Formulas of Szegő Type for the Periodic Schrödinger Operator

We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator H=-d2dx2+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H = -\frac{d^2}{dx^2}+V}$$\end{document} in dimension one. Admitting fairly general functions h with h(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h(0)=0}$$\end{document}, we study the trace of the operator h(χ(-α,α)χ(-∞,μ)(H)χ(-α,α))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h(\chi_{(-\alpha,\alpha)} \chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})}$$\end{document} and link its subleading behaviour as α→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha \to \infty}$$\end{document} to the position of the spectral parameter μ relative to the spectrum of H.