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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, Volume 28, Issue 4, Pages 462–473
DOI: https://doi.org/10.20537/vm180403
(Mi vuu651)
 

MATHEMATICS

Asymptotics of the Schrödinger operator levels for a crystal film with a nonlocal potential

M. S. Smetanina

Mozhga Branch, Udmurt State University, ul. Internatsional’naya, 88, Mozhga, 427790, Russia
References:
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.
Received: 30.08.2018
Bibliographic databases:
Document Type: Article
UDC: 517.958, 530.145.61
MSC: 35Q40, 35J10, 35P20
Language: Russian
Citation: M. S. Smetanina, “Asymptotics of the Schrödinger operator levels for a crystal film with a nonlocal potential”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:4 (2018), 462–473
Citation in format AMSBIB
\Bibitem{Sme18}
\by M.~S.~Smetanina
\paper Asymptotics of the Schr\"odinger operator levels for a crystal film with a nonlocal potential
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2018
\vol 28
\issue 4
\pages 462--473
\mathnet{http://mi.mathnet.ru/vuu651}
\crossref{https://doi.org/10.20537/vm180403}
\elib{https://elibrary.ru/item.asp?id=36873364}
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    Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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