Spectral shift function and Resonances near the low ground state for Pauli and Schrödinger operators

Abstract

We study the spectral shift function (SSF) $\xi (\lambda )$ and the resonances of the operator $H_V:=(\sigma \cdot (-i\nabla -\text{A}){)}^2+V$ 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 $\text{B}:=\text{curl}\text{A}$. 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.

30 pages