Related papers: Schroedinger Operators With Few Bound States

We consider the discrete Schr\"odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of…

We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in…

We consider the Schr\"odinger operator in ${\mathbb R}^n$, $n\geq 3$, with the electric potential $V$ and the magnetic potential $A$ being periodic functions (with a common period lattice) and prove absolute continuity of the spectrum of…

We show that the spectral measure of discrete Schr\"odinger operators $ (Hu)(n)= u({n+1})+u({n-1})+V(n)u(n)$ does not have singular continuous component if the potential $V(n)=O(n^{-1})$.

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…

We consider the Schr\"odinger operator on nanoribbons (tight-binding models) in an external electric potentials $V$. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the…

We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below…

We consider a class of singular Schr\"odinger operators $H$ that act in $L^2(0,\infty)$, each of which is constructed from a positive function $\phi$ on $(0,\infty)$. Our analysis is direct and elementary. In particular it does not mention…

We present the exact solution of the stationary Schr\"odinger equation equation for the potential $V=V_0/{\sqrt{x}}$. Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with…

We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schr\"odinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is…

In a first part of this paper we investigate the continuity (stability) of the spectrum of a class of non-local Schr\"odinger operators on varying the potentials. By imposing conditions of different strength on the convergence of the…

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the…

We study Schr\"{o}dinger operator $H=-\Delta+V(x)$ in dimension two, $V(x)$ being a limit-periodic potential. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this…

The subject of the paper are Schr\"odinger operators on tree graphs which are radial having the branching number $b_n$ at all the vertices at the distance $t_n$ from the root. We consider a family of coupling conditions at the vertices…

We consider discrete Schr\"odinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is always connected. This result is…

We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition…

It is known that the essential spectrum of a Schr\"odinger operator $H$ on $\ell^{2}\left(\mathbb{N}\right)$ is equal to the union of the spectra of right limits of $H$. The natural generalization of this relation to $\mathbb{Z}^{n}$ is…

We study the stationary scattering theory for the matrix Schr\"odinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption…

We consider the Hamiltonian $\hat {\mathrm{H}}_{\mu}$ of a system of three identical particles(bosons) on the $d-$ dimensional lattice $\Z^d, d=1,2$ interacting via pairwise zero-range attractive potential $\mu<0$. We describe precise…

We characterize the absolutely continuous spectrum of the one-dimensional Schr\"odinger operators $h=-\Delta+v$ acting on $\ell^2(\mathbb{Z}_+)$ in terms of the limiting behavior of the Landauer-B\"uttiker and Thouless conductances of the…