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In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"odinger operators in one dimension. These…
We consider perturbations of quasi-periodic Schr\"odinger operators on the integer lattice with analytic sampling functions by decaying potentials and seek decay conditions under which various spectral properties are preserved. In the…
We give an overview and extension of recent results on ergodic random Schr\"odinger operators for models on $\mathbb{Z}^d$. The operators we consider are defined on combinatorial or metric graphs, with random potentials, random boundary…
In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an…
We study multistate Schr\"odinger operators related to molecular dynamics. We consider potentials which do not necessarily decay and prove absence of the singular continuous spectrum and propagation estimates which mean the scattering at…
We compare the bottom of the spectrum of discrete and continuous Schr\"odinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the…
We first analyze the integrated density of states (IDS) of periodic Schr\"odinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with…
We prove that the integrated density of states (IDS) of random Schr\"{o}dinger operators with Anderson-type potentials on $L^2 (\R^d)$, for $d \geq1$, is locally H\"{o}lder continuous at all energies with the same H\"{o}lder exponent…
Spectral properties of Schr\"odinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for…
We discuss recent results on spectral properties of discrete alloy-type random Schr\"odinger operators. They concern Wegner estimates and bounds on the fractional moments of the Green's function.
This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…
In this paper we continue the study of the spectral gap of Schr\"odinger operators on large intervals and subject to Neumann boundary conditions. The main goal is to derive a lower bound on the spectral gap which is polynomial in the…
We study spectral properties of convolution operators $\mathcal L$ and their perturbations $H=\mathcal L+v(x)$ by compactly supported potentials. Results are applied to determine the front propagation of a population density governed by…
In this work we investigate the spectral statistics of random Schr\"{o}dinger operators $H^\omega=-\Delta+\sum_{n\in\mathbb{Z}^d}(1+|n|^\alpha)q_n(\omega)|\delta_n\rangle\langle\delta_n|$, $\alpha>0$ acting on $\ell^2(\mathbb{Z}^d)$ where…
We extend our results in \cite{hislop_marx_1} on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schr\"odinger…
We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…
We study the Schr\"odinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and…
We study the stationary scattering theory for the matrix Schr\"odinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption…
We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field and subject to an electric potential $v_0$ depending only on the variable along the magnetic field $x_3$. The operator $H_0$ has infinitely many eigenvalues of…
We show that there exist limit-periodic Schr\"odinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of P\"oschel.