Sobolev, A; (2019) Extreme eigenvalues of an integral operator. Journal of Spectral Theory , 9 (1) pp. 227-241. 10.4171/JST/246.

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Abstract

We study the family of compact operators Bα = V AαV , α > 0 in L 2 (R d ), d ≥ 1, where Aα is the pseudo-differential operator with symbol a (α) (ξ) = a(αξ), and both functions a and V are real-valued and decay at infinity. We assume that a and V attain their maximal values A0 > 0, V0 > 0 only at ξ = 0 and x = 0. We also assume that a(ξ) = A0 − Ψγ(ξ) + o(|ξ| γ ), |ξ| → 0, V (x) = V0 − Φβ(x) + o(|x| β ), |x| → 0, with some functions Ψγ(ξ) > 0, ξ 6= 0 and Φβ(x) > 0, x 6= 0 that are homogeneous of degree γ > 0 and β > 0 respectively. The main result is the following asymptotic formula for the eigenvalues λ (n) α of the operator Bα (arranged in descending order counting multiplicity) for fixed n and α → 0: λ (n) α = A0V 2 0 − µ (n)α σ + o(α σ ), α → 0, where σ −1 = γ −1 + β −1 , and µ (n) are the eigenvalues (arranged in ascending order counting multiplicity) of the model operator T with symbol V 2 0 Ψγ(ξ) + 2A0V0Φβ(x).

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