Perturbation of threshold of essential spectrum for waveguides with windows. II: Asymptotics
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 \(L^2(\Pi_{\omega_\varepsilon})\).
scientific article; zbMATH DE number 6532822
scientific article