Formulas of Szegő type for the periodic Schrödinger operator

Abstract

We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator $H=-\frac{d^2}{d{x}^2}+V$ 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 $H$.