Asymptotically exact solutions of Harper equation

TL;DR

This paper derives asymptotically exact solutions for the Harper equation, revealing the hierarchical spectral structure through string polynomial roots and Bethe equations, advancing understanding of incommensurate quantum systems.

Contribution

It introduces a novel method to solve the Harper equation exactly in the asymptotic limit using string polynomials and Bethe equations.

Findings

Roots of string polynomials classify spectrum

Hierarchical structure of the spectrum uncovered

Asymptotic solutions match numerical results

Abstract

We present asymptotically exact solutions of an incommensurate Harper equation---one-dimensional Schroedinger equation of one particle on a lattice in a cosine potential. The wave functions can be written as an infinite product of string polynomials. The roots of these polynomials are solutions of Bethe equations. They are classified according to the string hypothesis. The string hypothesis gives asymptotically exact values of roots and reveals the hierarchical structure of the spectrum of the Harper equation.