Abstract
Asymptotic expansions are constructed for negative eigenvalues λ −k ε of the Dirichlet problem with the density becoming negatively small, of order ε, in either a subdomain of fixed size, or a small, of diameter O(ε 1/m), neighborhood of an interior point. Such the eigenvalues lie far away from the coordinate origin and their order with respect to the small parameter is ε −1 in the first case and ε −m/(m+2) in the second one. The limit problems are formulated and investigated, the eigenvalues of which imply the limits as ε → +0 of the quantities −ελ −k ε and −ε m/(m+2) λ −k ε, respectively. Asymptotically precise estimates are obtained for the remainders in the expansions of the eigenvalues and eigenfunctions.
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 27, Part II, pp. 240–280, 2009.
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Nazarov, S.A. Asymptotics of negative eigenvalues of the Dirichlet problem with the density changing sign. J Math Sci 163, 151–175 (2009). https://doi.org/10.1007/s10958-009-9663-0
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DOI: https://doi.org/10.1007/s10958-009-9663-0