Related papers: Spectral Theory of Pseudo-Ergodic Operators
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…
We consider non-local perturbations $\Delta^\psi_G$ of sub-Laplacians on a step $2$ Carnot group $G$. The perturbations are by translation-invariant non-local operators acting along the vertical directions in $G$. We use harmonic analysis…
The necessary and sufficient conditions are given for a sequence of complex numbers to be the periodic (or antiperiodic) spectrum of non-self-adjoint Dirac operator.
In this paper we prove that a class of non self-adjoint second order differential operators acting in cylinders $\Omega\times\mathbb R\subseteq\mathbb R^{d+1}$ have only real discrete spectrum located to the right of the right most point of…
We study the pseudospectrum of a class of non-selfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small…
Let $(\cX, \rho)$ be a discrete metric space. We suppose that the group $\sZ^n$ acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a $\sZ^n$-periodic discrete metric space. We…
We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schr\"odinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schr\"odinger operators, assumed to have a local…
We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of…
The purpose of this note is to review some recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations.
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…
In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,\lambda)$. The purpose of the paper is to…
The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the…
The classical spectral theorem completely describes self-adjoint operators on finite dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for…
We study the essential spectrum and Fredholm properties of integral and pseudodiferential operators associated to (maybe non-commutative) locally compact groups G. The techniques involve crossed product C*-algebras. We extend previous…
We consider perturbations of quasi-periodic Schr\"odinger operators on the integer lattice with analytic sampling functions by decaying potentials and seek decay conditions under which various spectral properties are preserved. In the…
The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and…
We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form…
We consider ergodic families of Schr\"odinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a…
This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class…
We use C*-algebra theory to provide a new method of decomposing the eseential spectra of self-adjoint and non-self-adjoint Schrodinger operators in one or more space dimensions.