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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 2, Pages 28–64 (Mi ufa146)

Perturbation of an elliptic operator by a narrow potential in an $n$-dimensional domain

A. R. Bikmetov, R. R. Gadyl'shin

Bashkir State Pedagogical University, Ufa, Russia

Abstract: We study a discrete spectrum of an elliptic operator of the second order in an $n$-dimensional domain, $n\geq2$, perturbed by a potential depending on two parameters, one of the parameters describes the length of the support of the potential and the inverse of the other corresponds to the magnitude of the potential. We give the relation between these parameters, under which the generalized convergence of the perturbed operator to the unperturbed one holds. Under this relation we construct the asymptotics w.r.t. small parameters of the eigenvalues of the perturbed operators.

Keywords: elliptic operator, perturbation, matching of asymptotic expansions.

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Bibliographic databases:
UDC: 517.928+517.984

Citation: A. R. Bikmetov, R. R. Gadyl'shin, “Perturbation of an elliptic operator by a narrow potential in an $n$-dimensional domain”, Ufimsk. Mat. Zh., 4:2 (2012), 28–64

Citation in format AMSBIB
\Bibitem{BikGad12} \by A.~R.~Bikmetov, R.~R.~Gadyl'shin \paper Perturbation of an elliptic operator by a~narrow potential in an $n$-dimensional domain \jour Ufimsk. Mat. Zh. \yr 2012 \vol 4 \issue 2 \pages 28--64 \mathnet{http://mi.mathnet.ru/ufa146} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3432642} 

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This publication is cited in the following articles:
1. R. R. Gadyl'shin, S. V. Repjevskij, E. A. Shishkina, “On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 76–90
2. R. R. Gadylshin, A. A. Ershov, S. V. Repyevsky, “On asymptotic formula for electric resistance of conductor with small contacts”, Ufa Math. J., 7:3 (2015), 15–27
3. D. B. Davletov, D. V. Kozhevnikov, “The problem of Steklov type in a half-cylinder with a small cavity”, Ufa Math. J., 8:4 (2016), 62–87
4. I. Kh. Khusnullin, “Vozmuschenie volnovoda uzkim potentsialom”, Tr. IMM UrO RAN, 23, no. 2, 2017, 274–284
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