Related papers: Random Schr\"odinger operators with a background p…
As a prototype of an evolution equation we consider the Schr\"odinger equation i (d/dt) \Psi(t) = H \Psi(t), H = H_0 + V(x) for the Hilbert space valued function \Psi(.) which describes the state of the system at time t in space dimension…
We prove that, for a density of disorder $\rho$ small enough, a certain class of discrete random Schr\"odinger operators on $\Z^d$ with diluted potentials exhibits a Lifschitz behaviour from the bottom of the spectrum up to energies at a…
We use trace class scattering theory to exclude the possibility of absolutely continuous spectrum in a large class of self-adjoint operators with an underlying hierarchical structure and provide applications to certain random hierarchical…
The matrix Schr\"odinger equation is considered on the half line with the general selfadjoint boundary condition at the origin described by two boundary matrices satisfying certain appropriate conditions. It is assumed that the matrix…
\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct…
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)…
We consider Schr\"odinger operators in $\ell^2(\Z)$ whose potentials are obtained by randomly concatenating words from an underlying set $\mathcal{W}$ according to some probability measure $\nu$ on $\mathcal{W}$. Our assumptions allow us to…
We consider the 1d Schr\"odinger operator with random decaying potential and compute the 2nd term asymptotics of the density of states, which shows substantial differences between the cases $\alpha > \frac 12$, $\alpha < \frac 12$ and…
I consider random Schr\"odinger operators with exponentially decaying single site potential, which is allowed to change sign. For this model, I prove Anderson localization both in the sense of exponentially decaying eigenfunctions and…
The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…
We study the effect of non-negative potentials on the spectral gap of one-dimensional Schr\"odinger operators in the limit of large intervals. In particular, we derive upper and lower bounds on the gap for different classes of potentials…
We establish Anderson localization for 1-d discrete Schr\"odinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and…
We consider a two dimensional magnetic Schroedinger operator with a spatially stationary random magnetic field. We assume that the magnetic field has a positive lower bound and that it has Fourier modes on arbitrarily short scales. We prove…
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials…
Inverse scattering theory is extended to one-dimensional Schr\"odinger problems with near-boundary singularities of the form $v(z\to 0)\simeq -z^{-2}/4+v_{-1}z^{-1}$. Trace formulae relating the boundary value $v_0$ of the nonsingular part…
We study the spectral properties of a Schr\"odinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates,…
We show that fixed energy scattering measurements for the magnetic Schroedinger operator uniquely determine the magnetic field and electric potential in dimensions $n \geq 3$. The magnetic potential, its first derivatives, and the electric…
We study the region of complete localization in a class of random operators which includes random Schr\"odinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding…
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.
We solve inverse scattering problem for Schr\"odinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the S-matrix on some infinite sequence of positive real numbers. Using this…