# Perturbations of continuum random Schr\"odinger operators with applications to Anderson orthogonality and the spectral shift function

## TL;DR

This paper investigates how bounded, compactly supported perturbations affect multi-dimensional continuum random Schr"odinger operators, focusing on Anderson orthogonality and spectral shift functions within the localized regime, providing rigorous proofs and technical estimates.

## Contribution

It establishes the occurrence of Anderson orthogonality with non-zero probability in the localized regime and links the spectral shift function to spectral projection indices, supported by key exponential decay estimates.

## Key findings

- Anderson orthogonality occurs with positive probability in the localized regime.
- Spectral shift function is identified with the index of spectral projection pairs.
- Main technical estimate shows exponential decay of disorder-averaged Schatten norms.

## Abstract

We study effects of a bounded and compactly supported perturbation on multi-dimensional continuum random Schr\"odinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schr\"odinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11, 560-565 (2015)]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten $p$-norm of $\chi_{a}(f(H) - f(H^{\tau})) \chi_{b}$ in $a$ and $b$. Here, $H^{\tau}$ is a perturbation of the random Schr\"odinger operator $H$, $\chi_{a}$ is the multiplication operator corresponding to the indicator function of a unit cube centred about $a\in\mathbb{R}^{d}$, and $f$ is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for $H$.