Related papers: Schr\"{o}dinger operator with a complex steplike p…
By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schr\"odinger operators with periodic singular potentials.
We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the…
A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…
One-dimensional Schr\"odinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized…
In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an…
We study half-line discrete Schr\"odinger operators and their rank-one perturbations. We establish certain continuity and stability properties of the Fourier transform of the associated spectral measures. Using these results, we construct a…
The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the…
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.
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane.…
The generic simplicity of the spectrum of a Schr\"odinger-type operator on the n-dimensional torus is studied using the Rayleigh-Schr\"odinger perturbation theory. The existence of a perturbation potential of the Laplacian is proved and…
We construct a potential $V$ on $\RR^d$, smooth away from one pole, and a sequence of quasi-modes for the operator $-\Delta+V$, which concentrate on this pole. No smoothing effect, Strichartz estimates nor dispersive inequalities hold for…
We give examples of semiclassical Schr\"odinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the…
We consider the quasi-periodic Schr\"odinger operator $$ [H \psi](x) = -\psi"(x) + V(x) \psi(x) $$ in $L^2(\mathbb{R})$, where the potential is given by $$ V(x) = \sum_{m \in \mathbb{Z}^\nu \setminus \{ 0 \}} c(m)\exp (2\pi i m \omega x) $$…
This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation…
We analyze the spectrum of a discrete Schrodinger operator with a potential given by a periodic variant of the Anderson Model. In order to do so, we study the uniform hyperbolicity of a Schrodinger cocycle generated by the SL(2,R) transfer…
The generalized pseudospectral method is employed for the accurate calculation of eigenvalues, densities and expectation values for the spiked harmonic oscillators. This allows \emph{nonuniform} and \emph{optimal} spatial discretization of…
For the the Schr\"odinger operator $H=-\Delta+ V(x)\cdot$, acting in the space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we obtain some constructive conditions for discreteness of its spectrum. Basing on the…
We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schr\"odinger operator) $H=-d^2/dx^2+V$ to be a spectral operator of scalar type. The conditions show the remarkable fact that the property…
The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a…
The spectral analysis of discretized one-dimensional Schr\"{o}dinger operators is a very difficult problem which has been studied by numerous mathematicians. A natural problem at the interface of numerical analysis and operator theory is…