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We study effects of a bounded and compactly supported perturbation onmulti-dimensional continuum random Schr\"odinger operators in the region ofcomplete localisation. Our main emphasis is on Anderson orthogonality forrandom Schr\"odinger operators. Among others, we prove that Andersonorthogonality does occur for Fermi energies in the region of completelocalisation with a non-zero probability. This partially confirms recentnon-rigorous findings [V. Khemani et al., Nature Phys. 11, 560-565 (2015)]. Thespectral shift function plays an important role in our analysis of Andersonorthogonality. We identify it with the index of the corresponding pair ofspectral projections and explore the consequences thereof. All our results relyon the main technical estimate of this paper which guarantees separateexponential 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 ofthe random Schr\"odinger operator $H$, $\chi_{a}$ is the multiplicationoperator 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 boundedvariation with distributional derivative supported in the region of completelocalisation for $H$.

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