Formulas of Szegő Type for the Periodic Schrödinger Operator

Abstract

We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator H=-d2dx2+V in dimension one. Admitting fairly general functions h with h(0) = 0 , we study the trace of the operator h(χ(-α,α)χ(-∞,μ)(H) χ(-α,α)) and link its subleading behaviour as α→ ∞ to the position of the spectral parameter μ relative to the spectrum of H.