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asymptotics in the large coupling constant limit.

Mathematical Modelling of Natural Phenomena (1)

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Discrete Spectrum of the Periodic Schrödinger Operator with aVariable Metric Perturbed by a Nonnegative Potential
M. Sh. Birman, V. A. Sloushch
Journal:

Mathematical Modelling of Natural Phenomena / Volume 5 / Issue 4 / 2010

Published online by Cambridge University Press:

12 May 2010, pp. 32-53

Print publication:

2010
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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.