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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.

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
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[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.
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