Bender-Dunne Orthogonal Polynomials, Quasi-Exact Solvability and Asymptotic Iteration Method for Rabi Hamiltonian

TL;DR

This paper introduces a method using the asymptotic iteration technique to find quasi-exact solutions for the Rabi Hamiltonian, deriving eigenvalues, eigenfunctions, and Bender-Dunne orthogonal polynomials with nonpositive definite norms.

Contribution

It presents a novel approach to solving the Rabi Hamiltonian by connecting asymptotic iteration with Bender-Dunne polynomials, revealing their orthogonality properties.

Findings

Eigenvalues and eigenfunctions are obtained explicitly.

Bender-Dunne polynomials have nonpositive definite norms.

The method provides quasi-exact solutions for the Rabi Hamiltonian.

Abstract

We present a method for obtaining the quasi-exact solutions of the Rabi Hamiltonian in the framework of the asymptotic iteration method. The energy eigenvalues, the eigenfunctions and the associated Bender-Dunne orthogonal polynomials are deduced. The latter prove to have a nonpositive definite norm.