Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry

JOURNAL OF MATHEMATICAL PHYSICS 46, 012103 (2005)

Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry H. Panahia) Department of Physics, Guilan University, Rasht 41335-1914, Iran and Institute for Studies in Theoretical Physics and Mathematics, Teheran 19395-1795, Iran

M. A. Jafarizadehb) Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664, Iran; Institute for Studies in Theoretical Physics and Mathematics, Teheran 19395-1795, Iran; and Research Institute for Fundamental Sciences, Tabriz 51664, Iran (Received 2 August 2004; accepted 13 September 2004; published online 27 December 2004)

Two and three dimensional Hamiltonians with generalized and ordinary shape invariance symmetry have been obtained by Fourier transforming over some coordinates of the SU(3) Casimir operator defined on SU共3兲 / SU共2兲 symmetric space. It is shown that the generalized shape invariance of the two dimensional Hamiltonian is equivalent to SU(3) symmetry while in the three dimensional one, the ordinary shape invariance is equivalent to contracted SU(3) and there is one to one correspondence between the representations of the generalized shape invariance symmetry of the two (three) dimensional Hamiltonian and SU(3) [contracted SU(3)] Verma bases. © 2005 American Institute of Physics. [DOI: 10.1063/1.1827325]

I. INTRODUCTION

Exactly solvable potentials are the central and fundamental problems of mathematical physics, consequently they have attracted much interest both in theoretical physics and mathematics. There are many methods of obtaining exactly solvable potentials in quantum mechanics. The most powerful methods are the algebraic method,1 supersymmetric and shape invariant factorization of the Schrödinger equation in one and two or three dimensional exactly solvable models.2–5 In all these works it is shown that there is a close connection between the shape invariance symmetry of one or higher dimensional Hamiltonians and some rank one semisimple Lie algebra or higher rank nonsemisimple algebras, where this equivalence between the one dimensional shape invariant and the rank one semisimple Lie algebra has been shown in Ref. 6. Here in this work we introduce two and three dimensional Hamiltonians with a new kind of shape invariance. It is shown that the shape invariance symmetry of the two dimensional Hamiltonian is equivalent to SU(3) symmetry where we call it generalized shape invariance while the shape invariance associated with the three dimensional Hamiltonian is equivalent to contracted SU(3) ordinary shape invariance and there is one to one correspondence between the representation of the generalized shape invariance symmetry of two (three) dimensional Hamiltonians and SU(3) [contracted SU(3)] Verma bases. The paper is organized as follows: In Sec. II after introducing the parametrization of SU(3) Lie group we derive its right invariant vector fields and Casimir operator on SU共3兲 / SU共2兲. In Sec. III, using the Fourier transformation together with the coset reduction we obtain the two dimensional Hamiltonian Hl共m1 , m2兲 of charged particle on S2 sphere in the presence of an electric field. Section IV is devoted to SU(3) Verma basis,7 and their connection with eigenspectrum and de-

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012103-2

J. Math. Phys. 46, 012103 (2005)

H. Panahi and M. A. Jafarizadeh

generacy of the above Hamiltonian. In Sec. V we talk about the generalized shape invariance of this Hamiltonian. In Sec. V, through Inonu–Wigner contraction8 of SU(3), we obtain three dimensional ordinary shape invariance Hamiltonians. The paper is ended with a conclusion and one appendix. II. THE RIGHT INVARIANT VECTOR FIELDS AND QUADRATIC CASIMIR OPERATOR OF SU(3) GROUP OVER THE SU„3… / SU„2… COSET MANIFOLD

According to Ref. 9, we can parametrize an arbitrary element of the SU共3兲 / SU共2兲 coset manifolds in the following form:

冢

sin共␪兲

cos共␪兲ei共␩+␹1兲

0

冣

sin共␸兲e−i␩ sin共␪兲cos共␸兲ei共␩+␹2兲 , U = − cos共␪兲cos共␸兲ei共␹2−␹1兲 − cos共␪兲sin共␸兲e−i␹1 − cos共␸兲e−i共␩+␹2兲 sin共␪兲sin共␸兲ei␩

共2.1兲

where 0 ⬍ ␪, ␸ 艋 ␲ / 2 and 0 艋 ␩, ␹1, ␹2 ⬍ 2␲. Now, an arbitrary element of SU(3) group manifold can be obtained by multiplying the coset element given in (2.1) by an arbitrary element h of stability group SU(2) with the parametrization h = exp共i␭3␣兲exp共i␭2␤兲exp共i␭3␥兲 from left or right, where 3 ⫻ 3 Hermitian Gell–Mann matrices ␭i, i = 1 , 2 , . . . , 8 are defined as

冢 冣 冢 0 1 0

␭1 = 1 0 0 , 0 0 0

0 −i 0

␭2 = i 0

冢 冣 冢 0 0 1

␭4 = 0 0 0 , 1 0 0

冢

0 0

0 0

0 0 −i

␭5 = 0 0 i 0 0

冣

␭7 = 0 0 − i , 0 i 0

冣 冢

␭8 =

0 0

1

1

0

0

冣

␭3 = 0 − 1 0 , 0 0 0

0 , 0

冣 冢 冣

冑3

0 0 0

,

冢

␭6 = 0 0 1 , 0 1 0

1 0

0

0 1

0

0 0 −2

冣

共2.2兲

.

Usually the right invariant vector fields of SU(3) group manifold can be obtained from the right invariant su(3) Lie algebra valued one forms dG G−1 = e␮i d␰␮ ␭i with G = Uh,10 where e␮i are right invariant fiel-beins and ␰␮ = 共␪ , ␸ , ␩ , ␹1 , ␹2 , ␣ , ␤ , ␥兲 are coordinates of SU(3) group manifold. Then the right invariant vector field can be written as Ri = e␮i 共⳵ / ⳵␰␮兲, where e␮i are the inverse of the right invariant fiel-beins e␮i . Now, the SU(3) right invariant vector fields over the SU共3兲 / SU共2兲 coset manifold11,12 can be obtained simply by projecting the SU(3) right invariant vector fields over the principal bundle SU(3) to the base manifold SU共3兲 / SU共2兲, where it leads to the vanishing of the components which are tangent to SU(2) fiber manifold (stability group). Using the above prescription, after some lengthy and tedious calculation via Maple software we get the following expressions for the SU(3) right invariant vector fields over the SU共3兲 / SU共2兲 coset manifold:

冋

册

cot共␪兲 ⳵ 1 ⳵ ⳵ ⳵ +i , R± = e±i共␹1−␹2兲 ⫿cos共␸兲 ± cot共␪兲sin共␸兲 + i tan共␪兲cos共␸兲 2 ⳵␪ ⳵␸ ⳵␹1 cos共␸兲 ⳵␹2

冋

共2.3兲

cot共␪兲 ⳵ 1 ⳵ ⳵ Y ± = e±i␹1 ±sin共␸兲 ± cot共␪兲cos共␸兲 − i 2 ⳵␪ ⳵␸ sin共␸兲 ⳵␩ + 2i

册

cos2共␪兲 − sin2共␪兲sin2共␸兲 ⳵ cot共␪兲 ⳵ +i , sin共2␪兲sin共␸兲 ⳵␹1 sin共␸兲 ⳵␹2

共2.4兲

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012103-3

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

冋

冉

册

冊

1 ⳵ ⳵ ⳵ ⳵ X± = e⫿i␹2 ⫿ − i cot共␸兲 − + 2i cot共2␸兲 , 2 ⳵␸ ⳵␩ ⳵␹1 ⳵␹2

R3 = −

冉

冊

⳵ i ⳵ − , 2 ⳵␹1 ⳵␹2

R8 =

冉

共2.5兲

冊

i冑3 2 ⳵ ⳵ ⳵ − − , 2 3 ⳵␩ ⳵␹1 ⳵␹2

共2.6兲

where R± = 21 共R1 ± iR2兲, Y ± = 21 共R4 ⫿ iR5兲, and X± = 21 共R6 ± iR7兲. After some algebraic calculations one can show that, the above generators satisfy su(3) Lie algebra commutation relations. Also it is straightforward to show that the following defined su(3) quadratic Casimir operator commutes with all of the above generators C = 21 共R+R− + R−R+兲 + 21 共Y +Y − + Y −Y +兲 + 21 共X+X− + X−X+兲 + R23 + R28 ,

共2.7兲

where after calculations, we obtain the following expression (by ignoring the factor 41 ): C=−

冋

1 1 − sin2共␪兲cos2共␸兲 ⳵2 ⳵2 ⳵2 4 − cos2共␪兲sin2共␸兲 − cos2共␸兲 ⳵2 2 + 2 2 + 2 2 2 + ⳵␪ sin 共␪兲 ⳵␸ 3 sin 共␪兲sin 共␸兲 ⳵␩ sin2共␪兲cos2共␪兲sin2共␸兲 ⳵␹21

冉

+

1 2 ⳵2 ⳵2 ⳵2 ⳵2 − + − sin2共␪兲sin2共␸兲cos2共␸兲 ⳵␹22 sin2共␪兲sin2共␸兲 ⳵␩⳵␹1 ⳵␩⳵␹2 ⳵␹1⳵␹2

+

2共4 cos2共␪兲 − 1兲 ⳵ 2 cot共2␸兲 ⳵ + . sin共2␪兲 ⳵␪ sin2共␪兲 ⳵␸

册

冊

共2.8兲

Similarly we can calculate the SU(3) left invariant vector fields, where its structure constant, is minus the right invariant ones but its quadratic Casimir operator is the same as the right one. Here we need only the right invariant vector fields and we do not need to quote the left invariant ones here. Also one can show that the Casimir operator is the same as Laplace–Beltrami operator of adjoint invariant metric. III. REDUCTION OF THE CASIMIR OPERATOR TO TWO-DIMENSIONAL HAMILTONIAN OF A CHARGED PARTICLE ON S2 SPHERE

In order to reduce the SU(3) Casimir operator together with its right invariant vector fields defined on SU共3兲 / SU共2兲 coset manifold to a two-dimensional operator, first we eliminate the coordinate ␩ by Fourier transforming over the coordinate ␩ by kernel ei␩l, where after similarity transformation R → f −1共␪ , ␸兲Rf共␪ , ␸兲 with similarity function f −1共␪ , ␸兲 = sin共␪兲冑cos共␪兲sin共2␸兲 the right invariant vector fields and their quadratic Casimir operator take the following form:

冋

1 ⳵ ⳵ ⳵ R± = e±i共␹1−␹2兲 ⫿cos共␸兲 ± cot共␪兲sin共␸兲 + i tan共␪兲cos共␸兲 2 ⳵␪ ⳵␸ ⳵␹1 +i

册

cot共␪兲 ⳵ cos2共␪兲 − cos2共␸兲sin2共␪兲 ± , cos共␸兲 ⳵␹2 sin共2␪兲cos共␸兲

冋

共3.1兲

cos2共␪兲 − sin2共␪兲sin2共␸兲 ⳵ 1 ⳵ ⳵ Y ± = e±i␹1 ±sin共␸兲 ± cot共␪兲cos共␸兲 + 2i 2 sin共2␪兲sin共␸兲 ⳵␪ ⳵␸ ⳵␹1 +i

册

cot共␪兲 cot共␪兲 ⳵ cos2共␪兲 − sin2共␸兲sin2共␪兲 +l ⫿ , sin共␸兲 ⳵␹2 sin共␸兲 sin共2␪兲sin共␸兲

冋

共3.2兲

册

1 ⳵ ⳵ ⳵ + i cot共␸兲 + 2i cot共2␸兲 ± cot共2␸兲 + l cot共␸兲 , X± = e⫿i␹2 ⫿ 2 ⳵␸ ⳵␹1 ⳵␹2

共3.3兲

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012103-4

J. Math. Phys. 46, 012103 (2005)

H. Panahi and M. A. Jafarizadeh

R3 = −

冉

冊

⳵ i ⳵ − , 2 ⳵␹1 ⳵␹2

R8 = −

冉

冊

i冑3 ⳵ l ⳵ + − 2i . 3 2 ⳵␹1 ⳵␹2

共3.4兲

One can straightforwardly show that the above generators satisfy su(3) Lie algebra commutation relations 关R3,R±兴 = ± R±,

关R3,Y ±兴 = ± 21 Y ±,

关R8,Y ±兴 = ±

关R+,R−兴 = 2R3, 关Y +,Y −兴 = R3 + 冑3R8,

冑3 2

Y ±,

关R3,X±兴 = ± 21 X± ,

关R8,X±兴 = ⫿

关R±,Y ⫿兴 = ± X±,

冑3 2

X± ,

关R±,X⫿兴 = ⫿ Y ± ,

关X+,X−兴 = R3 − 冑3R8,

关Y ±,X±兴 = ⫿ R± .

共3.5兲

Also the Casimir operator reduces to

C=− +

冋

冉

1 1 1 − sin2共␪兲cos2共␸兲 ⳵2 ⳵2 ⳵ ⳵2 + cot共 ␪ 兲 + + ⳵␪2 ⳵␪ sin2共␪兲 ⳵␸2 sin2共␪兲sin2共␸兲 cos2共␪兲 ⳵␹21

冉

冊

1 ⳵2 ⳵2 ⳵ ⳵ l2 共4 − cos2共␪兲sin2共␸兲 − cos2共␸兲兲 + 2 − 2il + − 3 cos2共␸兲 ⳵␹22 ⳵␹1⳵␹2 ⳵␹1 ⳵␹2

册

冊

1 cot2共2␸兲 + tan2共␪兲 + cot2共␪兲 + . 4 sin2共␪兲

共3.6兲

For convenience we have denoted the reduced generators with the same notation of the preceding section. Now we eliminate the coordinates ␹1 and ␹2 through Fourier transformation over them with the kernel ei共m1␹1−m2␹2兲, then Casimir operator (3.6) reduces to the following Hamiltonian: Hl共m1,m2兲 = −

冉

冊

1 1 ⳵ ⳵ ⳵2 sin共␪兲 + 2 + Vl,m1,m2共␪, ␸兲 sin共␪兲 ⳵␪ ⳵␪ sin 共␪兲 ⳵␸2

共3.7兲

with Vl,m1,m2共␪, ␸兲 =

冉

m22 m21共1 − sin2共␪兲cos2共␸兲兲 1 + − 2m1m2 − 2l共m1 − m2兲 sin2共␪兲sin2共␸兲 cos2共␪兲 cos2共␸兲 +

冊

1 cot2共2␸兲 l2 共4 − cos2共␪兲sin2共␸兲 − cos2共␸兲兲 − tan2共␪兲 − cot2共␪兲 − . 3 4 sin2共␪兲

共3.8兲

On the other hand, the dynamical symmetric of motion of a charged particle in the presence of an external electromagnetic field on the symmetric spaces can give the following nonrelativistic Hamiltonian for motion of a charged particle on the two dimensional manifold with metric g␮␯ in the presence of static electromagnetic fields with vector potential Aជ and scalar potential V as13

H=−

1

冑g 共⳵␮ − iA␮兲共

冑gg␮␯共⳵␯ − iA␯兲兲 + V,

共3.9兲

where g is the determinant of metric g␮␯. Therefore, the Hamiltonian (2.8) can be interpreted as the Hamiltonian of a charged particle on S2 sphere with metric

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012103-5

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

g ␮␯ =

冉

1

0

0 sin2共␪兲

冊

共3.10兲

,

in the presence of an electric field with scalar potential (3.8). IV. THE ALGEBRAIC SOLUTION OF THE HAMILTONIAN BY USING VERMA BASES

Here in this section we try to solve our Hamiltonian algebraically, that is we will obtain its eigenspectrum by using the Verma bases of su(3) or A2 Lie algebra. According to Ref. 7, Verma bases of the irreducible representation space V共⌳兲 of A2 over C Lie algebra, where ⌳ = 共p , q兲 is the highest weight, consist of all vectors f a13 f a22 f a11兩p,q典,

共4.1兲

such that 0 艋 a1 艋 p,

0 艋 a 2 艋 q + a 1,

0 艋 a3 艋 min关q,a2兴,

共4.2兲

where ei, f i, hi, i = 1 and 2 are base of A2 Lie algebra satisfying the commutation relations 关ei, f i兴 = hi, for each simple root ␣1 = 共 relations

1 2,

冑3 / 2兲

关hi,ei兴 = 2ei,

关hi, f i兴 = − 2f i,

and ␣2 = 共

冑3 / 2兲.

1 2 ,−

i = 1,2

共4.3兲

Now, by comparing the commutation

关Y +,Y −兴 = R3 + 冑3R8,

关R3 + 冑3R8,Y ±兴 = ± 2Y ± ,

关X+,X−兴 = R3 − 冑3R8,

关R3 − 冑3R8,X±兴 = ± 2X±

共4.4兲

with (4.3) we have h1 = R3 + 冑3R8,

e 1 = Y +,

f1 = Y−,

h2 = R3 − 冑3R8,

e 2 = X +,

f 2 = X− .

共4.5兲

In an arbitrary representation of A2 with highest weight ⌳共p , q兲, the highest eigenweight satisfies ei兩p,q典 = 0, Therefore, the highest eigenfunction ␺ first order linear differential equations:

冋

sin共␸兲 −

冋

−

p,q

共4.6兲

i = 1,2.

共␪ , ␸ , ␹1 , ␹2兲 = 具␹2 , ␹1 , ␸ , ␪ 兩 p , q典 satisfies the following

cos2共␪兲 − sin2共␪兲sin2共␸兲 ⳵ cot共␪兲 ⳵ ⳵ ⳵ + cot共␪兲cos共␸兲 + 2i +i sin共2␪兲sin共␸兲 ⳵␪ ⳵␸ ⳵␹1 sin共␸兲 ⳵␹2

册

cot共␪兲 p,q cos2共␪兲 − sin2共␸兲sin2共␪兲 +l ␺ 共␪, ␸, ␹1, ␹2兲 = 0, sin共␸兲 sin共2␪兲sin共␸兲

册

⳵ ⳵ ⳵ + i cot共␸兲 + 2i cot共2␸兲 + cot共2␸兲 + l cot共␸兲 ␺ p,q共␪, ␸, ␹1, ␹2兲 = 0. ⳵␸ ⳵␹1 ⳵␹2

共4.7兲

共4.8兲

On the other hand, the highest eigenweight is the eigenstate of the Cartan subalgebra with R3 and R8 as its Gell–Mann basis,14 where the bases h␣1 and h␣2 are associated with simple roots ␣1 and ␣2, and they can be written in terms of these bases in the following form:15

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012103-6

J. Math. Phys. 46, 012103 (2005)

H. Panahi and M. A. Jafarizadeh

h␣1 = R3 + 冑3R8,

h␣2 = R3 − 冑3R8 .

共4.9兲

Writing the highest eigenweight in terms of fundamental weights, that is ␮ = p␮1 + q␮2 and considering the following relation between the simple roots and fundamental weights:

2

共 ␮ i, ␣ j 兲 = ␦ij , 共␣ j, ␣ j兲

共4.10兲

we can write 共R3 + 冑3R8兲␺ p,q共␪, ␸, ␹1, ␹2兲 = 具␮,R3 + 冑3R8典␺ p,q共␪, ␸, ␹1, ␹2兲 = 具␮,h␣1典␺ p,q共␪, ␸, ␹1␹2兲 = 2

共␮, ␣1兲 p,q ␺ 共 ␪ , ␸ , ␹ 1, ␹ 2兲 共 ␣ 1, ␣ 1兲

or 共R3 + 冑3R8兲␺ p,q共␪, ␸, ␹1␹2兲 = p␺ p,q共␪, ␸, ␹1, ␹2兲.

共4.11兲

Similarly by acting out the other basis of Cartan subalgebra on highest eigenweight we obtain 共R3 − 冑3R8兲␺ p,q共␪, ␸, ␹1␹2兲 = q␺ p,q共␪, ␸, ␹1, ␹2兲.

共4.12兲

The above eigenvalue equations imply the following ␹1 and ␹2 dependence of highest eigenweight ␺ p,q共␪ , ␸ , ␹1 , ␹2兲,

␺ p,q共␪, ␸, ␹1, ␹2兲 = e共i/3兲共2p+q+l兲␹1e−共i/3兲共p+2q−l兲␹2␾ p,q共␪, ␸兲.

共4.13兲

Substituting (4.13) in (4.7) and (4.8), we see that the integrability of the latter equation requires that p = q + l, that is for given values of the parameter l, only the representations 共q + l , q兲 are relevant to the eigenspectrum of our Hamiltonian. Hence, integrating Eqs. (4.7) and (4.8) we get the following expression for the highest eigenweight in these particular representations:

␺ q,l共␪, ␸, ␹1, ␹2兲 = ei共共q+l兲␹1−q␹2兲共cosq共␸兲sinq+1共␪兲cosq+l共␪兲冑cos共␪兲sin共2␸兲兲.

共4.14兲

Now, using the formula (4.1) we can obtain the lower eigenweights or Verma bases,

␺共a3,a2,a1兲共␪, ␸, ␹1, ␹2兲 = Y −a3X−a2Y −a1␺ q,l共␪, ␸, ␹1, ␹2兲,

共4.15兲

where 0 艋 a1 艋 q + l, 0 艋 a2 艋 q + a1, 0 艋 a3 艋 min关q , a2兴. In order to obtain eigenspectrum of the Hamiltonian (3.7), it is sufficient to eliminate ␹1 and ␹2 coordinates dependence of SU(3) Verma basis by Fourier transforming over them. This can be achieved simply by shifting the phase factor ei共共q+l兲␹1−q␹2兲 to the left-hand side of lowering operators X− and Y − in the above relation and integrating over the coordinates ␹1 and ␹2. Hence, we obtain q,l ␺共a3,a2,a1兲共␪, ␸, ␹1, ␹2兲 = ei共q+l−a1−a3兲␹1e−i共q−a2兲␹2␹q+l−a 共␪, ␸兲, 1−a3,q−a2

共4.16兲

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012103-7

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

where

冉兿 冉兿 a3

q,l ␹q+l−a 共␪, ␸兲 1−a3,q−a2

=

Y −共q + l − a1 − i + 1,q − a2兲

i=1

a1

⫻

冊冉兿 a2

X−共q + l − a1,q − i + 1兲

i=1

冊

冊

Y −共q + l − i + 1,q兲 共cosq共␸兲sinq+1共␪兲cosq+l共␪兲冑cos共␪兲sin共2␸兲兲,

i=1

共4.17兲 m K共i兲 = K共m兲K共m − 1兲 ¯ K共1兲 also the Fourier transand the operators appear in the products as ⌸i=1 formed operators have the following form:

冋

1 ⳵ ⳵ ⫿cos共␸兲 ± cot共␪兲sin共␸兲 − m1 tan共␪兲cos共␸兲 2 ⳵␪ ⳵␸

R±共m1,m2兲 =

+ m2

Y ±共m1,m2兲 =

册

cot共␪兲 cos2共␪兲 − cos2共␸兲sin2共␪兲 ± , cos共␸兲 sin共2␪兲cos共␸兲

冋

共4.18兲

cos2共␪兲 − sin2共␪兲sin2共␸兲 1 ⳵ ⳵ ±sin共␸兲 ± cot共␪兲cos共␸兲 − 2m1 2 ⳵␪ ⳵␸ sin共2␪兲sin共␸兲 + m2

X±共m1,m2兲 =

册

cot共␪兲 cot共␪兲 cos2共␪兲 − sin2共␸兲sin2共␪兲 ⫿ +l , sin共␸兲 sin共␸兲 sin共2␪兲sin共␸兲

共4.19兲

冋

册

1 ⳵ ⫿ − m1 cot共␸兲 + 2m2 cot共2␸兲 ± cot共2␸兲 + l cot共␸兲 , 2 ⳵␸

R3共m1,m2兲 = 21 共m1 + m2兲,

R8共m1,m2兲 =

冑3

冉

冊

2 m1 − m2 − l . 2 3

共4.20兲

共4.21兲

Now, by defining m1 = q + l − a1 − a3 and m2 = q − a2, we see that due to the inequalities (4.2), the parameters m1 and m2 must satisfy −q 艋 m1 艋 q + l and −共q + l兲 艋 m2 艋 q. Also it is straightforward to see that an arbitrary Verma basis is proportional to ei共m1␹1−m2␹2兲 or we can write

␺mq,l1,m2共␪, ␸, ␹1, ␹2兲 = ei共m1␹1−m2␹2兲␹mq,l1,m2共␪, ␸兲.

共4.22兲

Now, substituting (4.22) in (3.6) and Fourier transforming over the coordinates ␹1 and ␹2, the Casimir operator reduces to the required two dimensional Hamiltonian Hl共m1 , m2兲 given in (3.7). Therefore, the general eigenfunctions of the above Hamiltonian with eigenvalue E共q , l兲 = 31 共共q + l兲共2q + l + 3兲 + q共q + 3兲兲 (for the degeneracy of these eigenvalues see the Appendix) can be written as

冉兿 冉 a3

␹mq,l1,m2共␪, ␸兲

=

Y −共m1 + a3 − i + 1,m2兲

i=1

i=1

q+l−m1−a3

⫻

冊冉 兿 冊共

q−m2

兿 i=1

Y −共q + l − i + 1,q兲

X−共m1 + a3,q − i + 1兲

冊

cosq共␸兲sinq+1共␪兲cosq+l共␪兲冑cos共␪兲sin共2␸兲兲 . 共4.23兲

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012103-8

H. Panahi and M. A. Jafarizadeh

J. Math. Phys. 46, 012103 (2005)

FIG. 1. Diagram of seven possible allowed regions of eigenspectrum of Hamiltonian Hl共m1 , m2兲 in the 共m1 , m2兲 plane for given values of integer parameters q, l, m1, and m2 with l 艌 0. Horizontal lines mean application of the lowering operator Y − while the vertical line indicates the application of the lowering operator X−.

V. GENERALIZED SHAPE INVARIANCE SYMMETRY

In this section we show that the Hamiltonian Hl共m1 , m2兲 possesses a new kind of shape invariance symmetry, we call it generalized shape invariance. Obviously SU(3) symmetry of the Casimir operator before the reduction generates this special shape invariance symmetry. Using this symmetry we will obtain below the eigenspectrum of Hamiltonian Hl共m1 , m2兲, that is the eigenq,l 共␪ , ␸兲 corresponding to eigenvalue E共q , l兲 by consecutive application of lowering functions ␹m 1,m2 operators over the state with m1 = q + l and m2 = q. We will also obtain its degeneracy for given values of 共q + l , q兲, where it is the same as the one that can be obtained by using the inequalities (4.2) corresponding to Figs. 1 and 2.

FIG. 2. Diagram of seven possible allowed regions of eigenspectrum of Hamiltonian Hl共m1 , m2兲 in the 共m1 , m2兲 plane for given values of integer parameters q, l, m1, and m2 with l ⬍ 0. Horizontal lines mean application of the lowering operator Y − while the vertical line indicates the application of the lowering operator X−.

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012103-9

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

First we write the Hamiltonian Hl共m1 , m2兲 operator in terms of the Fourier transformed operators given in (4.18)–(4.21), 1 Hl共m1,m2兲 = 关R+共m1 − 1,m2 − 1兲R−共m1,m2兲 + R−共m1 + 1,m2 + 1兲R+共m1,m2兲 2 + Y +共m1 − 1,m2兲Y −共m1,m2兲 + Y −共m1 + 1,m2兲Y +共m1,m2兲 + X+共m1,m2 − 1兲X−共m1,m2兲 + X−共m1,m2 + 1兲X+共m1,m2兲兴 +

2 2 共m1 + m2兲2 3共m1 − m2 − 3 l兲 + . 4 4

共5.1兲

Now, using the following commutation relations, 关R3 + 冑3R8,Y ±兴 = ± 2Y ±,

关R3 − 冑3R8,Y ±兴 = ⫿ Y ± ,

关R3 − 冑3R8,X±兴 = ± 2X±,

关R3 + 冑3R8,X±兴 = ⫿ X± ,

共5.2兲

we can obtain the following relations: 共R3共m1 ± 1,m2兲 + 冑3R8共m1 ± 1,m2兲兲Y ±共m1,m2兲 − Y ±共m1,m2兲共R3共m1,m2兲 + 冑3R8共m1,m2兲兲 = ± 2Y ±共m1,m2兲, 共R3共m1 ± 1,m2兲 − 冑3R8共m1 ± 1,m2兲兲Y ±共m1,m2兲 − Y ±共m1,m2兲共R3共m1,m2兲 − 冑3R8共m1,m2兲兲 = ⫿ Y ±共m1,m2兲, 共R3共m1,m2 ± 1兲 − 冑3R8共m1,m2 ± 1兲兲X±共m1,m2兲 − X±共m1,m2兲共R3共m1,m2兲 − 冑3R8共m1,m2兲兲 = ± 2X±共m1,m2兲, 共R3共m1,m2 ± 1兲 + 冑3R8共m1,m2 ± 1兲兲X±共m1,m2兲 − X±共m1,m2兲共R3共m1,m2兲 + 冑3R8共m1,m2兲兲 = ⫿ X±共m1,m2兲.

共5.3兲

The above relations imply that the Hamiltonian Hl共m1 , m2兲 possesses shape invariance symmetry. Since, through the left action of the operators Y ±共m1 , m2兲 and X±共m1 , m2兲 on both sides of the following eigenvalue equations, q,l q,l Hl共m1,m2兲␹m ,m 共␪, ␸兲 = E共q,l兲␹m ,m 共␪, ␸兲, 1

2

1

2

冉 冉

q,l 共R3共m1,m2兲 + 冑3R8共m1,m2兲兲␹m ,m 共␪, ␸兲 = 2m1 − m2 − 1

2

q,l 共R3共m1,m2兲 − 冑3R8共m1,m2兲兲␹m ,m 共␪, ␸兲 = − m1 + 2m2 + 1

2

冊 冊

l q,l ␹ 共␪, ␸兲, 2 m1,m2

共5.4兲

l q,l ␹ 共␪, ␸兲, 2 m1,m2

we get

␹mq,l1±1,m2共␪, ␸兲 = Y ±共m1,m2兲␹mq,l1,m2共␪, ␸兲,

共5.5兲

␹mq,l1,m2±1共␪, ␸兲 = X±共m1,m2兲␹mq,l1,m2共␪, ␸兲.

共5.6兲

Therefore, the operators Y ±共m1 , m2兲 shift the parameter m1 by one unit or they push the unrenormalized eigenfunctions horizontally in Figs. 1 and 2, while the operators X±共m1 , m2兲 shift the

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012103-10

J. Math. Phys. 46, 012103 (2005)

H. Panahi and M. A. Jafarizadeh

parameter m2 by one unit or they push the eigenfunctions vertically in Figs. 1 and 2. Obviously the eigenfunctions vanish in the forbidden regions of Figs. 1 and 2. Using the relations (5.5) and (5.6), we obtain the following relations: q,l q,l Y +共m1 − 1,m2兲Y −共m1,m2兲␹m ,m 共␪, ␸兲 ⯝ ␹m ,m 共␪, ␸兲, 1

2

1

2

q,l q,l Y −共m1,m2兲Y +共m1 − 1,m2兲␹m −1,m 共␪, ␸兲 ⯝ ␹m −1,m 共␪, ␸兲 1

2

1

2

共5.7兲

and q,l q,l X+共m1,m2 − 1兲X−共m1,m2兲␹m ,m 共␪, ␸兲 ⯝ ␹m ,m 共␪, ␸兲, 1

q,l X−共m1,m2兲X+共m1,m2 − 1兲␹m ,m 1

2

2−1

1

q,l 共␪, ␸兲 ⯝ ␹m ,m 1

2

2−1

共␪, ␸兲,

共5.8兲

which indicate that the Hamiltonian Hl共m1 , m2兲 possesses the shape invariance symmetry. Actually the first pair of equations given in (5.7) imply the horizontal shape invariance while the second pair of equations given in (5.8) imply the vertical shape invariance symmetry in Figs. 1 and 2, respectively. Therefore, using this symmetry we can obtain the eigenfunctions of the isospectral Hamiltonians Hl共m1 , m2兲 with the eigenvalue E共q , l兲 simply by applying the lowering operators X− q,l and Y − over the highest weight ␹q+l,q 共␪ , ␸兲, namely we obtain all the eigenstates for the values of parameters m1 and m2 given in the allowed region of Figs. 1 and 2, such that these eigenfunctions vanish for the values of the parameters corresponding to the forbidden region. Also one can show that in this way we obtain exactly the same eigenspectrum that we have obtained in Sec. IV by using the Verma basis. Therefore, by consecutive application of lowering operators over highest eigenweight q,l q,l ␹q+l,q 共␪ , ␸兲, we can obtain an arbitrary eigenstate ␹m 共␪ , ␸兲, where the existence of different 1,m2 ordering of lowering operators or the different paths indicate the degeneracy of the Hamiltonian Hl共m1 , m2兲. It is straightforward to show that by this method we get exactly the eigenspectrum of the preceding section. For an illustration we explain below the real representation with q = 1 and l = 0 in detail. Using the relation (4.23), we get the following expression for highest eigenweight of (1,1) representation: 1,0 ␹1,1 共␪, ␸兲 = 2 sin2共␪兲cos共␪兲cos共␸兲冑cos共␪兲sin共2␸兲,

共5.9兲

then using (5.5) we get 1,0 1,0 ␹0,1 共␪, ␸兲 = Y −共1,1兲␹1,1 共␪, ␸兲 = − sin3共␪兲sin共2␸兲冑cos共␪兲sin共2␸兲.

共5.10兲

Now, we cannot lower the above eigenstate by acting the operator Y − on it, since the function 1,0 ␹−1,1 共␪ , ␸兲 corresponds to the values of parameters (m1 = −1, m2 = 1) which is not allowed for q = 1 and l = 0. Therefore in order to get lower eigenstates we must act out the operator X− on it, that is, we have 1,0 1,0 ␹0,0 共␪, ␸兲 = X−共0,1兲␹0,1 共␪, ␸兲 = 2 sin3共␪兲cos共2␸兲冑cos共␪兲sin共2␸兲.

共5.11兲

On the other hand, if we act out the lowering operator X− on the highest eigenstate, we will obtain 1,0 1,0 ␹1,0 共␪, ␸兲 = X−共1,1兲␹1,1 共␪, ␸兲 = − 2 sin2共␪兲cos共␪兲sin共␸兲冑cos共␪兲sin共2␸兲,

共5.12兲

where its further action will kill it, since the values of parameters (m1 = 1, m2 = −1) are not allowed, hence we cannot have the eigenstate corresponding to these values of parameters. Similarly the 1,0 共␪ , ␸兲 state to (m1 = 0, m2 = 0) or zero weight eigenstate, action of Y − will shift ␹1,0

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012103-11

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

1 1,0 ˜␹0,0 共␪, ␸兲 = Y −共1,0兲␹1,0 共␪, ␸兲 = 2共cos2共␪兲 − sin2共␸兲sin2共␪兲兲sin共␪兲冑cos共␪兲sin共2␸兲.

共5.13兲

We see that zero weight is degenerate but other eigenvalues are nondegenerate which is in agreement with the results of the preceding section. With the same procedure we can obtain the remaining part of the spectrum, that is we have 1,0 1,0 ␹0,−1 共␪, ␸兲 = X−共0,0兲␹0,0 共␪, ␸兲 = − 2 sin共2␸兲sin3共␪兲冑cos共␪兲sin共2␸兲, 1,0 1,0 ␹−1,0 共␪, ␸兲 = Y −共0,0兲␹0,0 共␪, ␸兲 = 2 sin共␸兲sin共2␪兲sin共␪兲冑cos共␪兲sin共2␸兲,

共5.14兲

and 1,0 1,0 1,0 ␹−1,−1 共␪, ␸兲 = Y −共0,− 1兲␹0,−1 共␪, ␸兲 = X−共− 1,0兲␹−1,0 共␪, ␸兲 = 2 sin共2␪兲cos共␸兲sin共␪兲冑cos共␪兲sin共2␸兲.

共5.15兲 We should remind that all eigenfunctions associated with the forbidden region of 共m1 , m2兲 plane vanish. For example, the action of the operators X−3Y − and Y −2X−2Y − on highest eigenstate leads to 1 1 vanishing functions ␹0,−2 共␪ , ␸兲 and ␹−2,−1 共␪ , ␸兲, respectively. Also one can show that by acting the operators X−Y −X−Y − and Y −X−Y −X− on the highest eigenstate we obtain eigenstates which are both 1,0 1,0 共␪ , ␸兲 = Y −X−2Y −␹1,1 共␪ , ␸兲. proportional to Verma base ␹−1,−1 VI. THREE DIMENSIONAL HAMILTONIAN WITH ORDINARY SHAPE INVARIANCE SYMMETRY

Here in this section we first make Inonu–Wigner contraction8 over the generators of su(3) Lie algebra given in (2.3)–(2.6), simply by making the change of coordinate ␪ = r / R and relating the new contracted generators to the old ones by R±c = 共1 / R兲R±, Rc3 = R3, X±c = X±, Y ±c = 共1 / R兲Y ±, Rc8 = R8. Then in the limit of R → ⬁ the set of su (3) bases reduces to

冋

册

1 ⳵ sin共␸兲 ⳵ ⳵ i R±c = e±i共␹1−␹2兲 ⫿cos共␸兲 ± + , 2 ⳵r r ⳵␸ r cos共␸兲 ⳵␹2

冋

冉

1 ⳵ cos共␸兲 ⳵ ⳵ ⳵ ⳵ i + − + + Y ±c = e±i␹1 ±sin共␸兲 ± 2 ⳵r r ⳵␸ r sin共␸兲 ⳵␩ ⳵␹1 ⳵␹2

冋

冉

册

冊

1 ⳵ ⳵ ⳵ ⳵ − i cot共␸兲 − + 2i cot共2␸兲 , X±c = e⫿i␹2 ⫿ 2 ⳵␸ ⳵␩ ⳵␹1 ⳵␹2 Rc3 = −

冉

冊

冉

冊

关Rc3,Y ±c 兴 = ± 21 Y ±c ,

关Rc3,X±c 兴 = ± 21 X±c ,

⳵ i ⳵ − , 2 ⳵␹1 ⳵␹2

Rc8 =

i冑3 2 ⳵ ⳵ ⳵ − − 2 3 ⳵␩ ⳵␹1 ⳵␹2

共6.1兲

冊册

,

共6.2兲

共6.3兲

共6.4兲

with the following commutation relations: 关Rc3,R±c 兴 = ± R±c ,

关Rc8,Y ±c 兴 = ± c 关R±c ,X⫿ 兴 = ⫿ Y ±c ,

冑3 2

Y ±c ,

关Rc8,X±c 兴 = ⫿

关X+c ,X−c 兴 = Rc3 − 冑3Rc8,

冑3 2

X±c ,

共6.5兲

关Y ±c ,X±c 兴 = ⫿ R±c .

Therefore, the generators R−c , R+c , Y −c , and Y +c commute with each other. Also the quadratic Casimir operator (2.8) reduces to

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012103-12

H. Panahi and M. A. Jafarizadeh

− Cc =

J. Math. Phys. 46, 012103 (2005)

1 1 ⳵2 1 ⳵2 ⳵2 3 ⳵ 2 cot共2␸兲 ⳵ + + + + 2 2 2 2 2 2 2 2 + 2 ⳵r r ⳵␸ r sin 共␸兲 ⳵␩ r ⳵r r ⳵␸ r sin 共␸兲 ⫻

冉

冊

1 ⳵2 ⳵2 ⳵2 ⳵2 ⳵2 + + 2 − 2 − 2 , ⳵␹1⳵␹2 ⳵␩⳵␹1 ⳵␩⳵␹2 ⳵␹12 cos2共␸兲 ⳵␹22

共6.6兲

˜ c = f −1共r , ␸兲Cc f共r , ␸兲 with f共r , ␸兲 = 1 / 冑r cos共␸兲 we get where after the similarity transformation C ˜ c = − 1 共R ˜ c ˜Rc + ˜Rc ˜Rc + Y c Y c + Y c Y c 兲 −C + − − + + − 2 − + = + +

1 ⳵ 2⳵ 1 1 1 ⳵ ⳵ ⳵2 r + 2 sin共␸兲 + 2 2 + 2 r ⳵r ⳵r r sin共␸兲 ⳵␸ ⳵␸ r sin 共␸兲 ⳵␩2 4r2 cos2共␸兲

冉

冊

1 1 ⳵2 ⳵2 ⳵2 ⳵2 ⳵2 −2 −2 , 2 + 2 +2 2 2 r sin 共␸兲 ⳵␹1 cos 共␸兲 ⳵␹2 ⳵␹1␹2 ⳵␩⳵␹1 ⳵␩⳵␹2 2

共6.7兲

where ˜Y ±c = Y ±c and ˜R±c = R±c ± 关1 / 4r cos共␸兲兴e±i共␹1−␹2兲. Now, by Fourier transforming over the coordinates ␹1 and ␹2 with the kernel ei共m1␹1−m2␹2兲, and the similarity transformation with function ei共m1−m2兲␩, the above Casimir operator reduces to H共m2兲 = −

冋

册

共m22 − 41 兲 1 ⳵ 2⳵ 1 1 ⳵ ⳵ ⳵2 r + sin共 ␸ 兲 + − . r2 ⳵r ⳵r r2 sin共␸兲 ⳵␸ ⳵␸ r2 sin2共␸兲 ⳵␩2 r2 cos2共␸兲

共6.8兲

Also after Inonu–Wigner contraction together with the Fourier transformation, the Casimir eigen共k兲 共k兲 = k 2␺ m provided that for finite l, we value equation 共C / R2兲␺ = 关E共q , l兲 / R2兴␺ reduces to H共m2兲␺m 2 2 let q → ⬁, R → ⬁ such that q / R = finite= k, therefore, we have a hierarchy of isospectral Hamiltonians labeled by the parameter m2 and one can show that this isospectral symmetry comes from the shape invariance symmetry of these Hamiltonians. To see this we first write the Hamiltonian H共m2兲 in terms of Fourier transformed and similarity transformed lowering and rising operators, A±共m2兲 =

B+ =

冉

冊

1 2m2 − 1 ⳵ sin共␸兲 ⳵ ⫿cos共␸兲 ± + , 2 ⳵r r ⳵␸ 2r cos共␸兲

冉

冊

1 1 ⳵ cos共␸兲 ⳵ ⳵ i sin共␸兲 + − + , 2 ⳵r r ⳵␸ r sin共␸兲 ⳵␩ r sin共␸兲

B− =

冉

共6.9兲

冊

1 ⳵ cos共␸兲 ⳵ ⳵ i − sin共␸兲 − − , 2 ⳵r r ⳵␸ r sin共␸兲 ⳵␩

in the following form: H共m2兲 = A+共m2兲A−共m2兲 + B+B− ,

共6.10兲

H共m2 − 1兲 = A−共m2兲A+共m2兲 + B+B− .

共6.11兲

Now, multiplying the eigenvalue equation 共k兲 共k兲 = 共A+共m2兲A−共m2兲 + B+B−兲␺m H共m2兲␺m 2

2

共6.12兲

from the left-hand side by the operator A−共m2兲 and using the fact that A±共m2兲 commute with B±, we obtain

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012103-13

J. Math. Phys. 46, 012103 (2005)

Two and three dimensional Hamiltonians

共k兲 共k兲 H共m2 − 1兲共A−共m2兲␺m 兲 = 共k2共A−共m2兲兲␺m 兲, 2

2

共6.13兲

共k兲 is the eigenfunction of H共m2 − 1兲 with the same eigenvalue k2, hence the therefore, A−共m2兲␺m 2 operator A−共m2兲 lowers the index m2 by one unit. The similarity one shows that A+共m2兲 raises m2 by one unit, respectively. Therefore, the Hamiltonian H共m2兲 possesses ordinary shape invariance symmetry with respect to parameter m2. For half-integer value of the parameter m2 we can obtain the continuous eigenspectrum of these Hamiltonians (since these Hamiltonians are positive definite), simply by acting these lowering and raising operators over the eigenfunction of the free particle as follows. Since for m2 = 21 the Hamiltonian H共m2兲 reduces to H共 21 兲 = −ⵜ2 with the eigenvalue E = k2 ជ 共kជ 兲 and eigenfunction ␺1/2 = eik.xជ and for Hamiltonian H共n2 + 21 兲 with non-negative integer n2, we ជ ជ have H共n2 + 21 兲␺n共k兲 = k2␺n共k兲 with 2

2

ជ ␺n共k2兲

n2

=

A+ 兿 j=1

冉 冊 j+

1 ikជ .xជ e . 2

共6.14兲

VII. CONCLUSION

Here in this work we have generalized the ordinary exactly solvable shape invariance Hamiltonians to Hamiltonians with non-Abelian type of shape invariance symmetry and an ordinary shape invariance one. Again it is shown that the new kind of shape invariance symmetry has its origin in group theory or better to say, the exact solvability of Hamiltonians are related in some way to Lie algebras or Lie groups. APPENDIX: DEGENERACY OF HAMILTONIAN Hl„m1 , m2…

In order to determine the degeneracy of Hamiltonian Hl共m1 , m2兲 for a given value of integer parameters m1 and m2, we should determine the range of variation of integer a3 by imposing the inequalities (4.2). For l 艌 0 and 0 艋 m2 艋 q we have the following three different regions for integer parameter m 1: (I) (II) (III) while (IV) (V) (VI) (VII)

−m1 艋 a3 艋 q − m2, m2 − m1 艋 q, −q 艋 m1 ⬍ 0, degeneracy= q − m2 + m1 + 1, 0 艋 a3 艋 q − m2, 0 艋 m1 ⬍ l + m2, degeneracy= q − m2 + 1, 0 艋 a3 艋 q + l − m1, l + m2 艋 m1 艋 q + l, degeneracy= q + l − m1 + 1, for −q − l 艋 m2 ⬍ 0 there are another four different regions for m1, −m1 艋 a3 艋 q, −q 艋 m1 ⬍ l + m2 ⬍ 0, − q − l 艋 m2 ⬍ 0, degeneracy= q + m1 + 1, −m1 艋 a3 艋 q + l − m1 + m2, l + m2 艋 m1 ⬍ 0, − q − l 艋 m2 ⬍ −l, degeneracy= q + l + m2 + 1, 0 艋 a3 艋 q, 0 艋 m1 ⬍ l + m2, −l 艋 m2 ⬍ 0, degeneracy= q + 1, 0 艋 a3 艋 q + l − m1 + m2, l 艋 m1 − m2 艋 q + l, 0 艋 m1 艋 q + l, −l − q 艋 m2 ⬍ 0, degeneracy = q + l − m1 + m2 + 1.

For l ⬍ 0 and 0 艋 m2 艋 q we have the following four different regions for integer parameter m 1: (I) (II) (III) (IV) while (V) (VI) (VII)

−m1 艋 a3 艋 q − m2, 兩l兩 ⬍ m2 − m1 艋 q, −q 艋 m1 艋 0, 0 艋 m2 艋 q, degeneracy= q − m2 + m1 + 1, −m1 艋 a3 艋 q − 兩l兩 − m1, m2 − 兩l兩 艋 m1 ⬍ 0, 0 艋 m2 ⬍ 兩l兩, degeneracy= q − 兩l兩 + 1, 0 艋 a3 艋 q − m2, 0 艋 m1 艋 m2 − 兩l兩, 兩l兩 艋 m2 艋 q, degeneracy= q − m2 + 1, 0 艋 a3 艋 q − 兩l兩 − m1, 0 艋 m2 − 兩l兩 艋 m1 艋 q − 兩l兩, 0 艋 m2 艋 q, degeneracy= q − 兩l兩 − m1 + 1, for −q + 兩l兩 艋 m2 ⬍ 0 there are another three different regions for m1, −m1 艋 a3 艋 q, −q 艋 m1 ⬍ m2 − 兩l兩, degeneracy= q + m1 + 1, −m1 艋 a3 艋 q − 兩l兩 − m1 + m2, m2 − 兩l兩 艋 m1 ⬍ 0, degeneracy= q − 兩l兩 + m2 + 1, degeneracy= q − 兩l兩 − m1 + m2 0 艋 a3 艋 q − 兩l兩 − m1 + m2, m1 − m2 艋 q − 兩l兩, 0 艋 m1 ⬍ q − 兩l兩, + 1.

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012103-14

H. Panahi and M. A. Jafarizadeh

J. Math. Phys. 46, 012103 (2005)

For given values of the integer parameters q, l, m1, and m2 the eigenspectrum of Hamiltonian Hl共m1 , m2兲 exists in the above seven regions of the 共m1 , m2兲 plane (see Fig. 1 for l 艌 0 and Fig. 2 for l ⬍ 0). Therefore for given values of m1 and m2 the eigenspectrum can be obtained simply by consecutive application of lowering operators over the highest eigenweight according to the paths shown in Figs. 1 and 2. 1

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