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

2018/08/16 by Giuseppe Cardone, Cardone, Giuseppe, Andrii Khrabustovskyi +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1808.05402

Abstract

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 ε. 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 ε→ 0 to a one-dimensional Schrödinger operator corresponding to a δ'-interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.

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