We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L , we prove that with high probability the eigenvalues below some threshold energy E_{sp} keep a distance of at least e^{-\mathrm{log} L)^\beta} for sufficiently large \beta > 1 . This implies simplicity of the spectrum of the infinite-volume operator below E_{sp} . Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E .

We prove full Szegő-type large-box trace asymptotics for selfadjoint Z d -ergodic operators Ω ∋ ω → Hω acting on L 2 (R d ). More precisely, let g be a bounded, compactly supported and real-valued function such that the (averaged) operator kernel of g(Hω) decays sufficiently fast, and let h be a sufficiently smooth compactly supported function. We then prove a full asymptotic expansion of the averaged trace Tr h(g(Hω) [−L,L] d ) in terms of the length-scale L.

For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(log L) β for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E. Contents 1. Introduction 1 2. Model and results 3 3. Clusters of eigenvalues 9 4. Proof of the level spacing estimates 13 5. Proof of the Minami-type estimate 24 6. Simplicity of spectrum and Poisson statistics 26 Appendix A. Properties of deformed Schrödinger operators 29 Appendix B. Eigenfunction decay for localized energies 30 Acknowledgements 32 References 32

We obtain a bound on the expectation of the spectral shift function for alloytype random Schrödinger operators on R d in the region of localisation, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the area of the surface where the boundary conditions are changed. As an application of our bound on the spectral shift function, we prove a reverse Wegner inequality for finite-volume Schrödinger operators in the region of localisation with a constant locally uniform in the energy. The application requires that the singlesite distribution of the independent and identically distributed random variables has a Lebesgue density that is also bounded away from zero. The reverse Wegner inequality implies a strictly positive, locally uniform lower bound on the density of states for these continuum random Schrödinger operators.

We study effects of a bounded and compactly supported perturbation on multi-dimensional 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, 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 χa(f (H) − f (H τ ))χ b in a and b. Here, H τ is a perturbation of the random Schrödinger operator H, χa is the multiplication operator corresponding to the indicator function of a unit cube centred about a ∈ 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.