Related papers: A New Class of Schr\"odinger Operators without Pos…

We provide examples of operators $T(D)+V$ with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the Fermi surfaces of constant kinetic energy $T$. We make the connection to…

In this note we study the Landis conjecture for positive Schr\"odin\-ger operators on graphs. More precisely, we prove a Landis-type result in the form of a decay criterion that ensures when $\mathcal{H}$-harmonic functions for a positive…

In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr\"odinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.

We prove a Wegner estimate for discrete Schr\"odinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially, no monotonicity assumption is required. This…

We improve the Lieb-Thirring type inequalities by Demuth, Hansmann and Katriel (J. Funct. Anal. 2009) for Schr\"odinger operators with complex-valued potentials. Our result involves a positive, integrable function. We show that in the…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

Given a Schr\"odinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The…

In this note, we prove weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at…

We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…

We study the low-energy asymptotics of the spectral shift function for Schr\"odinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue…

We consider a family of multi-dimensional Schr\"odinger operators $-\Delta+t V$ with a real $t$. The potential $V$ in our model decays at infinity in a special way, so that it satisfies a certain integral condition. We prove that the…

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function $N_L(E)$, the number of bound states of the operator $L = \Delta+V$ in $\R^d$ below $-E$. Here $V$ is a bounded potential behaving asymptotically…

We give a spectral description of the semi-classical Schrodinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a non-self-adjoint operator…

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 Schr\"odinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis…

In this paper we study absence of embedded eigenvalues for Schr\"odinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates…

The purpose of this paper is to give a systematic description of potentials decaying to zero at infinity, which generate eigenvalues at the edge of the absolutely continuous spectrum when combined with non-local operators defined by…

Eigenvalue behaviors of Schr\"odinger operator defined on $n$-dimensional lattice with $n+1$ delta potentials is studied. It can be shown that lower threshold eigenvalue and lower threshold resonance are appeared for $n\geq 2$, and lower…

The purpose of this paper is to study spectral properties of non-self-adjoint Schr\"odinger operators $-\Delta-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type…