ADSO - Approximation problems for Dirac and Schrödinger operators (FWF Project P33568)

The FWF Project P33568 is funded by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF) and supervised by Univ.-Prof. Dr. Jussi Behrndt and Dr. Markus Holzmann.

Project Team
Team at Graz University of Technology
Cooperation partners
Project Description

In many applications in science and engineering it is not possible to solve the underlying mathematical models exactly. Hence, suitable parameters in these mathematical models are replaced by idealized ones. The parameters should be chosen in such a way that the idealized model is easier accessible from a mathematical point of view and such that it still reflects the physical reality up to a reasonable level of exactness. To verify that the idealized models have similar properties as the original ones coming from applications is a difficult mathematical problem which is unsolved in many cases. It is the main goal of this project to justify the usage of several types of idealized models in a mathematically rigorous way. In the first part of the project so called Schrödinger operators with singular potentials are investigated. They play an important role in solid state physics to describe the propagation of particles in certain nano structures and also in the description of photonic crystals, which are already in use in computer systems as faster replacements for semiconductors. For these models there exist, under elementary assumptions, results which justify the replacement of the realistic parameters by the idealized singular potentials. It is one of the main goals in this project to extend these results to situations that appear in realistic applications in science and engineering. The second part is on so called Dirac operators, which are used in problems, where effects of the special theory of relativity play an important role. For instance, this is the case in the description of elementary particles like quarks or in the analysis of graphen, which appear in research for batteries, water filters or photovoltaic cells. For these problems the mathematical investigations are still at the very beginning. It is one of the main goals in this project to find elementary results on how parameters should be chosen in certain models such that the mathematical models reflect the physical reality in the correct way.

Related publications
  1. J. Behrndt, M. Holzmann, C. Stelzer, G. Stenzel:
    A class of singular perturbations of the Dirac operator: Boundary triplets and Weyl functions,
    Acta Wasaensia 462 (2021), 15-35.

  2. J. Behrndt, M. Holzmann, V. Lotoreichik, G. Raikov:
    The fate of Landau levels under δ-interactions,
    J. Spectral Theory (2022), 1203-1234; arXiv.

  3. J. Behrndt, M. Holzmann, M. Tusek:
    Spectral transition for Dirac operators with electrostatic δ-shell potentials supported on the straight line,
    Integral Equations Operator Theory 94 (2022), Art. 33 (13 pages); arXiv.

  4. J. Behrndt und A. Khrabustovskyi:
    Singular Schrödinger operators with prescribed spectral properties,
    J. Funct. Anal. 282 (2022), 109252, 49 pp.; arXiv.

  5. J. Behrndt, V. Lotoreichik und P. Schlosser:
    Schrödinger operators with δ-potentials supported on unbounded Lipschitz hypersurfaces,
    accepted for publication in Operator Theory: Advances and Applications; arXiv.

  6. P. Exner, M. Holzmann:
    Dirac operator spectrum in tubes and layers with a zigzag type boundary,
    Lett. Math. Phys. 112 (2022), Art. 102 (23 pages); arXiv.

  7. J. Behrndt, M. Holzmann, M. Tusek:
    Two-dimensional Dirac operators with general δ-shell interactions supported on a straight line,
    J. Phys. A 56 (2023), Art. 045201 (29 pages), arXiv.

  8. J. Behrndt, P. Schmitz, G. Teschl, C. Trunk:
    Relative oscillation theory and essential spectra of Sturm--Liouville operators, arXiv.

  9. J. Behrndt, M. Holzmann, G. Stenzel:
    Schrödinger operators with oblique transmission conditions in R^2,
    accepted for publication in Comm. Math. Phys.; arXiv.

  10. M. Holzmann:
    On the single layer boundary integral operator for the Dirac equation, arXiv.

  11. J. Behrndt, M. Holzmann, C. Stelzer, G. Stenzel:
    Boundary triples and Weyl functions for Dirac operators with singular interactions, arXiv.

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