Spectral shift functions and Dirichlet-to-Neumann maps

TL;DR

This paper links spectral shift functions of self-adjoint operators to Dirichlet-to-Neumann maps, providing explicit formulas for various elliptic and Schrödinger operators, enhancing understanding of spectral perturbations.

Contribution

It introduces an abstract operator-valued Titchmarsh--Weyl m-function approach to express spectral shift functions, applied to multiple classes of differential operators with explicit formulas.

Findings

Spectral shift functions are expressed via Dirichlet-to-Neumann maps.

Explicit formulas are derived for elliptic and Schrödinger operators.

Applications include operators with smooth, compact, and singular potentials.

Abstract

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schr\"{o}dinger operators with compactly supported potentials, and finally, Schr\"{o}dinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.