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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-and-passage geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense to the above singular Schrodinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.
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 pr
We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Lambda_q$, $q in mathbb{Z}_+$, and the perturbed operator $H_upsilon = H_0 + upsilondelta_Gamma$, where $Gamma s
Let $Omega_-$ and $Omega_+$ be two bounded smooth domains in $mathbb{R}^n$, $nge 2$, separated by a hypersurface $Sigma$. For $mu>0$, consider the function $h_mu=1_{Omega_-}-mu 1_{Omega_+}$. We discuss self-adjoint realizations of the operator $L_{mu
We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furt
We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (mathcal{L}u)(i) = -sum_{j sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i in V$, we defi