Formulas of Szegő type for the periodic Schrödinger operator

We prove asymptotic formulas of Szegő type for the periodic Schrödinger 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 $h(χ_{(-α,α)}χ_{(-\infty,μ)}(H)χ_{(-α,α)})$ and link its subleading behaviour as $α\to\infty$ to the position of the spectral parameter $μ$ relative to the spectrum of $H$.

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