The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials

TL;DR

This paper provides an asymptotic solution to the discretised harmonic oscillator, linking it to Mathieu functions and introducing a new class of generalized Hermite polynomials with applications in quantum systems.

Contribution

It introduces a novel asymptotic approach to solve the discretised harmonic oscillator and defines a new class of generalized Hermite polynomials related to Mathieu functions.

Findings

Explicit asymptotic solution for discretised harmonic oscillator

Introduction of generalized Hermite polynomials dependent on coupling parameter

Demonstrated convergence and orthogonality properties of the solutions

Abstract

We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables.