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Archive of Issues

Russia Izhevsk
Year
2018
Volume
28
Issue
4
Pages
462-473
 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. References Reed M., Simon B. Methods of modern mathematical physics. IV: Analysis of operators, New York: Academic Press, 1978. Mera H., Pedersen T.G., Nikolić B.K. Nonperturbative quantum physics from low-order perturbation theory, Physical Review Letters, 2015, vol. 115, issue 14, 143001. DOI: 10.1103/PhysRevLett.115.143001 Cornean H.D., Jensen A., Nenciu G. Metastable states when the Fermi Golden Rule constant vanishes, Communications in Mathematical Physics, 2015, vol. 334, issue 3, pp. 1189-1218. 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