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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 $V\bigl(x-(0,0,Nn_3)\bigr)$ 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 $N\to\infty$.
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
Linking options:
https://www.mathnet.ru/eng/mzm1663https://doi.org/10.4213/mzm1663 https://www.mathnet.ru/eng/mzm/v62/i5/p773
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