Self-adjoint operators

Lecture Notes in Mathematics Ed1ted by A Dold and B. Eckmann Senes Forschungsmst1tut fur Mathematik, ETH Zunch

433 William G. Faris

Self-Adjoint Operators

Springer-Verlag Berlin· Heidelberg· New York 1975

William G. Faris Battelle Institute Advanced Studies Center 1227 Carouge-Geneve/Suisse

Present Address: Dept. of Mathematics University of Arizona Tucson, AZ 85721/USA

Library of Congress Catalogmg in Publication Data

Faris, William G 1939Self~adjoint operators. (Lecture notes in mathematics ; 433) Includes btbliograph1cal references and index. 1" Selfadjoint operators. 2., Perturbation (t-1athemat1cs) 3. Quantum theory., I .. Title. II., Ser1es:

Lecture notes in mathemat1cs (Berlin) ; 433. QA3.L28 no. 433 [QA329.2] 510' .ss [515' .7i]

74-32497

AMS Subject Classifications (1970): 47-02, 47 A55, 81 A09, 81 AlO ISBN 3-540-07030-3 Springer-Verlag Berlin · Heidelberg · New York ISBN 0·387-07030-3Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1975. Pnnted in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE These lectures are primarily concerned with the problem of when the sum of two unbounded self-adjoint operators is self-adjoint.

The

sum fails to be self-adjoint when there is an ambiguity about the choice of boundary conditions.

There is then also an ambiguity in the

definition of functions of the sum. Kinetic energy

This problem is fundamental to quantum mechanics.

and potential energy are self-adjoint operators, and functions of either of these may be computed explicitly.

The sum is the total

energy, and the main task of quantum mechanics is to compute functions of it.

Thus the question of self-adjointness is the question of

whether this task is meaningful.

If it is, then among other things

the dynamics of the system is determined for all states and all times. There are obvious series expansions for certain important functions, such as the resolvent. not converge.

~he

point is that these series may

In order to demonstrate self-adjointness it is necessary

to develop alternative methods for approximating functions of the sum.

In the language of physics, the basic question is whether the force laws determine the motion.

The models currently used contain

elements which could lead to ambiguity.

For example, the charge on an

electron is considered to be squeezed down to one point. sequence its potential energy is unbounded. electrons occupy the same point?

As a con-

What happens when two

Their interaction energy is infinite,

and it is not clear how they will move. Actually, according to quantum mechanics it

lS

improbable that

two electrons will be at or even near the same point.

But there are

IV similar difficulties in the description of light by quantum field theory, and these are not so easy to resolve. whether this theory gives

una~~igous

The question of

predictions to an arbitrarily

high degree of accuracy remains open.

So it is worth while to examine

the mechanism for determining Lhe dynamics in better understood situations. The justification for these lectures is that there has been progress on the self-adjointness problem since the publication of Kato's book, Perturbation Theory for Linear Operators (1966).

This

progress has been stimulated largely by developments in quantum field theory.

While the results which depend on a series expansion have not

been significantly improved, those which exploit positivity are the heart of the recent developments.

This is perhaps because for many

physical systems one expects the total energy to be bounded below. While Kato's book is the basic reference for linear perturbation theory, there are other books with additional material on the quantum mechanical applications.

Those by Hellwig (1967) and Glazman (1965)

approach the subject with partial differential equation techniques. More recently new progress has been made on the determination of the spectra of quantum mechanical operators, and this is described in the books of Simon (l97la), Schechter ( 971), and JBrgens and Weidmann Recently a memoir of Chernoff (1974) has appeared.

(1973~

It contains a

valuable discussion of the addition problem which complements the present treatment. These lectures begin with a review of standard material on addition of self-adjoint operators. from the more recent developments.

The second part is a selection This includes a section on

v properties of eigenvalues, including uniqueness of the ground state. The third part is devoted to the classification of extensions of a Hermitian operator.

This

lS

standard material; however it is there

to illustrate what kind of ambiguity is possible when self-adjointness fails.

(After all, to understand the force of a theorem you have to

be able to imagine a situation where its conclusion fails.) part is a brief account of how self-adjointness

lS

The final

related to the

determination of a measure by its moments. The core of the lectures is the first two parts.

The theme

lS

the interplay between the two aspects of a quantum mechanical observable, as an operator and as a quadratic form.

In order to be

able to take functions of an observable it is necessary for it to be a self-adjoint operator.

But to add observables it is most natural

to add quadratic forms (that is, to add expectation values). Throughout, the theory is illustrated by the case of the Schrodinger equation for a non-relativistic particle in a given potential.

The emphasis is on results obtained by operator theory

rather than by partial differential equation methods.

(There is

almost no discussion of the case of ordinary differential equations, but this subject has recently been surveyed by Devinatz (1973).) There is a brief description of the applications to quantum field theory. These lectures are intended to be an introduction to one topic in operator theory.

They are not a complete treatment even of this

topic, but should be regarded as an invitation to the research literature.

In order to follow them it should be sufficient to know

real analysis and have some acquaintance with Hilbert space. spectral theorem is stated but not proved.

The

VI The lectures were first given (in a somewhat different form) at the Eidgenossische Technische Hochschule in Zurich during the spring of 1971.

I wish to thank Professor Barry Simon for references to the

literature, and Dr. Jean-Pierre Eckmann, Dr. Charles Stuart, and Dr. Lawrence Thomas for reading the manuscript.

I am especially grate-

ful to Miss Edeltraud Russo for her excellent secretarial work.

W. Faris

Geneva August, 1974

CONTENTS

Part I:

Forms and Operators

§l

Operators

§2

Forms

ll

§3

The addition problem

19

§4

Positive form sums

23

§5

Small form perturbations

25

§6

Small operator perturbations

30

§7

Strong convergence and approximation

38

Part II:

l

Operator Domains

§8

Order in Hilbert space

48

§9

Order and estimates

55

§10

Eigenvalues

54

§ll

Local conditions

75

§12

Commutators

79

Part III:

Self-adjoint Extensions

§13

Extensions of Hermitian operators

83

§14

The Friedrichs extension

89

§15

Extensions of semi-bounded operators

92

Part IV:

Moments

§15

Analytic vectors

§17

Semi-analytic vectors

References

99 106 109

Part I

§l

FORMS AND OPERATORS

OPERATORS In this section we review the basic facts about Hilbert space and

about self-adjoint operators acting in Hilbert space.

The goal is the

statement of the spectral theorem. A

Hilbert space is a vector space with a certain type of form

defined on it, so we begin with forms. Let

E be a complex vector space.

A sesquilinear form

mapping which assigns to each ordered pair

a complex number

S(f,g)

conjugate linear in where

a*

f

f

, g

S

is a

of elements of

E

in such a way that the correspondence is and linear in

g .

(Thus

S(af,g)

is the complex conjugate of the complex number

= a*S(f,g) a .)

There is a quadratic form associated with every sesquilinear form, the one which assigns to every

f

in

E the complex number

S(f,f)

The sesquilinear form may be recovered from the quadratic form by the polarization identity 4S(f,g)

S(g+f,g+f) + iS(g+if,g+if) - S(g-f,g-f) - iS(g-if,g-if)

Thus we sometimes also refer to the sesquilinear form as being defined on

E . The sesquilinear form

S(f,g)

= S(g,f)*

Proposition 1.1 .

for all

S f

is said to be Hermitian if and

g

in

E .

A sesquilinear form is Hermitian if and only if the

associated quadratic form is real.

- 2 -

Proof:

S(f,f)

If

quadratic form

is real, then

=

S(f,f)

S(f,g)

S(g,f)

and

*

have the same

S(f,f)* .

A sesquilinear form is called positive or strictly positive if its quadratic form is positive or strictly positive (except at zero). An inner product is a strictly positive sesquilinear form. customary to denote an inner product by

II f 11

quadratic form by

2



It is

and the associated

II f II ) •

(the square of the norm

An inner product satisfies the Schwarz inequality ll ~ llfllllgiJ .

I t follows

the supremum of the

ll

that the norm IJfll ~ l

with

llgjj

can be computed as

.

Another consequence of the Schwarz inequality is that if fixed, f ~

jective, since if



= 0 , and so

=

0

g = 0 .

for all

and

g

orthogonal we write

Define in

M

(i) (ii) (iii)

E , in particular

in f

is fixed, g...._....

E .)

=0



.

If

f

and

to be the set of

f

in

E

such that

The following assertions are evident: Mj_

is a closed subspace of

Mj_ = Mj_ If

MeN

g

are

f j_ g

M be a linear subspace of the inner product space Mj_

.

f

of an inner product space are called

orthogonal if their inner product

Let

E

E and functionals is in-

(Similarly, if

is a continuous linear functional on f

is

is a continuous conjugate linear functional on

This correspondence between elements of

Elements

g

then

Nj_ c Mj_ •

E

f j_ g

E .

for all

g

-

Notice also that i f Mj_

=

3 -

is dense in

M

E

'

then from (ii) we have

= { 0}

Ej_

f

If

is in

E

:12rojection of

ortho~onal

Mc E

and f

then

'

on

g

if

M

in

M

is called the

f = g + h

where

h

is ln

Mj_ •

A (complex) Hilbert SJ2ace is a complex vector space with an inner product, which is complete in the associated norm. Projection Theorem 1. 2 . closed linear subspace.

M.

projection on

Let

be a Hilbert space and

H

Then every element of

H = M0

That is,

Mc H

be a

H has an orthogonal

Mj_,

This fundamental existence theorem is stated without proof.

M be a linear subspace of

Let

theorem applied to

-

-

H .

j_

On the other hand, by the

M,H=MGM.

projection theorem applied to

Then by the projection

H = Mj_j_0Mj_ •

Mj_ ,

Since

Mc

Mj_j_

it follows that

M=


Mj_j_

M is a

It also follows from the projection theorem that if

Mj_

= {O}

.

In fact, i f

Mj_ =

{0} , then by (iv)

Riesz Representation Theorem 1.3 .

Let

for some element

g

of

H •

M

{O}j_= H.

H be a Hilbert space.

any continuous conjugate linear functional on f r---

H if and only if

H , then

linear subspace of the Hilbert space

Then

H is of the form

(In other words, the corres-

pondence between elements and functionals is surjective.) Proof:

M of

The kernel of the functional

H •

If

M = H we take

g

=

L 0

is a closed linear subspace Otherwise

Mj_

contains an

- 4 -

element c

h

~

(by the projection theorem).

0

is a complex number chosen so that ···-1

H , then

f - (Lf)*(Lg)·

= Lf (Lg)

-1

g

(Lg)

We take

g

.

If

= ch , where f

is in

M , so

is in

= Lf

Of course there is an analogous representation theorem for continuous linear functionals on Let

K be Hilbert spaces.

and

H

U : H-- K

H •

A linear transformation

is said to be unitary if it is bijective and preserves

inner products.

In other words, a unitary transformation is an

isomorphism of Hilbert spaces. Next we turn to the objects of main interest, operators. be a Hilbert space.

An operator

some linear subspace

V(A)

of

A H

A graph is a linear subspace of a graph is the space of

f

defines an operator if whenever

of

H

f 8 g

G

g

8

The domain

H

ln

A

belongs to

A graph A

and

V(A) A

f = 0 ,

g = 0 .

then

T : H 8 H- H 0 H

Let Then

T If

A•~

H .

A

f

H

is a linear transformation from

to

with

Let

=

is unitary and A

(TA)j_

T

2

be defined by

= (-g0f)

.

= -1

is a graph, its adjoint

= T(Aj_)

r(f8g)

A*

is the graph defined by

,

From the properties of the orthogonal complement we conclude that ( i)

A*

( ii)

A.'~

(iii)

If

AcE

A'"~'~'

= A.

(iv)

is closed (as a subspace of

= A* then

B 1'c A'~

H

0

H).

- 5 -

Here

A

is the closure of the graph

Proposition 1.4 and only if

Let

A

Corollary . A*

in

H

Then

8

H

A*

is an operator if

is densely defined.

if and only if

=

V(A)j_

Let

A

{O}

i f and only i f

if and only if

be a graph.

Then

A

Aj_n (H8{0})

{ 08 0}

V(A) = H • is an operator if and only

is densely defined.

A=

Proof: If that

be a graph.

A*n ({0}8H) = {o8o}

Proof:

if

A

A

A

A

(A*)* is an operator then

A

is said to be closable.

is densely defined and closable if and only if

operator which is densely defined.

Notice

A*

is an

Thus the class of densely defined

closable operators is well-behaved under the operation of taking the adjoint. Proposition l. 5

Let

is an operator.

A

Proof:

and

g

V(A''') = {g

Its domain

continuous in the norm of

V(A)

be a densely defined operator.

in

H}

and

in



H : f

t----

=

Then



is

for

in

f

A*

V
A*g = h

if and only i f

g 8 h j_T (f8Af)

i f and only if

g 8 hj_(-Af8f)

i f and only if

= (for all

f

in

V(A)) . Proposition 1.6 . kernel of Proof:

A*

A

be a densely defined operator.

Then the

is (range A)j_ .

The element

= 0

Let

for all

g

f

0

in

V(A)

,

precisely when

- 6 -

Corollary . that

A

Let

A

be a densely defined closed operator.

is injective and

H = (range A)

Proof:

~

Af

f

8 Af

from range A

h ~ A-lh

the same as the map A

is continuous.

Then

8 kernel A'~

The map

graph of

A-l

Assume

to the graph of

8 h , which is continuous.

A

is

Since the

is closed, so is range A •

An operator

A

is said to be Hermitian if

A c A'~ .

defined and

A

is densely

Notice that the operator

A

satisfies

A c A'''

A~'

A =

It is said to be self-adjoint if

if and only if







* , this equation says precisely that the form

for all

f

and

g

V(A) .

in

Since

is Hermitian. If

A

is a graph, its inverse

A-l

is the graph consisting of ;,

all

g

8 f

such that

Theorem 1.7 all

f

Let

and

g

A

f

8 g

A .

A-l

= A*-l .



=

Clearly

be an operator such that

V(A) .

in

is in

If there exists a complex number

that range A - z

=

Proof:

be the graph which is the adjoint of

Let

A* A-z

, so range A - z* an operator. (A-z)

-1

.~

=

c

H

and

then

H

H

implies

A

(A-z*)

H , V((A-z)

A - z (A-z)-

Thus

is self-adjoint. A .

l

~

= (A '-z)

-1

-1

-1*

Then

(A'~-z)-l

) = H

is

Thus '

, A = A''

The point of this theorem is that the property of

A

being self-

adjoint is related to the existence of the resolvent operator (A-z)-l : H-- H

such

Since

H

-1

=

A*-z

Since range

c (A -z)

range A - z*

z

for

(for a pair of complex conjugate z).

- 7 -

Let L

2

(M,~)

be a positive measure in the measure space

~

a

be a measurable complex function on

ponding multiplication operator g ~ag If

whenever

a

g

and

mult(a) ag

is real then mult(a)

Spectral Theorem 1.8 the Hilbert space operator.

Then

is a Hilbert space.

Let

by

M .

H

Let Then

A A

The corres-

M

acting in

are in

L

2

L

2

is defined

.

is self-adjoint (by Theorem 1.7). be a self-adjoint operator acting in is isomorphic to a multiplication of square

That is, there exists a Hilbert space

integrable functions, a unitary operator real measurable function

a

on

M

U :

H-. L 2 (M,~)

such that

and a

UA = mult(a)U

The spectral theorem will not be proved here. The point of this theorem is that any question about a single self-adjoint operator is a question about a function.

Thus in the

following we usually will regard assertions about a single selfadjoint operator as obvious and not needing proof.

The difficulties

begin with two (non-commuting) self-adjoint operators. Let

A

be a self-adjoint operator acting in

isomorphic to multiplication by some real function complex Borel measurable function on the reals.

H .

Then

A

is

a

Let

~

be a

Then

~(A)

is

defined as the operator which is isomorphic (by the same unitary operator) to multiplication by

~(a)

It may be proved that this

definition does not depend on the choice of the isomorphism. The most important of these functions

~

complex exponential, and indicator functions.

are the inverse, These give rise to the

-

8 -

resolvent, unitary group, and spectral projections of the self-adjoint operator. There is a set of complex numbers

z

such that the resolvent

(A-z)-lc: H-. H

is continuous.

spectrum of

The spectrum of a self-adjoint operator

A

Its complement is called the A

is a

closed subset of the real axis. If A is isomorphic to multiplication by a real function

ex , the spectrum of A is the essential range of ex

Multiplication by a function of absolute value one is a unitary operator.

Thus the complex exponentials of a self-adjoint operator

form a one-parameter unitary group.

There is a close relation between

the resolvent and the unitary group.

It is given by

+oo

(A-z)-lf = i

J-

exp(itz)exp(-itA)fdt

0

for

±Imz > 0

and the inversion formula

exp(-itA)f= lim(l+itA)-nf n

n-+oo

Let

S

be a Borel subset of the real line and write

indicator function of ment).

S

(which is

The spectral projection of

l A

on

S

and

0

1

for the

8

on its comple-

corresponding to

S

is

l

8

(A)

The resolvent may also be used to approximate spectral projections. For

E: > 0

let

o(E:;s) =

11

-1

2

2

(E:/(s +E: ))

Jbo(E:;s-x)dx converge to l if a is in the complement of [a,b]

s

As

is in

E:-. 0 , the functions

(a,b)

and to

Thus the operators

0

if

o(E:;A-x)

relevant to computation of certain spectral projections.

s are

But these

operators may be expressed in terms of resolvents by o(E:;A-x) = (21Ti)- 1 [
A

'where

z =X+ iE:

is shown to be self-adjoint, the proof

usually leads to an approximation procedure for certain functions of the operator.

These are most often the resolvent

(A-z)-l

for

z

9 -

-

for

exp(itA)

far from the real axis or the exponential

t

not too

These functions give a certain amount of useful information

large.

about asymptotic properties of the spectrum, but to find detailed spectral properties requires knowledge of the resolvent near the real This is a more

axis or of the large time limit of the exponential.

delicate matter, and is beyond the scope of these lectures.

There is

a huge amount of information on the determination of the spectrum, In fact, it might be said to be the main enterprise of

however.

theoretical physics.

EXAMPLES An example of a unitary operator from one Hilbert space to F : L 2 (Rn,dx)-. L (Rn,dk/(2~)n) 2

another is the Fourier transform

It is determined on a dense set by the formula Fg(k) h(x)

= g(k) = J exp(-ikx)g(x)dx

=f

Its inverse is determined by The importance of the Fourier trans-

exp(ikx)h(k)dk/(2TI)n

form is that it gives the spectral representation of translation invariant operators. Let

Consider first the one-dimensional case. A = -

and

f' A

and

J --2

in L2 such that defined on the subspace of f dx ' are absolutely continuous and f' and f" are in L2

let

Then

2

H = L CR,dx)

is self-adjoint. (A+c)

except when

c

~

-1

0 .

In fact

h(x) = ;c

_,2

(A+c)

J exp(-c

-1

21

Thus the spectrum of

is given by

lx-yl)h(y)dy A

is the positive real

axis. According to the spectral theorem, A multiplication operator.

is isomorphic to a

In this case the isomorphism is given by

f

- 10 -

the Fourier transform, and

A

is isomorphic to multiplication by

k2 •

In

n

dimensions the analogous operator is

A

= -~

It is

perhaps simplest to define this operator by means of the Fourier transform, so that by definition it is isomorphic to multiplication by

k

2

.

NOTES The most succint self-contained treatment of Hilbert space (including the spectral theorem) is §5 of the notes by Nelson (1969). The spectral theorem is also proved in Dunford and Schwartz (1963). A good treatment of Fourier transforms is Stein and Weiss (1971).

FORMS

§2

We now turn to the study of sesquilinear forms defined on a There will always be at least

linear subspace of a Hilbert space.

two forms under consideration, the given sesquilinear form and the (In quantum mechanics the

inner product of the Hilbert space.

corresponding quadratic forms give the expected energy and total The relation between these two forms is the central

probability.)

theme of this section. Let

We begin with an important example.

adjoint operator acting in the Hilbert space is the linear subspace of all

A

of

Q_(A) 1

II A 2 f II

The form of

is finite.

f

be a positive self-

A

H .

The form domain

in

by

Q (A)

is defined on

A

such that

H

1

1

S(f,g) =

V(A) of a self-adjoint operator

Notice that the operator domain A

consists of the

IIA!fll

2

f

in

H such that

~Af~

Since

is finite.

{' llfiiiiAfil , the form domain of a positive self-adjoint operator

is larger than its operator domain. It is illuminating to see what these objects look like in a spectral representation.

L 2 CM,~)

H and

that

V(A) Let

Hilbert space operator

a ?

A

0 .

A

Then

Q_(A) S(f,g)

is given by

corresponds to S

be a unitary correspondence between A

which establishes an isomorphism of

multiplication by and the form of

U

Let

L

corresponds to

0

?

2

= J (Uf)*a(Ug)d~

continuous in the norm of

Notice

L CM,(l+a )d~) .

E is dense in

Assume that

is defined on

(M,(l+a)~)

2

2

be a sesquilinear form on a linear subspace

H .

with

V(A) = {g H}

by

in

S(f,g)

E : f

H . t----

=

.

E of the

The associated S(f ,g)

is

- 12 -

In the above example, that of the form

S(f,g)

l

l

=

of a

positive self-adjoint operator, the associated operator is just the original operator

~

A : V(A)

H .

It is possible to characterize the forms of positive self-adjoint operators. Let

For this we need one more definition. S

be a positive form defined on the linear subspace

H •

of the Hilbert space an inner product on

Q.

Q = S(f,g) +

Notice that We say

S

Q

is closed if

is

Q is a Hilbert

space with this inner product. The form of a positive self-adjoint operator is closed. The next theorem states that there is a bijective correspondence between positive self-adjoint operators and densely defined, closed, positive forms. Form Representation Theorem 2.1 . be a dense linear subspace.

Let

Then the associated operator and

S

Proof:

is the form of

=H

range (A+l)

functional

f

~



there is an element

g

= Q = S(f,g)

(A+l)g

=h

S

be a closed positive form on

Q

is a positive self-adjoint operator

A Let

is self-adjoint it is sufficient to h

be an element of

H .

Then the

is a continuous conjugate linear functional

Q

on the Hilbert space

Q

A .

In order to show that

show that



A

H be a Hilbert space and

Let

Hence by the Riesz representation theorem in

Q such that

+ .

Thus

g

is in

V(A)

and

.

Next we show that orthogonal to

V(A)

V(A)

is dense in

Q

in the inner product of

Let

Q

h Since

Q be

in h

is in

H ,

- 13 -

h = (A+l)g , where

we may write

= = Q = 0

coincide on

V(A), and

Remark .

, so

h

=0

and

S

Q*

Q c H c Q*

Thus we may regard for the value of g

v

restriction of

Q).

Q*

in

A

A :

f

in

Q.

Ag

in

Q*

on

acting in

Q -. Q*

If

H

by

h

is an element of

A + l

S



is just the

Q : Ag

in

Q* .

= S(f,g)



associated with

V(A) = {g

to

surjective, and so the restriction

is in

A + l

:

: V(A) _. H

H}

Q _. Q*

.

is

is also sur-

This is what is needed for self-adjointness.

Q'"

The conjugate dual space llviiQ* = sup{ll

:

Q*

A + l

'

Q -Q'"

:

we is

may be expressed as

= <(A+l) -1 u,(A+l) -1 v>Q = <(A+l) -1 u,v>

Next notice that

=0

= Q

Since

Hence the operator

The inner product on



is a Hilbert space with the norm

II fiiQ _, l}.

II
Q*

with

A .

f ~

The Riesz representation theorem says that

unitary.

So the inner

and it is consistent to write

Q , define

is in

Then the operator

have

A

Q (the space of

be the conjugate dual space of

H , we may identify it with the element

jective.

of

It is possible to think of this construction in another

Let

If

Q(A)

is the form of

continuous conjugate linear functionals on of

But then

V(A)

is dense in both spaces.

V(A)

Q = Q(A)

products coincide on

is in

Q and on the form domain

The inner products on

way.

g

H

for all

is dense in h

in

Q*

H , then

In fact, if

=0

f

and so

is in f = 0

Q

- 14 -

The fact that Since

A + l

Q-- Q*

:

(A+l)V(A) = H

is dense in

V(A)

Q may now be seen as follows.

is unitary, this is equivalent to

being dense in

(A+l)Q

= Q* ,

which we have just seen

to be true. It is often easy to verify that a form is densely defined and positive.

To show that it is closed may be more difficult.

Is it

possible to produce closed forms at will by a process of completion? The answer is negative. Let

S

be a positive form on a linear subspace

Hilbert space

V .

on

Then

H •

V

= S(f,g)

V into

The inclusion map of

H

+

is an inner product

is continuous.

the Hilbert space which is the completion of

V

by continuity of the inclusion to a map from

Q to

then

Q may also be regarded as contained in Let

H

be a Hilbert space.

V of

linear subspace inner product closable if S

V

Let

= S(f,g)

+ .

Q is contained in

is the positive form defined on Let

form

A



be a positive for

f

and

not necessarily closed). its closure is the form of

H

is injective,

H

be a positive form on a

Then

S

V in the

is said to be

In that case the closure

H

Q by

Q be

Let

If the extension

Q be the completion of

Let

H

S

V of the

Q

= S(f,g)

S

of

+

self-adjoint operator and consider the g

in

V(A)

.

In fact, since A .

This form is closable (but V(A)

is dense in

Q(A) ,

- 15 -

EXAMPLES

H :: L 2 (IR,dx)

Let

and

k2

multiplication by

in the Fourier transform representation, the

s (f,g ) =

form of

A

is given by

If

g

is in the form domain

g

transform Hence

g

is isomorphic to

A

Since

is integrable: II g 11

1

J

2 -d I f- .c"kg k 2'1T

Q

~

'2= =

d f d 'dxg>
A , then its Fourier

of

2 2 II Ck +l) -! 11 11 (k +l) 2

~g 11 2

<

•

oo

is continuous and bounded.

Q*

The dual space inner product

Q

of

1

Q* = =

Q*

Thus the elements of

H in the

is the completion of

!

JJu(x)*expC-Ix-yl )v(y)dxdy

need not even be functions.

Next we give an example of a densely defined positive form which 2 is not closable. Let H = L CIR,dx) and let V be the form domain d2 for f and J-(k)dk Define S(f,g) = f(O)*g(O) = Jf(k)*dk of 2'1T 2'1T g g

The completion of contained in

H

sequence

in

f

n

v

lf
2 2 + J1f(x)l dx

In fact, for any complex number

v

f

such that

the completion is in fact into

in the norm

H0
H is that which sends

n

( 0)

::

z

and

f

n

there is a

z

-

is not

0

in

L2

So

and the natural map of the completion

(f,z)

into

f

•

There is an interesting application of the form representation theorem to a proof of a theorem of von Neumann.

The idea is to con-

sider a form which depends quadrat·ically on the operator Proposition 2.2 • if the form

S(f,g)

A : V(A)- H

The operator

=

on

V(A)

is closed if and only

is closed.

closable if and only if the form is closable.

A •

The operator is

- 16 -

Proof:

The mapping

f-. f

8 Af

from

inner products, and the injection of

to

H

into

H

V(A) V(A)

8

preserves

H

corresponds to

projection onto the first summand. Theorem 2.3 . A*A

Let

A

be a closed densely defined operator.

Then

a positive self-adjoint operator.

lS

Proof:

The form

defined.



on

V(A)

is positive, closed, and densely

By the form representation theorem the associated operator

is positive and self-adjoint. operator is just

But it is easy to check that this

A*A •

So far we have only considered forms of positive self-adjoint operators.

We now turn to forms of arbitrary self-adjoint operators.

Everything is as before - except that the convenient characterization of the forms is missing. If

H

is a Hilbert space and

its form domain IAI

Q(A) c H

If

V(A)

Q(A)

is larger than the operator

. 2

U : H-- L CM,~)

multiplication by The form of

is a self-adjoint operator,

is the form domain of the positive operator

As before, the form domain

domain

A

, then

a

is given by

clearly a Hermitian form. A

A

provides an isomorphism of Q(A)

corresponds to

S(f,g)

A

with

2

L CM,(l+lal )~)

= J (Uf)*a(Ug)d~

•

This is

The associated operator is just

: V(A) - . H .

Let

Q = Q(A)

Hilbert space

Q.

Thus we may regard

and let If

H

g

Q'~

is in

be the conjugate dual space of the H , then

as contained in

Q*

f

~



and write

is in

Q* .

for the

- 17 -

value of

If for

f

g

is in

Q

in

operator

Q_*

in

v

on

f

Q•

in

Q, we define This

A :

In the spectral representation

Ag

Q*

in

Q -. Q*

by



= S(f,g)

is an extension of the original

V(A) _,_ H •

A

The identifications

Q c H c Q''

may be confusing.

The point is

that a Hilbert space is always isomorphic to its conjugate dual space. Our

H with

conventions are such that the isomorphism of

the identity, while the isomorphism of

Q with

Q_*

is

H*

is

l + IAI

One may define a notion of closed form in the general setting. Let

H be a Hilbert space.

Qc H , Q

is dense in

Let

H, and the injection is continuous.

be the conjugate dual space of bounded Hermitian operator from Hermitian form be closed

if

S(f,g) A - z

complex conjugate

Q be another Hilbert space such that

= Q-

Q'~

Q

and identify

Q to

Q*

H c Q*

and let

The Hermitian form has range

Qt'

Let Let

S

Q*

A

be a

be the S

is said to

for some pair of

z

It is clear from Theorem 1.7 that the operator associated with a closed Hermitian form is self-adjoint.

However with this definition

it may be as difficult to check that the form is closed as to show directly that the operator is self-adjoint.

NOTES The basic results on forms originate with Friedrichs.

There is

a very complete discussion in Chapter VI of the book of Kato (1966). The form representation theorem is there called the second representation theorem.

(His first representation theorem is a slightly

18 -

weaker result for a wider class of forms.

The reason the result is

weaker is related to the pathology of square roots of non-self-adjoint operators discussed in Mcintosh (1972).)

There is also a brief

treatment in §7 of the notes of Nelson (1969). The definition of closed Hermitian form in the general case is due to Mcintosh (l970a) (l970b).

He poses the question as to whether

every closed Hermitian form is the form of its associated self-adjoint operator.

§3

THE ADDITION PROBLEM The main subject of these lectures is the addition problem:

When is the sum of two self-adjoint operators self-adjoint?

In order

to discuss this problem, we first need a definition of "sum". possibility is the operator sum: on

V(A)nV(B)

Let space

A

and

Then

B

S(f,g)

Q(A) n Q(B) .

be self-adjoint operators acting in the Hilbert

=

Assume that

and

Q(A) n Q(B)

V(A) n V(B) c V(C)

is in

Let V(C)

If

S .

on

Q(A)

and



on

is a Hermitian form on is dense in

Then

C

H

and let

C

be

is called the form sum of

B .

Proposition 3.1 .

Proof:



+

the operator associated with A

has an obvious definition

However there is a more general notion.

Consider their forms

H

Q(B) •

A + B

One

C

g and

Let

C

be the form sum of

A

and

B .

Then

•

be in Cg

V(A)

= Ag

n

V(B)

Then

=

S(f,g)

, so

+ Bg •

is the form sum of

A

and

B

even though in general it is possible that

we write V(A)

n

C

V(B)

=A + = {0}

The addition problem is important in quantum mechanics.

B

We now

summarize the principles of that subject. A quantum mechanical system is specified by a family of selfadjoint operators acting in a Hilbert space the observables of the system. by a unit vector in

H

H

these correspond to

A state of the system is determined

g

- 20 -

If

A

is a self-adjoint operator, then by the spectral theorem

there is a unitary transformation corresponds under If

f

U

element of

L (M, \l)

lity measure.

H

Ja -1 (E) lll

A 2

f

, then

a

A

on

a

f = Uf

Thus

M .

is an

is a probabi-

may then be regarded as a random

variable on this probability space.

ponding to

=l

II f II

with

The function

state determined by

such that

to multiplication by a real function

is a unit vector in 2

2 L (M, \l)

U : H -

Thus when the system is in the

, the probability that the observable corres-

has value in the set

E

of real numbers is

d)J

This probability may be expressed in a way which is independent of the representation by projection of

A

Notice that Q(A)



corresponding to a

, where

lE(A)

is the spectral

E •

has an expectation if and only if

f

is in

and in that case the expectation is

It has a second moment if and only if

f

is in

V(A)

A

is then

, and the

second moment is then

The uncertainty (or standard deviation) of M"'

2

2


= II
Since the spectrum of

A

is the essential range of

probability that the value is in the spectrum is whatever the state of the system.

J1il

2 d)J

a

, the

= = l ,

-

21 -

The most important observable of a quantum mechanical system is the energy, since it determines the time development of the system. If

H

is the self-adjoint operator corresponding to the energy, by

the spectral theorem it is isomorphic to multiplication by a real function.

Thus for each real

t

, exp(-itH)

is isomorphic to multi-

plication by a function of absolute value one, and hence is a unitary operator.

If

f

determines the present state of the system,

exp(-itH)f determines the state If

A

and

B

units of time later.

are self-adjoint operators, they are isomorphic

to multiplication operators, equivalence.

t

but not necessarily by the same unitary

Though in any state the corresponding observables may

have probability distributions, there may be no determination of a joint distribution. One important case of this is the kinetic energy potential energy

V .

H

0

and the

For each of these the isomorphism with a multi-

plication operator is explicitly known. distribution, but the total energy

H

They need not have a joint is supposed to be determined

by adding expectations:

for all

f

in

conditions on

Q(H H

0

)

and

and

V

0

a self-adjoint operator

H

Q(V) •

However it is necessary to find

so that this formula does indeed determine

- 22 -

EXAMPLE The Hilbert space which describes the states of a quantum mechanical particle moving in (Of course the case

=3

n

~n

may be taken as

H = L

n

,dx)

n .)

The Fourier transformation gives an isomorphism of

L (~n,dk(2TI)-n) .

(~

is of primary importance, but it is

illuminating to keep track of the dependence on

2

2

The points

k

with

H

have the interpretation of momenta.

In non-relativistic mechanics kinetic energy grows quadratically with k2

momentum, like in

H

is

H

0

The corresponding kinetic energy operator acting

= -~

The points

x

have the interpretation of position.

real measurable function on potential energy at operator acting in

x .

~n , V(x)

V

is a

may be regarded as giving the

Multiplication by

V

is the potential energy

H •

The natural question is then: adjoint?

If

It is precisely for these

For what V

has a well-determined solution (given by

V

is

H

= Ho

+ V

self-

that the Schrodinger equation exp(-itH)f ).

NOTES Chernoff (1970), (1974) has investigated the general addition problem.

He defines the sum of two self-adjoint operators by the

Trotter product formula.

He shows that when the sums exist the

operation is commutative but not necessarily associative. cancellation law also fails.

The

He also shows that if an operator can

be added to all self-adjoint operators, it must be bounded.

And

Goldstein (1972) has shown that the sum is discontinuous (with respect to strong resolvent convergence).

§4

POSITIVE FORM SUMS The following theorem says that there is usually no difficulty

adding positive self-adjoint operators. Theorem 4.1 •

Let

H

be a Hilbert space and

self-adjoint operators acting in dense in

Q_(H) = Q_(H

operator with Proof:

Let

0

)

= H0

H

n Q_(U)

Q = Q_(H ) n Q_(U)

Q

is a closed positive form on

Q_(H

0

)

be positive n Q_(U)

is

is a positive self-adjoint

.

The form

0

+ U

u

and

0

Assume that

H

H • Then the form sum

H

=

S(f,g)

+

By the form representation theorem

it is the form of a positive self-adjoint operator

H

with

Q_(H)

=Q

This theorem has an obvious generalization to operators which are bounded below.

(Just add a constant to each operator.)

EXAMPLE Let measurable function on 1 L

is in

measure.

Rn

H

0

= -~

and let

H •

of

C

Clearly

Assume that

U

locally on the complement of a closed set of zero H = H

Then

0

+ U

is self-adjoint and bounded below and

We have assumed that

complement of a closed set 00

be a real

which is bounded below.

To see this it is sufficient to show that in

U

U

QCH

0

)

n Q_(U)

is locally integrable on the

M of measure zero.

Let

F be the space

functions with compact support in the complement of F c Q_(H

0

)

n Q_(U)

and

F

is dense

is dense in

H •

M •

-

24 -

NOTES Sums of forms are discussed in Chapter VI of the book by Kato (1966).

(There is often confusion about the relation of the form

sum to the Friedrichs extension.

Kato shows on page 329 that the

form sum need not be the Friedrichs extension of the operator sum.)

§5

SMALL FORM PERTURBATIONS Another technique for defining sums of self-adjoint operators is

perturbation theory.

An operator of either sign can be added on,

provided that it is relatively small.

In this case self-adjointness

of the sum is proved by means of a convergent geometric series expansion. In order to obtain convergence, it is necessary to have a notion of what is meant by relatively small.

For this we need some basic

facts about norms.

E be a complex vector space with inner product

Let

II f

norm

2

II

defined on

E .

w(S)

and

= sup{

•

Let

S(f,g)

be another sesquilinear form

Define the sesquilinear form norm by

llsll = sup{IS(f,g)l by



1

:

llfll

~ l , hll ~ l}

I S(f ,f) I : ~ f II ~ l} .

and the quadratic form norm The sesquilinear form is said

to be bounded if its sesquilinear form norm is finite. Proposition 5.1 . is Hermitian, then Proof:

I t is always true that

w(S)

I t is clear that

= llsll

w(S) ~

llsll

~ 2w(S)

.

If

•

w(S)
On the other hand, we see from the polarization identity that

This implies that If

S

IISII <;: 2w(S)

is Hermitian, then

. S(h,h)

4ReS(f,g) = S(f+g,f+g) - S(f-g,f-g)

.

is always real. It follows that

Hence

S

- 26 -

llsll ~ w(S).

This implies that Let

and

E

F

linear (or conjugate linear) transformation II All

sup{ll Af II : II fll

::

~

A : E -F

is

The transformation is said to be bounded

l}

i f its norm is finite.

The norm of a

be inner product spaces.

I t is easy to see that

A

is bounded if and

only if it is continuous.

E is an inner product space, its conjugate dual

Recall that if

E*

space

E

consists of the bounded conjugate linear functionals on

The value of

v

E*

in

on

f

E is denoted

in

.

There is a

natural bijective norm preserving correspondence between bounded sesquilinear forms

S

E and bounded operators

on

A : E-

E* given by

S(f,g) ::: Theorem 5.2 .

Let

H

Hilbert space

H .

Let

be a self-adjoint operator acting in the

0

W be a Hermitian form on

there exist positive constants ± ~ a 0

H = H

0

+ V

a

and

for all

f

b in

with QCH

0

a

QCH <

l

0

)

such that

such that

Then the form sum

)

is self-adjoint. 1

Proof:

Assume We have

b > 0 Q c H c Q*

Hermitian operator

V(H)

= {f

in

Q

H Hf

and give and

H

QCH 0

= H0

acting in is in

H}

)

jjfiiQ = II
the norm

+ W :

Q-

Q* .

Define the

H by restricting to To show that

H : V(H) -

H

is

self-adjoint, it is sufficient to show that range

(H-z) ::: H

for

z

imaginary and sufficiently large. that

H :

Q-

Q*

satisfies range

For this it is sufficient to show (H-z) =

Q*

-

27 -

This would be established if we could show the existence of (H-z) -l = [1 + ( H - z) -lW] -l ( H - z) -l ·. 0

ll '<- ;,

0

_,_

b y a serles . . expanslon.

ll '<-

Q _,_ Q

(H -z)-lW :

For this it is enough to show that

0

has norm

strictly less than one. ±

But by assumption

It follows that Q _,_ Q'''

bounded by

a .

On the other hand

morphism of Hilbert spaces, so by

a .

is an operator with norm

I H I + b : Q _,_ Q'' o

Q _,_ Q

( IH I+b)-lW : 0

is an isohas norm bounded

Thus the norm of

(H -z)-lW 0

=

(H -z)- CIH l+b)(IH l+b)-~: Q - Q 1

0

(l+b/lzl )a< l

0

, for

is bounded by

0

lzl

sufficiently large.

EXAMPLES 2 n H = L ( IR , dx ) .

Let U

and

0

=

-to

is in

Q_(H

for some

n = 2).

0

)

p ?

H = -to + V

Then

n Q_(U) n

IRn .

Assume

H

is dense in

(and where

2

V = U + W , where

and let

W are real measurable functions on

bounded below and that

if

H

Let

p

?

l

±W

'ii

a

a(H +U+c) 0

Write The

l

<

Lp

For since and so

if

W

p

n

=

l

U

±W

~

a(H +b) 0

is bounded below it follows that

w is a small form perturbation of H0 Woo

is bounded and

Wp

may be chosen arbitrarily small.

U

But then by

~ + 1 = l . p

+

is in

Holder's inequality where

q

W

, p > l

is self-adjoint and bounded below and

W = Woo + Wp , where

norm of

is

Assume that

All that is needed to prove this is an estimate for some

U

-

l

l

l n

28 -

in terms of

Notice that

2q;.. 2

Thus the bound on



follows from Sobolev inequalities.

H0

We see in the next example that for

+

W to be a self-adjoint

operator it is not at all necessary for the form

W to be the form of

a self-adjoint operator. d2 A = --2 Recall that if dx then its Fourier transform ! Let is ln Ll 2

H = L ( IR ,dx)

Let

,

Q_(A)

function in

L

and

p(-k) = PCk)~'

with

=I I f(k) 1'iHk-r)g(r)dk/2TI



w

Define

which satisfies the hypotheses of the theorem.

then

iJ

Hence

-

~

by

d2 + W is 2 dx

is the Fourier transform of a real function

W is multiplication by

operator.

"

Q'~

Q -

H •

a self-adjoint operator acting in If

be a

w is a Hermitian form

Then

dr/211

is in

f

~

in

, and hence is a self-adjoint

However there are cases where

W is not a multiplication

operator. For instance, let line and set W :

Q-

at

p

Q*

b

be a real number and

GCk) = b exp(ipk) • is given by

defined by

Wg

< f' 6 >

p

=

bg6

Then p

,

where

p

a point on the

= bf(p)*g(p)

•

Thus

is the Dirac measure

= f(p)1: •

In this example, the functions g ln the domain of the selfd2 adjoint restriction A + w = - - - + b6 have a slope discontinuity at p 2 dx p equal to bf(p) Thus, for such g' (A+W)g is in L2 even though it is false in general that The resolvent of venience, set

p

=0

•

Ag

or

Wg

is in

L2

A + W can be computed explicitly. Then

For con-

- 29 -

(A+W+c) -

-1

g(x)

= (A+c) -1 g(x)

1

1

l

(

(b/(2c 2 +b))exp(-c 2 lxl) J

--1

1

exp(-c 2

2c 2

IYI

)g(y)dy

has a pole at c = Thus the b2 The corresponding self-adjoint operator has an eigenvalue -ll If

b < 0

then

eigenfunction is

(A+W+c)-l

exp(;blxl)

Another example is

O(k)

= ni

sign(k) ,

that the inverse Fourier transform of X

- 2- 2 , for x +s

rri exp<-slkl )sign(k)

is

Hence

s > 0

(

To analyze this, note

*

x - g(x)dx = p.v. = lim J f(x) - 2 2 s+O x +s

jr

f(x) *l xg(x)dx

This

principal value integral is not the form associated with multiplication by

l X

NOTES Theorem 5.2 is related to what Kato (1966) calls the pseudoFriedrichs extension.

However it is really not so much an extension

theorem as a perturbation theorem. The theorem may not be strengthened to allow d2 A = --2 and H = A + 6 , w = -A , then ±W .:;; H 0 0 dx is not a closable form.

a = l but

For i f H

0

+

w=

Sobolev inequalities are treated in Stein (1970). The class of Schrodinger operators for which the form sum is self-adjoint and bounded below has most recently been studied in detail by Schechter (1972),

6

§

6

SMALL OPERATOR PERTURBATIONS The more traditional approach to perturbation theory is in

terms of operator sums rather than form sums.

The hypotheses of the

theorems are stronger (second order rather than first order in the operators), but so are the conclusions. We begin by reviewing the relation between first order and second order estimates.

First we need a preliminary remark.

Then we

can discuss the order relation between self-adjoint operators.

H is a Hilbert space and

If

its norm is the norm of the form c"' : H - H

= = "

is

that

A

and

Q(A) c Q(B)

Proposition 6.1 .

B

be self-adjoint operators.

Proof: that so

~



B 2 ~A 2 Let c



and

B

II

B~ fll

II A-~B~ II

II

.:i

.,

A~ fll

l

in

~

A

Q_(A)

means .

IIBfii~JIAfll

and

be self-adjoint operators which are

B

~

A

if and only if

1

'

B

be a real number which is strictly less than

Then

2

f

V(B)::>V(A)

By adding a constant we may assume 2

is also

Then

for all

meansthat

A

bounded below and let the lower bounds.

c"'

Thus

II c"' II = II c II

and

Noticethat

is a bounded operator,

The form of the adjoint



...

bounded and in fact Let

c : H -H

c = 0

.

1

that is, IIB2A-211 ~ l

That is, IIA -~g II

.,

_1

II B

2

g II

(A-c)

'

-1

~

Then 1

But

(B-e) B

~

1

-1

says

A 1

1

(B2A-2)"'=> A-2B2

-1 B-1 A <>

The next proposition is the main result about this order relation: second order estimates imply first order estimates. Proposition 6.2 •

Let

A

Then

and

B

0 <> B

~

be positive self-adjoint operators A •

- 31 -

Proof:

First consider the special case when there exists a

2 0 < c "" B2 <> A A-1

=

I:

2 1f

(A2+t2)-l

Then

(A 2 +t 2 )- 1 dt A-1

we see that

,

and so

B <> A

In the general case we have every

s

Hence

0

>

B

Theorem 6.3 .

Let

H

Hilbert space

H •

Let

with

V(W)

with

a

H ::: H

2

Proof:

V(H

l

<

0

)

t 0

b

~a

Hence

W(H -z)

H

= H0

-1

0

)

2 l (A +s) 2

_.

w2

~ a 2 CH 02 +b 2 ) .

z = ±ib .

and set 2

<(H -z)

-1

0

2

f,(H +b

H

2

0

and

b

= V(H 0 )

V(H) Then

)(H -z)

-1

0

exists.

a

H

Then the operator sum

f> =

has norm bounded by

and

Thus

is self-adjoint.

Notice that

implies

in turn implies that

for

W be a self-adjoint operator acting in

= (H 0 -z)- 1 [l+W(H 0 -z)-l]-l + W : V(H

"'

A , by the monotone convergence theorem.

~

: H _. H

0

(H-z)-l

2 l 0 <> B <> (B +s) 2

Assume that there exist constants

•

B-1

be a self-adjoint operator acting in the

0

such that

Assume

so using

.

W is a self-adjoint operator with

+

0

::>

'

with

and the corresponding representation for

B-1

"'

(B2+t2)-l

""

c

±W <>

a(

I H I +b) • 0

',12

"

which

"'

Thus the hypotheses of

Theorem 6.3 are stronger than those of Theorem 5.2 conclusion is also, since we can identify

But the

V(H) •

EXAMPLES Let function on

2 n H = L ( IR ,dx)

IRn •

and

Assume that

H

0

= -to •

w is in

Let

W be a real measurable for some

p

~

n 2 (and

- 32 -

p ;;:. 2

n = l '2' or 3

if

Lp

The

q

l

?

2-

=w

+ wp

00

norm of

inequality l

w

w p

-

where

'

if

n = 4).

w

is bounded and

Then

H = H + w 0

is

00

is in

Lp

w p

By Holder's

may be chosen arbitrarily small.

I Wfll2 = II wP liP I

2 n

2

>

V(H) = V(H ) 0

self-adjoint with Write

p

'

f llq

The bound on

l l - +p q

=

in terms of

II

where

'

II fllq

l 2

.

Notice that

<-ll+ b H

II

follows from

Sobolev inequalities. Next we turn to a class of results where the domain of the self-adjoint operator is not determined quite so explicitly. Let

Proposition 6.4

be a Hermitian operator.

Then

A

is

A is also Hermitian.

closable and Proof:

A

Since

A c A{'

Proposition 6.5 .

and

Let

is closed, A

Ai'

A

be Hermitian and

is closable and A

A c A'~ .

be self-adjoint with

1

Then Proof:

A = A~ , A 1 1

Since

is closed.

Thus

A c A1

Since

AcA , 1

A~ c A'" , that is, A c A'~ 1 A Hermitian operator if

A

is said to be essentially self-adjoint

A is self-adjoint. Since

adjoint

A = A** , A Hermitian operator


if and only if

Proposition 6.6 .

If

A

A{'

A

is essentially self-

is Hermitian

(A'" c A;";) •

is essentially self-adjoint, then it has a

unique self-adjoint extension. Proof:

If

Al

is self-adjoint' then

A cAlc A'~ = A* = A

-

33 -

EXAMPLES (l)

Let

domain

H

= L2 (Rn,dx)

and let

A

be the Laplace operator with functions with compact

V(A)

support.

Then

is essentially self-adjoint, and its closure is

A

~

the self-adjoint Laplace operator

•

The following proof of this result uses an important technique, invariance under a unitary group.

c"'com in

Let

f

is translation invariant, the translates

c"'

Let

com

g

c"'com

be orthogonal to

Since

be in f

of

a

f

are also

in the inner product of

It follows that

1

Since the Fourier transform of an L function, g*(l+k

4

)f = 0

Since

•

never zero, it follows that

(2)

Let

H = L2 (Rn,dx)

Let

g

f

=0

and let

oo n E = Ccom(R -0) , the space of

support not intersecting the origin. Q(A)

and when it is dense in

function determines the may be chosen so that

, g

=0

A=-~ ""

C

be the Laplace operator.

functions with compact E

is dense in

(When the latter holds, A

E is essentially self-adjoint.)

restricted to

is

.

We ask when

V(A)

f

In order to discuss

this question it will be convenient to introduce the language of tempered distributions. Let

be the space of rapidly decreasing

S

c"'

functions on

(Rapidly decreasing means that each derivative goes to zero at l TXT •)

infinity faster than every polynomial in elements of and

q, xqDpg

S n

is said to converge to - . xqDpg

uniformly.

g

in

S

A sequence if for every

of p

-

34 -

A tempered distribution is a continuous conjugate linear functional on S

If

S

, the value of

If

=

on

is a tempered distribution and g

will be written ~n

is a function on

u

is integrable,

v

v

then

(g*udx

u

g

is in

. 2 -m m , ( l+x ) u

such that, for some

defines a tempered distribution by

It is a fact that this tempered distribution

J

determines the function (almost everywhere). The nice features of tempered distributions stem from the fact that the Fourier transform maps

onto

S

S

Thus if

•

tempered distribution, the Fourier transform = , where

by v

g

v

of

v

v

is a

may be defined

is an arbitrary element of

Then

S

is also a tempered distribution. For example, the Fourier transform of the Dirac measure = g(O)*

defined by 6

= g(O)* =

is given by

corresponds to the function Differentiation sends

Thus

1 .

into

S

distribution, its derivative

6

Dpv

S

•

Thus if

v

is a tempered

may be defined by

This may be expressed in terms of Fourier transforms as transform of If

= <(-ik)Pg,v> Dp6

For example the Fourier

corresponds to the polynomial

2

H = L (~n,dx)

and

functions

f

in

this that

s

is dense in

such that

H

Q •

A

= -to J1fl

2

'

consists of the

Q

then

2
v

whose Fourier transforms correspond to functions 2



2 1 Cl+k )- dk <

=

oo

•

The value of

= Jr*vdk(2rr)-n .

I t is clear from

o.*

Hence the elements of

identified with certain tempered distributions

J1vi

(ik)p

v

on

f

may be

in fact those

v

such that

is then

The Fourier transforms of the elements

-

Q*

of

35 -

Q*

are thus functions, even if the elements of

are only

tempered distributions. Similarly, V

I

~

If I

2

o +k

4

) ctk <

oo

consists of the functions such that

v*

Its conjugate dual space

•

may be identified

with those tempered distributions which have Fourier transforms satisfying

J1vl

2

4


for all

But since

v

f

•

is dense in Q, consider v in

E

In order to see when = 0

oo

E •

in

~

2

But

v

v*

in

flv1

2

v

=0

.

oo

E , then v

,

flv
so

That is, v 2

1

2


is <

00

•

J

0

E is dense in

is zero on

4 1 Cl+k )- dk <

=

v

measure.

6

Q* ,

is also in

this implies

To see when v

with

is a distribution with support at one point, it is a

a polynomial. n

Qf'

It is sufficient to show that

linear combination of derivatives of a

If

v

V , we use the same argument. v

is a polynomial.

must be zero if

If

But since

n? 4 •

J

If

A

is an operator, its numerical range consists of the set

of complex numbers If

A



such that

g

is in

V(A)

llgll = l .

with

is a Hermitian operator, its numerical range is a subset

of the real numbers. Proposition 6.7

Let

A

be an operator and let

number which is at a strictly positive distance range of for all

f

Proof;

Let

II (A-z) g II V(A)

'

A

~

Then

A- z

in range g

be in

is injective and

d

be a complex from the numerical

1 II
~ (1/d)llfll

(A-z) V(A)

with

II gil

I I = I - zl

II (A-z)hll ~ dll hll

z

.

Let

f

=

=

l

Then

~

d

Thus for all

(A-z)h

.

h

in

This gives the estimate

- 36 -

Proposition 6.8 .

Let

A

be a Hermitian operator and let

z

be a

complex number which is at a strictly positive distance from the numerical range of are dense in Proof:

A .

Assume that range

Then

H

A

Then

A

z

(A-z)-l

and

(A-z*)-l

are closed. H •

Theorem 1.7 implies that

Theorem 6.9 •

Let

H

Hilbert space

H •

Let

V(W)

::>

V(H

0

)

0

and range

A

is self-adjoint.

W be a self-adjoint operator acting in

Assume that there exists a constant

•

H

= H0

b

H

such

+ W with domain

is essentially self-adjoint.

)

Proof:

Assume

b

t

and set

0

norm bounded by one.

Write

to show that range (l+T) If

f

orthogonal to

Hence

(l+T)f

(l+T)f •

=0

=

Then

±ib •

T :: W(H -z) 0

H - z = [l+T] (H -z)

has

It is sufficient

0

is dense in

-1

H •

(l+T)

Since

(l+T)f - f

(l+T)f

=

, then in particular it is

= Tf

, we have

•

Then z

z

is orthogonal to range

Next we show that

and

A- z

be a self-adjoint operator acting in the

0

Then the operator V(H

It follows

Since they are dense by assumption, they are

both equal to

with

A - z*

Hence by

are continuous.

from the Corollary to Proposition 1.6 that range A - z*

range

is at a strictly

A

positive distance from the numerical range of Proposition 6.7

and

is essentially self-adjoint.

A be the closure of

Let

A - z

(H-z)g

is not real, g = 0 •

=

0

implies

(l+T)f

Hence

f

=

0 0

f

=0

Since

Let H

is Hermitian

- 37 -

Thus the only element orthogonal to range So range

(l+T)

(l+T)

is

f

=0

•

is dense.

NOTES There is an alternative proof of Proposition 6.2 in Kato's book (1966) (on page 292). Theorem 6.3

is the classic perturbation theorem of Rellich.

Theorem 6.9 is taken from Chapter V of Kato's book (1966). may

WUst (1971) showed that the hypothesis be replaced by

JjWfjj ~ IJH 0 fjj + bJJ fjj •

(This has been generalized to

Banach spaces by Chernoff (l972a) and Okazawa (1971).)

Wust (1972)

also has other results on borderline cases. There is a discussion of tempered distributions in the book by Stein and Weiss (1971).

The characterization of distributions with

support at a point may be found in the treatise of Schwartz (1966).

§7

STRONG CONVERGENCE AND APPROXIMATION In previous sections we have seen that the sum of two self-

adjoint operators is defined under rather general conditions. instance, if H

= H0

+ U

H

and

0

U

For

are positive self-adjoint operators, then

is self-adjoint in circumstances where

being a small perturbation of

H

0

in any sense.

U

is far from

The question arises

whether such a sum may be approximated by small perturbations. Thus there would be a possibility of calculating approximations to the resolvent and other functions of

H .

The purpose of this section is to show that this can be done. We construct a sequence Uk The resolvent of Hk

= H0

of bounded operators which approximate

+ Uk

U.

may be computed by an uniformly con-

vergent series expansion in powers of is then supposed to converge strongly (that is, pointwise) to the resolvent of

H .

The basic problem is thus to develop techniques to prove strong convergence.

We begin with a review of the various kinds of con-

vergence and the basic facts about them.

This will lead to a series

of results about when weak convergence and certain order relationships imply strong convergence. Consider a Hilbert space

H

There are two important notions

of convergence for elements of

H

We say

norm

II fn -fll _.

for all vergence.

g

in

fn-. f

fn _. f

weakly if

strongly if the

0

We say

-.

H

Strong convergence obviously implies weak con-

-

Proposition 7.1 • then

fn -

f

If

fn -

strongly.

f

39 -

weakly, and if

II fll

lim supll f n II ,

?-

(Weak convergence and no loss of norm

implies strong convergence.) Proof:

Then

less than

llfn-fll

llfll

2

2

= llfnll

2

- 2llfll

2

+ llfll

- - + llfll 2

=0

2

has lim sup

•

There are at least three important types of convergence for bounded operators

A : H -

H

bounded sets) if the norm if

A f - Af



Anf- Af



for each

n

uniformly (on

-A

We say

in

f

weakly for each

converge to

A

IIA n -All- 0 •

strongly for each

n

weakly if

We say

We say

H •

f

in f

A- A n A

n

strongly

-A

H , that is, the numbers

and

g

in

H •

Uniform

convergence implies strong convergence which in turn implies weak convergence. The following four propositions deal with convergence of functions of bounded operators. Proposition 7.2 •

If

An- A

there exists a constant AnBn -

AB

strongly.

The first two are elementary. strongly, Bn- B 11An11 ~ M

M with

strongly, and

for all

n , then

(Multiplication is strongly continuous on

bounded sets.) Proposition 7.3 •

If

A

n

-A

weakly.

weakly, then

(The

adjoint is weakly continuous.) A bounded normal operator is a bounded operator such that

AA'~ g

= A 1'A. in

(In other words, A

is normal if

1

IIA 'gll

=

IIAgll

for all

H •

Proposition 7.4 • An- A

Let

strongly, then

An

and

A~- A*

A

be bounded normal operators. strongly.

If

- 40 -

Proof:

If

n

n

That is, A'''g- A*g

weakly. A

strongly, then

A -A

are normal, IIA~gll

A

= IIAngll-

weakly, so g

in

H •

Thus there

IIAgll

n

But since

Since this holds for

H , A*n -A* strongly as operators.

in

g

-A

A*g -A*g strongly.

is no loss of norm, so all

n

weakly for all

n

and

A

The spectral theorem for bounded normal operators states that any such operator is isomorphic to multiplication by a bounded complex function on some

L

2

space.

A Borel measurable subset

S

of the complex plane will be said to be of spectral measure zero if the set on which this function takes values in

S

is of measure

zero. Proposition 7.5 . and let Let

~

R

Let

Rn

be a sequence of bounded normal operators

be a bounded normal operator such that

Proof:

strongly.

be a bounded Borel measurable function on the complex plane

which is continuous except on a closed set of zero.

Rn-- R

~(Rn)

Then

--

~(R)

R

spectral measure

strongly.

By the principle of uniform boundedness the

Thus it is sufficient to consider

~

II R II are bounded. n

on a compact disc

D

in the

complex plane. First we show that the result is true for any continuous on

D

The set of

~

which is

for which the result is true is a

uniformly closed algebra, by Proposition 7.2 assumption.

~

It contains

z

by

Further, it is closed under complex conjugation, by

Proposition 7.4 •

Hence by the Weierstrass approximation theorem it

contains all continuous functions. Next we show that the result extends to certain discontinuous functions

~

.

Let

y

be a continuous function which is zero on the

- 41 -

~

discontinuities of ~(Rn)y(R)

•

Since both

= ~(R )[y(R)-y(R )] + ~(R )y(R) n n n n

~(R)y(R)

= y(R)f

g

•

vectors are dense in

~Y

for all vectors

of

H •

to be a continuous function which is

0

on the discontinuities of

l n

from them.

~

and

l

at a distance greater than

Since the discontinuities are of spectral measure zero,

the dominated convergence theorem implies that in

g

Thus it is sufficient to show that such

To see this, choose

f

are continuous,

converges strongly to

~(Rn)g-. ~(R)g

In other words,

the form

and

y

y(R )f _. f

for all

n

H •

The next two propositions show how one can discuss convergence of functions of self-adjoint operators. Let

be a Hilbert space.

H

adjoint operators acting in

Let

An

be a sequence of self-

We say that the

H •

A

n

converge to

in the sense of strong resolvent convergence if for some away from the spectra of the

A

and

n

z

A

(bounded

A)

strongly. Proposition 7.6 • Let

A

Let

An

be a sequence of self-adjoint operators.

be another self-adjoint operator. E c V(A )

subspace

such that

n

is dense in

V(A)

,

then

A

n

A f - . Af n

_.A

Assume that there is a for all

f

in

E •

If

E

in the sense of strong resolvent

convergence. Proof:

Let

f

be in

E and set

h = (A-z)f

Then

(A-z)-lh - (A -z)-lh = (A -z)-l(A -A)(A-z)-lh = (A -z)-l(A -A)f _. 0 n n n n n But since

E is dense in

V(A)

,

(A-z)E

is dense in

H .

Since we

have convergence on a dense set and a uniform bound, we have convergence on all of

H

-

~

If and

A

42 -

is a bounded Borel measurable function on the real line, ~(A)

is a self-adjoint operator, then

operator.

(Of course how

~

is a bounded normal

is defined off the spectrum of

A

doesn't matter.) A Borel measurable subset

E

of the line will be said to be of

spectral measure zero (with respect to the self-adjoint operator

Proposition 7.7 •

Assume that the sequence

A

n

of self-adjoint

operators converges to the self-adjoint operator strong resolvent convergence.

Let

~

A

in the sense of

be a bounded Borel function

on the real line which is continuous except on a closed set of spectral measure zero. Proof: ~(R)

Let

=

=

~(t)

~(An)

~

R = (A-z)

-1

, so that

is continuous except on a set of

measure zero (which may include zero).

A

strongly.

-.~(A)

and set

~(z+l/t)

Then

~(A)

Then

A )

Thus if we set

R R

n

spectral

= (An -z)-l

The next three results are concerned with when weak convergence (that is, convergence of forms) implies strong convergence. The conditions are stated in terms of the order relation between selfadjoint operators. Proposition 7.8 with Proof:

Rn

~

R .

Let If

I

2

1

(R-Rn) 2 ll

II (R-Rn)~fll = Theorem 7. 9 below.

Let

Let An

and

Rn-. R ~

II (R-RnHil

Rn

A

R

be bounded self-adjoint operators

weakly, then

Rn-. R

strongly.

1

II (R-Rn) 2 fll

and

_. 0 •

be a self-adjoint operator which is bounded

be a sequence of self-adjoint operators such that

- 43 -

A -s A

Assume that there is a subspace

n

that

-. A

Q.(A) , then Proof:

_.A

n

for every

.., c

-1

-1

However the

A-lf n

-1

and for

f

I
-1 f,g> I n

-1

that

in

and

H

E is dense in

If

in

g

-1

~

1

are bounded in -1

<> -1

-An )f,Ag>-. 0 Hence

V(A)

such

0 < c <> A-s An.

E

we have

1

1

<(A -A)A f,A f> 2 <(A -A)g,g>2 -s 2 <(A -A)g,g> 2 ~ 0 . n n n n n n

<>

-1

n

in the sense of strong resolvent convergence.

n

<(A

E

in

for all

We may assume without loss of generality that

Then

-1

f

E c Q.(An)

.

So since

for all

A-l_,_ A-l

g

in

weakly.

n

A-l-. A-l

Q.(A) , because

E is dense in Q.(A) Since

Q.(A) ,

, in particular for A-l n

<S

g

in

A-l , it follows th

strongly.

n

EXAMPLE H = o

Let L1

be a sequence of positive

and

-~

U n

~

0

functions which have support in a ball

(centered at some fixed point) with radius which goes to zero with Let

H

= Ho

+ U

Then

n

Hn - . H

0

n.

in the sense of strong resolvent

convergence. To see this, let

E be the space of

Coo

support in the complement of the center point. is dense in

Q.(H ) . 0

Since

the preceding theorem shows that resolvent convergence.

functions with compact If

H -- H n o Hn - . H

n

2 , then

E

as forms on

E

~

in the sense of strong

- 44 -

The drama of this example is that there is no restriction on the magnitude of

U

as

n

Nevertheless functions of of

H

varies, only on the size of its support.

n

Hn

converge to the corresponding functions

In particular the solutions

0

equations converge to Theorem 7.10 .

Let

exp(-itHn)

exp(-itH ) . 0

A

be an increasing sequence of self-adjoint

n

An~

operators which are bounded below, so that be a self-adjoint operator such that f

is in

Q(An)

of the Schrodinger

and

Then

A

n

A

~

A

and such that whenever

is bounded, then

An _. A

Let

An+l

f

is in

Q(A)

and

in the sense of strong resolvent

convergence. Proof:

We may assume that

0 < c

An

~

-1

decreasing and bounded below (by A

), it follows that the

have a weak limit, hence a strong limit. below by

Since this limit is bounded

A-l , it must have trivial kernel.

Call the limit

B-l ~ A-l , which implies that

Thus

B-l

Since

n

An_. B

in the sense of strong resolvent convergence, we need only

show that

B

=A

.

The trick is to notice that by Proposition 7.7 strongly implies that Q = Q(B)

in

are

Since the

strongly.

We will show that for

g

Q

in

A-1 _. B-1 n

Consider the space J.

An2 g -

1

A2 g

strongly

H .

If

g

is in

Q

then

B~g

is in

H

If

f

is in

Q

then

1

An2 f

is also in

n

=

H J.
n '

(since

A n

_J. _1 (B 2 -A 2

n

~

J. )B 2 g>

B).

_. 0

are uniformly bounded, it follows that

Hence 1

Since the

IIA n2 g II ~ II Bg II But weak

- 45 -

convergence and no loss of norm implies strong convergence.

In particular, if

g

is in

follows by assumption that Hence

A

extends

B .

g

But

Q(B) , - . .

is in B

~

So

Q(A)

A , so

A

and

=B



It

=

.

.

Now we are almost ready for the main approximation theorem. First we review the basic results on form sums that we have obtained so far. Let U

~

0

~

H

0

0

be a self-adjoint operator acting in

be another self-adjoint operator

said to satisfy condition

A

form small if there exists Theorem 7.11 .

Let

H

a

operators acting in

H

n Q(U)

)

particular

and

=U

and

U

is

H

W is said to be ±W

such that

H0

"" 0

~

a(H +b) 0

be a self-

W be self-adjoint Assume that

+ W

Assume that

whose form is the form of Q(U)

V

U

is dense in

Then

b

Then

H •

U

~

0

W is form small (in

Then there exists a self-adjoint operator

0

)n

Let

H •

A

Q(W):, Q(H ) ) •

0

and

< l

and set

and satisfies condition

Proof:

0

be a Hilbert space and

adjoint operator acting in

Q(H) = QCH

Q(H

W be a self-adjoint operator.

Let

H

if

acting in

Let

H •

H

H

0

+ V

= (H 0 +U)

+ W .

Also

is bounded below, in fact, H

~

-b .

This theorem summarizes the main conclusions of Theorems 4.1

and 5. 2. If operators where

A

is a self-adjoint operator, the corresponding truncated

Ak

lA I > k

are defined by

Ak = A

where

IAI ~ k , Ak

=0

- 46 -

Approximation Theorem 7.12 ~

acting in

H

Let

acting in

H

Assume that

is form small. operators.

U

Let and

0

U

Set

0

W~ 0

be a self-adjoint operator

be self-adjoint operators

satisfies condition

A

and that

W

be the corresponding truncated Then

V : U + W , Vk : Uk + W

Hkr : H0 + Vkr- Hk :: H0 + Vk k-

0

wr

Let

~

H

as

r -

and

oo

Hk -

H

H

+ V

0

as

in the sense of strong resolvent convergence.

oo

Proof:

Since

Hk

~

Hkr , the first result follows from Theorem 7.9 .

For the second, note that Theorem 7.10 .

Consider

bounded.



Then

follows that

f

is in

Hk f

~

in

H .

Thus we may hope to apply

Q(Hk)

is bounded. Q(U) .

Hence

= QCH 0 ) Since f

with

-

is in

Q(H)

, it

= Q(H 0 )

n Q(U)

The theorem indeed applies.

NOTES The standard reference on strong resolvent convergence and on when convergence of forms implies strong resolvent convergence is Chapter VIII of Kato's book (1966). Wlist (1973) has given a convergence theorem which doesn't depend on the assumption of semi-boundedness.

This is applied to Dirac

operators by Schmincke (1973). Parrot (1969) has given examples which illustrate some of the difficulties with more general definitions of limit. The example of the Schrodinger operator with a potential with small support is due to Friedman (1972).

potentials

Un

He shows that if the

are positive and the capacity of the support of

Un

- 47 -

approaches zero, then resolvent sense.

H

0

+ Un

approaches

H

0

in the strong

However he gives an example when

n

3

that this may fail if the positivity condition is dropped. does not contradict Proposition 7.6, since until

n

~

4.)

to show (This

E is not dense in

V(H 0

A paper of Schonbek (1973) has further discussion.

)

Part II

§8

OPERATOR DOMAINS

ORDER IN HILBERT SPACE We have seen that there is not much difficulty in adding positive

self-adjoint operators. of the sum.

Now we turn to a more detailed examination

In particular, we would like to see to what extent we

can get information about the operator sum. The interest in knowing about the operator sum may be illustrated Let

H

operators and consider their operator sum

H

by the following theorem of Trotter.

Let if

H H

+ V

0

be the closure of the operator sum. is self-adjoint, then

and

0

V

be self-adjoint D(H ) n D(V) .

on

0

The theorem states that lim(exp(-i!H )exp(-i!V))nf n o n

exp(-itH)f

n+oo

for all

in

f

(In the application to quantum mechanics this

H .

formula is closely related to the Feynman path integral.) One of the main tools will be the partial order relation in a Hilbert space

2 L (M, \l)

•

It is more convenient to discuss order for

real functions than for complex functions, so we begin with the reduction to real Hilbert space. A real Hilbert space is a real vector space with an inner product such that the space is complete.

Most of the theory of

complex Hilbert spaces carries over to real Hilbert spaces.

Notice,

however, that there is no polarization identity. Let

W be a complex Hilbert space.

conju!liation i f anti-unitary and

Tf

f

T

'

then

f

and

T2

l

is called real.

If

A

'")

'

T : W _. W is a a''T(f) + b•'
(T(af+bg)

is anti-linear

(

Then

If

f

is in

, w

is an operator acting

-

W such that

in

TAf = ATf , then

V(A)

49 -

f

in

implies that

A

is called real.

Tf

V(A)

is in

and

The set of real elements in a complex Hilbert space forms a real Hilbert space.

If

space invariant.

A

is real, then

A

leaves this real Hilbert

Questions about real operators in a complex Hilbert

space may usually be reduced to questions about operators in a real Hilbert space. In the rest of this section we will take the Hilbert space

H

to be a real Hilbert space. Let

H

L

2

An operator

(M, \l)

positivity preserving if

~

f

0

A : H _. H

implies

Af

~

is said to be

0

Since

A

is

linear, if it is positivity preserving it is also order preserving, that is, f

~

g

implies

~

Af

Ag . L2

Notice that the set of positive functions in cone.

is a closed

Thus the property of being positivity preserving is preserved

under strong limits.

This fact will be used in the proof of the

following proposition. Proposition 8.1

Let

operator acting in

H

preserving for all

c

on

M .

Set

V

Assume that

=U

preserving for all Proof: 0

~

uk

Let ~

k

uk and

H c

and -r

easy to verify that

Let

is positivity

0

w

satisfies condition

A

H

+ V .

0

?-

0

Then

(H+c)

0

~

and

-1

be real functions W is form small.

is positivity

sufficiently large. be the truncations of

wr ~

u

(H +c)

be a self-adjoint

0

~

0

-1

and

0

>

H

Let

Assume that

U

+ W and

2

L (M, \l)

H

wr

~

Let

0

(Hkr+c)

-1

Hkr

u

and

Ho + uk + wr

w

,

so that

I t is

is positivity preserving for

c

-

sufficiently large. (Hkr+c)

-1

50 -

In fact, we may expand this as

(H +c+k) o

-1, oo [ -1] n , and each term is Ln-o (k-Uk-W )(H +c+k) r o

positivity preserving. -b

If

is less than the lower bound of c > b

then we may take

0

~

Hkr

'

and expand

-l.Loo [ -l]n (Hkr+c) n=O (c-b)(Hkr+c)

(Hkr +b)-1

w .:: H0 + wr

+

H

positivity preserving for all such

b .

Hence

+b)-1

(Hkr

is

The conclusion thus follows

from the approximation theorem (Theorem 7.12). The following proposition gives a way of verifying condition In fact, the conclusion is stronger than condition Proposition 8.2 •

Let

Let

H

e

Let

self-adjoint operator acting in

H e

that

(H +c)

-1

be a function such that

oo •

<

V(H

We will see in fact that

V(H

Since

<e,Ue>

0

)

is dense in

Q(H

0

)

,

Then

0

)

H

0

QCH

n Q(U)

c 0

0

~

0 .

>

)n

H

be a

Assume

0

0

is positivity preserving for all

0

Proof:

such that

H

A •

be an element of

which is strictly positive almost everywhere.

A .

Q(U)

Let

U

~

is dense

is dense in

V(H 0

)

this implies the conclusion of

the theorem. Let

00

L (e)

to see that

{f

Loo (e)

lhl .l e .

let

c

±f .:: te (H +c) 0

t} •

In fact, if

H •

Since

for some

e > 0

It is easy

h .l Loo (e) , then

almost everywhere, this

h = 0 .

Next observe that fact, i f

L 2 : ±f ~ te

is dense in

h.le sign(h) , so implies that

in

-1

'

(H +c)

then

00

L (e)

-1

'

±(H +c) 0

00

L (e)

leaves

0

-1

f

~

t(H +c)

it follows that

-1

0

CcV(H

invariant. e

= tc 00

0

)n

-1

e

L (e)

In I f we

However

0

- 51 -

since

U

l

L (M,e

is in

2

lJ)

00

L (e) c Q_(U) , and so

,

C is dense in

Thus it is sufficient to show that 00

L (e)

is dense in

C

H , so

(H +c)

-1

0

00

L (e)

V(H ) •

But

0

is dense in

V(H ) 0

Theorem 8. 3

Let

2 L CM,)J) .

H

Let

e

is strictly positive almost everywhere. adjoint operator acting in (H +c)

-1

2

real function in

2

L (M,e lJ)

Let

V(H

00

00

Let

L (e)

{f

is dense in

H ·

0

)

c +

0

>

which

H

be a self-

0

0

Assume that

0

Let

U .

Then

U H

be a

0

~

is

n V(U) .

L 2 : ±f ~ te

in

0

0

H

H

~

H

H e

such that

H

essentially self-adjoint on

L (e)

Let

is positivity preserving for all

0

Proof:

be an element of

First we show that the form sum

for some

H

H

0

+ U

t} .

As before,

is a self-adjoint

In fact, <e,Ue> ~ llei!IIUell < oo, so Proposition 8.2

operator. applies.

Hence

dense in

H •

Q_(H ) n Q_(U)

is dense in

0

Q_(H

Now we prove that the restriction of

H

0

)

to

,

which in turn is

V(H

0

)

n V(U)

is

essentially self-adjoint._ We know that that

o

~

(H+c)

uk

(Hk+c)

k

~

-1

e

-1

e

c

-1

(H +c)

-1

0

±f

~

e

is positivity preserving.

In fact, if we set

e -

(H 0 +c)

we obtain

It follows that if

-1

We now show

H

, we have

k - . oo

we let

~

(H+c)

te , then

E = (H+c)-lLoo(e).

-1

Then

-1

c

e

-1

e

(H+c)-le ~ c- 1 e .

(H+c)-l ±(H+c)

Uk(Hk+c)

-1

f

leaves ~

t(H +c) 0

00

00

L (e) -1

e

~

invariant. tc

-1

e .

EcV(H)nL (e)cV(H)nV(U)

Let

In fact,

If

- 52 -

E c V(H ) n V(U)

We will see that

V(H)

If

g

E , then

is in

Hg

- Thus

E

E

Ug

V(H 0

)

(H+c)

-1

H

are in f

But then

V(H 0

is in

)

•

E is dense in

Finally we show that H , and so

and

is dense in

E

is continuous, so

is also contained in

dense in

and that

0

00

L (e)

V(H) .

00

But

is dense in

L (e)

V(H)

=

is -1

(H+c)

H

EXAMPLE Let Then

T

L2 (~ n ,dx) .

H

Define

T : H _. H

by

f(x)* .

Tf(x)

is a conjugation and the real elements defined by

the real functions.

Fourier transform representation is effect of

T

t

The corresponding operator

FTF-l

Tf(k)

T

in the

Thus the

is to reverse momenta.

The operator

-~

is clearly real with respect to

T •

Notice

that this is a reflection of the fact that the kinetic energy even in the momentum Let

0

serving for

at some point

x .

is

-1

is positivity pre-

0

S

is in

Since

2

It is enough to show that this is true on a

dense set, such as f

(H +c)

We will see that

c > 0

Assume

k

k .

= -~ .

H

somewhere.

are

g

S

f ~ 0

with

is also in But then

and

S , g

f(x)

g = (H +c)-lf 0

is negative

must take its minimum value

(-Hc)g(x)

~

cg(x) < 0 , which

~

0

is a contradiction. Let ~n

.

H

Assume that

Let U

is in

H

0

2

= -~

and

U

L <~n,exp(-2alxl )dx)

be a function on for some

a <

oo

-

Then

H

H

0

+

U

53 -

V(H

is essentially self-adjoint on

0

) n V(U)

This

•

may be proved by means of the following trick. Let

Y

be a real function on

ffin

which is constant on a ball

centred at the origin and is zero elsewhere. adjoint with the same domain as

Y

suitable choice of negative eigenvalue. -c

2

(where

c

>

set

-~

, since

Y -~

we may arrange that

-~

Then

+

Y

e

We may choose it as negative as we please.

2

Then

e

e(x)

so that

We now use our freedom to choose that e(x) so

c

"

u

>

a

This implies that

k exp (-a I xI) is in

2

n

Hence 2

L (ffi , e ( x) dx)

-~

+

Y

Let and

and zero is an eigenvalue of

H1

be the eigenfunction of

choose the sign of

By a

has a strictly

in place of

It is sufficient to prove the theorem with Let

is self-

is bounded.

0) be the most negative eigenvalue of + c

Y

+

e

0

>

with

0 •

-~.

We may

0

for all

c

arbitrarily large to require

X

satisfies an estimate

f U(x) 2 e(x) 2 dx
k 2f U(x) 2 exp(-2alx I )dx<

"' '

and Theorem 8.3 is applicable.

NOTES A proof of the theorem of Trotter mentioned in the beginning may be found in §8 of the notes by Nelson (1969). The essential self-adjointness theorem (Theorem 8.3) is based on the fact that there is an auxiliary Banach space that is left invariant.

This type of theorem arose in quantum field theory in

the work of Rosen (1970).

An abstract formulation appeared in an

important paper of Segal (1970).

The theory was extended by Simon

and Hoegh-Krohn (1972) in their survey of hypercontractive semigroups.

-

54 -

The theory presented here is essentially the specialization of this work to the case when it is sufficient to deal with contractive semigroups.

This specialization was applied in quantum field theory by

Hoegh-Krohn (1971).

It was also studied by Semenov (1972).

The first application of these ideas to Schrodinger operators was by Simon (l973a). (l972b), (1973).

The trick used in the example is due to Faris

§9

ORDER AND ESTIMATES In order to obtain information about operator sums it would be

convenient to be able to pass from one second order estimate to another.

There are inequalities which allow this; for example it is

always true that if

A

and

B

are self-adjoint then

However even when A2 + B2

is not so easy to estimate

difficulty is that the cross terms

A

and

B

in terms of AB + BA

are positive it (A+B)

2

The

need not be positive.

In this section we see how this difficulty arises and how it may be avoided in certain cases by use of order properties.

The

main result (Theorem 9.2) is illustrated by an example from quantum field theory. 00

We write Anf

is in

C (A)

for the space of all vectors

H for all

n

=

0,1,2,3, . . . .

Let

Assume that

A

A

2

V(A+B)

0

00

C (A)cV(B)

AB + BA >- -a - bA 2 Then

~

and

= V(A)n

V(B)

may give

V(A+B) . B

~

0

be self-adjoint operators.

and is dense in

for some

00

C (A)

The point is that it is not necessary

to identify a dense subspace of Theorem 9.1

such that

The following theorem

shows that an estimate of the cross terms on information about the domains.

f

a

and

b

<

Assume also that

V(B)

l

as forms on

Coo(A) .

and there is an estimate

+ B2 ~ c(A+B) 2 + d .

Proof:

We need only the estimate which shows that

V(A+B) c

V(A)

n

V(B)

•

Since

ABA

AB(l+A/n) + (l+A/n)BA ~ - a(l+A/n) It follows that

is positive, we have 2

- bA

AnB + BAn ~ - a - bA~

2

•

Let

A

as forms on

A(l+A/n)

n 00

C (A) .

-1

.

Since

- 56

An

00

C (A)

is bounded and

holds on

V (B)

V(B)

is dense in

An + B

increases to the form sum

(A +B+l)-lh-. (A+B+l)-lh

for all

n

u

V(A+B)

be in

(A +B+l) n

n

-1

h

are bounded by in

V(A)

n

V(B)

u = (A+B+l)

Then

u

2

and

operator acting in

H

preserving for all

c

H

+ u

0

for some Proof: set

if

Let

0 :< (Hk+c) k -. 0 :<

oo

-1

f

Now let

'

IIWCH+c)- fll

f

~

-1

0

±IWI CH+c)-lf ~

Hence

0

(H +c)

lwl (H+c)-lf ~

1

0

Then

We see that

•

h

H

Set

and

II Bu n 11

in

IIA n u n 11

2

Let

Let QCH

0

)

U

2

kil (A+B+l)u 11 ~

H

0

u

n Q(U)

be a real measurable H

is dense in

w2 :< a 2 CH +b)2 0 where

u

(Hk+c)

-1

is

2

is positivity

0

0

f- (H +c) 0

0 ~

Let

otherwise, and

0

'

such that

w2 :< a 2 CH+b) 2

Then u :< k

M

is positivity preserving and so

~

f

(H +c)

(H +c)-lf

-1

0

f

Hence

0

0

Since

lwl CH+c)- 1fl ~ 1

IIW
11

-1

IWI (H +c)-lf

be arbitrary.

1

Uk(Hk+c)

(H+c)-lf ~

1

IIWCH+c)-

-1

IWI
~

~ IIWCH 0 +c)- 1

1

1

1fl

, we see that

IIW
.

2

be a self-adjoint

0

(H +c)-l ~

Let

we have

0

~

and

kllh11

Assume that

be equal to

uk

for some

be a real measurable function on

>

Ho + uk

Hk f

b

h

V(B)

~

2

>

w

and

+ 11Bull

2

L (M, \l)

such that

Let a

2

l!Aull H

M

is in

n

H , by Theorem 7.10

in -1

A + B ,

It follows from Fatou's lemma that

Let

function on

h

Then

kll h 11

Theorem 9.2 .

H

as forms on

In other words,

.

Since

u

, the same form estimate

V(B)

II~

Let

- 57 -

EXAMPLES and

~n

such that

H

Let

0

Thus

u

+

w

U 9

0

be a function on

is essentially self-adjoint on

w be a real function on

Let

v

+ U

0

H

V(H

2

a CH +b)

such that

H

0

+ V

)

n V(U)

2

0

W is a small operator perturbation of

Then

0

H0

with Let

is essentially self-adjoint on

2

W .,; a

In fact, by Theorem 9.2 small operator perturbation of

H

0

2

(H +U+b)

2

0

U .

+

, so

Hence

W

H

0

is also a

+ V

is essentially self-adjoint on V(H )nV(W)nV(U)cV(H )nV(V) 0

0

The rest of this section is devoted to an example from quantum field theory.

The basic idea is to treat fields as quantum mechanical

observables and to try to write quantum mechanical equations of motion in terms of the fields.

The difficulty is that the require-

ments of relativistic invariance force a very singular behavior of the interaction terms involving the fields. These difficulties are

much less severe

dimensions than in three space dimensions.

in one and two space The most progress so

far has been in constructing solutions in the one space dimensional case.

In this case it has been possible to construct a relativistic

theory.

A preliminary step in this construction is the construction

of a theory on a finite interval, essentially the quantum theory of a non-linear vibrating string. We begin with the construction of the field. periodic interval of length multiples of

(2n/L)

and let

Let

L

r+

r

Consider a

be the set of all integer

be the set of strictly positive

-

58 -

1

integer multiples of

(2n/L)

.

The functions

(l/L)

2

and

1

J.

(2/L) 2 cos(kx)

(2/L) 2 sin(kx) , where

and

k

r

ranges over

+

form an orthonormal basis for the functions on the interval.

If

u

is a real function, it has a Fourier series expansion u(x) coefficients. Let

M

be the space of all sequences of real numbers.

Let the

sequences be labeled in the same way as the coefficients of a Fourier series, so that a typical sequence is

r

over

n

-J. 2

0

where

k

ranges

+

Let or

{x ,xk,yk}

be the product measure on

v

2

exp(-y )dy

on each coordinate.

functions

n

- J.2

2

exp(-x )dx

The Hilbert space describing

H ~ L 2 (M,v)

the state of the field is

Let

M which is

Each of the coordinate H

is a multiplication operator in m

be a positive real number and set

~(k)

The field is defined by the Fourier series expansion Thus

$(X)

the Fourier coefficients in its expansion are multiplication operators. For each fact, let

f

x, $(X) and

g

is a densely defined Hermitian form. be functions in

finitely many coordinates.

are zero for

L (M,v)

which depend only on

Then the matrix elements sufficiently large.

In



and

Hence the sum defining

is a finite sum.

While the field average

k

2

J~$(x)u(x)dx

$(X)

$

and

x

is not an operator, its

with a sufficiently nice function

multiplication operator. expansions of

at a point

u

u

is a

This may be seen from the Fourier series In fact,

- 59 -

J>

_1

and

yk

<

field converges in

1

The functions

+ Ek(akxk+bkyk))J(k) , 2

1 (M,v)

are orthogonal in

2 2 -1 Ek(ak+bk))J(k)

if

-J.

ax)J(O)~ 0 0

(x)u(x)dx

and all have the same norm.

, the series expression for the averaged

00

2

to a measurable function.

The operators corresponding to the time derivatives of the Fourier coefficients of the field are

!(

Py· k

i

=

rr(x)

a -y ) ayk k 1

p

xk -

1 a -x ) I ( axk k

and

The time derivative field is defined as

·

1

1

1

(l/1)~p )J(0)~ + (2/1) 2 Ek(pxkcos(kx)+pyksin(kx))J(k» 2

0

Again

this may be interpreted as a densely defined Hermitian form. The energy for the linear field is given by the formal expression

H

0

This may be interpreted in the Fourier series representation as a limit of partial sums, and evaluated as a2 )J ( 0) [ -~ --+x
H

0

0

fx-] 0

a2 a2 a l + E lJ (k) [ _! --2+Xk - - - - - y k 2 a xk 2 a 2 k axk yk

Similarly, the momentum carried by the field in the is

-J

1

a~~ x

direction

a~k-yk

rr(x)V,P(x)dx Lkkf(xk a:k). We now examine these 0 expressions to see what kind of objects they represent. P

a

xax

First consider the operator 1 2 (R,rr - ~ exp(-x 2 )dx) a*

2

-1

a

~(2x---)

ax

and a

This operator factors as 2-t_~_

the commutation relation polynomials

(a*)nl

eigenvalues

n

The

ax

operators

a'~a

aa>'•

+ l

a

and

a l

acting in a~'a

N

and

ax

satisfy Hence the with

N

It is known that these Hermite

0 ' l ' 2' 3' ...

polynomials form a basis for

a*

0

n 2- 2 (2x-~)nl are eigenvectors of

, where

2

1 (R,11

-

1

~exp(-x

2

)dx)

, so

N

is a

-

60 -

self-adjoint operator and its spectrum consists of the positive integers. Next let us consider the case of two factors.

2 2 exp(-(x +y ))dxdy) . The energy operator may be written 1 a2 1 a2 a -2 ax2 + x ax 2 ay2 + Y ay and the momentum operator is L

is as

2

(R

~ (x2--y l

ay

2

The Hilbert space

,n

-1

_a_) ax

= !i

It is useful to write these operators in complex form. z =

X

+ iy

a !(2_-i aay) iy and let = X az = 2 ax a a a + z Define N = -a z''' and + az az

-

zi:

and

a lc . az"' = 2 a;z+l a a a N + z·lt a z;, az a z•~

a~)

N +

-

The operators

N

and

N + N +

Then the energy operator is

the momentum operator is

Let

N +

N

and

the operators for the number of particles moving in the

and

+x

are

and

-x

direction. Let us now examine as

a

1 '

N+ = a~a+

= (z*-aaz)

basis for N +

and

L

and

2

(~

N

,n

a_ -1

and

N

,

where

-

a•'•a -

N

, and 2

N+

= az;,

.

in more detail.

..

a

a" = (z-az;,) ' a+ = +

exp(-z*z)dxdy)

m

and

consisting of eigenvectors of

L:kk(N+k-N-k)

simplify to N k

= a*a k k

H = with

.

If we sum over all

L:k\l

(k)Nk

k

in

r

N+

and

N

are the

n .

Now let us return to the entire field.

p

aza ,

form a

The vectors

The corresponding eigenvalues of

positive integers

They factor

and

P "

k

L:kkNk

have spectrum

The

in

energy may be

r

these expressions

The operators and are

0,1,2,3, ...

interpreted as the number of particles with momentum

k

The

energy of a particle is related to the momentum by the relativistic formula

)J(k) =

~ •

- 61 -

It follows from the above analysis that the vectors i·

(a~

1

)

nl

'·

... (a~j)

vectors of

nj l

H

and

0

2 L (M,v)

form a basis for P .

adjoint operators.

consisting of eigen-

Thus in particular

H

and

0

P

are self-

The total energy and momentum may be thought of

either as total field energy and momentum or total particle energy and momentum.

~~(x)u(x)dx

Of course the field observables

and the

)

particle number observables

Nk

do not commute.

The non-linear interaction that has been most studied in this type of model is a certain function of the field.

As long as

it is an operator of multiplication by a real function

v

2 L (M,v)

is in

The operator H l 0

=0

exp(-V)

and H

preserving for all

<

oo

such that p <

for all

is in

00

is positive, self-adjoint and satisfies

0

(H +c)

It is not difficult to see that

.

V

L

c

-1

0

is positivity

0 •

>

The crucial ingredient in the study of this model is an infinite dimensional analog of the Sobolev inequality.

H = H

Write

0

+ V

= H0

+ U + W where

the positive and negative parts of and

exp(-W)

is in

the inequality

I

g if

LP(M,v) 2 1

dv


I

If

v

for all I

2

This reads

~

U

U

Then p

<

oo

log If I dv- II f

•

and

0

is in

Insert

ll~logll f

11

2 +II f

W~ 0

are

2

L (M,v)

g = -W/am

in

ll~logli exp (g) 11 2 •

It follows that -W

Thus if

m

clear that H

~

H

0

+ V

>

0 , then U

W


aH 0 + logll exp ( -W) is form small.

satisfies Condition A .

is a self-adjoint operator.

11

2 am

From Proposition 8.2 it is Hence by Theorem 7.11

- 62 -

One can also pose the question of whether

H :: H

is

+ V

0

essentially self-adjoint when restricted to the intersection of the domains.

That this is true for

It has been shown that to

H

H0

,

so Theorem 9.2

= H0

+

U

+

W

H

0

+ U

follows from Theorem 8.3 .

is actually operator small with respect

implies that it is also true for

W

NOTES Glimm and Jaffe (1969) gave an abstract theorem on self-adjointness of the operator sum based on the analysis of cross terms in The idea is to use the double commutator identity If

B

~

0 , then

problem is to estimate the double commutator.

TBT

~

0 , so the

There is an interesting

extension of the Glimm-Jaffe technique in a paper by Konrady (1971). The cross terms in the case of differential operators have been studied in a series of papers by Everitt

~nd

Giertz (1972)

).

The abstract Theorem 9.1 presented here is adapted from an recent paper by Okazawa (1973).

There is related material in a paper by

Gustafson and Rejto (1973). A first attempt to use arguments based on order to allow negative perturbations may be found in Faris (l972b). idea of Theorem 9.2 is due to Davies (1973).

The basic

Faris (1973) showed

that this technique leads to second order estimates.

The relation

to the maximum principle was pointed out by H. Weinberger. Protter-Weinberger (1967) book on the maximum principle.)

(See the In the

application to quantum fields this represents an alternative to, or at least a variant of the approach based on hypercontractive semigroups mentioned before.

- 63 -

The origin of these techniques is closely connected with the development of quantum field theory.

The model described here was

treated rigorously by Nelson (1966).

He obtained estimates which

showed that the negative part of the interaction is form small. part of the estimates involving inequality and to the

H

0

logarithmic

The

are related to a hypercontractive inequality used here.

The

development of the hypercontractive inequality is surveyed by Nelson (1973); the fact that it is equivalent to logarithmic inequalities was discovered by Gross (

).

His paper also treats the applications.

The construction of a fully relativistic model was undertaken by Glimm and Jaffe.

They have described their earlier work in several

surveys (1971) (1972).

The recent work on this subject tends to

emphasize space-time symmetry and give less empasis to the Hamiltonian. There are surveys of some of this in the notes edited by Velo and Wightman (1973).

§10

EIGENVALUES The conditions for the definition of the form sum of two self-

adjoint operators are too general to allow us to say much about the spectrum.

However a somewhat stronger but still not very restrictive

condition implies certain properties of the eigenvalues.

This

condition guarantees continuity of the eigenvalues in the coupling constant.

Also in the presence of an order structure in the Hilbert

space it allows an application of the Perron-Frobenius theorem to the form sum.

This gives a method for proving that the lowest eigen-

value has multiplicity one (the ground state is unique). Let space H

H

0

H

;:.

be a self-adjoint operator acting in the Hilbert

0

u

Let

u

Then

;:. 0

be another self-adjoint operator acting in Q_(H ) n Q_(U)

is said to satisfy condition B i f

is

0

Notice that condition B differs from condition A only in one respect.

In condition A

In condition B

Q_(H

Let where

\

H

0

)

required to be dense in Q_(H ) 0

(with respecc is dense in

0

H

H

implies condition A . and

is real.

\

lS

Q_(H )).

W be self-adjoint operators and set Assume that

This makes it plausible that if value of

n Q_(U)

it is required to be dense in

to the Hilbert space norm of condition B

0

, then

E\

H\

~

H0 + \W,

H\f\



is finite at any particular

is differentiable there.

-

If Q(H\)

=:

65 -

W is a small form perturbation of Q_(H

0

)

c Q(W) , so



that in this case the eigenvalue

H

then

,

0

is indeed finite. E\

f\

is in

So we expect

depends differentiably on

\

.

The situation is not so nice for large perturbations; the eigenvalues need not even be continuous in the coupling constant. However the next theorem shows that condition B

implies continuity.

First we need a definition of the increasing sequence of eigenvalues at the bottom of the spectrum.

Let

operator which is bounded below.

Let

En

=:

E .:;: E 1 2

~

E

dimension ;, n}

The sequence

It is always true that

H < E

En

Let

H

which are bounded below. Let

E

> 0

other hand, H < En g

H .:;: c

inf{c ~

3

...

H , then

is increasing. If

H .:;: E

E

is not

n

and

H

H ~ H , then

If

H~

Then

E

+

n

E

n

.:;:

En

has dimension ~ n .

E

has dimension < n .

On the

Thus there is a unit vector

En

Let

~

H

0

0

+

E

H

0

Since

Then

+ AU

be a self-adjoint operator.

H

0

~

space of dimension

H\

,

n

on which

necessarily contained in space of dimension

n

E on

~

as

Eon

E\n -

Let

E\n H

0

Q_(Ho)

contained in

.:;:

E

on +

\

-

E

> 0 E

) 0

n Q_(U)

Let

Let \

~

U ;, 0 0

and

0 There is a subThis subspace is

By condition B Q_(H

It

•

be a self-adjoint operator satisfying condition B

Proof:

n

be two self-adjoint operators

En.:;: .:;: .:;:

Theorem 10.2 .

H\

~

has dimension

n

in the first space which is orthogonal to the second space.

follows that

set

has

is an eigenvalue.

Proposition 10.1 .

Proof:

be a self-adjoint

has dimension < n

n

a limit point of the spectrum of and

H

there is a subon which

-

66 -

On the other hand, the subspace on which has dimension < n . with

~

EAn Since

Thus we may choose a vector



~



is finite, this is less than

sufficiently small

+ A 0

~

g

in

HA < EAn QCH

0

)

n Q(U)

E0 n + 2s + A . E

on

+ 3s

for

A .

For the application of the Perron-Frobenius theorem we need certain facts about invariant subspaces.

Let

A

be a bounded

operator acting in the Hilbert space

H .

onto

leaves invariant the subspace

some closed subspace.

if and only if

AP

~

Then

PAP •

A

commutes with

reduces If

PA(l-P) P .

be the projection

1 - P .

0 , so

AP

If ~

A

also leaves this

PAP + PA(l-P)

~

PA , that is,

In this case we say that the closed subspace

A . A

is a bounded operator which leaves a closed subspace

invariant, its adjoint variant.

~

P

The projection onto the orthogonal

complement of the subspace is invariant, then

A

Let

A*

leaves the orthogonal complement in-

Thus an invariant closed subspace for a self-adjoint

operator actually reduces the operator. Let

2 H ::: L (M, \l

)

•

A configuration projection is a projection

which is a multiplication operator.

The range of such a projection

is a configuration subspace. A configuration projection is multiplication by the indicator function

1

3

of some measurable subset

ScM

configuration subspace may be identified with

The associated 2 2 L (S) c L (M) .

- 67 -

H

Consider a Hilbert space bounded operator.

Then

A

2 L (M, \l)

=:

Let

A : H _. H

be a

is said to be indecomposable if it leaves

invariant no non-trivial configuration subspace. The following result is a Hilbert space version of the PerronFrobenius theorem.

Notice that the cone of positive elements may

have empty interior. Let

Theorem 10.3 self-adjoint operator. value of A

A .

Assume that

Assume also that

A

A

A : H -. H ~

a

where

be a bounded

a

is an eigen-

is positivity preserving.

is indecomposable if and only if the eigenvalue

a

Then

has multi-

plicity one and the corresponding eigenspace is spanned by a function u

which is strictly positive almost everywhere.

Proof:

Assume that

vector with

Au

A

is indecomposable and let Since

au

~

A

u

be a unit

is positivity preserving, it also

preserves the real functions, so we may assume that Then

a



~

"'



so

'

positive and negative parts of Let 2 L CS±)

then

2 L CS±)

in u

'

s±

~

0 •

Since

A

Hence either

u±



~

~

Let

a lui

I t follows that

u

be the set where

A lui

=:

vanishes. a

~

,

u

or

-u

Au±

au±

~

is in

Af ;,. 0

so

is indecomposable, either

be the

u±

f ;. 0

If

0

is real.

u

u+

~

0

is also

or

is strictly positive almost

everywhere. This reasoning applies to any real eigenvector. strictly positive functions can be orthogonal. multiplicity one.

So

But no two a

must have

- 68 -

Assume now on the other hand that almost everywhere, and that 2

2

L (S) c L (M)

commutes with

one, Pu

=u

in turn implies

Hence

=1

P

P

APu

But since or

~

P

u

0

a

A , and so

has multiplicity

almost everywhere, this

0

>

>

Let

reduces

Since

aPu

~

u

be the projection

is self-adjoint, L (S)

Pu = 0

or

A , and let 2

A

P

au , where

~

has multiplicity one.

be invariant under Since

A

a

Au

0 •

In order to apply this theorem to the resolvents of unbounded operators, it will be useful to consider some facts about projections which commute with self-adjoint operators. P in

is a projection, to say that V(H)

implies that

self-adjoint and

P

Pf

P

If

H

is an operator and

commutes with

is in

V(H)

commutes with

and

H ~

HPf

H , then

H

means that

PHf

and

If

P

self-adjoint operator and P

commutes with

H

c

Thus if

is not in the spectrum of

if and only if

is a self-adjoint operator acting in

P

commutes with

is

H

may be

simultaneously realized as multiplication operators in some by the spectral theorem for commuting operators.

f

L H

2

space,

is a

H

then

(H+c)-l. When H

L 2 (M,)J) , we say that

is

H

indecomposable if it commutes with no non-trivial configuration projection.

Thus a semi-bounded self-adjoint operator A ~ (H+c)-l

decomposable if and only if sufficiently large

c .

Proposition 10.4

Let

H

operator acting ln

H

Let

and

w

on A

M with and

u

~

0

w is form small.

~

""

2

L (M, \l)

u

and

Set

v

;;:

is in-

is indecomposable for

Let

H

0

~

0

be a self-adjoint

w be real measurable functions

Assume that

0

H

u + w and

u H

satisfies condition ~

H

0

+

v

Let

Fl 0

be the self-adjoint operator whose form is the closure of the form of

-

H

0

restricted to

then

H

Let P

is in

H

Q(H)

is indecomposable,

if and only if

g

P

in

for all

be a projection. Q(H)

f

and

implies g

in

Pg

Q(H) •

The only thing worth proving is that the condition on the P

commutes with

H and

is in and

HPg



Let

commutes with

H , then

Q(H) .

P

Since

In particular,for

g

0

g

~

Since by definition

P

implies

Pg

in

Q(H)

=

0

0

Q
Pg

is in

f

and

we have

and Pg

in

Q(H

Q
p

g

0

)

p

and that

P

in

V , it

in

Q(H) .

Q(H)

and

Q(H)

sense on

g

it follows that

0

is actually a

Assume that

H0

(H +c) 0

f

and

g

in

A0

commutes with Let

H •

g

If

Thus the relation

Let

operator acting in

f

must in fact be true for all

This implies that Theorem 10.5 .

Q(H)

is dense in

Q


Then

V(H)

Hence

for all

is continuous in the

continuous operator on

)

•

for all

0

is in

0

is in

be a configuration projection.



Pg

Q(H

=

g

commutes with the multiplication operator

=

This says that



Assume

=



follows that

~

H .

= PHg>.

Proof of Proposition:

in

H

=

and

forms implies

V(H)

H0

If

be a self-adjoint operator and

commutes with

Proof:

Hg

Q(H)

is indecomposable.

Lemma Then

69 -

>--

-1

0

be a self-adjoint is positivity

- 70 -

preserving for on

M

with

A

and

w

c

u

0

>~

0

Let

and

w .::

>

is form small.

u

w

and

0

be real measurable functions

u

Assume that

v

Set

u + w and

;::

H

satisfies condition

v

+

H

;::

0

fl 0

Let

be the self-adjoint operator associated with the closure of the form of

H

Q_(H) :: Q_(H ) n Q_(V)

restricted to

0

indecomposable.

If

H

~

b , where

is an eigenvalue of

b

has multiplicity

u

which is strictly positive almost everywhere.

Proof:

Let

=

A

(H+c)

By Proposition 8.1 shows that

A

one

b

A

-1

c > -b .

, where

saying that the eigenvalue

to show that

H

H , then

b

of

(b+c)

=:

-1

.

Proposition 10.4

Hence the eigenvalue

a

of

A

But this is equivalent to

H

has multiplicity one.

H0 = H0

is satisfied, then

and it is enough

Sometimes Proposition 8.2 and

is indecomposable.

0

A .::: a

Then

is positivity preserving.

has multiplicity one, by Theorem 10.3

B

is

and the eigenspace is spanned by a function

is indecomposable.

If condition

H0

Assume that

0

Theorem 10.4 give a convenient way to establish condition

B

and

1

and

indecomposability.

EXAMPLES Let H e 0

=

H

~e

'

Let in

1 L

= L 2 CIR,dx)

~

0

H

2 ll d 2l - --+x J 2l dx2

::

0

exp(-~x

2

Then

H

0

~

~

)

be a measurable function on

IR •

If

U

is locally

on the complement of a closed set of measure zero, then

condition

A

operator for B .

e(x) =

where U

and

is satisfied. A

In fact, if

at the origin.

~

0.)

(Thus

HA

= H0

+ AU

is a self-adjoint

However this is not sufficient for condition

4 U(x) = (l/x ) , then However for all

A

>

0

U

is locally in

1 L

except

the lowest eigenvalue of

-

3

is

71 -

HA

Also, the lowest eigenvalue of either reason condition

B

has multiplicity two.

L

(~,exp(-x

holds. that

2

)dx)

at zero. So for

must fail.

It follows from Proposition 8.2 that if l

A

Hence the eigenvalue is not continuous in

~2

(that is, <e,Ue>

U

is in

B

is finite), then condition

In fact, it is not difficult to see directly that it suffices U

1

be locally in

L

Now let us examine the continuity of the eigenvalue in the higher dimensional case. n

~

We say that

3 •

Let

K

K

has capacity zero if whenever

tempered distribution with support contained in
electrostatic energy codimension Let

H

2

(-to)

K

v = 0

v> , then

v

is a

and with finite A surface of

(or less) has capacity zero. L2('on,dx) u'

be a function on

~n .

,n~

3 ,an d

Assume that

complement of a compact set is satisfied.

-1

~n , where

be a compact subset of

K

Let U

u

~

0

is locally integrable on the

of capacity zero.

B

Then condition

(As a consequence the eigenvalues are continuous in

the coupling constant.) Proof:

We must show that the space of

support in the complement

of

K

Coo

functions with compact

are dense in

Q.(H ) • 0

It

is

sufficient to show that they are dense in the space of functions with
~u>

<

Let

oo

which vanishes on

v

be a continuous linear functional on this space

Coo

com

(Kc) •

Then

electrostatic energy supported on zero, v

0 .

v K

This proves the density.

is a distribution of finite Since the capacity of

K

is

-

72 -

Next we turn to uniqueness of the ground state in the higher dimensional case.

Here the condition is that the perturbation

doesn't erect a barrier between different regions. Let on K

H

0

~

-~

and let

U

~

0

be a function

which is locally integrable on the complement of a closed set of measure zero.

Assume that the complement of

H

Then the ground state of Proof:

~

H

0

+

U

K

is connected.

(if it exists) is unique. (For simplicity we

This is an application of Theorem 10.5 .

are considering a case where the perturbation is positive.)

fl 0

thing we have to check is that Let

S

Nn S

x

~n

in

Let

P

lS , where

S

near

x

such that

P

Sc

u = l

x

not in x .

the commutativity gives

~lS

lS

H0

commutes with

is continuous near

suppS u suppSc = !Rn

that

or

P = 0

S N

is

of

x ,

is a closed

P = l

P

~n .

is multiplication Let

Let

Then ~

then

~lsu

x , so Kc

u

The first

S

as c K . u

be a function in

is in ls~u ~

D(H

0

0 .

)

and so

But this

x cannot belong to is connected.

as

Since

aS ~ suppS n suppSc c K , this implies that

and

suppSc c K or

,

K .

Now we also use the assumption that

suppS c K

S

is the complement of

near

implies that

Then

is a measurable subset of

To see this, consider (Kc)

The support of

be a configuration projection.

assertion is that if

com

The support of

~n .

as= suppSn suppSc , where

Coo

Rn .

such that for all neighborhoods

has strictly positive measure.

subset of

by

is indecomposable.

be a measurable subset of

the set of all

The only

.

Since

K

is of measure zero, this implies

This completes the proof.

- 73 -

The theorems on uniqueness of the ground state also apply to the quantum field example. not until the limit

L -.

Thus there is no degeneracy - at least ro

is taken!

NOTES The proof of continuity of the eigenvalues in the coupling constant given here is based on a simple variational argument.

It

provides an alternative to an argument in Kato's book (1966; Chap. VIII, §3).

He shows that in the situation of Theorem 7.9 the strong

resolvent convergence is enough to imply continuity of the eigenvalues. The importance of the question of continuity of the eigenvalues may be seen from the case of a large perturbation for which the Taylor series expansion for the eigenvalues is defined.

The size of

the remainder term depends on the estimates which prove continuity. (There is a discussion of this in a paper by Simon (l97lc).) The example of discontinuity of the eigenvalues is due to Klauder and is discussed further by Simon (l973d). The application of the Perron-Frobenius theorem in quantum mechanics is due to Glimm and Jaffe (1971).

Segal (1971) showed how

this type of result could be deduced from an abstract perturbation theorem.

The theory was generalized to a von Neumann algebra setting

by Gross (1972) and to ordered Hilbert space by Faris (l972a). (Simon (l973e) showed that certain conditions related to indecomposability which are used in this work are actually equivalent.)

The proof

of the present version (Theorem 10.5) of the perturbation theorem follows another paper of Faris (l972b).

However the statement of the

- 74 -

theorem is slightly more general.

Simon remarked that this allows

the example to be formulated with a topological condition for indecomposability, and he gave a proof of indecomposability using Wiener path integrals. Gross

has given a different sort of criterion for

uniqueness of the ground state, based on analytic vector ideas. Gross (1972) has also proved an existence theorem for the ground state.

He is able to avoid strong compactness assumptions by the

use of order properties. The notion of capacity used in the example is not completely standard (due to the use of distributions instead of measures). However in the present context it is equivalent to the more usual formulation.

(This is discussed in a paper of Deny (1950).)

LOCAL CONDITIONS

§ll

The results on operator sums that are obtained in the context of Hilbert space theory all involve global conditions on the In order to obtain a result involving only local

perturbation.

conditions it is necessary to develop a more specialized theory.

In

this section the theory of distributions on Euclidean space is applied to give a particularly sharp result for Schrodinger operators. functions on

Coo

Coo (Rn) , the space of com

Consider

Rn

which

The elements of this space are called test

have compact support. functions. K c Rn

Let

coo com

functions in

Dpg

00

in

gn n

-

C (K) DPg

00

C (K)

be a compact set and

which have support in

is said to converge to

g

be the subspace of A sequence of elements

K in

00

if for every

C (K)

uniformly. Coo com

A distribution is a conjugate linear functional on that for every compact 00

is continuous.

C (K)

the value of

v

on

will be written

v



~

v

C~om ,



is said to converge to a for every test function

if

A distribution

vn

is in

g

is a distribution and

v

If g

such

KeRn , the restriction of the functional to

A sequence of distributions distribution

p

is said to be positive if

g

~

0

g .

implies

This notion of positivity defines a partial order on

0

the space of distributions. If

u

is a locally integrable function on

over any compact subset

K c Rn

= Jg*u dx , where

g

Rn , it is integrable

Thus it defines a distribution by

is a test function.

A basic lemma of

- 76 -

distribution theory is that this distribution determines the function (almost everywhere). If

v

is a distribution, its derivative ~v

(Thus The space

DPv

is given by

is contained in the space

Coo com

is defined by

S •

=

<~g,v>

.)

Recall that a

tempered distribution is a continuous conjugate linear functional on S .

Every tempered distribution defines a distribution, by restric-

tion.

It is a fact that

is dense in

Coo com

determines the tempered distribution.

S , so this distribution

Thus the tempered distributions

may be identified with certain distributions. the positive elements of

It is also a fact that

are dense in the positive elements of

Coo com

S , so the two possible notions of positivity of a tempered

distribution coincide. There are certainly locally integrable functions that do not determine tempered distributions - for example an exponential function.

However the fact that such functions are excluded can

actually be an advantage, as we shall now see. The space functions.

OM

is defined as the space of slowly increasing

(Slowly increasing means that each derivative is bounded

IX I • )

at infinity by some polynomial in operate on

s ,

is in If

Coo

s

by multiplication.

then

M

is in

distributions.

is also in

OM , then

In fact

The elements of

That is, i f


is in

OM OM

and

g

s


;

acts on the space of tempered < , so this corresponds

to a multiplication operator in the Fourier transform representation. For example

corresponds to

, so

- 77 -

leaves the space of tempered distributions invariant. 1-~

Notice that

has no inverse as an operator on the space of distributions. The inequality needed for the proof of the main theorem is

developed in the following two lemmas. Preparatory Lemma .

Fix

a real function such that ~s

(u) :;:. s

Proof:

1

0

>

E

u

and set

and

~u

~

s(x)

2 2 . /x +s

1 L

are in

Let

locally.

u

be

Then

(u)~u

If

u

is smooth we have

~s(u) = s"(u)(Vu) 2 + s 1 (u)~u:;:. s 1 (u)~u In the general case let such that

up-. u

up

be a sequence of smooth functions

~up-. ~u

and

s(up) - . s(u) ~s(up)-+ ~s(u)

1 L

locally in

Since L1

locally in

Hence

as distributions.

Now assume that the

up

also converge to

s 1 (uP)~uP-. s 1 (u)~u

We will show that

u

almost everywhere.

locally in

1 L

But The first

term goes to zero since

s'

is bounded, while the second term goes

to zero by the dominated convergence theorem. Since inequalities are preserved under distribution limits, the general case follows from the smooth case. Let

Lemma . 1 L

u

locally.

Proof:

Let

s (x) _. lxl E

Hence

~s

E

be a real function such that

Then

E

>

0

u

and

~u

are in

~lui :;:. sign(u)~u and set

uniformly and

(u)-. ~lui

s

(x) E

=

s~(x)-.

.;;z::z

Notice that

sign x

as distributions and

pointwise as s

1

E

(u)~u

-.

E-.

sign(u)~u

0

-

11

locally in

78 -

(by the dominated convergence theorem).

The result

follows from the previous lemma. Assume that

Theorem 11.1 -~ + U

Then

U ~ 0

(on the space of

Coo

L 2 (~n) .

and is locally in

functions with compact support)

is essentially self-adjoint. Proof:

It is sufficient to show that the image of the

-~ + U + l

with compact support under Let range. H

f

be a real function in (-~+U+l)f

Then

~ (U+l)f

~

0

~If!~ ~f sign f

(1-~)jfi ~ 0 .

~

2

1

(1-~)-l

2

which is orthogonal to this Thus

so by the lemma,

CU+l)f sign f ~ CU+Uifl:::. ifl

But

1

funccions

in the sense of distributions. L1

is locally in

is dense in

Coo

Hence

acts on the space of tempered distrilfl ~ 0 , f ~ 0 .

butions and preserves positivity, so

NOTES The classic treatise on distributions is by Schwartz (1966). The theorem of this section is due to Kato (1972).

The

technique is applied to local singularities by Simon (l973b) and by Kalf and Walter (1973).

(Kalf and Walter (1972) and Schmincke

(1972) had earlier results based on partial differential equation methods.)

Simon (l973c) has also given an application of the

technique to magnetic vector potentials. If

u

>,-

0

the operator sum

is only locally in -~

+ U

l

L

, then one cannot expect that

is essentially self-adjoin·t.

But Kato

(1974) has shown that his inequality may still be used to obtain information about the domain of the form sum

-~

+ U .

§12

COMMUTATORS Another approach to proving self-adjointness is through If a Hermitian operator almost commutes

commutation properties.

with a self-adjoint operator, this may force it to be self-adjoint. An elementary result of this nature is presented here. In addition, an example is given in which the fact that the Hermitian operator has certain commutation properties with respect to a unitary group implies that it is self-adjoint.

The result is

rather special, but is interesting in that it is not limited to second order differential operators. Theorem 12.1

be a Hermitian operator and

H

Let

V(N) c V(H)

(i)

c

and for some constant

in

f

and all

V(N)

±i{ -} <' c

(ii)

is essentially self-adjoint.

H

Without loss of generality we may assume that

Proof:

a positive

Assume that

self-adjoint operator.

Then

N

use the fact that range of Let

H

We

1 .

is sufficiently large.

lbl

be orthogonal to the range of

f

~

is essentially self-adjoint provided that the

is dense whenever

H- bi

N

particular we have from

(i)

H - bi


that

-1

Then in

f> - <(H-bi)N

-1

f,f>

=:

It follows from this and from (ii) that ±2b
-1

f>

this implies

=:

±i{
=:

-1

0 .

f,f> -
-1

f>}

~

c
-1

f>

If

2lbl > c ,

0.

-

80 -

EXAMPLE The following example is a self-adjoint Schrodinger operator which is not bounded below.

In order to emphasize the commutation

p. = -i ---

properties we write

and

ax.

J

= multiplication

qk

J

[pj,qk] = -iojk

These operators have the commutation relation

H = L 2 (IR n ,dx)

Let

measurable function on U(x) ~ -cx by

U

2

!Rn

operator.

Let

Coo H

c

U(q)

will be written as (of

2

=

u

Let

-t,

which is locally in

for some constant

coo com

space

= p

and

by

be a real

2 1

and satisfies

The operator of multiplication Thus

p

2

acting on the

+ U(q)

functions with compact support) is a Hermitian

be its closure.

The assertion is that

H

is

self-adjoint. To prove this, we choose an auxiliary self-adjoint operator N N in

~

~

The choice we make is

0

p

2

+ cq

2

~

0

N = p

2

+ U(q) + 2cq

We know from Theorem ll.l that

2

Clearly c

00

com

is dense

V(N)

We wish to apply Theorem 12.1 .

In order to verify hypothesis

(i) it is sufficient to show that the inequality in

Coo com

This follows

from the double commutator identity 2 22 H = (N-2cq )

=

2 2 2 24 N - 2cq N - N2cq + 4c q

2 4 N - 4cL: .q.Nq. + 4cq - 2cL:. [q.,[q. ,N]] J

J

J

J

J

J

2 2 2 N - 4cL.q. (N-cq )q. + 2cn .<_ N + 2cn J

J

J

To verify hypothesis (ii) it is sufficient to show that the estimate on the commutator holds for vectors in

coo com

This is

-

81 -

also a simple computation: ±l. [ H,NJ

±l. [ p 2 ,2cq 2]

~

±4c ( pq+qp )

~

4c ~ ( p 2 +cq 2 ) < 4 c ~ N

~

In the proof of Theorem 12.1 the resolvent of the auxiliary operator played an important role.

For the next theorem certain

unitary groups are used in place of the resolvent. H

Let x .

=

Proof: and

.

d P = -i dx

Let

q

and

= multiplication

by

Set exp(-~ist)exp(itp)exp(isq)

and

JJu(s,t)W(s,t)dsdt If

f

Let

1f

H and

is in

a s iat + 2

qW(u) = W(xu)

Theorem 12.2 .

Let

in the variables acting on

S .

H

p

q

A

We have

for all

u

Let

'

Assume that

and W(u)f

so

A

is in

pW(u) = W(1fu)

Then

2

W(u)f

lS in

s

be an elliptic polynomial A

0

?

as an operator

is essentially self-adjoint.

We may assume that

We must show that

t

-

pnW(u)f = W( 1f n u)f

= L 2 (IR , dx) .

and

Then

a = ias

X

L2

are in

2

S(IR ) , then

lS in

u

and

Hence

qnW(u)f = W( Xn u)f

Proof:

=

exp(~ist)exp(isq)exp(itp)

W(s,t) W(u)

2 L (IR, dx)

~

A ;:. l

.

Consider

g

in

H

with

g .lAS •

g = 0 .

= 0

for all

u

in

S , by the lemma.

= = JJ(Lu)(s,t)dsdt , where the same polynomial in

1f

the sense of distributions.

and

x •

This says that

L

But

L

=

is 0

in

- 82 -

Now we appeal to the regularity theorem for elliptic partial differential equations. function of 1

2

for all

s

n , so

and g

The conclusion is that In particular

t

is in

But then it follows that



n

n

and

p g

is a are in

q g

S



0

'

so

This result has an obvious extension to

0 •

g n

dimensions.

NOTES This type of commutator theorem originates with Glimm and Jaffe (l972b).

The present version is due to Nelson (1972).

The proof

and the application follow a paper of Faris and Lavine (1974). (Kalf (1973) has shown that the application may also be treated by partial differential equation methods.) The proof of Nelson (1972) actually gives a stronger result: ±i[H,N] ~eN ,

(the analytic vectors), and then

H

is essentially self-adjoint.

Yakimov (1974)

and

McBryan (1973)

also have improvements on the theorem.

The result on elliptic polynomials in

p

and

case of a theorem of Nelson and Stinespring (1959).

q

is a special The elliptic

regularity theorem used in the proof may be found in the book of Dunford and Schwartz (1963).

(The exact reference is to the last

sentence of Corollary 4 on page 1708.) Chernoff (1973) has given a rather different approach to selfadjointness questions based on finite propagation speed.

Kato (1973)

has used this to obtain a result for Schrodinger operators similar to Theorem 12.1 .

SELF-ADJOINT EXTENSIONS

Part III

§13

EXTENSIONS OF HERMITIAN OPERATORS In this section we review the standard theory of extensions of

Hermitian operators.

The main conclusion is that if a Hermitian

operator is not essentially self-adjoint, then either it has no self-adjoint extensions or it has infinitely many.

In the latter

case they are parametrized by unitary operators. Let A*

lS

A

be a densely defined operator with adjoint

a closed operator, its domain

the inner product Now let

A

v

=

VCA*)

Since

is a Hilbert space with

+ .

be a Hermitian operator.

continuous linear functional on

A* .

VCA*)

A boundary value is a

which vanishes on

V(A)

•

A

boundary condition is a condition obtained by setting a boundary value equal to zero. If

A

is a Hermitian operator, and

extension of specify Since

A 1 V(A)

specify

D

only if

c VCA ) c 1

A ::

is a restriction of

A* .

i f and only i f

VCA ) c V(A~') 1

by imposing a set of boundary conditions.

be a Hermitian operator and set

.

Thus to

as closed subspaces, it is possible to

V(Af')

ker(A*+i) D +

A 1

is a self-adjoint

it is enough to specify the subspace

VCA ) 1

Let and

A , then

A1

D

::

D +

Then

A

If

0

D

::

A

0

D = ker (A'~-i) +

is essentially self-adjoint if and is closed, then The spaces

the deviation from self-adjointness.

D+

A and

is self-adjoint D

thus measure

They are called the deficiency

spaces, and their dimensions are the deficiency indices of

A .

-

If

e

is in

Hence for

e1

D

and

+

and

e

in

2

From this we see that Proposition 13.1 .

f

D

+

Let

is in

D+

.l

and

D f

and

1

A*(e+f) = i(e-f)

, then f

in

2

we have

D

in the norm of

D

A

84 -

be a closed Hermitian operator.

Then

V
Let

g

+ = 0 0

V(A)

be orthogonal to

Thus

•

for

f

E

v (A)

'

V
in .;,

so

A* . Then and

is in

A g

V(A * ) = V(A) 8 ker(A'~2 +l) , by the projection

theorem. Now consider l

g

,..

= 2i((A'+i)g1 ker(A '

that

Let B

2

A

Hermitian:

f1

= D+ 8

by

B(g,h)

and

Corollary

Proof:

g

+l)

f

2

in

We may write it as is in

D± , this shows

D

Define the sesquilinear form

= - .

and D

.

Since

= -B(h,g)*.

B(g,h)

B(u 1 +e 1 +f 1 ,u +e +f ) 2 2 2

where

ker(A'

(A'~-i)g)

Notice that if and

.., 2

in

be a Hermitian operator.

V
on

+l)

g

Then

B

is skew-

This form is called the boundary form. are in

and

, then

= 2i(<e 1 ,e 2 >-)

All boundary conditions are of the form is a fixed element of

B(g,h)

=0

,

D 8 D +

It follows from the Riesz representation theorem that the

most general boundary condition is of the form fixed element

f

of

D+ 8 D

But

v = 0

for some

-

85 -

'/* v=+

0 .

Thus we may set

g

It is worth noting here that there are two possible norms on the deficiency spaces

V
D± , namely that of

original Hilbert space.

and that of the

However they differ only by a constant

factor. Proposition 13.2 .

be a Hermitian operator.

Proof:

Consider

VCA ) 1

Then

V(A)

A"''<e+f) = i (e-f)

A

A

Since

D

to

A 1

V(A)

8

Then

is self-adjoint i f and only i f

But i f then

e

and

e

A 1

be

A 1

D+ = H

and

be a Hermitian extension of range(A -i) = H 1

A and

which amounts to the requirement that and

(A -i) (e+f) 1

if and only if

Let

H •

range(A -i) =>D 1 +

Let

D

is closed, range(A-i)

'

But since

and the graph of a unitary

8 D

range(A +i) = H 1

is in

e + f

is self-adjoint if and only

range(A+i) Al

such that

B(e+f,e+f) = 0

Then

VCA 1 ) is the direct sum of

Proof:

in

D

be a closed Hermitian operator.

A c A c A* 1

D+

be an

B(e+f,e+f) = 2i
Let

operator from

f

and

is Hermitian when

'

an operator with

D +

A1

and the graph of an isometry from

to a linear subspace of

+

in

e

Al

Theorem 13.3 .

D

Let

is Hermitian if and only if

Then

is the direct sum of

a linear subspace of

if

A

A c A c A* 1

operator with VCA 1 )

Let

range(A +i) 1 f

are in

-2if and

and f

::>

D+

D_ and

D

with

e + f

in

V
(A +i)(e+f) = 2ig , so this is true 1

can be arbitrary elements of

Thus the assertion follows from Proposition 13.2 .

D+

and

D

- 86 -

Theorem 13.4 .

Let

A

be a closed Hermitian operator.

not self-adjoint, then either

A

If

A

is

has no self-adjoint extensions or

infinitely many self-adjoint extensions, according to whether the deficiency indices are unequal or equal. Proof:

D

Assume

D

and

+

unitary operators from

D

+

are not both zero. to

Then there are no

, or infinitely many, according

D

to whether the dimensions are unequal or equal. Corollary . extension.

A

Proof:

Let

A

be a Hermitian operator with a unique self-adjoint

Then

A

is essentially self-adjoint.

also has a unique self-adjoint extension, and

A

closed, so Let

with

A c A c A* 1

Proof~

A

Since

*

A c Ai* c A

be a Hermitian operator. Then

B(g,h) = 0

and

AcA c 1

h

g

in

P.* , it follows by taking adjoints that V(A*) , then l

It follows that

h

is in

V(A~;

B(g,h)



for all

g

= - =

Proposition 13.5 .

Let

A

in 0

l

VCA ) , that is, if and only if 1

for all

g

in

VCA ) . 1

A, so that

B(u,h) = 0

for all

u

Let

A 1

VCA ) = V(A)0 UcV(A*), 1

is the graph of a unitary operator from

are given by

= A*h .

if and only if

the boundary conditions defining the set of VCA ) 1

A*h

be a closed Hermitian operator.

be a self-adjoint extension of U

is in

h

VCA ) 1

is in

=

be an operator

Al

if and only i f

l

h



where

V (A'~)

is in

for all

Let

Thus i f

.

is

is self-adjoint.

Lemma

V
A

h in

D+

to

D

which belong to U .

Then

87 -

-

The set

Proof: VCA ) 1

u

and

and only i f for all

u

is in

lS

VCA ) 1

in

g

in

is in

VCA ) 1

'

if

that is,

u

in

Let

W be a Hilbert space and

Let

Proposition 13.6 . conjugation.

for all

B(u,h)

0 = B (g, h)

u

h

Thus by the lemma

f

where

g = f + u

consists of all

VCA ) 1

A

T : W-- W be a

be a Hermitian operator which is real with

respect to the conjugation

A

Then

T .

has a self-adjoint

extension. Proof:

T

The conjugation

D

, so the deficiency

~+ = [O,oo) .

We will always con-

maps

D

onto

+

indices are equal.

EXAMPLE Let d

sider L

2

dx

H = L 2 (~+,dx) , where

such that

f'

is also in

L

2

2

.

with the additional restriction that

d2 with no restrictions at zero. 2 dx is not Hermitian, in fact the boundary form is

The adjoint A*

that B(f,g)

deficiency spaces

e~(O)

A*

is

1 = - = f(O)*g•

The kernel of

where

d defined on f A - ---- dx 2 ' f(O) = 0 and f' (0) = 0 .

Set

is a closed Hermitian operator.

A

Then

f

to be defined on absolutely continuous functions

A* - z D±

(0)

-

is spanned by

(corresponding to

Notice

f' (O)*g(O) l

exp(-(-z) 2 x)

Thus the

z = ±i) are spanned by

are exponential functions satisfying

and

e±

-

It follows that u = e+ + exp(ie)e_ equivalent to

cos

B(e±,h)

88 -

= h' (0) + exp(±i~)h(O)

The boundary condition 8

2

h '
B(u,h)

Set is thus

0

o

Thus we see that the most general boundary condition defining a self-adjoint extension is of the form and

b

real (and not both zero).

ah(O) + bh' (0)

=

0

with

a

From this point of view there is

no reason to choose one extension over any other.

NOTES The theory of extensions of Hermitian operators may be found in the book of Dunford and Schwartz (1963).

THE FRIEDRICHS EXTENSION

§14

In this section we see that a Hermitian operator which is Among these there

bounded below always has self-adjoint extensions. is a maximal one,

the Friedrichs extension.

Friedrichs Extension Theorem 14.1 . acting in a Hilbert space and

f

Proof:

II f II~

in

g

=

be the completion of

V

=



Q -H

since

g

in

Q

Hence if

Since

v

is dense in

1f

Q

the functional

V(A)

is continuous in the norm of

is continuous, we have

in

f

is injective, so that we

We must show that

The point is that for fixed ~

extends by continuity to

Qc H .

may identify

f

with respect to the norm

V(A) V(A) c H

The inclusion

f>

1 : Q- H

a map

, defined for

Then the form

H

, is closable.

V(A)

Q

Let

be a positive operator

A

Let

0

'

then

this implies

= <1f,CA+Ug> for al

v

v f

= =

0

Thus

H .

Hence

0

\

v

in

g

for all

:

Q-H

is injective. Thus any positive Hermitian operator

A

acting in

determines a closed densely defined positive form. hand, for any such form the associated operator This self-adjoint operator of

AF

AF

H

On the other is self-adjoint.

is called the Friedrichs extension

A . The Friedrichs extension of a positive Hermitian operator is

thus a positive self-adjoint operator.

One can define the Friedrichs

extension of an arbitrary semibounded Hermitian operator.

If

A

~

d

is such an operator, then its Friedrichs extension is a self-adjoint

-

operator

~

AF

90 -

d , with the same lower bound.

Notice that if

already self-adjoint, then the Friedrichs extension of

A

A

is

is

A

itself. The Friedrichs extension is maximal among semibounded selfadjoint extensions. Proposition 14.2 . and let

AF

Let

A

be a positive densely defined operator

be its Friedrichs extension.

Let

A1

self-adjoint extension which is bounded below. fact, the form of Proof:

Let

-c

A

extends the form of

1

QCA )

the same norm

1

in the norm

V(A)

Q(AF)





Hence

1

=

for all

f

AF .

In

A

By

1

is the

, or what is the same,

= .

is the completion of the larger space

1

~

A

be strictly less than the lower bound of

the completion in the norm hand

Then

AF

the construction of the Friedrichs extension, completion of

be any other

in

Q(AF) c QCA ) 1

Q(AF)

On the other

V CA 1 )

:::>

V (A)

in

and

.

EXAMPLE Let f ( 0) = 0 extension

H

= L2 (1R+,dx)

and

f'

is in

A

Then

f ' ( 0) = 0

with boundary conditions

be A

~

0 , so

A

has a Friedrichs

AF

Notice that completion

and

Q of H

<JLf ~f> dx 'dx

for all

f

in

V(A)

thus consists of the

f

in

H

V(A)

and such that

f(O)

=0

is not preserved in the completion.)

(The condition

.

The

such that f' (0)

=0

- 91 -

To find the self-adjoint operator

Q is

continuous on f(O) = 0

satisfies in

H .

Thus

AF , we ask for which Since every

H

f

f Q

in

the only additional restriction is that d2 2

AF

with the boundary condition

f(O)

in

f"

=

is

0 .

dx

It is interesting to see what self-adjoint operator is associated with the form also in

H

continuous on

<_£_f _i_f> dx 'dx

defined on all

We ask for which H

Since

it is necessary that

f' (0)

f


=0

f

in

H

such that

f'

among these is >

= - - g(O)*f'(O) , we see that Hence the operator is

d2

dx the boundary condition

f' (0)

is

2

0

The Friedrichs extension is treated briefly in Nelson (1969) and more extensively in Kato (1966).

with

EXTENSIONS OF SEMI-BOUNDED OPERATORS

§15

is a semi-bounded self-adjoint operator with

A

of

AF

extension

The Friedrichs

be a semi-bounded Hermitian operator.

A

Let

The purpose of this section is to classify

the same lower bound.

which are bounded below by

A

of

A 1

all self-adjoint extensions

We will find that they are parametrized by

some fixed constant.

This has the advantage that the closed positive

closed positive forms.

forms are a partially ordered set. We consider for simplicity the case self-adjoint extensions A

deficiency space of Let

~

A 1

on some linear subspace of of

Let

H •

K

positive self-adjoint operator acting in

function is

[o,oo]

with values in

measure space

+oo

s- 1

then so is

ever neither

s

Theorem 15.1

= ker G

K

S

is the form of a Hence by the spectral

S

is a closed positive form,

Note that

where-

SS-l = l

are zero.

nor A

Let A*

(The set on which the

[o,oo] .)

be a Hermitian operator such that

A

~

d

There is a bijective correspondence between A 1

positive self-adjoint extensions forms

be the closure of the domain

(In fact both are isomorphic to the form defined

by a function with values in

N

and classify all

K~ .)

corresponds to

It follows from this that if

Let

0

is isomorphic to the form defined by a function on some

S

theorem

>

be a closed positive form

S

Then by the representation theorem

S .

d

ker A* .

be a Hilbert space and let

H

~

It is useful to define the

0

N

as

A

defined in

of

A

N , given by -1 A1

-1 -1 . AF + G

and positive closed

>

0 .

93 -

-

is also a positive

defines a positive closed form, so

-1

AF

~

d

-1

~

A 1

Since

closed form.

closed form. A 1

Thus since

~

Since and

AF ,

AF

-1

~

d

>

-1

-

= range A c range A , A A = l 1 1 1

AF

are both extensions of

=

-1 -1 AF + G

N~

is also a -1

A1

is never zero, neither is

Since

N .

be a positive closed form in

G

-1 is positive and bounded, A 1 closed form.

on

A

is a positive closed form which is zero on

G-l

0 , so

is also a positive

AF

N

On the other hand, let Then

But

Hence

is bounded.

Then

A .

be a positive self-adjoint extension of

A 1

Let

Proof:

Since positive

a

is

, so

a densely defined positive closed form, that is, a self-adjoint operator. N~

G-1

Since

N~ c range Al

and since

'

N~

is zero on

'

this is

'

l

Corollary .

Let

Assume that

A A1

extension

A

on

on

= AlAl

N~

Hence

be a closed Hermitian operator with

Al

A

of

Take

G = 0

satisfies

A

~

d

>

0

Then there is a self-adjoint

is not self-adjoint.

which is minimal among all positive self-adjoint A 1

extensions, and zero is an eigenvalue of

Al

= AlAF

A

extends

Proof:

-1

-1

--1 A A 1

on

-1 -1 Al = AF

not self-adjoint, N ¢. 0

N on

G-1

Then

N~

and

on Al

0

N

'

so the extension N

on

Since

is

A

and so zero is an eigenvalue.

Notice that the Friedrichs extension corresponds to the choice G

oo

•

If

A

is a closed Hermitian operator with

self-adjoint, then for every

c

<

extension

of

A

A - c

which is bounded below by

~

m

which is not Thus we

m , Nc

may apply the above construction to

A

to produce a self-adjoint c

and which has

c

as

-

an eigenvalue.

94 -

It follows in particular that if a semi-bounded

Hermitian operator has a unique semi-bounded extension, then the operator is essentially self-adjoint. Pro:12osition 15.2 A

d > 0

;.

Proof: u

in

V(A)

u

=

be in

u

is also in But

N

Q(AF)

this says that

is dense in

V(A)

Let

N = ker A1'

Let

and

Then

let

G

Let

in

v

, so this implies that

Q_(AF)

and

We have

the norm ker A

1


= ker G

f + u

u
in -1

1

'

h>

.

AF

where

w

g

and

is in

u

'

is in

g

N

in the sense that Q(Al)

'

with

g = f + u

Q(G) -1

h> = +
g = A -l h 1

-1

h

h> for all

-1

Q_(Al ).

, f

is orthogonal to

h

-- A-lh F , an d 1

u

=

=

g + w

representation in fact holds in general.

=

f + (u+w)

Then

+

are restricted to be orthogonal to

ker A , then 1

in

in the completion with respect to

Assume now that

and set

G

and

and the form equality becomes

Notice that If

-1

1

A ;. d > 0

be the corresponding closed positive form in

This continues to hold for all

=

V(A)

for all

= 0

Q_(AF)

in

v

be a positive self-adjoint extension of

Al

is the direct sum of

Al

Proof:

g

fol' all

be a Hermitian operator such that

A

= + f

= 0

Then

'

Q
Then

0 .

Theorem 15.3

A

be a Hermitian operator such that

A

N = ker A*

Let

Let

If

Let

ker A 1

, and so the

95 -

-

EXAMPLE Let

H

L 2 CIR+,dx)

;:

g ( 0)

boundary conditions

k > 0

Let ;:

0

g' ( 0)

'

d2 --2 + k2 with the dx We wish to find all

A

and 0

;:

;:

positive self-adjoint extensions. 2

Let

ker A* .

N

Since

is spanned by

conditions, N

d + k 2 with no boundary ---2 dx Thus N is one-dimenc exp(-kx)

A*

sional. A1

The form of

be a positive self-adjoint extension.

A1

Let

has the representation

i

=f + u

g

where

=

with

with

2 + k ~

0

i

+

~

oo

=

0

and

and

= c exp(-kx) .

u(x)



=

ilc1

2

f + u

g

for some constant

may be made explicit by writing

= g(x) - g(O)exp(-kx) + g(O)exp(-kx)

u(x) = g(O)exp(-kx).

Thus

It is not difficult to compute that the form of

=

2 2 + k + (i-k)lg(O) 1 .

The operator

+ k 2 + h(O);:((i-k)g(O)-g' in

h .

d

2

---- + k 2 dx g'(O) = (i-k)g ( 0) .

Thus

condition

The case

is

A

1

i

=

oo

i

< k , then

2

acting on

(0))

is A

1

is

A

1

g(O)

= 0 .

If

i

is continuous

with the boundary

g

gives the Friedrichs extension

with the boundary condition while if

A1

such that

g

defined on all

Here

•

The decomposition g(x)

f(O)

~

k

AF then

has an eigenvalue strictly less than

k

2

-

96 -

(This shows, incidently, that the Friedrichs extension

AF

is not

characterized by the fact that it has the same lower bound as

A.)

EXAMPLE 2

H = L2 (IR + ,dx)

Let

which vanish near zero.

Let Let

-~ + a2 , defined on functions

A

0

A

A

0

dx

Then

x

d2

= ----2

A*

+

dx

a 2 x

with

no boundary conditions. The behavior of these operators depends on the value of the parameter l)

a . 3

a

~

4

We distinguish three cases:

2)

l 4

3)

a <

~

3 a <

4

l

-4

Cases l and 2 are distinguished by the fact that

A

is positive.

This may be seen by noting that i f f vanishes near zero, then 2 d l 0 ~ <(~-~)f (~-~)f> = Thus A ::::. 0 so A 'l- 0 • 2 dx 2x ' dx 2x 0 ' ' dx 2 4x It follows that the Friedrichs extension AF ~ 0 exists. Thus A has a canonical extension which is a positive self-adjoint operator. Case l is distinguished by the fact that

A

is already self-

adjoint; there is no choice at all of self-adjoint extension. may be seen by computing

N

= ker(A*+l) .

This

The equation

-f" + ~f + f = 0 may be solved in terms of Hankel functions, but x2 there is a more elementary way to see what is happening. When x ~ the solutions behave like in

L

2

c exp(x) + c exp(-x) 1 2

at infinity are of the form

hand, such a solution behaves like

.

oo

The only solutions

f(x) ~ c exp(-x)

.

On the other when

x

~

0.

-

Here

and

rl

then

rl

and

: ;,. 2

L2

is not in

f

In case 2 r

;:

rl

Hence

l

=2

are the solutions of r (r-l) ;: a r2 l Since X is not in r2 ~ -2

['2

3

and so 3

'

>

2

N

l

>

rl

2

=

L2

is in

L2

4

'

near zero,

>

f(x)

l

(except when

-2 d

'V

,

l

X~

a

,

+ d x 2 log 2

X

;:

l

-4 '

near zero).

is one dimensional, and there are many

N

'

3 a ::;,.

If

0

> r2

and the solution

f(x)

97 -

self-adjoint extensions.

The Friedrichs extension is characterized rl by the fact that functions in its domain vanish at x = 0 like x A similar analysis in case 3 shows that the solutions of rl r2 (A*-i)f = 0 behave at the origin like d x + d x where r 1 2 1 r

2

both satisfy

of solutions and

Re r A

=

~

and

Hence there is a one dimensional space But what is worse, there

is not self-adjoint.

is no natural way to select a self-adjoint extension. adjoint extensions are characterized by a phase

The self-

exp(ie)

, and this

is an additional piece of information needed to specify the operator.

EXAMPLE It is worth looking at an example in and consider the operator

A

on functions which vanish near the origin. radial distance from the origin.

o~ I

c2...+n- 2 HII 2 = <2...f 2...f> ar r ar 'ar

positive.

Let

dimensions.

n

= -to

0

Here

-

(n-2)

r

=

2

1

acting

4

2 r

lxl

is the

Since 2


1

'2 f > r

~

Hence it has a Friedrichs extension

0

AF

, A

0

is

which is also

positive. If dense in

n

~

2 , then the functions which vanish near the origin are

Q(t,)

,

so

AF

~

-to

However notice the curious fact that

- 98 -

when

n

l

functions in

Q(AF)

so for functions in

n :: l

In fact, when

, it is false that

vanish at the origin, which is not necessarily d2

Q(--)

dx

2

The same reasoning shows that if n 2 (n-2) l function on !Rn with W(x) ? 2 4

~

2

,

then

w is a real

and if

r

In other words, when

n

?

a shallow attractive potential doesn't

3

produce negative eigenvalues.

(However when

n

=l

or

2

the argument

fails; an attractive potential always produces a negative eigenvalue.)

NOTES The theory of extensions of semi-bounded operators is classic work of Krein (1947).

The theory was extended by Birman (1956).

Nelson (1964) found a canonical choice of boundary condition for l a that makes it a non-self-adjoint operator. The a < -4 x dx relationship between Nelson's extension and the self-adjoint extensions

---2 + -2 '

is discussed by Radin (

).

There is a rigorous general theory of the relation between boundary conditions and self-adjoint extensions of ordinary differential operators.

This Weyl theory justifies the procedure of examining the

behavior of the solutions at the two end points separately. a good discussion in the book of Dunford and Schwartz (1963).

There is

MOMENTS

Part IV

§16

ANALYTIC VECTORS If

A

moments of

is a self-adjoint operator and

f

A

= 1,2,3, . . . .

are the numbers

, n

hope that if one knows the moments of this would determine

A .

A

is a unit vector, the One might

for sufficiently many

f

This procedure might also give a way of

showing that a Hermitian operator is essentially self-adjoint.

In

fact this is so, but it is necessary to make precise what is meant by sufficiently many Let

be an operator.

00

space V(A)

A

f

C (A)

The space of

consisting of all

for all

= 0,1,2,3, ...

n

then the moments The space

Cw(A)

consisting of all

The space

f

f

B(A)

in If

such that

H

f

Coo vectors for

is a

Anf

If

is in A ,

are all defined. 00

of analytic vectors is the subspace of such that there exists constants

a

C (A)

and

00

of bounded vectors is the subspace of

I!Anf!l

b

C (A) abn .

.$

B (A) c Cw (A) c Coo (A) .

A

is a self-adjoint operator, it is clear from the spectral

theorem that A

is the

Coo vector for

consisting of all vectors which satisfy an estimate Clearly

A

B(A)

is dense in

(The converse is also true: if

H •

is a closed Hermitian operator and

is self-adjoint. h = (z-A)g range(z-A)

' ::>

where B (A)

range(z-A) = H

In fact, i f g = (z-A) is dense in

is in

h -1

B(A) B(A)

zL:

H

and since

'

and

(A/z)nh

h =

n

is dense in

then

H

I z! > b

'

then

Thus A

is closed,

HoweveJc> we will see later that there is a better

A

- 100 -

in

A

H , then

A

Let

is self-adjoint.)

be a Hermitian operator acting in

in

D(f)

be the closure of

M(f)

Akf , k = 0,1,2, ...

Then the restriction

H

M(f) .

is a Hermitian operator acting in

A : D(f)-- M(f)

In fact,

This restricted operator has a self-adjoint extension. let

. by be deflned

T : D(f) -- D(f)

=

* kf , [ckA

T

Then

is a

is said to be a determining vector if

f

is essentially self-adjoint.

A : D(f) _. M(f)

and

A

conjugation which commutes with The vector

TLckA k f

be the extension by continuity.

T : M(f) -- M(f)

f

and consider

H

be the linear span of the

D(f)

Let Let

is dense

Cw(A)

is a closed Hermitian operator and

A

result: if

(This is equivalent

to the uniqueness of the self-adjoint extension.) be a Hermitian operator acting in

A

Let

Proposition 16.1 .

D(f) , where

Assume that the union of the vector, is dense in Let

Proof:

in

u

and a

be in

h

H •

determines, there is a u .

dense in

D(f) h

such that

f

But since

h .

approximates

(A±i)g

range A±i

This shows that

H

Proposition 16.2 .

Let

analytic vector for

A

Proof:

approximates

u

approximates

(A±i)g

Hence

in

g

f

Then there is a determining vector

such that

D(f)

is a determining

is essentially self-adjoint.

A

Then

H •

f

H

Let

f

A : D(f)-- M(f)

be in

A

Then every

be a Hermitian operator.

is a determining vector for Cw(A) •

A •

The Hermitian operator

has a self-adjoint extension

A 1

acting in

M(f) •

- 101 -

Since of

f

is in

Cw(A) , there is an

exp(itA )f = exp(itA)f 1 Let

g

be in

verges for

ltl

Then

Since

is uniquely determined by A , so

A

acting in

Theorem 16.3 .

Let

If

A

ltl

<

=

E

A 1

•

exp(itA)g

is dense in

Hence

also con-

M(f) , exp(itA ) 1

is uniquely determined by

is essentially self-adjoint on

be a Hermitian operator.

set of analytic vectors, then Proof:

such that the expansion

exp(itA )g 1

D(f)

A

M(f) A

0

converges for

D(f) .

< c:

>

E

A

If

A

D(f) .

has a dense

is essentially self-adjoint.

has a dense set of analytic vectors, then

dense set of determining vectors, and so

A

A

has a

is essentially self-

adjoint. The rest of this section is devoted to the relation between the analytic vectors of two different operators. analytically dominates

X

if

We say that

Cw(A) c Cw(X) .

In order to get

analytic dominance, we will need estimates on commutators. and

A

are operators, we write

of taking the commutator of

A

(adX)A with

XA - AX

X

A

If

X

for the operation

The following theorem

requires second order estimates on commutators. Theorem 16.4 constants

a

Let b

and

A

and c

X

be operators.

such that for all

I Xu II~

eli Au II

and Then

A

analytically dominates

X .

Assume that there are u

in

V(A)

- 102 -

Proof: rr (u)

We first bound of

n

II Xnu II

in terms of a linear combination

2

II Au II, !IA ull, ... , IIAnull

In order to apply the commutator

bounds, it is convenient to first prove that will follow from this that Since

and try to bound the sum.

Define

ell AXju II Hence

rr (u) n

~

we may write

This gives

" 0 (u) = II ull

~=l[~Jabkk!rrn-k+l (k)

for

rrj+l(u)

rrn(u)

n

inductively by

err (Au) + I n

rrn+l (u)

II Xnu II ~ rr (u)

,n Lk=O [n) k «adX) k A)X n-k u ,

XnAu

It

0,2, ... ,n-l

j

is the desired bound.

'

and

I t follows that i f

then

ell AXn II ~ rrn+l (u)

The only difficulty is that it

is defined in terms of a complicated recursion relation. The recursion relation for

l

-,rr (u) n. n

involves a convolution, so tn it is convenient to introduce the transform ~(t,u) -,rr (u) . n n. n

=I

The relation then becomes d

dt~(t,u)

where

=

v(t) = abt(l-bt)-l .

multiplication by Write d

v(t)

c~(t,Au)

+

d

v(t)dt~(t,u)

,

Thus the convolution is replaced by

and the shift by differentiation. Then

~(t,u)

d

when l dtpr(t) = c Pr-l (t) + v(t)dtpr(t) . Since pr(O) s -1 it follows that p (s) = c r (1-v(t)) p Ct)dt Since r J0 r r- 1 . c K(s)r this may be solved to glve pr(s) = r! , where s r r 1 ds) = (l-v(t))- dt Hence ~(t,u) = I c K;s) IIArull 0 r.

I

r

;;. l

,

- 103 -

If we put together the information we have up to now, we obtain the estimate

=

v(O)

Since

s , and hence

the right hand side converges for sufficiently small u

Thus

so does the left hand side.

in order to conclude that

restricted to

A

and

X

, it

Cw (A) c Coo (A)

Since

is sufficient to obtain the estimates for Coo (A)

is a self-adjoint

A

invariant.

Coo (A)

leaves

X

operator and

Cw(X) .

is in

One situation which may occur is that

Cw(A)

is in

u

Hence if

s-- 0 .

as

0 , K(s) - - 0

Cw (A) c Cw (X)

We take such a This theorem says

situation as the setting for the next theorem.

that first order estimates on commutators imply analytic dominance. Theorem 16.5 .

H

Let

~

be a self-adjoint operator.

l

Coo(H)

a Hermitian operator which leaves X~ cH

H

(adX)nH ~ abnn!H

and

Q c H c Q'~

Let 2

,

be defined by

Then

v II

Hermitian and

(adX)nH

H : Q -

Let w = Hu

u

be in

Then

Q'~

II u IIQ = II H2 u

II

and

is an isomorphism.

Since

X

is

is either Hermitian or skew-Hermitian, it

f o ll ows f rom t h e h ypot h eses t h at are operators bounded by

Coo(H) .

as quadratic forms on

_, II v IIQ* = II H

Assume that

invariant.

X •

analytically dominates

Proof:

be

X

Let

c

and

X :

ll ~

abkk!

--

ll ~

*

an d

(ad X) kH :

ll ~

~ --

respectively.

Cw(H). We wish to estimate

IIXnull

Set

and write

The point is that it is much easier to estimate

II XnwiiQ'~ .

Simply

ll ~

*

- 104 -

apply the previous theorem to the operators

Q* in

The conclusion is that

w

X

II [xn ,H- ] w IIQ

that

Write [xn ,H]

q

so that

b

<:;

yq

Let

and

s

acting

>

0

q .

be a small We will see

qnn!

<:;

[xn ,H- 1]

= -H-l [Xn ,Hj H-l

Xn-kH-1 = H-1 Xn-k + rLX n-k ,H -1]

and

This gives two terms to be estimated.

The estimate is done by induction. 1

II [xm ,H- ] w IIQ

<:;

qmm!

II [ Xn ,H -1] w IIQ

y

for

m

<

n .

Assume that

Then

<:;

n [nJ Ik=l k ab k k! (rs n-k (n-k)! +q n-k (n-k)!)

~

q n!y(l-y)

n

-1

a(r+l)

<:;

n

q n!

is chosen sufficiently small. These estimates combine to give a bound for

that

y <:;

X

= In [nJ ( (adX)kH)Xn-k . Insert the second identity in the k=l k The resulting factor on the right may be expressed as

first.

if

acting in

r s mm.'

<:;

The remaining term is more difficult.

1

H

is an analytic vector for

Q* , that is, there is an estimate

constant and choose

and

u

is in

Cw(X)

IIXnu~

which proves

.

NOTES The theorem on analytic vectors and self-adjointness is due to Nelson (1959).

The proof given here follows Simon (1972); it is a

modification of techniques of Nussbaum (1965). There is a notion of quasi-analytic vector which generalizes the notion of analytic vector.

Nussbaum (1965) has proved a stronger

theorem relating quasi-analytic vectors to self-adjointness.

- 105 -

The original theorem on analytic dominance is also due to Nelson (1959).

Goodman (1969) has shown that the first estimate in

the hypothesis may be weakened.

The other theorem on analytic

dominance, which allows form estimates in the hypotheses, is due to Sloan (

).

Sloan

also be weakened.

has shown that his first estimate may

- 106 -

§17

SEMI-ANALYTIC VECTORS Let

A

be a Hermitian operator.

vectors consists of the

IIAnfll

~ abn(2n)!

f

The space of semi-analytic

00

in

C (A)

which satisfy an estimate

for some constants

a

and

b

(depending on

f).

Every analytic vector is semi-analytic. Lemma .

Let

constant.

f

be a semi-analytic vector for

Then

f

is semi-analytic for

A

and let

c

be a

A + c

Proof:

Proposition 17.1 •

Let

A

be a semi-bounded Hermitian operator.

Then every semi-analytic vector for A

A

is a determining vector for

•

Proof:

Let

f

be a semi-analytic vector for Akf

in ear span of the Let

A

0

and

let

be the restriction of

acting in Since

D(f)

to

Let

D(f)

be the closure of

M(f)

A

A

be the

D(f)

We must shovJ that

A

is semi-bounded, it has a self-adjoint extension.

0

A

0

~

m

>

0

According to the theory

.

of extensions of semi-bounded operators it is enough to show that there is a unique self-adjoint extension f

A 1

is a semi-analytic vector, if

~

0 . t

is sufficiently

small the series

,

cos(t A2 )g l

converges for all determined by

0

is essentially self-adjoint.

M(f)

the lemma we may assume that

Since

A

A

0

g

ln

D(f)

Thus

is uniquely

By

- 107 -

d2 1 - - cos(t A2 )h 2 dt l is also uniquely determined by A The value of

Theorem 17.2

Let

A

at

t

0

Thus

is

0

be a semi-bounded Hermitian operator.

there exists a dense set of semi-analytic vectors for

A .

Assume

Then

A

is essentially self-adjoint.

EXAMPLE The restriction of a self-adjoint operator to a dense set of analytic vectors need not be essentially self-adjoint.

The difficulty

is that the vectors need not be analytic for the restriction. Let Let E

H

Nc V

A

be an unbounded self-adjoint operator with domain H

be a closed hyperplane which is dense in

B(A) n N

By the lemma below, E

On the other hand, E

is not dense in

set of analytic vectors for

N

is dense in

V .

So

V .

Let and hence in

E is a dense

A , but the restriction of

A

to

E is

not essentially self-adjoint. Lemma . Let

Let

V be a Hilbert space and

B be a dense linear subspace of

N be a closed hyperplane. V

Then

Bn N

is dense in

N . NOTES The proof of the theorem on semi-analytic vectors is adapted from Simon (l97lb).

There is a more general notion, that of Stieltjes

vector, and Nussbaum (1969) proved a stronger theorem relating Stieltjes vectors to self-adjointness.

Chernoff (l972a) has discussed

the relation between various self-adjointness results of this type. The idea for the example is due to Simon (unpublished).

The density

- 108 -

lemma may be found in the book Dynamical Theories of Brownian Motion, by E. Nelson (Princeton University Press, Princeton, N.J., 1967). Masson and McClary (1972) have applied these notions to quantum mechanical Hamiltonians.

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