Abstract
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 \({h(\chi_{(-\alpha,\alpha)} \chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})}\) and link its subleading behaviour as \({\alpha \to \infty}\) to the position of the spectral parameter μ relative to the spectrum of H.
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Pfirsch, B., Sobolev, A.V. Formulas of Szegő Type for the Periodic Schrödinger Operator. Commun. Math. Phys. 358, 675–704 (2018). https://doi.org/10.1007/s00220-018-3106-z
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DOI: https://doi.org/10.1007/s00220-018-3106-z