Related papers: Half-line eigenfunction estimates and singular con…
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…
This paper is dedicated to $L^p$ bounds on eigenfunctions of a Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-\Delta_g)^{\alpha/2}$ is defined…
We study the spectrum of operators in the Schwartz space of rapidly decreasing functions which associate each function with its composition with a polynomial. In the case where this operator is mean ergodic we prove that its spectrum…
We consider Schr\"odinger operator in dimension $d\ge 2$ with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum…
In this paper we study a connection between finite-gap on one energy level two-dimensional Schrodinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators.…
We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials…
We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and…
In this paper we study the boundedness in weighted variable Lebesgue spaces of operators associated with the semigroup generated by the time-independent Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^d$, where $d>2$ and the…
We study one-dimensional Schr\"{o}dinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class…
This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal…
We prove that the spectrum of a Schrodinger operator that is periodic in certain directions and super-exponentially decaying in the others is purely absolutely continuous.
We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan-Wigner transformation of the spin…
One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schr\"odinger operators is the method of gauge transform. In this paper, we extend this method to an abstract…
Here we show that for Schr\"{o}dinger operator with decaying random potential with fat tail single site distribution, the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schr\"{o}dinger…
We consider the Schr\"odinger operator with a periodic potential on a quasi 1D continuous periodic model of armchair nanotubes in $\R^3$ in a uniform magnetic field (with amplitude $B\in \R$), which is parallel to the axis of the nanotube.…
We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and…
We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More…
We consider discrete one-dimensional Schroedinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of…
We study the discreteness of the spectrum of Schrodinger operators which are defined on N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their…
In this article we study the semi-classical distribution of complex zeros of the eigenfunctions of the 1D Schr\"odinger operators for the class of polynomial potentials of even degree, when an energy level E is fixed.