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

Giuseppe Cardone, Andrii Khrabustovskyi*

*Corresponding author for this work

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)1320-1342
Number of pages23
JournalJournal of mathematical analysis and applications
Volume473
Issue number2
DOIs
Publication statusPublished - 15 May 2019

Keywords

  • Neumann Laplacian
  • Norm resolvent convergence
  • Operator estimates
  • Spectral convergence
  • Thin waveguide
  • δ-interaction

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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