Spectral Theory for Schrödinger Operators with δ-Interactions Supported on Curves in R³

Creators: Behrndt, Jussi; Frank, Rupert L.; Kühn, Christian; Lotoreichik, Vladimir; Rohleder, Jonathan

Abstract

The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with δ-interactions supported on closed curves in R^3. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten–von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.

Additional Information

© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Communicated by Jan Derezinski. Open access funding provided by Graz University of Technology. Jussi Behrndt, Christian Kühn, Vladimir Lotoreichik, and Jonathan Rohleder gratefully acknowledge financial support by the Austrian Science Fund (FWF), Project P 25162-N26. Vladimir Lotoreichik also acknowledges financial support by the Czech Science Foundation, Project 14-06818S. Rupert Frank acknowledges support through NSF Grant DMS-1363432. The authors also wish to thank Johannes Brasche and Andrea Posilicano for helpful discussions and the anonymous referees for their helpful comments which led to various improvements.

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