Bikmetov, A. R.; Gadyl’shin, R. R. On the spectrum of the Schrödinger operator with large potential concentrated on a small set. (English. Russian original) Zbl 1169.35044 Math. Notes 79, No. 5, 729-733 (2006); translation from Mat. Zametki 79, No. 5, 787-790 (2006). This paper studies the simple eigenvalues of Schrödinger operators on a bounded domain of \(\mathbb{R}^n\) when the potential is large, smooth and increasingly concentrated on a small set. Its main result gives an explicit formula for the first order term of the asymptotic expansion of these eigenvalues as perturbations of simple eigenvalues of the Laplace operator. A corresponding asymptotic formula for the eigenfunctions is also provided. Reviewer: Lyonell Boulton (Edinburgh) Cited in 1 Document MSC: 35P05 General topics in linear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation 35B20 Perturbations in context of PDEs 47F05 General theory of partial differential operators Keywords:Schrödinger operator; spectrum of the Schrödinger operator; Dirichlet boundary condition; Neumann boundary condition; Banach space; holomorphic operator-valued function PDFBibTeX XMLCite \textit{A. R. Bikmetov} and \textit{R. R. Gadyl'shin}, Math. Notes 79, No. 5, 729--733 (2006; Zbl 1169.35044); translation from Mat. Zametki 79, No. 5, 787--790 (2006) Full Text: DOI References: [1] O. A. Oleinik, G. A. Iosif’yan, and A. S. Shamaev, Mathematical Problems of the Theory of Strongly Inhomogeneous Elastic Media [in Russian], Izd. Moskov. Univ., Moscow, 1990. [2] O. A. Oleinik and J. Sanchez-Hubert, and G. A. Yosifian, Bull. Sc. Math. Ser. 2, 115 (1991), 1–27. [3] V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow, 1976. · Zbl 0342.35052 [4] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Heidelberg, 1966; Russian translation: Mir, Moscow, 1972. · Zbl 0148.12601 [5] M. A. Lavrent’ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow, 1987. [6] A. M. Il’in, Matching Asymptotic Expansions of the Solutions of Boundary-Value Problems [in Russian], Nauka, Moscow, 1989. [7] R. R. Gadyl’shin, Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 132 (2002), 97–104. [8] L. D. Landau and E. M. Livshits, Theoretical Physics, vol. 3, Quantum Mechanics: Nonrelativistic Theory [in Russian], Nauka, Moscow, 1989. [9] B. Simon, Ann. Phys., 97 (1976), 279–288. · Zbl 0325.35029 [10] M. Klaus, Ann. Phys., 108 (1977), 288–300. · Zbl 0427.47033 [11] R. Blankenbecler and M. L. Goldberger, and B. Simon, Ann. Phys., 108 (1977), 69–78. [12] M. Klaus and B. Simon, Ann. Phys., 130 (1980), 251–281. · Zbl 0455.35112 [13] D. E. Pelinovskii and K. Salem, Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 122 (2000), 118–128. [14] P. Zhevandrov and A. Merzon, ”Asymptotics of eigenfunctions in shallow potential wells and related problems,” in: Advances in Mathematical Sciences: Asymptotic Methods for Wave and Quantum Problems (M. V. Karasev, Editor), Amer. Math. Soc. Translations, Ser. 2, Amer. Math. Soc., Providence, R.I., 2003, pp. 235–284. · Zbl 1140.81396 [15] F. Bentosela, R. M. Cavalcanti, and P. Exner, and V. A. Zagrebanov, J. Phys. A, 32 (1999), 3029–3039. · Zbl 1055.81522 [16] R. R. Gadyl’shin, Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 138 (2004), 41–54. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.