Related papers: Resonances under Rank One Perturbations
We discuss quantum graphs consisting of a compact part and semiinfinite leads. Such a system may have embedded eigenvalues if some edge lengths in the compact part are rationally related. If such a relation is perturbed these eigenvalues…
We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics…
Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank…
The influence of oscillatory perturbations on autonomous strongly nonlinear systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time, and their frequency increases according to a power law.…
We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schr\"odinger-type differential operator in $L^2(\mathbb{R};\mathbb{R}^n)$, with an asymptotically periodic potential. The studied…
The Friedrichs model~\cite{Friedrichs} is revisited to obtain precise results about the asymptotic behaviour (the so-called Breit-Wigner formula~\cite{Breit}) of a resonance near an embedded eigenvalue and the ``spectral concentration"…
The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the…
We consider Dirac, Pauli and Schr\"odinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of…
We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that…
The paper is devoted to the problem of resonances in one-dimensional disordered systems. Some of the previous results are reviewed and a number of new ones is presented. These results pertain to different models (continuous as well as…
We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this…
We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing…
We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues. In the one-dimensional case we prove a novel version of the Sturm oscillation theorem: if the dependence on the eigenvalue…
In this paper we consider a resonance problem, in a generic regime, in the consideration of relaxation of ground states of semilinear Schrodinger equations. Different from previous results, our consideration includes the presence of…
Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary…
The paper deals with the Neumann spectral problem for a singularly perturbed second order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue…
We consider a quadratic operator pencil with a small periodic perturbation multiplied by the spectral parameter. It is motivated, in particular, by a one-dimensional Klein-Gordon equation with a time-parity-symmetric perturbation. We study…
We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We…
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schr\"odinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the…
We study the hybrid type of rank one perturbations in $\mathbb{R}^2$ and $\mathbb{R}^3$, where the perturbation supported by a circle/sphere is considered together with the delta potential supported by a point outside of the circle/sphere.…