Superintegrability and quasi-exactly solvable eigenvalue problems
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ELEMENTARY PARTICLES AND FIELDS
Superintegrability and Quasi-Exactly Solvable Eigenvalue Problems* E. G. Kalnins1) , W. Miller, Jr.2) , and G. S. Pogosyan3), 4) Received November 19, 2007
Abstract—We show the intimate relationship between quasi-exact solvability, as expounded, for example, by A. Ushveridze, and separation of variables as it applies to specific quantum Hamiltonians. This approach is generalized to apply to finite solutions of quantum Hamiltonians. PACS numbers: 05.50.+q, 05.70.Fh, 75.40.Mg DOI: 10.1134/S1063778808050220
1/4 − b2 1/4 − c2 − − Ψ = EΨ. y2 z2
1. INTRODUCTION For purposes of this paper, we define exactly solvable quantum-mechanical systems as those for which the corresponding eigenfunctions can be completely determined in terms of hypergeometric functions (or equivalently, by classical polynomials). For such systems, the corresponding eigenvalues are also exactly calculable [1]. More generally, we can study “quasiexactly solvable” quantum systems for which some of the eigenvalues are calculable by algebraic means as also are the corresponding eigenfunctions. In this case, the algebraic eigenfunctions will, in general, arise from the solution of ordinary differential equations with more than three regular singularities, such as Heun’s equation [2]. The concept of algebraic or “quasi-exact” solvability can be thought of as a generalization of the relations satisfied by the zeros of classical polynomials [3, 4]. For example, Legendre polynomials P (x) ≈ ni=1 (x − θi ) (usually associated with exactly solvable problems) have zeros which satisfy the algebraic system 1 = 0. (2 + 1)θi + 2 θj − θi j = i
As a further example, consider the superintegrable quantum Hamiltonian eigenvalue problem [5] 1/4 − a2 HΨ = − Λ + ω 2 (x2 + y 2 + z 2 ) − x2 ∗
The text was submitted by the authors in English. Department of Mathematics, University of Waikato, Hamilton, New Zealand. 2) School of Mathematics, University of Minnesota, Minneapolis, USA. 3) ´ Departamento de Matematicas, CUCEI, Universidad de ´ Guadalajara, Jalisco, Mexico. 4) Yerevan State University, Yerevan, Armenia. 1)
¨ This Schrodinger eigenvalue problem can be solved via variable separation in more than one way because it is a “superintegrable” system. We can choose to do this in spheroconical coordinates, viz., (u − e1 )(v − e1 ) , (e1 − e2 )(e1 − e2 ) (u − e2 )(v − e2 ) , y2 = r2 (e2 − e1 )(e2 − e1 ) (u − e3 )(v − e3 ) , z2 = r2 (e3 − e2 )(e3 − e1 )
x2 = r 2
where e1 < u < e2 < v < e3 . In these coordinates, ¨ separated solutions of the Schrodinger equation have the form ω (ωr 2 )Λ(u)Λ(v), Ψ = r exp − r 2 L+1/2 n 2 where Lαn (z) is a Laguerre polynomial, s a b c (λ − θi ), Λ(λ) = (λ − e1 ) (λ − e2 ) (λ − e3 ) i=1
and θi satisfy b+1 c+1 2 a+1 + + + = 0, θi − e1 θi − e2 θi − e3 θi − θj j = i
λ = u, v. The separation equations for the function Λ(λ) are d dΛ + − ( + 1)λ + µ P (λ) P (λ) dλ dλ a2 − 1/4 + (e2 − e1 ) λ − e1 b2 − 1/4 + (e1 − e2 ) × (e2 − e3 ) λ − e2
+ (e1 − e2 )(e1 − e3 )
925
926
KALNINS et al.
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