Generalized Bethe Ansatz Equations for Hofstadter Problem

Abstract

The problem of diagonalization of the quantum mechanical Hamiltonian, governing dynamics of an electron on a two-dimensional triangular or square lattice in external uniform magnetic field, applied perpendicularly to the lattice plane, the flux through lattice cell, divided by the elementary quantum flux, being rational number, is reduced to generalized Bethe ansatz like equations on high genus algebraic curve. Our formulae for the trigonometric case, where genus of the curve vanishes, contain as a particular case recent result of Wiegmann and Zabrodin.