Spectral shift function for operators with crossed magnetic and electric fields

TL;DR

This paper derives a formula for the spectral shift function's derivative for operators with crossed magnetic and electric fields, establishing a limiting absorption principle and estimates under certain spectral conditions.

Contribution

It introduces a new representation formula for the spectral shift function derivative in the context of magnetic and electric field operators with potential.

Findings

Derived a representation formula for the spectral shift function derivative.

Established a limiting absorption principle for the operator.

Provided an estimate for the spectral shift function derivative away from the spectrum.

Abstract

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y), \: B > 0, \epsilon > 0$. We establish a limiting absorption principle for $H(B, \epsilon)$ and an estimate ${\mathcal O}(\epsilon^{n-2})$ for $\xi'(\lambda; B, \epsilon)$, provided $\lambda \notin \sigma(Q)$, where $Q = (D_x - By)^2 + D_y^2 + V(x,y).$