Related papers: Whittaker-Hill equation and semifinite-gap Schroed…
We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then…
Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency…
We prove sharp lower bounds on the spectral gap of 1-dimensional Schr\"odinger operators with Robin boundary conditions for each value of the Robin parameter. In particular, our lower bounds apply to single-well potentials with a centered…
Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the…
In this paper we obtain sharp Lieb-Thirring inequalities for a Schr\"odinger operator on semi-axis with a matrix potential and show how they can be used to other related problems. Among them are spectral inequalities on star graphs and…
In this paper we investigate the spectrum and spectrality of the one-dimensional Schrodinger operator with a periodic PT-symmetric complex-valued potential.
We consider Schr\"odinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be…
The paper studies the Hill--Schr\"odinger operators with potentials in the space $H^\omega \subset H^{-1}\left(\mathbb{T}, \mathbb{R}\right)$. The main results completely describe the sequences arising as the lengths of spectral gaps of…
We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These…
We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in $\bx$, and (ii) the perturbation $B$ has order less than $2m$.…
Ordinary differential operators with periodic coefficients analytic in a strip act on a Hardy-Hilbert space of analytic functions with inner product defined by integration over a period on the boundary of the strip. Simple examples show…
We study continuum Schr\"odinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters.…
We study here class of 1D spectral-meromorphic (s-meromorphic) OD operators $L=\partial_x^n+\sum_{n-2\geq i\geq 0}a_{n-2-i}\partial_x^i$ with meromorphic coefficients $a_j$ near $x\in R$ such that all eigenfunctions $L\psi=\alpha\psi$ are…
We address the problem on the right definition of the Schroedinger operator with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with…
Schr\'{o}dinger's equation with distributional $\delta$, or $\delta'$ potentials has been well studied in the past. There are challenges in simultaneously addressing some of the inherent issues of the system: The functional operator cannot…
This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so called…
We consider Schr\"odinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We determine trace formulas for the Schr\"odinger operators. The proof is based on the…
We discuss convergence properties of the spectral shift functions associated with a pair of Schrodinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches…
For a class of discrete quasi-periodic Schroedinger operators defined by covariant re- presentations of the rotation algebra, a lower bound on phase-averaged transport in terms of the multifractal dimensions of the density of states is…
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…