Sharp asymptotics of the quasimomentum

TL;DR

This paper derives the high-energy asymptotics of the quasimomentum and related functions for a Schrödinger operator with a periodic potential in Sobolev spaces, enhancing understanding of spectral properties.

Contribution

It provides new asymptotic formulas for the quasimomentum and Titchmarsh-Weyl functions for periodic Schrödinger operators with potentials in Sobolev spaces.

Findings

Asymptotic formulas for quasimomentum at high energy

Asymptotics of Titchmarsh-Weyl functions and Bloch functions

Extension to potentials in Sobolev spaces

Abstract

We consider the Schr\"odinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy.