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Mat. Zametki, 1997, Volume 62, Issue 5, Pages 773–781 (Mi mz1663)  

On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator

Yu. P. Chuburin

Physical-Technical Institute of the Ural Branch of the Russian Academy of Sciences

Abstract: Let $V(x)$, $x=(s_1,x_2,x_3)$, be a potential periodic in $x_1,x_2$ and exponentially decreasing as $|x_3|\to\infty$, and let $V_N(x)$ be the sum of shifts $V(x-(0,0,Nn_3))$ over all integer $n_3$. We prove that the spectrum and eigenfunctions (not necessarily in the class $L^2$) of the Schrödinger operator with potential $V_N$, considered in a box, approximate the spectrum and eigenfunctions of the operator with potential $V$ and, for the negative part of the spectrum, the approximation converges exponentially in $N\to\infty$.

DOI: https://doi.org/10.4213/mzm1663

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English version:
Mathematical Notes, 1997, 62:5, 648–654

Bibliographic databases:

UDC: 517.984.56
Received: 20.03.1996

Citation: Yu. P. Chuburin, “On approximation of the “Membrane” Schrödinger operator by the “Crystal” operator”, Mat. Zametki, 62:5 (1997), 773–781; Math. Notes, 62:5 (1997), 648–654

Citation in format AMSBIB
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\paper On approximation of the ``Membrane'' Schr\"odinger operator by the ``Crystal'' operator
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\yr 1997
\vol 62
\issue 5
\pages 773--781
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\jour Math. Notes
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\pages 648--654
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