Related papers: Absence of embedded eigenvalues for Riemannian Lap…

We study the Schr\"{o}dinger operator with a constant magnetic field in the exterior of a compact domain in euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin…

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we…

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…

In this note, we study some properties of threshold resonant states or threshold eigenfunctions for discrete Schr\"odinger operators. We mainly prove two theorems. First, we prove that resonant states at the elliptic threshold have the same…

We establish Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds $(\Omega,\g)$ with boundary, for both the compact case and the case that $\Omega$ is the exterior of a smooth, non-trapping obstacle in Euclidean…

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the…

We review some recent results on eigenvalues of fractional Laplacians and fractional Schr\"odinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $\sigma_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the…

Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace…

We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schr\"odinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate…

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr\"odinger operator with a constant magnetic field and a random potential which…

This article is an expanded version of the plenary talk given by Evans Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace operators and Schr\"odinger operators with potentials containing curvature on certain regions of…

An analogue of Rellich's theorem is proved for discrete Laplacian on square lattice, and applied to show unique continuation property on certain domains as well as non-existence of embedded eigenvalues for discrete Schr{\"o}dinger…

Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique continuation result for differential forms, satisfying a certain first order differential inequality, on Riemannian manifolds with empty boundary. The present paper…

We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its…

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…

I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than…

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is…

We consider a Schr\"odinger Operator with a matrix potential defined in $L_2^m(F)$ by the differential expression\begin{equation*} L(\phi(x))=(-\Delta+V(x))\phi(x) \end{equation*}and the Neumann boundary condition, where $F$ is the $d$…