Sextic anharmonic oscillators and orthogonal polynomials

Abstract

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.

11 pages

References in corpus (6)

• Asymptotic iteration method for eigenvalue problems
• Construction of exact solutions to eigenvalue problems by the asymptotic iteration method
• Perturbation theory in a framework of iteration methods
• New Quasi-Exactly Solvable Sextic Polynomial Potentials
• Type A N-fold Supersymmetry and Generalized Bender-Dunne Polynomials
• A recursion technique for deriving renormalized perturbation expansions for one-dimensional anharmonic oscillator

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