Related papers: Rigorous results on Schroedinger operators with ce…
We study discrete quasiperiodic Schr\"odinger operators on $\ell^2(\zee)$ with potentials defined by $\gamma$-H\"older functions. We prove a general statement that for $\gamma >1/2$ and under the condition of positive Lyapunov exponents,…
In this paper we consider two classes of random Hamiltonians on $L^2(\RR^d)$ one that imitates the lattice case and the other a Schr\"odinger operator with non-decaying, non-sparse potential both of which exhibit a.c. spectrum. In the…
We study fluctuations of polynomial linear statistics for discrete Schr\"odinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth…
Here, the Morgan type uncertainty principle and unique continuation properties of abstract Schredinger equations with time dependent potentials are obtained in Hilbert space valued function classes. The equations include linear operator in…
We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…
We study continuum Schr\"odinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters.…
We prove quantitative spectral inequalities for the (anisotropic) Shubin operators on the whole Euclidean space, thus relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole space…
This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…
We study the scattering properties of Schr\"{o}dinger operators with bounded potentials concentrated near a subspace of $\mathbb{R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal…
We prove approximate controllability of the bilinear Schr\"odinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and…
A periodic Schr\"odinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions…
We characterize the spectrum of one-dimensional Schr\"odinger operators H=-d^2/dx^2+V with quasi-periodic complex-valued algebro-geometric potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the…
We survey some aspects of the theory of the integrated density of states (IDS) of random Schroedinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the…
We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition…
In this paper, we study the multidimensional lattice Schr\"odinger operators with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative Green's function estimates, the arithmetic version of Anderson (and dynamical)…
The scattering matrix of the Schrodinger operator with smooth short-range electric and magnetic potentials is considered. The asymptotic density of the eigenvalues of this scattering matrix in the high energy regime is determined. An…
We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schr\"odinger operators on the line defined with a potential generated by this system and a H\"older sampling function, have almost…
We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schr\"odinger operators on the discrete line or…
We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schroedinger operators with magnetic fields and scalar potentials, the first one relying on the eigenvalue counting function of…
We construct multidimensional almost-periodic Schr\"odinger operators whose spectrum has zero lower box counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.