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›1mApplication of nonunitary representations

of dynamical groups in quantum-mechanical calculations›m

A.V.Sergeev›?4ma)›?m, A.I.Sherstyuk

›3mS.I.Vavilov State Optical Institute

Tuchkov per. 1, St.-Petersburg, 199034 Russia›m


Finite-dimensional non-unitary representations of a non-

compact dynamical Lie group are used for derivation of exact

solutions of the Schrödinger equation. These solutions

correspond to non-physical values of quantum numbers, but

they may serve both for verification of perturbation theory

expansions and for subsequent extrapolation into the

physical region of quantum numbers. The developed formalism

has been applied to spherically and axially-symmetric

modified Coulomb potentials, the Sturmian eigenvalue problem

and an anharmonic oscillator.

PACS numbers: 03.65.Fd, 03.65.Ge







›1m1. Introduction›m

Elaboration of new methods for high order perturbation

theory (PT) calculations acquire recently a special

significance in connection with the development and

application of various methods for the generalized summation

of divergent series in quantum mechanics and field theory.

The problem is considerably simplified if the dynamical

symmetry of the unperturbed system is taken into account.

The use of various "selection rules" arising from the

symmetry of the system leads to highly convenient recursion

relations between the terms of PT series.Œ1,2‹

Here we examine additional opportunities arising

from complex extensions of Lie algebra corresponding

to the dynamical group of the system concerned. As

the operators in quantum mechanics are mostly infinite

ones we deal with the representations of non-compact Lie

groups and Lie algebras u(p,q) corresponding to them.Œ3‹ All

the unitary irreducible representations of such algebras

attached to the discrete series are infinitely dimensional.

However choosing the Lie algebra of U(p+q) compact group as

the complex extension of u(p,q) Lie algebra we come to study

of finite-dimensional irreducible representations of u(p+q)

algebra coinciding with non-Hermitian irreducible

representations of u(p,q) algebra. They correspond to

finite-dimensional non-unitary representations of U(p,q)

group. In this way we proceed to solve the original problem


in a finite basis which enable us to obtain exact solutions.

On solving specific problems in quantum mechanics this

technique enable us to derive strict relations for the energy

and expectation values of coordinate as functions of quantum

numbers arising from the symmetry of the Hamiltonian. These

relations are explicitly fulfilled after the analytical

continuation in the region of non-physical quantum numbers.

Efficiency of the present approach is demonstrated on an

example of one-particle modified Coulomb potential. The

simpliest case of the spherically-symmetric potential is

investigated in Sec. 2. Besides the purely algebraic

approach we use an approach based on an explicit solution of

the differential equations for eigenfunctions in coordinate

space (Sec. 3). The non-physical values of quantum numbers

are proved to be formally brought into relation with the

irregular (at the origin) solutions of the Schrödinger

equation. More general case of the potential with an axial

symmetry is considered in Sec. 4. Exact non-physical

solutions are obtained particularly for Stark and Zeeman

effects in a hydrogen atom.

›1m2. Modified Coulomb potential and O(2,1) group›m

The class of spherically-symmetric modified Coulomb



›11`V(r) = rŒ-1‹ (-Z + ŽS v‹›3mi›mŒrŒ›3mi›m‹)›55`(1)



is of considerable interest for solving many physical

problems. In eq. (1) r is the radial coordinate, Z and v‹›3mi›mŒ

are constants. The Yukawa potential V(r)=-Z rŒ-1‹ exp(-Žlr),

the funnel-like potential V(r)= -Z rŒ-1‹ + Žlr and a number of

others fall under this category.

The Schrödinger equation for the radial function

multiplied by r

›z›5`Ž' 1 dŒ2‹ ›3ml›m(›3ml›m+1) Ž)

›5`Ž&-Œ_ ___‹ + Œ______‹ + V(r) - EŽ& P(r) = 0›55`(2)

›5`Ž( 2 drŒ2‹ 2rŒ2‹ Ž*›5z

may be represented in the form of an algebraic equation

containing only generators of O(2,1) Lie group T‹1Œ, T‹2Œ,


{(1/4)(T‹3Œ+T‹1Œ) + 2(T‹3Œ-T‹1Œ)[V(2(T‹3Œ-T‹1Œ)) - E]}Y = 0,›55`(3)

where T‹1Œ = -(d/dr)r(d/dr) + sŒ2‹/4r - r/4,

T‹2Œ = (›3mi›m/2)[r(d/dr) + (d/dr)r],

T‹3Œ = T‹1Œ + r/2, s=2›3ml›m+1, Y(r) = rŒ-1/2‹P(r).›55`(4)

The operators T‹1Œ, T‹2Œ, T‹3Œ obey the commutation relations

[T‹1Œ,T‹2Œ]=›3mi›mT‹3Œ, [T‹2Œ,T‹3Œ]=-›3mi›mT‹1Œ, [T‹3Œ,T‹1Œ]=-›3mi›mT‹2Œ.›55`(5)

It follows from (5) that T‹kŒ (k = 1, 2, 3) form the basis of Lie

algebra of noncompact O(2,1) group (or SU(1,1) group locally


isomorphic to it) and at fixed ›3ml›m realize the irreducible

representation of o(2,1) algebra with the Casimir operator

Q = -T‹1Œ Œ2‹ - T‹2Œ Œ2‹ + T‹3Œ Œ2‹ = (sŒ2‹-1)/4 = ›3ml›m(›3ml›m+1).

The eigenfunction system corresponding to one of the

operators T‹kŒ may be chosen as the basis of the

representation. The operator T‹3Œ has a purely discrete

spectrum of eigenvalues n = ›3ml›m+1, ›3ml›m+2, ..., and its

eigenfunctions y‹nŒ coincide (within a common factor) with

the "Sturmian" functions of a hydrogen-like atom.Œ4‹ The action

of T‹kŒ on the basic function y‹nŒ has the

simple form:

(T‹1Œ ± ›3mi›mT‹2Œ)y‹nŒ = (c‹n›3ml›m ŒŒ± ‹)Œ1/2‹ y‹n±1Œ,

T‹3Œ y‹nŒ = n y‹nŒ›55`(6)

where c‹n›3ml›m ŒŒ± ‹ = (n-›3ml›m-1/2±1/2)(n+›3ml›m+1/2±1/2).

After the scaling x = (2/n)r the recursion relations

between the expansion coefficients for the energy and the

wave function may be derived as the consequence of eqs. (3),

(5), (6). This method was used in Refs. 1, 2, 5 for

calculation of high orders of PT for the screened Coulomb

potential. In the general case of the potential (1) it was



E‹n›3ml›mŒ = -ZŒ2‹/2nŒ2‹ + v‹1Œ + (1/2Z)[3nŒ2‹ - ›3ml›m(›3ml›m+1)] v‹2

+ (nŒ2‹/2ZŒ2‹)[5nŒ2‹ + 1 - 3›3ml›m(›3ml›m+1)] v‹3

+ (nŒ2‹/8ZŒ3‹)[35nŒ4‹ + 25nŒ2‹ - 30nŒ2‹›3ml›m(›3ml›m+1) + 3›3ml›mŒ2‹(›3ml›m+1)Œ2‹‹

- 6›3ml›m(›3ml›m+1)] v‹4Œ + (nŒ2‹/8ZŒ4‹)[-7nŒ4‹ - 5nŒ2‹ + 3›3ml›mŒ2‹(›3ml›m+1)Œ2‹] v‹2ŒŒ 2‹‹

+ (nŒ4‹/8ZŒ5‹)[-45nŒ4‹ - 63nŒ2‹ + 14nŒ2‹›3ml›m(›3ml›m+1)‹

+15›3ml›mŒ2‹(›3ml›m+1)Œ2‹ + 10›3ml›m(›3ml›m+1)] v‹2Œv‹3Œ + ...›55`(7)

and an analogical expansion for expectation values. Here

n=1, 2, 3, ... is the principal quantum number, ›3ml›m is the

azimuthal quantum number, 0Ž<›3ml›mŽ approximation EŒ(0)‹= -ZŒ2‹/2nŒ2‹ coincides with the Coulomb

energy. PT corrections represent polynomials in n and ›3ml›m.

For the further purposes it is convenient to regard

su(1,1) algebra as the complex extension of the real Lie

algebra of SU(2) compact group locally isomorphic to the

group of three-dimensional rotations. Its generators L‹kŒ (k=

1, 2, 3) are the self-adjoint operators obeing the relations

[L‹1Œ,L‹2Œ]=-›3mi›mL‹3Œ, [L‹2Œ,L‹3Œ]=-›3mi›mL‹1Œ, [L‹3Œ,L‹1Œ]=-›3mi›mL‹2Œ,

Q' = L‹1Œ Œ2‹ + L‹2Œ Œ2‹ + L‹3Œ Œ2‹ = (sŒ2‹-1)/4, s=1,2,3,...›55`(8)

Let us consider three non-Hermitian operators

T' ‹kŒ=exp(›3mi›mŽj‹kŒ)L‹kŒ. It is no trouble to prove that at Žj‹1Œ=Žj‹2Œ=Žp/2

and Žj‹3Œ=0 they obey commutation relations (5). So their

matrix representation in the basis of eigenfunctions

corresponding to one of the operators

T' ‹1Œ=›3mi›mL‹1Œ, T' ‹2Œ=›3mi›mL‹2Œ, T' ‹3Œ=L‹3Œ›55`(9)


(or L‹kŒ) realizes at fixed s the finite-dimensional (s-

dimensional) non-Hermitian representation of su(1,1)

algebra. Particularly the basis of such representation is

formed by the eigenvectors of the operator T' ‹3Œ

corresponding to eigenvalues n = -›3ml›m, -›3ml›m+1, ..., ›3ml›m, where

›3ml›m = (s-1)/2, and the formulae for the matrix elements of

the operators T' ‹kŒ (k=1, 2, 3) are the same as for T‹kŒ and

they follow from (6).

Now let us consider eq. (3) in which T‹1Œ and T‹3Œ are

substituted by T' ‹1Œ and T' ‹3Œ. Since the relations (3), (5), (6)

hold both for the self-adjoint representation (4) and for

the representation (9) the dependence of its solution upon

n, ›3ml›m is the same in both cases. But now the problem is

considerably simplified because eq. (3) is equivalent to the

finite system of linear equations for s components of the

vector Y in the space of the irreducible representation of

su(2) algebra :


›20`ŽS R‹›3mi›mjŒ Y‹jŒ = 0›55`(10)


Denoting L‹±Œ = L‹3Œ±›3mi›mL‹1Œ = T' ‹3Œ±T' ‹1Œ,


v(x) = xV(x) = ŽS v‹kŒ xŒk‹, v‹0Œ = -Z we obtain


R = L‹+Œ/4 + v(2L‹-Œ) - 2EL‹-Œ.

It is convenient to choose the eigenvectors of the operator



y' ‹kŒ = (2L‹-Œ)Œk-1‹ y' ‹1Œ, k = 1,2,...,s,›55`(11)

as the basis of the representation, where y' ‹1Œ corresponds

to the minimal eigenvalue (L‹2Œy' ‹1Œ = -›3ml›my' ‹1Œ). In this basis the

matrix elements assume the form

(L‹-Œ)‹›3mi›mjŒ = (1/2)Žd‹›3mi›m,j+1Œ, (L‹+Œ)‹›3mi›mjŒ = 2›3mi›m(s-›3mi›m)Žd‹›3mi›m,j-1Œ,


R‹›3mi›mjŒ = Œ1 _‹‹ 2Œ›3mi›m(s-›3mi›m)Žd‹›3mi›m,j-1Œ + ŽS v‹kŒŽd‹›3mi›m,j+kŒ- EŽd‹›3mi›m,j+1Œ›55`(12)


(really an index k in eq. (12) is limited to s-1). The

determinant of the matrix R is a polynomial in E and v‹kŒ (k=0,

1, ..., s-1). The energy is determined from the condition

det R = 0 (13)

and represents a multi-valued algebraic function in v‹0Œ, ...,

v‹s-1Œ. Specifically for purely Coulomb potential we have

EŒ(0)‹= -v‹0 ŒŒ2‹/2nŒ2‹, where n = -›3ml›m, -›3ml›m+1, ..., ›3ml›m and nŽ=0. The

solution of eq. (13) approaching EŒ(0)‹ at the Coulomb limit

represents the energy level with non-physical quantum numbers

and generates the expansion coefficients (7) at |n|Ž<›3ml›m.

For example at n = ›3ml›m = 1/2 (s=2)

E‹n›3ml›mŒ= -2ZŒ2‹+ v‹1,›55`(14)

and at n = ›3ml›m = 1 (s=3)


E‹n›3ml›mŒ= -ZŒ2‹/2 + v‹1Œ + v‹2Œ/2Z,›55`(15)

It follows from (14) and (15) that all coefficients in (7)

with an exception of a few first ones vanish for such

values of n, ›3ml›m. Hence we obtain exact relations for the

expansion coefficients (7) at any order of PT. The

established relations may be used to check the

expressions for these coefficients.

We list also the expressions for E‹n›3ml›mŒ at s=4 and s=5:

E‹1±1/2,3/2Œ = (2/9) ZŒ2‹{-5

± 4[1 + (27/8) v‹2Œ/ZŒ3‹ + (81/32) v‹3Œ/ZŒ4‹]Œ1/2‹} + v‹1Œ,›55`(16)

E‹3/2±1/2,2Œ = (1/16) ZŒ2‹[-5 + 12v‹2Œ/ZŒ3‹ ± 3(1 + 24v‹2Œ/ZŒ3‹

+ 64v‹3Œ/ZŒ4‹ + 16v‹2 ŒŒ2‹/ZŒ6‹ + 64v‹4Œ/ZŒ5‹)Œ1/2‹] + v‹1Œ.›55`(17)

By an expansion of the functions (14) - (17) in powers of

v‹1Œ, v‹2Œ, v‹2Œ, v‹4Œ it is possible to find the values of the

polynomials in eq. (7) for six pairs of (n, ›3ml›m). Using the

determined values it is not difficult to fit the

coefficients of the polynomials in front of v‹2Œ, v‹3Œ, v‹4Œ,

v‹2 ŒŒ2‹, v‹2Œv‹3Œ. By this way the energy may be extrapolated into

the physical region n>›3ml›m at any order of PT. So we obtain

an alternative method for making the expansion (7).

According to the Hellmann - Feynman theorem

= ŽoE/Žov‹kŒ, k=0,1,2,...›55`(18)


So the expectation values may be expanded in powers of

v‹2Œ, v‹3Œ, ..., the expansion coefficients representing

polynomials in n, ›3ml›m.Œ6‹ In the region of

non-physical quantum numbers they may be determined by

differentiation of the non-physical energy. For example

at n = ›3ml›m = 1/2 =4Z, =0 for jŽ>1 and at n = ›3ml›m =1

=Z+v‹2Œ/2ZŒ2‹, =1/2Z, =0 for jŽ>2.

›1m3. The solution of the problem at |n|Ž<›3ml›m›1m in the

coordinate representation›m

Let us consider a singular solution of the

differential eq. (2) having the expansion at rŽ}0

P(r) = c‹1Œ rŒ-›3ml›m‹ + c‹2Œ rŒ-›3ml›m+1‹ + ...›55`(19)

After the substitution of (19) into eq. (2) we obtain an

infinite system of linear equations for coefficients c‹1Œ,

c‹2Œ, ...

›z ŽB

ŽS R‹›3mi›mjŒ c‹jŒ = 0, ›3mi›m = 1,2,...›55`(20)


where R‹›3mi›mjŒ are defined by eq. (12) as before. Since R‹›3mi›mjŒ=0

at j>›3mi›m+1 for ›3mi›mŽ›3mi›m for ›3mi›m=s the subsystem of

the first s eqs. of (20) contains only unknown quantities

c‹1Œ, ... c‹sŒ and coincides with the system (10) if we put

c‹›3mi›mŒ = Y‹›3mi›mŒ. So in both cases the spectrum of eigenvalues is

the same. The basis (11) obviously coicides with the set


{rŒ-›3ml›m-1/2‹, rŒ-›3ml›m+1/2‹, ..., rŒ›3ml›m-1/2‹} since 2L‹-Œ=r.

Putting ›3mi›m = s+1, s+2, ... in eq. (20) we obtain


c‹›3mi›m+1Œ = ›4mŒ 2 ›mŽ+‹‹ Ž,Œ ŽS v‹kŒ c‹›3mi›m-kŒ - Ec‹›3mi›m-1Ž. ŒŒŽ-‹›55`(21)

›8`Œ›3mi›m(›3mi›m-s)‹ k=0›5z

It means that the coefficient c‹s+1Œ may be chosen

arbitrarily and c‹s+2Œ, c‹s+3Œ and so on may be succesively

determined by eq. (21).

We list for example the wave function of the several

non-physical states at Z=1:

P‹(n=›3ml›m=1/2)Œ = rŒ-1/2‹ + 2rŒ1/2‹ + ›3mO›m(rŒ3/2‹),

P‹(n=›3ml›m=1)Œ = rŒ-1‹ + 1 + (1/2)(1+vŒ2‹‹ 2Œ)r + ›3mO›m(rŒ2‹),

P‹(n=1±1/2,›3ml›m=3/2)Œ = rŒ-3/2‹ + (2/3)rŒ-1/2‹

+ (2/9)(-1±2D)rŒ1/2‹ + (2/81)(-26-27v‹2Œ±28D)rŒ3/2‹ + ›3mO›m(rŒ5/2‹),

P‹(n=3/2±1/2,›3ml›m=2)Œ = rŒ-2‹ + (1/2)rŒ-1‹ + (1/16)(1+4v‹2Œ±D‹1Œ)

+ (1/32)[-1-4v‹2Œ±(5/3)D‹1Œ]r + (1/256)[-5-48v‹2Œ-64v‹3Œ

+ 48v‹2 ŒŒ2‹+96v‹4Œ±(17/3+12v‹2Œ)D‹1Œ]rŒ2‹ + ›3mO›m(rŒ3‹),

where D = [1+(27/8)v‹2Œ+(81/32)v‹3Œ]Œ1/2‹,

D‹1Œ = (1+24v‹2Œ+64v‹3Œ+64v‹4Œ+16v‹2 ŒŒ2‹)Œ1/2‹.


Let us dwell in more detail upon calculation of the

matrix elements of various operators at the non-physical

quantum numbers. Here we confine ourselves to diagonal

matrix elements (expectation values) of operators rŒk-1‹

(kŽ>0) defined in Sec. 2 as the functions ŽoE/Žov‹kŒ

analytically continuated into the region of non-physical

quantum numbers in every order of PT. The convential

definition of the expectation value by means of integration

obviously is not suitable in the present case because the

singular wavefunctions (19) are no longer square-integrable.

The problem arises how to generalize the convential scalar

production in a Hilbert space of square-integrable functions

on the case of non-physical quantum numbers in order to

express the expectation values in terms of wavefunctions.

Note in this connection that according to eq. (12) the

matrix R is not symmetrical which reflects the fact that the

Hamiltonian is no more a self-adjoint operator. But the

matrix R became symmetrical after the transposition of its


R Œp‹‹›3mi›mjŒ = R Œp‹‹j›3mi›mŒ = R‹s+1-›3mi›m,jŒ = H‹›3mi›mjŒ - ES‹›3mi›mjŒ,



›11`H‹›3mi›mjŒ = (1/2)(›3mi›m-1)(j-1)Žd‹s,›3mi›m+j-2Œ + ŽS v‹kŒ Žd‹s,›3mi›m+j+k-1Œ,


›11`S‹›3mi›mjŒ = Žd‹s,›3mi›m+jŒ.

So the subsystem consisting the first s eqs. (20)

reduces to a generalized eigenvalue problem with a weight


S for some self-adjoint operator:


›11`ŽS (H‹›3mi›mjŒ - ES‹›3mi›mjŒ)c‹jŒ = 0, ›3mi›m = 1,2,...,s.›55`(22)


Hence we obtain an analogue to the Hellmann - Feynman


›zŽoE Ž1qqŒ s‹qŽ5 Ž1qqŒ s‹qŽ5 ŽoH‹›3mi›mjŒ

Œ___‹ Ž7 ‹‹qqqŒŒ c‹›3mi›mŒ S‹›3mi›mjŒ c‹jŒ = Ž7 ‹‹qqqŒ Œ c‹›3mi›mŒ Œ____‹ c‹jŒ.›55`(23)

Žov‹kŒ Ž2 ‹‹›3mi›m,j=1 ŒŒŽ6 Ž2 ‹‹›3mi›m,j=1 ŒŒŽ6 Žov‹kŒ›5z

Let us introduce an operation in the space of functions

of the type (19)

›z Ž1qq Œs‹qŽ5 2›3ml›m

{P‹1Œ,P‹2Œ} = Ž7 c‹›3mi›mŒ Œ(1)‹ S‹›3mi›mjŒ c‹jŒ Œ(2)‹ = ŽS c‹›3mi›mŒ Œ(1)‹ cŒ(2)‹ ‹2›3ml›m+1-›3mi›mŒ,›55`(24)

›11`Ž2qqqŽ6 ›3mi›m=1


where c‹›3mi›m ŒŒ(1)‹ and c‹j ŒŒ(2)‹ (›3mi›m, j = 1, 2, ..., s) are the

coefficients in the expansion of functions P‹1Œ and P‹2Œ

correspondingly. Then eq. (23) may be rewritten in a more

clear form:

›10`›z = Œ›4mŽoE ›m‹ = Œ›4m{P,r P}›m Œk-1‹‹›55`(25)

›19`ŒŽov‹kŒ {P,P}›5z

As a result we find the operation (24) to be an analogue

of the scalar production at nonphysical quantum numbers (this

operation is not a new scalar production in the strict sense

of the term because the weight operator S is not positively


definite). Obviously {P‹1Œ, P‹2Œ} coincides with the coefficient

before rŒ-1‹ in the Laurant expansion of the product P‹1ŒP‹2Œ.

The operation (24) also may be defined as

{P‹1Œ,P‹2Œ} = (1/2Žp›3mi›m) ŒŽ$ ‹‹Ž%Œ›19`›5wO›w P‹1Œ(z)P‹2Œ(z)dz,

where the integration is carried out in a positive direction

over a small cycle in a complex plane embracing the origin.

The method expounded here is highly suitable for

calculating the expectation values as it do not require the

estimation of integrals. Let us calculate for example the

Coulomb expectation values.

In the case of the Coulomb potential V(r) = -Z/r the non-

physical solutions are represented by the singular

(at r Ž}0) functions

P‹n›3ml›mŒ(z) = exp(-Zr/n) rŒn‹ ‹2ŒF‹0Œ(-n+›3ml›m+1,-n-›3ml›m;-n/2Zr)›55`(27)

(at rŽ}ŽB P‹n›3ml›mŒŽ}0 for n>0 and P‹n›3ml›mŒŽ}ŽB for n<0). The corresponding

eigenenergies are E‹n›3ml›mŒ = -1/2nŒ2‹.

Further we shall use the relation

P‹n›3ml›mŒ(r) = A‹n›3ml›mŒ P‹-n,›3ml›mŒ(r) + Q‹n›3ml›mŒ(r)›55`(28)

following from the known formula for the confluent

hypergeometric function (see Ref. 7, eq. (7.6.3)). In

eq. (28)

A‹n›3ml›mŒ = (-1)Œn-›3ml›m‹ (n/2Z)Œ2n‹ (›3ml›m+n)!/(›3ml›m-n)!


is a constant and

Q‹n›3ml›mŒ(r) = (-n/2Z)Œn-›3ml›m-1‹ (›3ml›m+n)!/(2›3ml›m+1)! rŒ›3ml›m+1‹ exp(-Zr/n)

ŽK ‹1ŒF‹1Œ(-n+›3ml›m+1;2›3ml›m+2;2Zr/n) = ›3mO›m(rŒ›3ml›m+1‹)

is a regular (at rŽ}0) solution of the Schrödinger eq.

(2). Its analytical continuation into the physical region

n>›3ml›m produces the wavefunction for a hydrogen-like atom

which is regular at the origin and at infinity.

Using eq. (28) we find

{P‹n›3ml›mŒ,rŒk-1‹P‹n›3ml›mŒ} = A‹n›3ml›mŒ{P‹n›3ml›mŒ,rŒk-1‹P‹-n,›3ml›mŒ} + {P‹n›3ml›mŒ,rŒk- 1‹Q‹n›3ml›mŒ}›55`(29)

For kŽ>0 the secound term in r. h. s. of eq. (29) is

eliminated. So according to eq. (25)

›z›10`{P‹n›3ml›mŒ,rŒk-1‹P‹n›3ml›mŒ} {P‹n›3ml›mŒ,rŒk-1‹P‹-n,›3ml›mŒ} Z

= Œ_____________‹ = Œ_______________‹ = Œ__‹ f‹kŒ›55`(30)

›12`{P‹n›3ml›mŒ,P‹n›3ml›mŒ} {P‹n›3ml›mŒ,P‹-n,›3ml›mŒ} nŒ2‹›5z

where f‹kŒ (k = 0, 1, ..., 2›3ml›m) are the coefficients of the

expansion in powers of 1/r of the function

f(r) = P‹n›3ml›mŒ(r) P‹-n,›3ml›mŒ(r)

= ‹2ŒF‹0Œ(-n+›3ml›m+1,-n-›3ml›m;-n/2Zr) ‹2ŒF‹0Œ(n+›3ml›m+1,n-›3ml›m;n/2Zr)

= 1 + (nŒ2‹/Z)rŒ-1‹ + (nŒ2‹/2ZŒ2‹)(3nŒ2‹-›3ml›mŒ2‹-›3ml›m)rŒ-2‹ + ...,


and f‹kŒ = 0 for k>2›3ml›m. Particularly = Z/nŒ2‹,

= (1/2Z)(3nŒ2‹-›3ml›mŒ2‹-›3ml›m) and so on. Such formulae may be

directly continuated into the region of physical values

n>›3ml›m and they coincide with the results of Grant and

Lai.Œ6‹ The fact that these expressions vanish at

nonphysical values |n|Ž<›3ml›m and ›3ml›m easy verification of their correctness.

Apart from diagonal matrix elements let us calculate for

example an off-diagonal radial matrix element:


= Œ_______________________________‹

›18`{P‹n,›3ml›m-1Œ,P‹n,›3ml›m-1Œ}Œ1/2‹ {P‹n›3ml›mŒ,P‹n›3ml›mŒ}Œ1/2‹


›11`= Œ___________ __________________________________‹

›13`AŒ1/2‹‹ n,›3ml›m-1Œ AŒ1/2‹‹ n›3ml›mŒ {P‹n,›3ml›m-1Œ,P‹-n,›3ml›m-1Œ}Œ1/2‹ {P‹n›3ml›mŒ,P‹-n,›3ml›mŒ}Œ1/2‹

›3e›13`Z Ž+ ›3ml›m+nŽ-Œ1/2‹

›11`= Œ__‹ Ž&-Œ___‹Ž& g‹k+1Œ,

›13`nŒ2‹ Ž, ›3ml›m-nŽ.›5z

where g‹›3mi›mŒ (›3mi›m = 0, 1, ..., 2›3ml›m-1) are the coefficients of the

expansion in powers of 1/r of the function

g(r) = P‹n,›3ml›m-1Œ(r) P‹-n,›3ml›mŒ(r)

= ‹2ŒF‹0Œ(-n+›3ml›m,-n-›3ml›m+1;-n/2Zr) ‹2ŒF‹0Œ(n+›3ml›m+1,n-›3ml›m;n/2Zr)

= 1 - (n/Z)(›3ml›m-n)rŒ-1‹ - (3nŒ3‹/2ZŒ2‹)(›3ml›m-n)rŒ-2‹

- (nŒ3‹/2ZŒ3‹)(›3ml›m-n)(5nŒ2‹-›3ml›mŒ2‹+1)rŒ-3‹ + ...,

g‹›3mi›mŒ=0 at ›3mi›mŽ>2›3ml›m, and kŽ>0. Particularly


= -(1/n)(nŒ2‹-›3ml›mŒ2‹)Œ1/2‹,

= -(3n/2Z)(nŒ2‹-›3ml›mŒ2‹)Œ1/2‹,

= -(n/2ZŒ2‹)(5nŒ2‹-›3ml›mŒ2‹+1)(nŒ2‹-›3ml›mŒ2‹)Œ1/2‹.

Such formulae are also true for the physical values n>›3ml›m

and they agree with the results of Shertzer.Œ8‹

Without any trouble one can transfer the developed

technique of non-physical quantum numbers on the Sturmian

approach to the problem when one searches for allowable

values of the charge at fixed energy. At the same time the

Sturmian approach for the modified Coulomb potential is

equivalent to the Schrödinger approach for the anharmonic

oscillator. These results are given in Appendix.

›1m4. Generalization on axially-symmetric potentials›m

A generalization of the potential (1) is represented by

an axially-symmetric potential of the type


›11`V(r,z) = (1/r) ŽS v‹›3mi›mkŒrŒ›3mi›m‹zŒk‹,›55`(31)


where the coordinate z is directed along the axis of the

symmetry. As before Z = -v‹00Œ is Coulomb charge.

Let us introduce parabolic coordinates Žx = r+z, Žh = r-z,

and rewrite the potential (31) in the form



V = Œ___‹f(Žx,Žh), f(Žx,Žh) = ŽS f‹›3mi›mkŒŽxŒ›3mi›m‹ŽhŒk‹,›55`(32)


where f‹00Œ=v‹00Œ, f‹10Œ=(v‹10Œ+v‹01Œ)/2, f‹01Œ=(v‹10Œ-v‹01Œ)/2,

f‹11Œ=(v‹20Œ-v‹02Œ)/2 and so on. In parabolic coordinates the

Schrödinger equation multiplied by r takes the form

›zŽ' Ž+Žo __‹ŽxŒŽo __ Žo __‹ŽhŒŽo __Ž- m _Œ2‹ -1 -1

Ž(Œ-‹Ž,ŽoŽx ŽoŽxŒ+‹ŽoŽh ŽoŽhŽ. Œ+ ‹4 Œ(Žx +Žh )›5z

‹‹›20`+ f(Žx,Žh) - ŒE _ ‹‹2Œ(Žx+Žh)ŒŽ) ‹‹Ž*Œ Žq(Žx,Žh) = 0›55`(33)

and contains only the operators T‹1Œ(Žx), T‹3Œ(Žx), T‹1Œ(Žh),

T‹3Œ(Žh) defined by eqs. (4) in which s=|m| and r is

substituted by Žx or Žh.

We search for non-physical solutions of eq. (33) in

the form


Žq(Žx,Žh) = (ŽxŽh)Œ-s/2‹ ŽS c‹›3mi›mjŒ ŽxŒ›3mi›m‹ ŽhŒj‹.›55`(34)


Substituting eq. (34) into eq. (33) we arrive to the system

of linear equations for the coefficients c‹›3mi›mjŒ

(›3mi›m+1)(s-›3mi›m-1)c‹›3mi›m+1,jŒ + (j+1)(s-j-1)c‹›3mi›m,j+1Œ


›1`+ ŽS f‹›3mi›m'j'Œ c‹›3mi›m-›3mi›m',j-j'Œ - (E/2)(c‹›3mi›m-1,jŒ+c‹›3mi›m,j-1Œ) = 0›55`(35)



where it is assumed that c‹›3mi›mjŒ = 0 at ›3mi›m<0 or j<0.

If s is a positive integer the infinite system of eqs.

(35) for 0Ž<›3mi›m,j<ŽB is separated on the closed finite

subsystem of eqs. for ›3mi›m, j Ž< s-1 and on the infinite

subsystem for another (›3mi›m, j). By putting the determinant of

the finite sybsystem equal to zero one can find

the non-physical energy levels. In the purely Coulomb case

E = -ZŒ2‹/2(n‹1Œ+n‹2Œ)Œ2‹ where the quantum numbers n‹1Œ and n‹2Œ vary

from -(s-1)/2 to (s-1)/2 unlike for the physical states

when n‹›3mi›mŒ = p‹›3mi›mŒ + (s+1)/2 and p‹›3mi›mŒ = 0,1,2,... In the general

case of arbitrary potential V(r,z) at s=2 there is a single

energy level with n‹1Œ = n‹2Œ = ±1/2

E = -ZŒ2‹/2 + v‹10Œ + f‹11ŒZŒ-1‹ - vŒ2‹‹ 01Œ/2ZŒ2‹.›55`(36)

At s=3 there are three levels with n‹1Œ=0 and n‹2Œ=±1,

n‹1Œ=±1 and n‹2Œ=0, and n‹1Œ=n‹2Œ=±1 found from the cubic equation

[8(E-v‹10Œ) + 1 + 16(vŒ2‹‹ 01Œ+v‹02Œ-v‹20Œ)] {[2(E-v‹10Œ)+1]Œ2‹ - 4vŒ2‹‹ 01Œ}‹

- 4[2(E-v‹10Œ)+1][3s‹02Œ-3vŒ2‹‹ 20Œ+5vŒ2‹‹ 02Œ + 4(3s‹12Œ+6f‹22Œ+5v‹02Œv‹20Œ‹

-4v‹01Œd‹12Œ+4vŒ2‹‹ 01Œs‹02Œ)] + 3(9vŒ2‹‹ 11Œ+3vŒ2‹‹ 20Œ+3vŒ2‹‹ 02Œ-10v‹02Œv‹20Œ‹

-12v‹01Œv‹11Œ-48v‹11Œd‹12Œ) - 32 vŒ2‹‹ 02Œ(3v‹02Œ-v‹20Œ) + 96[v‹01Œd‹12Œ-s‹02Œd‹12Œ‹

+ (v‹20Œ-v‹02Œ)s‹12Œ + f‹22Œ(v‹20Œ-3v‹02Œ) + 2dŒ2‹‹ 12Œ] + 48v‹01Œd‹12Œ(v‹20Œ-7v‹02Œ)‹

- 128v‹01Œ[v‹01Œ(s‹02Œ+s‹12Œ+2f‹22Œ) - 2d‹12Œv‹02Œ] = 0,›55`(37)

where s‹02Œ = f‹02Œ+f‹20Œ = (1/2)(v‹02Œ+v‹20Œ),‹

s‹12Œ = f‹12Œ+f‹21Œ = (1/4)(3v‹30Œ-v‹21Œ),‹

d‹12Œ = f‹12Œ-f‹21Œ = (1/4)(3v‹03Œ-v‹21Œ), f‹22Œ = (1/8)(3v‹40Œ-v‹22Œ+3v‹04Œ),


the eq. (37) being written for the case Z=1. At arbitrary s

there are s(s-1)/2 levels. If the potential has a spherical

symmetry (v‹0›3mi›mŒ = 0) the energy (36) is converted into (15)

and the solutions of eq. (37) are converted into (15) and


As an illustration of the above approach let us

consider an exactly solvable model with a potential V(r,z)

= -1/r + Žlz/r. It follows from eqs. (36) and (37) that

E‹1/2,1/2Œ = -1/2-ŽlŒ2‹/2 at s=2 and E‹1,0Œ = -1/2+Žl, E‹01Œ = -1/2-Žl,

E‹11Œ = -1/8-2ŽlŒ2‹ at s=3. These four nonphysical energies agree

with the general formula for the energy independent on m

›z›14`1Ž+ 1 + 4n‹2ŒŒ 2‹ Žl Ž-Œ2‹

E‹n›?5m1›?m,n›?5m2›?mŒ = Žl - Œ_‹Ž/Œ_____________________‹Ž0

›14`2Ž,n‹2Œ[1+4Žl(n‹2ŒŒ 2‹-n‹1ŒŒ 2‹)]Œ1/2‹+n‹1ŒŽ.

›3e›8`1 Ž+n‹1Œ-n‹2ŒŽ-

= -Œ_________‹ + Ž&Œ_____‹Ž&Žl - 2n‹1Œn‹2ŒŽlŒ2‹ - 4n‹1Œn‹2Œ(n‹1ŒŒ 2‹-n‹2ŒŒ 2‹)ŽlŒ3‹ + ›3mO›m(ŽlŒ4‹).

2(n‹1Œ+n‹2Œ)Œ2‹ Ž,n‹1Œ+n‹2ŒŽ.›5z


For another example we take the potential V(r,z) = -1/r - Ez

+ (HŒ2‹/8)(rŒ2‹-zŒ2‹) which describes a hydrogen atom in parallel

electric (›1mE›m) and magnetic (›1mH›m) fields. At s=2 we obtain E

= -1/2. In this case the energy shift (from the Coulomb

energy) vanishes and all the coefficients in the PT

expansion in powers of E and H go to zero. For instance

in the expansion of the Stark energy

E‹n›?5m1›?mn›?5m2›?mmŒ = -1/2(n‹1Œ+n‹2Œ)Œ2‹ + (3/2)(n‹1ŒŒ 2‹-n‹2ŒŒ 2‹)E‹

+ (1/16)(n‹1Œ+n‹2Œ)Œ4‹[-14(n‹1ŒŒ 2‹+n‹2ŒŒ 2‹)-40n‹1Œn‹2Œ+9mŒ2‹-19]EŒ2‹ + ›3mO›m(EŒ3‹)


all the polynomials in n‹1Œ, n‹2Œ, m before various powers

of E go to zero at n‹1Œ=m‹2Œ=1/2 and m=2. At s=3 the non-

physical energies are determined from eq.

(2E+1)Œ2‹(8E+1) - 6(2E+1) HŒ2‹ + 27 EŒ2‹ = 0,

and at s=4 they are determined from eq.

(2E+1)Œ3‹ (8E+1)Œ2‹ (18E+1) - 90(2E+1)Œ2‹ (8E+1)(6E+1) HŒ2‹‹

+ 9(2E+1)(346E+77) HŒ4‹ + 324 HŒ6‹ + 72(2E+1)(280EŒ2‹+94E+13) EŒ2‹‹

- 648(2E-5) EŒ2‹ HŒ2‹ - 34992 EŒ4‹ = 0.

›1m5. Summary›m

In the present paper the method is proposed to calculate

the bound states of quantum systems by means of PT taking

into account the symmetry of the unperturbed Hamiltonian.

The use of the recursion relations for the matrix elements

following from the symmetry enable one to present the

dependence of PT corrections on the quantum numbers as

simple analytical expressions (polynomials) admitting the

analytical continuation generally on arbitrary complex-

valued quantum numbers.

In the spherically-symmetric case investigated

in detail it appeared to be useful to examine

the behavior of PT expressions in the non-physical region

of quantum numbers n, ›3ml›m (|n|Ž<›3ml›m) because the expressions for

PT corrections at any order are essentially simplified in

this region. The developed approach appears to be suitable

both for verification of PT terms determined by another


methods and for direct calculation of its values for

arbitrary state by extrapolation of the obtained expressions

into the physical region of quantum numbers.

Mathematically the states corresponding to |n|Ž<›3ml›m arise

during the consideration of finite-dimensional non-unitary

representations of a dynamical group (O(2,1) group at present

case). Such states have no physical interpretation because

the corresponding solutions tend to infinity at r=0. The

approach applied here may be generalized on another

Hamiltonians having some non-compact Lie group as a group

of dynamical symmetry. As an example there has been

considered a case of an axially-symmetric potential. The

system of the hydrogen-like Sturmian functions in the

non-physical region in connection with the perturbative

problem for a spherically-symmetric oscillator was

investigated in Appendix.


One of us (A.I.Sherstyuk) wishes to thank Professor

B.R.Wybourne for useful discussion. This work was supported

in part by a grant from firm "Vopros".

›1mAppendix. Sturmian approach to the problem›m

Sometimes it is suitable to rewrite the Schrödinger

equation (2) in the form

›z›11`Ž' 1 dŒ2‹ ›3ml›m(›3ml›m+1) ZŽ)

›11`Ž&-Œ_ ___‹ + Œ______‹ + W(r) - Œ_‹Ž& P(r) = 0›55`(A.1)

›11`Ž( 2 drŒ2‹ 2rŒ2‹ rŽ*›5z

where W(r) = ŽnŒ2‹/2 + w‹1Œr + w‹2ŒrŒ2‹ + ..., Žn = [-2(E-v‹1Œ)]Œ1/2‹,›1;80"s›1`›"z›21mŽn is ›3;21mnu›m›2"z›15;80"s

w‹1Œ=v‹2Œ, w‹2Œ=v‹3Œ and so on. At fixed energy the charge in

eq. (A.1) assumes quantized values


Z‹n›3ml›mŒ = nŽn + (1/2ŽnŒ2‹)[3nŒ2‹-›3ml›m(›3ml›m+1)]w‹1Œ - (n/8ŽnŒ5‹)[34nŒ2‹- 18›3ml›m(›3ml›m+1)+5]w‹1ŒŒ 2

+ (n/2ŽnŒ3‹)[5nŒ2‹-3›3ml›m(›3ml›m+1)+1]w‹2Œ + ...›55`(A.2)

where n = ›3ml›m+1, ›3ml›m+2, ... and the wavefunctions (within some

limitations on the potential) form the complete system in

the functional space with a scalar product


›10`(P‹1Œ, P‹2Œ) = Ž? P‹1Œ(r)rŒ-1‹P‹2Œ(r)dr›55`(A.3)


In the Sturmian approach the non-physical quantum number n

takes the values n = -›3ml›m, -›3ml›m+1, ... ›3ml›m including zero

unlike the Schrödinger approach. As before the charge is

expressible in the asymptotic form (A.2) but it may be also

determined from an algebraic equation. For example Z‹00Œ = 0

at s=1, Z‹±1/2,1/2Œ = ±Žn/2 at s=2 and Z‹-1,1Œ, Z‹0,1Œ, Z‹1,1Œ

represent the roots of the cubic equation Z(ZŒ2‹-ŽnŒ2‹) - w‹1Œ = 0

at s=3.

Let us assume that all the non-physical values of the

charge are different. Then the eigenfunctions corresponding

to them are orthogonal ones:

= Žd‹nn'Œ,›55`(A.4)

where the scalar product of two non-physical functions PŒ(1)‹

and PŒ(2)‹ of the type

PŒ(›3mi›m)‹(r) = c‹1ŒŒ (›3mi›m)‹rŒ-›3ml›m‹ + c‹2ŒŒ (›3mi›m)‹rŒ-›3ml›m+1‹ + ... + c‹sŒŒ (›3mi›m)‹rŒ›3ml›m‹ + ›3mO›m(rŒ›3ml›m+1‹)›55`(A.5)

is defined as



= ŽS c‹kŒŒ (›3mi›m)‹cŒ(2) ‹‹s-k+1Œ = {PŒ(1)‹,rŒ-1‹PŒ(2)‹}.›55`(A.6)


In addition, any function in the form (A.5)

may be expanded over the system of functions P‹n›3ml›mŒ

with n = -›3ml›m, -›3ml›m+1, ... ›3ml›m:


P(r) = ŽS Œ-1‹ P‹n›3ml›mŒ(r) + ›3mO›m(rŒ›3ml›m+1‹).›55`(A.7)


Notice that eq. (A.1) after the substitution of the

variable r = r'Œ2‹ is transformed into the Schrödinger equation

for a spherically-symmetric anharmonic oscillator

›zŽ' 1 dŒ2‹ ›3ml›m'(›3ml›m'+1) Ž)

Ž&-Œ_ ____‹ + Œ________‹ + V'(r') - E'Ž& P'(r') = 0›55`(A.8)

Ž( 2 dr'Œ2‹ 2r'Œ2‹ Ž*›5z

where ›3ml›m' = 2›3ml›m +1/2, V'(r') = 4r'Œ2‹ W(r'Œ2‹)‹

= ŽwŒ2‹r'Œ2‹/2 + v'‹ 1 Œr'Œ4‹ + v'‹ 2 Œr'Œ6‹ + ...,‹

Žw=2Žn, v'‹ ›3mi›mŒ = 4w‹›3mi›mŒ (›3mi›m=1,2,...), E'=4Z, P'(r') = r'Œ-1/2‹ P(r'Œ2‹).

This fact allows one to obtain exact solutions of eq. (A.8)

at non-physical half-integer values of ›3ml›m' = 1/2, 3/2, ...

The existence of such solutions for an anharmonic oscillator

was firstly mentioned by Zinn-Justin.Œ9‹



Œ1‹A.I.Sherstyuk and A.M.Shkol'nik, Izv.Akad.Nauk SSSR Ser.

Fiz. ›1m41›m, 2648 (1977).

Œ2‹A.Bechler, Ann.Phys. ›1m108›m, 49 (1977).

Œ3‹A.O.Barut and R.Raczka, ›3mTheory of Group Representations

and applications›m (Polish Scientific Publishers, Warszawa, 1977).

Œ4‹P.P.Pavinsky and A.I.Sherstyuk, Vestn. LGU ›1m22›m,

11 (1968) (in Russian).

Œ5‹A.V.Sergeev and A.I.Sherstyuk, Sov.Phys.JETP ›1m55›m,

625 (1982).

Œ6‹M.Grant and C.S.Lai, Phys.Rev.A ›1m20›m, 718 (1979).

Œ7‹Y.L.Luke, ›3mMathematical functions and their approximations›m

(Academic, New York,1975).

Œ8‹J.Shertzer, Phys.Rev.A ›1m44›m, 2832 (1991).

Œ9‹J.Zinn-Justin, J.Math.Phys. ›1m22›m, 511 (1981).


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