Formulas of Szeg\H{o} type for the periodic Schr\"odinger operator

TL;DR

This paper derives Szeg"H{o}-type asymptotic formulas for the trace of certain operators related to the one-dimensional periodic Schr"odinger operator, linking spectral properties to asymptotic behavior as the interval size grows.

Contribution

It establishes new Szeg"H{o}-type asymptotic formulas for the periodic Schr"odinger operator in one dimension, connecting spectral data with trace asymptotics.

Findings

Derived asymptotic formulas for the trace operator as interval size increases.

Linked subleading behavior to the spectral position of the parameter.

Extended Szeg"H{o} theory to periodic Schr"odinger operators.

Abstract

We prove asymptotic formulas of Szeg\H{o} type for the periodic Schr\"odinger operator $H=-\frac{d^2}{dx^2}+V$ in dimension one. Admitting fairly general functions $h$ with $h(0)=0$, we study the trace of the operator $h(\chi_{(-\alpha,\alpha)}\chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})$ and link its subleading behaviour as $\alpha\to\infty$ to the position of the spectral parameter $\mu$ relative to the spectrum of $H$.