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 $\Psi (x)=x^{\gamma }{P}_d(x)\mathcal{A}(x)$, where $\mathcal{A}(x)>0$ is known in closed form, and $P_d(x)$ is a polynomial to be determined. That is $\frac{\Psi (x)}{x^{\gamma }\mathcal{A}(x)}={\sum }_{n=0}^{\infty }{a}_n{x}^n$ 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 $\mathcal{A}$: $\Psi (x)={\sum }_{j=0}^{\infty }{\Omega }_j{\mathcal{P}}^{(j)}(x)\mathcal{A}(x)$, where $⟨{\mathcal{P}}^{(j_1)}|\mathcal{A}|{\mathcal{P}}^{(j_2)}⟩={\delta }_{j_1,j_2}$, and ${\Omega }_j=⟨{\mathcal{P}}^{(j)}|\Psi ⟩$. 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 $j\ge d+1$); 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 $\Phi =x^{\gamma }\mathcal{A}(x)\Psi (x)$ {\it Bessis}-representation (i.e. a spontaneous change in the degrees of freedom of the system).

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References in corpus (1)

• Bound states for the quantum dipole moment in two dimensions