On the Quasi-Exact Solvability of the Confluent Heun Equation
Abstract
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{\mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2014
- DOI:
- 10.48550/arXiv.1406.2643
- arXiv:
- arXiv:1406.2643
- Bibcode:
- 2014arXiv1406.2643G
- Keywords:
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- Mathematical Physics
- E-Print:
- 16 pages, 4 figures. V2: references added, typos corrected