$\delta'$-interaction as a limit of a thin Neumann waveguide with transversal window

TL;DR

This paper studies how a thin waveguide with a small connecting window behaves as the window shrinks, showing it converges to a one-dimensional Schrödinger operator with a delta-prime interaction, with spectral implications.

Contribution

It demonstrates the convergence of the Neumann Laplacian on a thin waveguide to a delta-prime interaction operator as the window size tends to zero, including spectral convergence and rate estimates.

Findings

Convergence of the Neumann Laplacian to a delta-prime interaction operator.

Spectral convergence of the operators.

Quantitative estimates of the convergence rate.

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 $\varepsilon$. 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.