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We consider a waveguide-like domain consisting of two thin straight tubular domains connected through a tiny window. The perpendicular size of this waveguide is of order $varepsilon$. Under the assumption that the window is appropriately scaled we prove that the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense as $varepsilonto 0$ to a one-dimensional Schrodinger operator corresponding to a $delta$-interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.
This paper is concerned with the study of theexistence/non-existence of the discrete spectrum of the Laplaceoperator on a domain of $mathbb R ^3$ which consists in atwisted tube. This operator is defined by means of mixed boundaryconditions. Here we
We demonstrate how to approximate one-dimensional Schrodinger operators with $delta$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with room-
We consider the self-adjoint Schrodinger operator in $L^2(mathbb{R}^d)$, $dgeq 2$, with a $delta$-potential supported on a hyperplane $Sigmasubseteqmathbb{R}^d$ of strength $alpha=alpha_0+alpha_1$, where $alpha_0inmathbb{R}$ is a constant and $alpha_
Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_ell : = H_{0, ell} - V$, $ell =D,N$, where the scalar potential $V$ is non neg
Let $Lambdasubset mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Delta u= u u$ in $Lambda$ and va