# Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach

## TL;DR

This paper develops a semiclassical trace formula for hermitian systems of pseudodifferential operators and applies it to analyze the spectral shift function for matrix-valued Schrödinger operators, providing precise asymptotics and expansions.

## Contribution

It introduces a general stationary approach to derive semiclassical trace formulas and spectral shift function asymptotics for systems with energy crossings.

## Key findings

- Weyl type semiclassical asymptotics with sharp remainder for spectral shift function
- Full asymptotic expansion of the derivative of the spectral shift function
- Effective treatment of potentials with energy-level crossings

## Abstract

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schr\"odinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.