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Discrete Spectrum of the Periodic Schrödinger Operator with aVariable Metric Perturbed by a Nonnegative Potential

Published online by Cambridge University Press:  12 May 2010

M. Sh. Birman  and
V. A. Sloushch
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Abstract

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

Keywords

periodic Schrödinger operatordiscrete spectrumspectral gapsasymptotics in the large coupling constant limit.

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References

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