Related papers: Poisson eigenvalue statistics for random Schr\"odi…

Suppose some random resource (energy, mass or space) $\chi \geq 0$ is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume ${\Bbb E}\chi =\theta <\infty $ and suppose the amount of the individual…

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

We consider square-integrable functionals of Poisson point processes for which the variance upper bound provided by the classical Poincar\'{e} inequality is suboptimal, a phenomenon known as superconcentration. In this paper, we establish a…

Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific…

The Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p}+V$ on high tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry is studied under the assumption of non-degeneracy of the curvature…

In these paper we study the adjacency matrix of some infinite graphs, which we call the shift operator on the $L^p$ space of the graph. In particular, we establish norm estimates, we find the norm for some cases, we decide the triviality of…

We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields…

We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large…

This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…

In the first part of this manuscript a relationship between the spectrum of self-adjoint operator matrices and the spectra of their diagonal entries is found. This leads to enclosures for spectral points and in particular, enclosures for…

We present a general method to obtain the exact rate function $\Psi_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[\mathcal{I}_N[a,b]=kN] \asymp e^{-N\Psi_{[a,b]}(k)}$ that a $N \times N$ sparse random matrix has…

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. Using random walks and exit times, we associate to certain subgraphs (domains) a pair of sequences, each of which…

We develop a new approach for the Anderson localization problem. The implementation of this method yields strong numerical evidence leading to a (surprising to many) conjecture: The two dimensional discrete random Schroedinger operator with…

We consider one-dimensional random Schr\"odinger operators with a background potential, arising in the inverse problem of scattering. We study the influence of the background potential on the essential spectrum of the random Schr\"odinger…

In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of…

We discuss the spectral properties of singular Schr\"odinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case the discrete spectrum is nonempty if the star arms are…

The Novikov-Shubin invariant associated to a graph provides information about the accumulation of eigenvalues of the corresponding adjacency matrix close to the origin. For a directed graph these eigenvalues lie in the complex plane and…

We consider the ensemble of adjacency matrices of Erd\H{o}s-R\'{e}nyi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk…

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy and $n\geq 5$ is odd. In particular, we show that if there is an…

We prove some local estimates on the trace of spectral projectors for random Schr\"odinger operators restricted to cubes $\Lambda \subset R^d$. We also present a new proof of the spectral averaging result based on analytic perturbation…