Related papers: Eigenfunctions and Quantum Transport with Applicat…
We consider a natural class of extensions of the Anderson model on $\mathbb Z^d$, called random block Schr\"odinger operators (RBSOs), defined on the $d$-dimensional torus $(\mathbb Z/L\mathbb Z)^d$. These operators take the form…
We consider two models of one-dimensional discrete random Schrodinger operators (H_n \psi)_l ={\psi}_{l-1}+{\psi}_{l +1}+v_l {\psi}_l, {\psi}_0={\psi}_{n+1}=0 in the cases v_k=\sigma {\omega}_k/\sqrt{n} and v_k=\sigma {\omega}_k/ \sqrt{k}.…
We consider discrete one-dimensional Schr\"odinger operators whose potentials are generated by H\"older continuous sampling along the orbits of a uniformly hyperbolic transformation. For any ergodic measure satisfying a suitable bounded…
We analyze the robustness of optimally controlled evolution equations with respect to spatially localized perturbations. We prove that if the involved operators are domain-uniformly stabilizable and detectable, then these localized…
This paper studies the delocalized regime of an ultrametric random operator whose independent entries have variances decaying in a suitable hierarchical metric on $\mathbb{N}$. When the decay-rate of the off-diagonal variances is…
This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum…
We develop a new approach for the Anderson localization problem. The implementation of this method yields strong numerical evidence leading to a (surprising to many) conjecture: The two dimensional discrete random Schroedinger operator with…
We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian random…
We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…
This paper considers the family of Schr\"odinger operators on $\ell^2(\mathbb{Z})$ given by independent but not necessarily identically distributed and possibly unbounded potentials. We assume a finite exponential moment and allow the…
We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional…
A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schro\"edinger equation in momentum representation on the case of deformed Heisenberg…
It is known that the eigenfunctions of a random Schr\"odinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that…
We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner…
Let H be a Schr\"odinger operator defined on an unbounded domain D in R^d with Dirichlet boundary conditions (D may equal R^d in particular). Let u(x,E) be a solution of the Schr\"odinger equation (H-E)u(x,E)=0, and let B_R denote a ball of…
We prove under certain assumptions that there exists a solution of the Schrodinger or the Heisenberg equation of motion generated by a linear operator H acting in some complex Hilbert space H, which may be unbounded, not symmetric, or not…
We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields…
We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schr\"odinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength…
Let (M,g) be a n-dimensional compact Riemannian manifold. We consider the magnetic deformations of semiclassical Schrodinger operators on M for a family of magnetic potentials that depends smoothly on $k$ parameters $u$, for $k \geq n$, and…
It is reported a combined numerical approach to study the localization properties of the one-dimensional tight-binding model with potential modulated along the prime numbers. A localization-delocalization transition was found as function of…