Asymptotics of spectral gaps of the 1D schrodinger operator with Mathieu potential

Abstract

The one-dimensional Schrödinger operator L(y) = -yn + v(x)y, considered on R with π[pi]-periodic real-valued potential v(x), is self-adjoint, and its spectrum has a gap-band structure- the intervals of continuous spectrum are separated by spectral gaps. In this thesis, we study the asymptotic behaviour of the spectral gaps of L. In the case of the Mathieu potential v(x) = 2a cos (2x), we give an alternative proof of the result of Harrell-Avron-Simon about the precise asymptotics of the lengths of spectral gaps.