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arxiv: 1808.05402 · v1 · pith:47MZLTP7new · submitted 2018-08-16 · 🧮 math.SP · math-ph· math.AP· math.MP

δ'-interaction as a limit of a thin Neumann waveguide with transversal window

classification 🧮 math.SP math-phmath.APmath.MP
keywords windowconvergencedeltadomaininteractionneumannprovethin
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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 $\varepsilon\to 0$ to a one-dimensional Schr\"odinger 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.

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