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

Abstract

Abstract. We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L 2 (R d ) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

A positive density analogue of the Lieb–Thirring inequality

“…The main idea is now that, for the question of boundedness, each Bloch function u n (ξ, x) can be replaced by the corresponding plane wave exp(ix • ξ). Arguments of this sort have been carried out in a similar context in [37,3,2,8], for instance. When ∇ ξ λ n (ξ) = 0 for all ξ such that λ n (ξ) = µ, the Fermi surface is smooth and can be locally replaced by a sphere.…”

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