[Paper Review] Exact and quasi-exact solvability of two-dimensional superintegrable quantum systems. I. Euclidean space

E. G. Kalnins, Willard Miller+1 more|ArXiv.org|Dec 11, 2004

Quantum Mechanics and Non-Hermitian Physics57 references3 citations

TL;DR

This paper introduces a new classification of solvability in two-dimensional superintegrable quantum systems in Euclidean space, distinguishing between exactly solvable (ES) and quasi-exact solvability (QES) based on the nature of solutions to the Schrödinger equation. It shows that separation of variables in these systems yields both ES problems—solved via hypergeometric functions—and QES problems—characterized by polynomial solutions satisfying higher-order recurrence relations, with explicit examples for N=2 and N=3 systems demonstrating quantized spectra through algebraic equations.

ABSTRACT

In this article we show that separation of variables for second-order superintegrable systems in two-dimensional Euclidean space generates both exactly solvable (ES) and quasi-exactly solvable (QES) problems in quantum mechanics. In this article we propose the another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schroedinger equation, can be expressed in terms of hypergeometrical functions $_mF_n$ and is QES if the Schroedinger equation admit polynomial solutions with the coefficients satisfying the three-term or more higher order of recurrence relations

Motivation & Objective

• To redefine and classify exactly solvable (ES) and quasi-exactly solvable (QES) quantum systems in two-dimensional Euclidean space.
• To establish a systematic framework for identifying ES and QES problems arising from separation of variables in second-order superintegrable systems.
• To demonstrate that polynomial solutions in separated equations correspond to QES systems when coefficients satisfy three-term or higher-order recurrence relations.
• To show that energy and separation constants are quantized via algebraic equations in QES systems, particularly for generalized Lamé equations.
• To distinguish between full, partial, and higher-order QES systems based on the number of intervals and recurrence structure.

Proposed method

• Define ES systems as those whose Schrödinger equation solutions are expressible in terms of generalized hypergeometric functions ${}_{m}F_{n}$.
• Define QES systems as those admitting polynomial solutions where coefficients satisfy three-term or higher-order recurrence relations.
• Apply separation of variables to second-order superintegrable systems in 2D Euclidean space to derive ordinary differential equations.
• Analyze the generalized Lamé equation (192) for N=2 and N=3, showing that finiteness of eigenfunctions leads to polynomial solutions and quantized energy spectra.
• Use recurrence relations to derive algebraic constraints on energy and separation constants, with the number of required determinant conditions reduced via overcomplete systems.
• Classify systems as Nth-order QES when polynomial solutions are determined by (N+1)-term recurrence relations and spectra are found numerically from algebraic equations.

Experimental results

Research questions

• RQ1 How can exactly solvable and quasi-exactly solvable quantum systems be systematically distinguished in two-dimensional superintegrable systems?
• RQ2 What role do recurrence relations of order three or higher play in characterizing quasi-exact solvability?
• RQ3 How does the requirement of finiteness of wavefunctions across multiple intervals lead to quantization of energy and separation constants?
• RQ4 What is the connection between polynomial solutions of the Schrödinger equation and the structure of recurrence relations in separated variables?
• RQ5 Can higher-order QES systems be classified based on the number of recurrence terms and the solvability of resulting algebraic systems?

Key findings

• For N=2, the generalized Lamé equation yields a first-order quasi-exactly solvable system where polynomial solutions (Lamé polynomials) satisfy a three-term recurrence relation and lead to quantized energy levels $E = \ell(\ell+1)/R^2$.
• For N=3, the system becomes second-order quasi-exactly solvable, with polynomial solutions derived from a four-term recurrence relation, and the spectrum of separation constants $\lambda_1(R)$ and $\lambda_2(R)$ determined by solving two algebraic equations.
• The requirement of finiteness of eigenfunctions in all intervals $(a_1,a_2), (a_2,a_3), (a_3,a_4)$ leads to polynomial solutions and quantization of energy and separation constants.
• In the N=3 case, the overcomplete system of homogeneous equations for expansion coefficients is reduced to two necessary and sufficient determinant conditions for nontrivial solutions.
• The paper establishes a hierarchy of solvability: exactly solvable, quasi-exactly solvable, partially quasi-exactly solvable, and non-exactly solvable, based on recurrence structure and solution finiteness.
• The classification extends to Nth-order QES systems, where polynomial solutions are governed by (N+1)-term recurrence relations and spectra are obtained numerically from algebraic equations.