Related papers: Agmon estimates for Schr\"odinger operators on gra…
The Agmon estimate shows that eigenfunctions of Schr\"odinger operators, $ -\Delta \phi + V \phi = E \phi$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x: V(x) > E…
We construct an expansion in generalized eigenfunctions for Schrodinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
We provide in this Letter a two-point generalisation of the Agmon estimate for Schr\"odinger operators on graphs recently established by S. Steinerberger. It reduces to his estimate when the two points belong to different sets separated by…
The Agmon estimate for multi-dimensional discrete Schr\"{o}dinger operators is studied with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schr\"{o}dinger…
We consider non-self-adjoint electromagnetic Schr\"odinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic…
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…
In this work we obtain weighted boundedness results for singular integral operators with kernels exhibiting exponential decay. We also show that the classes of weights are characterized by a suitable maximal operator. Additionally, we study…
For the discrete Schr\"odinger operator we obtain sharp estimates for the number of negative eigenvalues.
We estimate the number of small eigenvalues of Schr\"odinger operators on Riemannian vector bundles over geometrically finite manifolds.
An important result by Agmon implies that an eigenfunction of a Schr\"{o}dinger operator in $\mathbb{R}^n$ with eigenvalue $E$ below the bottom of the essential spectrum decays exponentially if the associated classically allowed region $\{x…
We report our results on the scaling limit of the eigenvalues and the corresponding eigenfunctions for the 1-d random Schr\"odinger operator with random decaying potential. The formulation of the problem is based on the paper by…
We consider Schr\"odinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional…
This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our…
The aim of this paper is to derive Agmon's type exponential estimates for solutions of elliptic systems of partial differential equations on $\sR^n$. We show that these estimates are related with the essential spectra of a family of…
We study Schr\"odinger operators on compact finite metric graphs subject to $\delta$-coupling and standard boundary conditions. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the…
In this article, we investigate systems of generalized Schr\"odinger operators and their fundamental matrices. More specifically, we establish the existence of such fundamental matrices and then prove sharp upper and lower exponential decay…
In this paper we focus on the validity of some fundamental estimates for time-degenerate Schr\"{o}dinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison…
We prove a dispersive estimate for periodic discrete Schr\"odinger operators on the line with optimal rate of decay. Additionally, by standard methods, we deduce dispersive estimates for the discrete nonlinear Schr\"odinger equation with…
We study Schr\"odinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schr\"odinger operators on general weighted graphs.
We review recent probabilistic results on covariant Schr\"odinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between…