ISSN 1063-7788, Physics of Atomic Nuclei, 2008, Vol. 71, No. 5, pp. 925–929. c  Pleiades Publishing, Ltd., 2008. ELEMENTARY PARTICLES AND FIELDS Superintegrability and Quasi-Exactly Solvable Eigenvalue Problems * E. G. Kalnins 1) , W. Miller, Jr. 2) , and G. S. Pogosyan 3), 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. INTRODUCTION For purposes of this paper, we define exactly solv- able 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 sys- tems, the corresponding eigenvalues are also exactly calculable [1]. More generally, we can study “quasi- exactly 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 equa- tions 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 gen- eralization of the relations satisfied by the zeros of classical polynomials [3, 4]. For example, Legendre polynomials P  (x) ≈  n i=1 (x − θ i ) (usually associ- ated with exactly solvable problems) have zeros which satisfy the algebraic system (2 + 1)θ i +2  j = i 1 θ j − θ i =0. As a further example, consider the superintegrable quantum Hamiltonian eigenvalue problem [5] H Ψ=  − Λ+ ω 2 (x 2 + y 2 + z 2 ) − 1/4 − a 2 x 2 ∗ The text was submitted by the authors in English. 1) Department of Mathematics, University of Waikato, Hamil- ton, New Zealand. 2) School of Mathematics, University of Minnesota, Min- neapolis, USA. 3) Departamento de Mat ´ ematicas, CUCEI, Universidad de Guadalajara, Jalisco, M ´ exico. 4) Yerevan State University, Yerevan, Armenia. − 1/4 − b 2 y 2 − 1/4 − c 2 z 2  Ψ= EΨ. This Schr ¨ odinger 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., x 2 = r 2 (u − e 1 )(v − e 1 ) (e 1 − e 2 )(e 1 − e 2 ) , y 2 = r 2 (u − e 2 )(v − e 2 ) (e 2 − e 1 )(e 2 − e 1 ) , z 2 = r 2 (u − e 3 )(v − e 3 ) (e 3 − e 2 )(e 3 − e 1 ) , where e 1 <u<e 2 <v<e 3 . In these coordinates, separated solutions of the Schr ¨ odinger equation have the form Ψ= r  exp  − ω 2 r 2  L +1/2 n (ωr 2 )Λ(u)Λ(v), where L α n (z) is a Laguerre polynomial, Λ(λ)=(λ − e 1 ) a (λ − e 2 ) b (λ − e 3 ) c s  i=1 (λ − θ i ), and θ i satisfy a +1 θ i − e 1 + b +1 θ i − e 2 + c +1 θ i − e 3 +  j = i 2 θ i − θ j =0, λ = u, v. The separation equations for the function Λ(λ) are  P (λ) d dλ  P (λ) dΛ dλ +  − ( + 1)λ + µ +(e 1 − e 2 )(e 1 − e 3 ) a 2 − 1/4 λ − e 1 +(e 2 − e 1 ) × (e 2 − e 3 ) b 2 − 1/4 λ − e 2 +(e 1 − e 2 ) 925