Quasi-exactly solvable models based on special functions

TL;DR

This paper introduces a systematic method to extend quasi-exactly solvable models using special functions, demonstrating how certain Hamiltonians become solvable at specific parameters by linking them to special function-based operators.

Contribution

It presents a novel systematic approach to extend QES systems by constructing invariant subspaces using special functions, with applications to physical Hamiltonians.

Findings

Construction of finite-dimensional invariant subspaces using special functions

Identification of parameter values where the two-photon Rabi Hamiltonian is QES

Connection between QES operators and hypergeometric functions

Abstract

We suggest a systematic method of extension of quasi-exactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As a example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasi-exactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function.