Spectral shift function and Resonances near the low ground state for Pauli and Schr\"odinger operators

TL;DR

This paper investigates the spectral shift function and resonances near zero energy for Pauli and Schrödinger operators with magnetic fields, providing a new representation and insights into their singularities and trace formulas.

Contribution

It introduces a novel representation of the SSF derivative involving holomorphic functions and harmonic measures, advancing understanding of spectral properties near the ground state.

Findings

Representation of SSF derivative as a sum involving holomorphic functions and resonances

Analysis of singularities of SSF at zero energy

Derivation of a local trace formula for the operator

Abstract

We study the spectral shift function (SSF) $\xi(\lambda)$ and the resonances of the operator $H_V := \big( \sigma \cdot (-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $\sigma := (\sigma_1,\sigma_2,\sigma_3)$ are the $2 \times 2$ Pauli matrices and $V$ is a hermitian potential decaying exponentially in the direction of the magnetic field $\textbf{B} := \text{curl} \hspace{0.6mm} \textbf{A}$. We give a representation of the derivative of the SSF as a sum of the imaginary part of a holomorphic function and a harmonic measure related to the resonances of $H_V$. This representation warrant the Breit-Wigner approximation moreover we deduce information about the singularities of the SSF at the origin and a local trace formula.