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Russia Izhevsk
Year
2018
Volume
28
Issue
4
Pages
462-473
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Section Mathematics
Title Asymptotics of the Schrödinger operator levels for a crystal film with a nonlocal potential
Author(-s) Smetanina M.S.a
Affiliations Udmurt State Universitya
Abstract We consider a three-dimensional Schrödinger operator for a crystal film with a nonlocal potential, which is a sum of an operator of multiplication by a function, and an operator of rank two (“separable potential”) of the form $V=W (x) +\lambda _1(\cdot,\phi _1)\phi _1+\lambda _2(\cdot,\phi _2)\phi _2 $. Here the function $W(x)$ decreases exponentially in the variable $x_3$, the functions $\phi _1(x)$, $\phi _2(x)$ are linearly independent, of Bloch type in the variables $x_1,\,x_2$ and exponentially decreasing in the variable $x_3$. Potentials of this type appear in the pseudopotential theory. A level of the Schrödinger operator is its eigenvalue or resonance. The existence and uniqueness of the level of this operator near zero is proved, and its asymptotics is obtained.
Keywords Schrödinger equation, nonlocal potential, eigenvalues, resonances, asymptotics
UDC 517.958, 530.145.61
MSC 35Q40, 35J10, 35P20
DOI 10.20537/vm180403
Received 30 August 2018
Language Russian
Citation Smetanina M.S. Asymptotics of the Schrödinger operator levels for a crystal film with a nonlocal potential, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 462-473.
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