Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
- Department of Mathematics, University of Waikato, Hamilton (New Zealand)
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose 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 hypergeometric functions {sub m}F{sub n} and is QES if the Schroedinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set.
- OSTI ID:
- 20768743
- Journal Information:
- Journal of Mathematical Physics, Vol. 47, Issue 3; Other Information: DOI: 10.1063/1.2174237; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
Similar Records
Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory
Unified theory of exactly and quasiexactly solvable ''discrete'' quantum mechanics. I. Formalism