Quasi Exactly Solvable Difference Equations

TL;DR

This paper introduces new quasi exactly solvable difference equations in discrete quantum mechanics by deforming known exactly solvable Hamiltonians, resulting in systems with a finite number of exactly computable eigenvalues and eigenfunctions.

Contribution

It presents explicit examples of quasi exactly solvable difference equations derived from deformations of classical solvable Hamiltonians, extending the class of solvable models in discrete quantum mechanics.

Findings

Finite number of exactly calculable eigenvalues and eigenfunctions

Deformation of classical Hamiltonians to create quasi exactly solvable systems

Extension of known solvable models to difference equations

Abstract

Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.