Analytically solvable Hamiltonians for quantum systems with a nearest neighbour interaction

TL;DR

This paper identifies specific conditions under which quantum harmonic oscillator chains with nearest neighbour interactions are analytically solvable, introducing three new solvable models with explicit spectra.

Contribution

It demonstrates that systems with interaction matrices matching Jacobi matrices of orthogonal polynomials are analytically solvable, and introduces three new such Hamiltonians based on Krawtchouk, Hahn, and q-Krawtchouk polynomials.

Findings

Derived explicit spectra for the new Hamiltonians.

Established conditions for analytical solvability based on orthogonal polynomial matrices.

Presented properties and spectra of the three new models.

Abstract

We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest neighbour interaction of the form $-q_r q_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle, such systems are always numerically solvable and involve the eigenvalues of the interaction matrix. In this paper, we investigate when such a system is analytically solvable, i.e. when the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This is the case when the interaction matrix coincides with the Jacobi matrix of a system of discrete orthogonal polynomials. Our study of possible systems leads to three new analytically solvable Hamiltonians: with a Krawtchouk interaction, a Hahn interaction or a q-Krawtchouk interaction. For each of these cases, we give the spectrum of the Hamiltonian (in analytic form) and discuss some typical properties of the spectra.