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2D Schrödinger operators with singular potentials concentrated near curves (2007.10761v1)

Published 21 Jul 2020 in math.SP, math-ph, and math.MP

Abstract: We investigate the Schr\"{o}dinger operators $H_\varepsilon=-\Delta +W+V_\varepsilon$ in $\mathbb{R}2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $\gamma$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-\Delta+W+a\partial_\nu\delta_\gamma+b\delta_\gamma$, where $\delta_\gamma$ is Dirac's $\delta$-function supported on $\gamma$ and $\partial_\nu\delta_\gamma$ is its normal derivative on $\gamma$. Assuming that the operator $-\Delta +W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $\gamma$ for the eigenfunctions $u+=\alpha u-$, $\alpha\, \partial_\nu u+-\partial_\nu u-=\beta u-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $\gamma$.

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