A geometric approximation of δ-interactions by Neumann Laplacians. (25th October 2021)

• Record Type: Journal Article
• Authors: Khrabustovskyi, Andrii; Post, Olaf
• Is Part Of: Journal of physics. Volume 54:Number 46(2021)
• Publication Date: 2021-10-25
• Subjects: δ-interaction -- singularly perturbed domains -- Neumann Laplacian -- norm resolvent convergence -- operator estimates -- spectral convergence; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ac2d52
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We demonstrate how to approximate one-dimensional Schrödinger operators with δ -interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with 'room-and-passage' geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in generalised norm resolvent sense to the above singular Schrödinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.