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We study the discrete spectrum in the gaps of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. We are interested in the asymptotics (for large coupling constant) of the number of eigenvalues that have been “born” (or have “died”) at the edges of the gap. The high-energy (Weyl) asymptotics and the threshold asymptotics are distinguished. At the right edge of the gap the competition between the Weyl contribution and the threshold contribution to the asymptotics is possible. § 0. Introduction 1. Let A be a selfadjoint elliptic periodic second order operator in L2(R), d ≥ 2, given by the expression A = −div g(x)∇ + p(x), and let V be an operator of multiplication by a function V (x) ≥ 0 decaying at infinity. Let the interval (λ−, λ+) be a gap in the spectrum of A. We put A±(α) = A∓αV (x), α > 0. By N+(α, λ+) we denote the number of eigenvalues of the operator A+(t) that have been ”born” at the point λ+ as the coupling constant t has been growing from 0 to α. The function N−(α, λ−) is defined similarly for the operator A−. We are interested in the asymptotics of these functions as α →∞ (in the large coupling constant limit). The corresponding asymptotics may be rather diverse, depending both on A and V . They were studied in a number of papers. First of all, we mention [B3, B4, BL] and especially the survey [B2] and the references therein. Another approach to the class of problems under discussion was proposed in [Iv]. The asymptotic behavior of N±(α, λ±) depends on dimension d and on the character of decay of V . The situation for d ≥ 3 differs substantially from that for d = 2. If d ≥ 3 and V ∈ Ld/2(R), the function N+(α, λ+) has the Weyl asymptotics N+(α, λ+) ∼ (2π)−dωdαd/2 ∫ V (det g)−1/2 dx, α →∞; (0.1) here ωd is the volume of the unit ball in R. If V 6∈ Ld/2(R), then the estimate N+(α, λ+) = O(α) is violated, and N+(α, λ+) can have an arbitrary order of growth greater than d/2. Essentially, the asymptotics (0.1) has a “high-energy” origin, while the behavior of N+(α, λ+) with V 6∈ Ld/2(R) is determined by the “threshold” effect near the edge of the gap of the unperturbed operator A. (See discussion in [B2, §2].) 1991 Mathematics Subject Classification. Primary 35P20.

On discrete spectrum in the gaps of a two-dimensional periodic elliptic operator perturbed by a decaying potential