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Spectral shift functions and Dirichlet-to-Neumann maps.

Jussi Behrndt1, Fritz Gesztesy2, Shu Nakamura3

  • 11Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria.

Mathematische Annalen
|April 2, 2019
PubMed
Summary
This summary is machine-generated.

This study introduces a novel method to calculate the spectral shift function using an operator-valued Titchmarsh-Weyl m-function. This approach is applied to elliptic partial differential and Schrödinger operators, yielding explicit spectral shift function formulas.

Keywords:
35J1535P2547A1047A4047A5547B1047B2547F0581Q10Primary 35J10Secondary 35P20

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Area of Science:

  • Mathematical Physics
  • Spectral Theory
  • Partial Differential Equations

Background:

  • The spectral shift function quantifies changes in the spectrum of self-adjoint operators under perturbation.
  • Abstract operator theory provides a framework for studying spectral properties of various operators.

Purpose of the Study:

  • To express the spectral shift function using an abstract operator-valued Titchmarsh-Weyl m-function.
  • To apply this general result to specific classes of operators, including elliptic PDEs and Schrödinger operators.

Main Methods:

  • Utilizing the abstract operator-valued Titchmarsh-Weyl m-function.
  • Applying the method to self-adjoint realizations of second-order elliptic partial differential operators.
  • Analyzing Schrödinger operators with compactly supported potentials.

Main Results:

  • An explicit formula for the spectral shift function is derived.
  • The Dirichlet-to-Neumann map is employed to determine the spectral shift function for specific operator types.
  • The general theory is successfully applied to concrete problems in mathematical physics.

Conclusions:

  • The Titchmarsh-Weyl m-function provides a powerful tool for spectral analysis.
  • The Dirichlet-to-Neumann map offers an effective method for explicit spectral shift function computation.
  • This work unifies spectral shift function calculations across different operator classes.