# Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function

### Adrian Dietlein

Ludwig-Maximilians-Universität München, Germany### Martin Gebert

University of California, Davis, USA### Peter Müller

Ludwig-Maximilians-Universität München, Germany

A subscription is required to access this article.

## Abstract

We study effects of a bounded and compactly supported perturbation on multidimensional continuum random Schrödinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schrödinger 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 (2015), 560–565]. 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 Schrödinger 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 $H$.

## Cite this article

Adrian Dietlein, Martin Gebert, Peter Müller, Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function. J. Spectr. Theory 9 (2019), no. 3, pp. 921–965

DOI 10.4171/JST/267