + B(x)]. The Dirac equation is the system

a'U' = CU",

a"U" = CU',

(1.1)

in which the unknowns U' and U" are vectors. Let us require that the two components of each of these vectors be lagrangian functions, the four of them defined on a single compact lagrangian manifold V in Z(3) = E3 Q E3. The Dirac equation (1.1) is evidently equivalent to i. the system [C2

- a" o a'] U' = 0

in the unknown U', ii. the system

[C2 - a'oa"]U" = 0

(1.3)

in the unknown U". The calculation of C2 - a" o a' and C2 - a' o a" is easy and standard. In order to state the result, let C2

H(x,p)=21 p+B(x)I2+ 2 -2Az(x), 21

J'(x, P) = 2o[t n B(x)] + 01A.],

J" = 2a[" A B - 2a [Ax where

a n B = curl B; Z[C2 - a" o a'] is the matrix associated to the 2 x 2 matrix

H+1J'; V

250

IV,§2,1

z [C 2 - a' o a"] is the matrix associated to the 2 x 2 matrix

H + 1J". V

These matrices are not self-adjoint; but one is the adjoint

Remark 1.1.

of the other.

Let W be the hypersurface in E3 Q E3 given by the equation

Notation.

W: H(x, p) = 0.

(1.5)

Let #3' and /3" be the lagrangian amplitudes of U' and U":

#':V-CZ

fl":V-'C2.

By theorem 4 of II,§3, equation (1.2) is equivalent, mod(l/v2), to the conditions : W

df +J'13'=0, where d/dt is the Lie derivative 2' in the direction of the characteristic vector x of V Similarly, (1.3) is equivalent, mod(1/v2), to the conditions:

VC W dg-

+ J11 9

11

0.

( 1 . 7)

By (1.1), /3' and /3" satisfy the two equivalent relations

{A(x) + a[p + B(x)] }/i' = C$",

{A - a[p + B] } f3" = C/'

(1.8)

on V. The equivalence of (1.2) and (1.3) proves the equivalence of the equations (1.6) and (1.7). Relation (1.8) evidently transforms the solutions of one of these equations into the solutions of the other. The definition (1.5) of W means that there exists a function

p:W -R+ such that A = C cosh 2p,

p + B = C sinh 2p.

IV,§2,1

251

Let

i

C sink 2p

a[ p+ B],

so that

C2

=1

(1.10)

by (1.8) of §1. It follows that

Ce ±2pT = A ± a [ p + B]. Relation (1.8) connecting fl' and ji" is then written

#" = e2ptf'. It means that there exists a function

#:V_+ C2 such that

p"=e"fl.

fl'=e-PTY,

The equivalent relations (1.6) and (1.7) may then be written

d +JfJ=0,

(1.11)

where the 2 x 2 matrix J is given by J = epT

d (e - PT) +

eptJ'e-pt =

e - pT

d (epT) + e-PTJ"epT.

(1.12)

dt

dt

Let us show how this definition of J is equivalent to (1.14), which makes the following theorem evident.

Solving the systems (1.2) and (1.3) (which are equivalent to the Dirac equation) mod(1/v2) on V is equivalent to solving the system

THEOREM 1.

aU=0mod12 V

(1.13)

in two unknowns, where a is the matrix of lagrangian operators associated

to the 2 x 2 matrix

H+1J. V

252

IV,§2,1

H is defined by (1.4)1 and J by

J(x,P)=2Or[a Remark 1.2.

A

B]+2A+Ca[(p+ B)AAx]..

(1.14)

1 J, and thus a, are self-adjoint, which shows by remark 1.1 V

the advantage of substituting (1.13) for (1.2)-(1.3).

Proof of (1.14). By the definitions (1.4) and (1.12) of J', J", and J, it suffices to prove the following formulas, where < , > is the scalar product

in E3 and curl B = a/ax A B: 2

fe ° dt(e PT) + e ,dt(ePT)}

2(A

_

- C)a[curlB

A i 2A+Ca[(p+ B)AAx]

+ 2