Annales Henri Poincaré

, Volume 18, Issue 4, pp 1305–1347 | Cite as

Spectral Theory for Schrödinger Operators with \(\varvec{\delta }\)-Interactions Supported on Curves in \(\varvec{\mathbb {R}^3}\)

  • Jussi Behrndt
  • Rupert L. Frank
  • Christian Kühn
  • Vladimir Lotoreichik
  • Jonathan Rohleder
Open Access


The main objective of this paper is to systematically develop a spectral and scattering theory for self-adjoint Schrödinger operators with \(\delta \)-interactions supported on closed curves in \(\mathbb {R}^3\). 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.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Authors and Affiliations

  • Jussi Behrndt
    • 1
  • Rupert L. Frank
    • 2
  • Christian Kühn
    • 1
  • Vladimir Lotoreichik
    • 3
  • Jonathan Rohleder
    • 4
  1. 1.Institut für Numerische MathematikTU GrazGrazAustria
  2. 2.Mathematics 253-37 CaltechPasadenaUSA
  3. 3.Department of Theoretical Physics, Nuclear Physics InstituteCzech Academy of SciencesŘežCzech Republic
  4. 4.Institut für MathematikTU HamburgHamburgGermany

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