Related papers: Resonances under Rank One Perturbations
Non-autonomous perturbations of isochronous systems in the plane are considered. It is assumed that the intensity of perturbations decays with time, and the frequency is asymptotically constant with the limiting value satisfying a resonance…
First-order perturbative calculation of the frequency-shifts caused by special relativity is performed for a charged particle confined in a Penning trap. The perturbed motion is approximated by the Jacobian elliptic functions which describe…
The self-adjoint matrix Sturm-Liouville operator on a finite interval with a boundary condition in the general form is studied. We obtain asymptotic formulas for the eigenvalues and the weight matrices of the considered operator. These…
In the paper, Sturm--Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their…
We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We…
The dynamics of self-oscillatory extended systems, resonantly forced at a frequency close to that of the natural oscillations (1:1 resonance), is shown to be universally described by a complex Ginzburg-Landau equation containing an…
We give a comprehensive treatment of Sturm-Liouville operators with measure-valued coefficients including, a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl-Titchmarsh theory. We avoid previous…
We study spectral properties of one-dimensional singular perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for…
We study analytically as well as numerically the dynamics of a quantum map near a quantum resonance of an order q. The map is embedded into a continuous unitary transformation generated by a time-independent quasi-Hamiltonian. Such a…
We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schr\"odinger and Dirac-type operators. Special…
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 investigate the eigenvalues of perturbed spherical Schr\"odinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of…
We study linear perturbations of Donoghue classes of scalar Herglotz-Nevanlinna functions by a real parameter $Q$ and their representations as impedance of conservative L-systems. Perturbation classes $\mathfrak M^Q$, $\mathfrak…
The article studies the Dirichlet and Dirichlet-Neumann problems for the Sturm-Liouville equation perturbed by an integral operator with a convolution kernel. Sharp asymptotic formulas for the eigenvalues of these problems are found. The…
In this article, we study a two-parameter family of rotating rank-one maps defined on $\textbf{S}^1\times [1, 1+b]\times \textbf{S}^1$, with $b\gtrsim 0$, whose dynamics is characterised by a coupling of a family of planar maps exhibiting…
This is a survey article. We consider different problems in connection with the behavior of functions of operators under perturbations of operators. We deal with three classes of operators: unitary operators, self-adjoint operators, and…
We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their…
Finite rank perturbations of diagonalizable normal operators acting boundedly on infinite dimensional, separable, complex Hilbert spaces are considered from the standpoint of view of the existence of invariant subspaces. In particular, if…
We reconstruct the rank-one, singular (point-like) perturbations of the $d$-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schr\"{o}dinger operators with regular potentials centred around…
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…