On Time-Dependent Quasi-Exactly Solvable Problems

Abstract

In this paper we demonstrate that there exists a close relationship between quasi-exactly solvable quantum models and two special classes of classical dynamical systems. One of these systems can be considered a natural generalization of the multi-particle Calogero-Moser model and the second one is a classical matrix model.

Shape Invariant Potentials in "Discrete Quantum Mechanics"

“…The present problems could be considered as those of finding associated polynomial solutions of Bethe ansatz like equations. They occur in many branches of theoretical physics, for example, the quasi-exactly solvable single and multi-particle quantum systems [18] on top of the well-known integrable spin chains [19].…”

Time-dependence in non-Hermitian quantum systems

“…While many QES models have been studied in stationary settings, little was known for time-dependent systems before our work [12]. So far a time-dependence has only been introduced into the eigenfunctions in form of a dynamical phase [98,99]. However, no QES systems with explicitly time-dependent Hamiltonians have been considered up to now.…”

Quantum Inozemtsev model, quasi-exact solvability and 𝒩-fold supersymmetry

“…. , M should all vanish [23], which results in a set of rational ("Bethe ansatz" type) equations for {ξ k }'s:…”

“…The equivalence of quasi-exact solvability and N -fold supersymmetry is generally established. Other related notions, the "Bethe Ansatz" type equations [23], ODE spectral equivalence [24,25] and Bender-Dunne polynomials [26] are simply explained from the new point of view. Section four deals with the quasi-exact solvability of a single particle trigonometric BC type Inozemtsev model.…”