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We study various spectral theoretic aspects of non-self-adjoint operators. Specifically, we consider a class of factorable non-self-adjoint perturbations of a given unperturbed non-self-adjoint operator and provide an in-depth study of a…
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding…
The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…
We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of…
We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…
Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…
We survey results that have been obtained for self-adjoint operators, and especially Schr\"odinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus…
We define the concept of instability index of an isolated eigenvalue of a non-self-adjoint operator, and prove some of its general properties. We also describe a stable procedure for computing this index for Schroedinger operators in one…
We study ergodic Schr\"odinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which…
We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…
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…
A number of results on radial positive definite functions on ${\mathbb R^n}$ related to Schoenberg's integral representation theorem are obtained. They are applied to the study of spectral properties of self-adjoint realizations of two- and…
For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial…
Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral…
We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$…
We make a spectral analysis of discrete Schroedinger operators on the half-line, subject to complex Robin-type boundary couplings and complex-valued potentials. First, optimal spectral enclosures are obtained for summable potentials.…
This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…
We consider some compact non-selfadjoint perturbations of fibered one-dimensional discrete Schr\"odinger operators. We show that the perturbed operator exhibits finite discrete spectrum under suitable\- regularity conditions.
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…
Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric…