Related papers: Spectral estimates for the Schr\"odinger operators…
We study spectral properties of the Schroedinger operator with an imaginary sign potential on the real line. By constructing the resolvent kernel, we show that the pseudospectra of this operator are highly non-trivial, because of a blow-up…
We prove dispersive estimates for the linear Schr\"odinger evolution associated to an operator -\Delta + V, where the potential is a signed measure of fractal dimension at least 3/2.
The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the…
We consider Schr\"odinger operators with periodic electric and magnetic potentials on periodic discrete graphs. The spectrum of such operators consists of an absolutely continuous (a.c.) part (a union of a finite number of non-degenerate…
This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic…
We consider discrete Schr\"odinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schr\"odinger operators are well…
Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…
Starting from the semi-classical spectrum of Schr\"odinger operators $-h^2\Delta+V$ (on $\mathbb{R}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express…
We show that spectral Hausdorff dimensional properties of discrete Schr\"oodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations,…
We continue our study of the dependence of the density of states measure and related spectral functions of Schr\"odinger operators on the potential. Whereas our earlier work focused on random Schr\"odinger operators, we extend these results…
We show that Schr\"odinger-type operators on discrete connected periodic graphs do not have flat bands for generic potentials.
We show that if we start from a symmetric lower semi-bounded Schr\"odinger operator $\mathcal{H}$ on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a…
We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter…
We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an…
In this article we present comparisons between the spectrum of a one-dimensional Schr\"odinger operator for a particular periodic potential and for its restriction to a finite number of sites. We deduce from this finite, but large, number…
I consider 4-dimensional Schr\"odinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(\mu^{-1}h^{-3})$ and the principal part $\asymp…
In the present paper our aim is to explore some spectral properties of the family two-particle discrete Schr\"odinger operators $h^{\mathrm{d}}(k)=h^{\mathrm{d}}_0(k)+ \mathbf{v},$ $k\in \T^\mathrm{d},$ on the $\mathrm{d}$ dimensional…
In this paper we continue the study of the spectral gap of Schr\"odinger operators on large intervals and subject to Neumann boundary conditions. The main goal is to derive a lower bound on the spectral gap which is polynomial in the…
Consider the one-dimensional discrete Schr\"odinger operator $H_{\theta}$: $$(H_{\theta} q)_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n \ , \quad n\in Z \ ,$$ with $\omega\in R^d$ Diophantine, and $V$ a real-analytic function on $ T^d=(…
This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…