Schrödinger operators and singular perturbations
(FWF project 25162-N26)

Team at Graz University of Technology

Cooperation partners

Project summary

The analysis and spectral theory of Schrödinger operators with δ-potentials have attracted an enormous attention in the last decades. This field in mathematical physics is particularly challenging, as it crosses boundaries between operator theory, mathematical analysis and partial differential equations. Point interaction models (also called zero range or δ-interaction models) serve as rough approximations of more complicated interactions in quantum systems, and are regarded as solvable models in the sense that the spectrum, eigenfunctions, resonances and scattering quantities can be computed explicitly. This project makes a new step towards more realistic models and goes far beyond the present state of the art in the theory of singular and supersingular δ-interaction models. Instead of δ-interactions supported on finite (or discrete) point sets we will investigate δ-interactions which are supported on general curves, surfaces and manifolds of arbitrary co-dimension. This requires advanced analytic tools, in particular, refined methods from operator theory and partial differential equations. Our main objective is to develop an abstract perturbation approach to infinite dimensional singular perturbations of selfadjoint operators, and to provide a thorough and in-depth spectral analysis of Schrödinger operators and more general elliptic partial differential operators with singular and supersingular δ-potentials supported on manifolds. Furthermore, it is planned to investigate closely related scattering and multidimensional inverse problems, to analyse bound states of Schrödinger operators with δ-interactions on curves and special surfaces, and to discuss some explicitly solvable models. The proposed project is of fundamental nature since only certain cases of δ-perturbations supported on curves and manifolds have been understood by the time, and a general operator theoretic approach and a comprehensive spectral analysis of singular and supersingular δ-interactions – as is our vision – does not exist until now.

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
  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, Christian Kühn and Jonathan Rohleder
    Eigenvalues of Schrödinger operators and Dirichlet-to-Neumann maps
    Proc. Appl. Math. Mech. (2013), 517–518.
  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. Jussi Behrndt, Matthias Langer, and Vladimir Lotoreichik
    Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators
    J. London. Math. Soc. (2) 88 (2013), 319–337.
  6. Jussi Behrndt and Till Micheler
    Elliptic differential operators on Lipschitz domains and abstract boundary value problems
  7. Vladimir Lotoreichik
    Note on 2D Schrödinger operators with δ-interactions on angles and crossing lines
    Nanosystems: Phys. Chem. Math. 4 (2013), 166–172.
  8. Vladimir Lotoreichik
    Lower bounds on the norms of extension operators for Lipschitz domains
    to appear in Operators and Matrices.