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Physical applications of second-order linear differential equations that admit polynomial solutions. (English) Zbl 1200.81049

Summary: Conditions for the second-order linear differential equation

i=0 3 a 3,i x i y '' + i=0 2 a 2,i x i y ' - i=0 1 τ 1,i x i y=0

to have polynomial solutions are given. Several applications of these results to Schrödinger’s equation are discussed. Conditions under which the confluent, biconfluent and general Heun equation yields polynomial solutions are explicitly given. Some new classes of exactly solvable differential equations are also discussed. The results of this work are expressed in such a way as to allow direct use, without preliminary analysis.

MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
34L40Particular ordinary differential operators
35J10Schrödinger operator
33D50Orthogonal polynomials and functions in several variables
35C11Polynomial solutions of PDE