A Moments' Analysis of Quasi-Exactly Solvable Systems: A New Perspective on the Sextic Anharmonic and Bender-Dunne Potentials

Abstract

There continues to be great interest in understanding quasi-exactly solvable (QES) systems. In one dimension, QES states assume the form $\Psi(x) =x^\gamma P_d(x) {\cal A}(x)$, where ${\cal A}(x) > 0$ is known in closed form, and $P_d(x)$ is a polynomial to be determined. That is ${{\Psi(x)}\over {x^\gamma{\cal A}(x)}} = \sum_{n=0}^\infty a_nx^n$ truncates. The extension of this "truncation" procedure to non-QES states corresponds to the Hill determinant method, which is unstable when the {\it reference} function assumes the physical asymptotic form. Recently, Handy and Vrinceanu introduced the Orthogonal Polynomial Projection Quantization (OPPQ) method which has non of these problems, allowing for a unified analysis of QES and non-QES states. OPPQ uses a non-orthogonal basis constructed from the orthonormal polynomials of ${\cal A}$: $\Psi(x) = \sum_{j=0}^\infty \Omega_j {\cal P}^{(j)}(x) {\cal A}(x)$, where $\langle {\cal P}^{(j_1)}|{\cal A}|{\cal P}^{(j_2)} \rangle = \delta_{j_1,j_2}$, and $\Omega_j = \langle {\cal P}^{(j)}|\Psi\rangle$. For systems admitting a moment equation representation, such as those considered here, these coefficients can be readily determined. The OPPQ quantization condition, $\Omega_{j} = 0$, is exact for QES states (provided $j \geq d+1$); and is computationally stable, and exponentially convergent, for non-QES states. OPPQ provides an alternate explanation to the Bender-Dunne (BD) orthogonal polynomial formalism for identifying QES states: they correlate with an anomalous kink behavior in the order of the finite difference moment equation associated with the $\Phi = x^\gamma {\cal A}(x) \Psi(x)$ {\it Bessis}-representation (i.e. a spontaneous change in the degrees of freedom of the system).