Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems

Abstract

The construction of analytic solutions for quasi-exactly solvable systems is an interesting problem. We revisit a class of models for which the odd solutions were largely missed previously in the literature: the anharmonic oscillator, the singular anharmonic oscillator, the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schrödinger system from the kink stability analysis of ${\varphi }^6$-type field theory. We present a systematic and unified treatment for the odd and even sectors of these models. We find generic closed-form expressions for constraints to the allowed model parameters for quasi-exact solvability, the corresponding energies and wavefunctions. We also make progress in the analysis of solutions to the Bethe ansatz equations in the spaces of model parameters and provide insight into the curves/surfaces of the allowed parameters in the parameter spaces. Most previous analyses in this aspect were on a case-by-case basis and restricted to the first excited states. We present analysis of the solutions (i.e. roots) of the Bethe ansatz equations for higher excited states (up to levels $n$=30 or 50). The shapes of the root distributions change drastically across different regions of model parameters, illustrating phenomena analogous to phase transition in context of integrable models. Furthermore, we also obtain the $sl(2)$ algebraization for the class of models in their respective even and odd sectors in a unified way.

References in corpus (11)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Quasi-Exactly Solvable Systems and Orthogonal Polynomials
• A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
• Exact polynomial solutions of second order differential equations and their applications
• Unified derivation of exact solutions for a class of quasi-exactly solvable models
• Hidden algebra of the $N$-body Calogero problem
• The generalized quantum isotonic oscillator
• Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
• Generalized quantum isotonic nonlinear oscillator in d dimensions
• New quasi-exactly solvable class of generalized isotonic oscillators
• Extended Nikiforov-Uvarov method, roots of polynomial solutions, and functional Bethe ansatz method

Cited by in corpus (1)

• Dirac Equation with Space Contributions Embedded in a Quantum-Corrected Gravitational Field