Physical applications of second-order linear differential equations that admit polynomial solutions

Abstract

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.

13 pages, no figures

References in corpus (6)

• Analytical solutions of the Bohr Hamiltonian with the Morse potential
• Criterion for polynomial solutions to a class of linear differential equation of second order
• Polynomial Solution of Non-Central Potentials
• Energies and wave functions for a soft-core Coulomb potential
• Soft-core Coulomb potentials and Heun's differential equation
• Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations

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• On Polynomial Solutions of Linear Differential Equations with Applications
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