Perturbation of threshold of essential spectrum for waveguides with windows. II: Asymptotics
type: Item
instance of: scholarly article
title: Perturbation of threshold of essential spectrum for waveguides with windows. II: Asymptotics
zbMATH Open document ID: 1335.35162
author: Denis Borisov
published in: Journal of Mathematical Sciences (New York)
publication date: 2016-01-19T00:00:00Z
review text: Let \(x = (x^\prime, x_3)\) and \(x^\prime = (x_1, x_2)\) be the Cartesian coordinates in \({\mathbb R}^3\) and \({\mathbb R}^2\), respectively, and let \(\omega\) (window) be a bounded connected domain in \({\mathbb R}^2\) with \(C^\infty\) boundary. Set \(\Pi_\omega = \{x : x_3 \in (-d, 0) \cup (0, \pi)\} \cup(\omega\times\{0\})\), where \(d \leq \pi\). Consider the magnetic Schrödinger operator \({\mathcal H}_\omega=(i\nabla + A)^2+V\) in \(\Pi_\omega\) with a Dirichlet boundary condition, where the magnetic and electric potentials \(A = A(x) = (A_1(x),A_2(x),A_3(x))\) and \(V = V(x)\) are assumed to be infinitely differentiable and compactly supported. The essential spectrum \(\sigma_{\mathrm{ess}}({\mathcal H}_\omega)=[1,+\infty)\). In the previous paper of the same author (see [J. Math. Sci., New York 205, No. 2, 141--181 (2015; Zbl 1325.35119); translation from Probl. Mat. Anal. 77, 19--54 (2014)]), it is proved that there are critical window shapes such that an indefinitely small enlargement of the window size (\(\omega\to\omega_\varepsilon\)) generates one or several eigenvalues emerging from the threshold of the essential spectrum. In the paper under rewiew, the author finds complete asymptotic expansions (and analyzes their structure) for the spectral parameters \(k_\varepsilon\) and also nontrivial solutions \(\psi_\varepsilon\) to the boundary value problem \[ ((i\nabla+A)^2+V)\psi_\varepsilon=(1-k^2_\varepsilon)\psi_\varepsilon\;\text{in }\;\Pi_{\omega_\varepsilon}, \quad \psi_\varepsilon=0\;\text{on }\;\partial\Pi_{\omega_\varepsilon}, \] in the space \(L^2(\Pi_{\omega_\varepsilon})\).
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47N50
zbMATH DE Number: 6532822
zbMATH Keywords: waveguides
zbMATH Keywords: bound-state asymptotic estimates
zbMATH Keywords: discrete spectrum
zbMATH Keywords: resonance
zbMATH Keywords: Schrödinger operator
MaRDI profile type: Publication
full work available at URL: s10958-015-2580-5
OpenAlex ID: W2169357999
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DOI: 10.1007/S10958-015-2580-5
description: scientific article; zbMATH DE number 6532822