Sextic anharmonic oscillators and orthogonal polynomials

Details Sextic anharmonic oscillators and orthogonal polynomials Sextic anharmonic oscillators and orthogonal polynomials

2006-05-18

arxiv.org/abs/math-ph/0605057v1

J. Phys. A 39, 8477 - 8486 (2006)

Physics

Mathematical Physics

11 pages

Scientific paper

10.1088/0305-4470/39/26/014

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.

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