A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential

Abstract

A quantum system is quasi-exactly solvable (QES) when a subset of its eigenvalues can be obtained in closed form, or they are the roots of closed form expressions. In one dimension, QES states assume the form , involving a known positive reference function and a polynomial P(x), so that the Taylor expansion of truncates at a finite order. For non-QES states this truncation procedure, for suitable reference functions, can provide approximate values for the eigenvalues. It corresponds to the Hill determinant method, which is known to be unstable when assumes the controlling asymptotic form of the physical state. For this reason, it cannot be used to simultaneously generate exact QES states while approximating non-QES states. To address these limitations, the orthogonal polynomial projection quantization (OPPQ) method was developed (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202; 2013 J. Phys. B: Atom. Mol. Opt. Phys. 46 115002). It is demonstrated in this paper that OPPQ can directly and transparently provide, at the same time, both algebraic solutions for QES states and convergent, numerically stable, approximations for non-QES states. Within the OPPQ analysis for the wavefunction representation , the Bender–Dunne energy orthogonal polynomials correspond, exactly (up to a numerical factor), to the energy dependent power moments ν(p) = ∫dx xpΦ(x). Within this perspective, the existence of QES states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the ν's, suggesting a change in the number of degrees of freedom. This was first noted through the implementation of the eigenvalue moment method, the first application of semidefinite programming analysis to quantum operators (Handy and Bessis 1985 Phys. Rev. Lett. 55 931). This moments' perspective also reveals additional properties for the non-QES states of the same symmetry as the QES states: their lower order ν-moments must be zero. We demonstrate our results in the context of the two sextic potentials: Vsa(x) = x6 + mx2 + bx4 and Vss(x) = x6 + mx2 + b/x2.