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Zbl 1200.81049
Ciftci, Hakan; Hall, Richard L.; Saad, Nasser; Dogu, Ebubekir
Physical applications of second-order linear differential equations that admit polynomial solutions.
(English)
[J] J. Phys. A, Math. Theor. 43, No. 41, Article ID 415206, 14 p. (2010). ISSN 1751-8113; ISSN 1751-8121/e

Summary: Conditions for the second-order linear differential equation $$\left(\sum_{i=0}^3 a_{3,i}x^i\right) y''+ \left(\sum_{i=0}^2 a_{2,i}x^i\right) y'- \left(\sum_{i=0}^1 \tau _{1,i}x^i\right) y=0$$ 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 2000:
*81Q05 Closed and approximate solutions to quantum-mechanical equations
34L40 Particular ordinary differential operators
35J10 Schroedinger operator
33D50 Orthogonal polynomials and functions in several variables
35C11

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