Schrödinger operators with δ- and δ'-interactions on manifolds


Project teams
Czech team
 Austrian team

nuclear

 TU Graz
(Nuclear Physics Institute, Czech Academy of Sciences)
Prof. Dr. Pavel Exner (supervision), Dr. Petr Siegl, Dr. Milos Tater and M. Sc. Michal Jex 		(Institute of Computational Mathematics, Graz University of Technology)
Prof. Dr. Jussi Behrndt (supervision), Dr. Vladimir Lotoreichik (supervision), Dr. Jonathan Rohleder and Dipl.-Math. Christian Kühn.

Motivation


Schrödinger operators with δ-interactions on curves and special hypersurfaces have been studied intensively in the last decades. Many of their analytic and spectral properties are well understood at the present moment. In contrast to that, spectral problems for Schrödinger operators with δ-interactions on manifolds of higher co-dimension have been studied only very little. These more singular δ-interactions have applications in quantum many-body problems, where classical regular potentials are approximatively replaced by equivalent δ-potentials. The corresponding Schrödinger operators may have unexpected properties, e.g. the spectrum is not necessarily bounded from below. These properties are still not investigated in-depth. Another class of more singular interactions are δ'-interactions on curves and hypersurfaces. The treatment of such type of interactions is still an open problem. Even a rigorous general definition of the corresponding self-adjoint Schrödinger operator does not exist until now.


Project goals


  • Investigation of Schrödinger operators with δ'-interactions on Lipschitz hypersurfaces.
  • Development of an approach to Schrödinger operators with δ-interactions on manifolds of higher co-dimensions.
  • Analysis of many-body systems with δ-interactions.


Related publications

  1. Jussi Behrndt, Pavel Exner and Vladimir Lotoreichik
    Schrödinger operators with δ and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
    submitted.
  2. Jussi Behrndt, Pavel Exner and Vladimir Lotoreichik
    Essential spectrum of Schrödinger operators with δ-interactions on the union of compact Lipschitz hypersurfaces
    Proc. Appl. Math. Mech. (2013), 523–524.
  3. Jussi Behrndt, Pavel Exner and Vladimir Lotoreichik
    Schrödinger operators with δ-interactions supported on conical surfaces
    to appear in J. Phys. A: Math. Theor.
  4. Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
    Schrödinger operators with δ and δ'-potentials supported on hypersurfaces
    Ann. Henri Poincaré 14 (2013), 385–423.
  5. Vladimir Lotoreichik
    Note on 2D Schrödinger operators with δ-interactions on angles and crossing lines
    Nanosystems: Phys. Chem. Math. 4 (2013), 166–172.


Visits in 2013

  1. Vladimir Lotoreichik, Rez, 8 July - 29 July.
  2. Petr Siegl, Graz, 2 September - 8 September.
  3. Michal Jex, Graz, 23 September - 19 October.
  4. Christian Kühn, Rez, 30 September - 11 October.
  5. Jussi Behrndt, Rez, 20 November - 1 December.
  6. Pavel Exner, Graz, 16 December - 22 December.


Visits in 2014

  1. Vladimir Lotoreichik, Rez, 12 May - 25 May.
  2. Michal Jex, Graz, 22 August - 12 September.
  3. Petr Siegl, Graz, 1 September - 12 September.
  4. Jonathan Rohleder, Rez, 8 October - 17 October.


Events

Mini-Workshop: Schrödinger operators with δ-interactions on manifolds, 24 September 2013.
Program.