Bound states of weakly deformed soft waveguides

• URL: http://arxiv.org/abs/2211.01989v1
• Date: Thu, 3 Nov 2022 16:56:39 GMT
• Title: Bound states of weakly deformed soft waveguides
• Authors: Pavel Exner, Sylwia Kondej, Vladimir Lotoreichik
• Abstract summary: We consider the two-dimensional Schr"odinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region. In the critical case $int_mathbbR f,mathsfd x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $varepsilon > 0$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we consider the two-dimensional Schr\"odinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly supported continuous function. We prove that if $\int_{\mathbb{R}} f \,\mathsf{d} x > 0$, then the respective Schr\"odinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove that the discrete spectrum is empty for all sufficiently small $\varepsilon > 0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\varepsilon > 0$.

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