Related papers: A Criterion for Hill Operators to be Spectral Oper…
We derive necessary and sufficient conditions for a one-dimensional periodic Schr\"odinger (i.e., Hill) operator H=-d^2/dx^2+V in L^2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the…
In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential…
A number of topics in the qualitative spectral analysis of the Schr\"odinger operator $-\Delta + V$ are surveyed. In particular, some old and new results concerning the positivity and semiboundedness of this operator as well as the…
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…
Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely,…
We consider Schr\"odinger operators with periodic potentials in the positive quadrant for dim $>1$ with Dirichlet boundary condition. We show that for any integer $N$ and any interval $I$ there exists a periodic potential such that the…
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 consider discrete Schr\"odinger operators with real periodic potentials on periodic graphs. The spectra of the operators consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain…
The determination of the spectrum of a Schr\"odinger operator is a fundamental problem in mathematical quantum mechanics. We discuss a series of results showing that Schr\"odinger operators can exhibit spectra that are remarkably thin in…
We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…
Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…
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 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…
A scalar valued random field is called operator-scaling if it satisfies a self-similarity property for some matrix E with positive real parts of the eigenvalues. We present a moving average and a harmonizable representation of stable…
We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an…
We consider the Schr\"odinger operator $H = -\Delta + V$ in a layer or in a $d$-dimensional cylinder. The potential $V$ is assumed to be periodic with respect to some lattice. We establish the absolute continuity of $H$, assuming $V \in…
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…
Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $\tau = - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in…
This work is a continuation of our previos paper \cite{Zel1}, where for the the Schr\"odinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some constructive sufficient conditions for…