Related papers: Spectral statistics of random Schr\"{o}dinger oper…
We investigate some bounds for the density of states in the pure point regime for the random Schr\"{o}dinger operators $H^{\omega}=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nq_n(\omega)$, acting on $\ell^2(\mathbb{Z}^d)$, where…
We discuss spectral properties of the one-dimensional Schr\"odinger operator with a potential of the form $\sum V(n)\delta(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval…
We investigate the spectral properties of the Schr\"odinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$…
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…
This paper sets out to study the spectral minimum for operator belonging to the family of random Schr\"{o}dinger operators of the form $H\_{\lambda,\omega}=-\Delta+W\_{\text{per}}+\lambda V\_{\omega}$, where we suppose that $V\_{\omega}$ is…
We study Schr\"{o}dinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions…
The paper studies the Hill--Schr\"odinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of…
We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values…
We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use…
Here we show that for Schr\"{o}dinger operator with decaying random potential with fat tail single site distribution, the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schr\"{o}dinger…
In this work, we study the spectral statistics for Anderson model on $\ell^2(\mathbb{N})$ with decaying randomness whose single site distribution has unbounded support. Here we consider the operator $H^\omega$ given by $(H^\omega…
In the paper we study the behaviour of the lengths of spectral gaps $\{\gamma_{q}(n)\}_{n\in \mathbb{N}}$ in a continuous spectrum of the Hill-Schr\"{o}dinger operators $$S(q)u=-u"+q(x)u,\quad x\in\mathbb{R},$$ with 1-periodic real-valued…
We analyze spectral properties of two mutually related families of magnetic Schr\"{o}dinger operators, $H_{\mathrm{Sm}}(A)=(i \nabla +A)^2+\omega^2 y^2+\lambda y \delta(x)$ and $H(A)=(i \nabla +A)^2+\omega^2 y^2+ \lambda y^2 V(x y)$ in…
The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schroedinger operator H = Delta+V. Here Delta is the discrete Laplacian and V is a random potential. It is well known that under certain…
The spectrum of random ergodic Schr\"odinger-type operators is almost surely a deterministic subset of the real line. The random operator can be considered as a perturbation of a periodic one. As soon as the disorder is switched on via a…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
We consider Schr\^odinger operators $H_\alpha$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_\alpha, \lambda)$ when $\lambda$ goes to $0$ and the $L^1\to L^\infty$ dispersive estimates…
We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…
We consider operators with random potentials on graphs, such as the lattice version of the random Schroedinger operator. The main result is a general bound on the probabilities of simultaneous occurrence of eigenvalues in specified distinct…
For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…