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Bound States in a Locally Deformed Waveguide: The Critical Case

Authors:	Exner P Vugalter SA

Institution:	(1) Nuclear Physics Institute, Academy of Sciences, 25068 e, Prague;(2) Doppler Institute, Czeck TEchnical University, Behova´ 7, 11519 Prague, Czech Republic;(3) Nuclear Physics Instittute, Academy of Sciences, 25068 e, Prague, Czech Republic;(4) Radiophysical Research Institute, B. Pecherskaya 25/14, 603600 Nizhni Novgorod, Russia

Abstract:	We consider the Dirichlet Laplacian for astrip in $\mathbb{R}^2$ with one straight boundary and a width $a(1 + \lambda f(x))$ , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, $\int {_{ - b}^b } f(x)dx = 0$ , the operator has nobound statesfor small $\left\| \lambda \right\|{\text{ if }}b < (\sqrt 3 /4)a$ .On the otherhand, a weakly bound state existsprovided $\left\\| {f\prime } \right\\| < 1.59a^{ - 1} \left\\| f \right\\|$ . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies $- c_1 \lambda ^4 \leqslant \varepsilon (\lambda ) - (\pi /a)^2 \leqslant - c_2 \lambda ^4$ for all $\left\| \lambda \right\|$ sufficiently small.

Keywords:	quantum waveguides Dirichlet Laplacians bound states
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