Spectral shift functions and Dirichlet-to-Neumann maps.

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.