English

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

Spectral Theory 2018-08-17 v1 Mathematical Physics Analysis of PDEs math.MP

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\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 ε0\varepsilon\to 0 to a one-dimensional Schr\"odinger operator corresponding to a δ\delta'-interaction of a non-negative strength. We estimate the rate of this convergence, also we prove the convergence of spectra.

Keywords

Cite

@article{arxiv.1808.05402,
  title  = {$\delta'$-interaction as a limit of a thin Neumann waveguide with transversal window},
  author = {Giuseppe Cardone and Andrii Khrabustovskyi},
  journal= {arXiv preprint arXiv:1808.05402},
  year   = {2018}
}

Comments

26 pages, 2 figures