Related papers: Pointwise bounds for $L^2$ eigenfunctions on local…

- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on…

The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to…

We study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant…

For Laplace-Beltrami operators associated to metrics which are long range perturbations of the flat one, we prove estimates for powers of the resolvent as the spectral parameter goes to zero. We also discuss applications to the local energy…

We study the eigenvalues and eigenfunctions of a differential operator that governs the asymptotic behavior of the unsupervised learning algorithm known as Locally Linear Embedding when a large data set is sampled from an interval or disc.…

Given a compact Riemannian manifold $M$ of dimension $m\geq 2$, we study the space of functions of $L^2(M)$ generated by eigenfunctions of eigenvalues less than $L\geq 1$ associated to the Laplace-Beltrami operator on $M$. On these spaces…

We study the $2k$-th moment at the central point of the family of symmetric square $L$-functions attached to holomorphic Hecke cusp forms of level one, weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$…

We prove sharp uniform $L^p$-bounds for low-lying eigenfunctions of non-self-adjoint semiclassical pseudodifferential operators $P$ on $\mathbb{R}^{n}$ whose principal symbols are doubly-characteristic at the origin of $\mathbb{R}^{2n}$.…

Let $\pi$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $\pi$ in the $t$-aspect. More generally, we address the spectral aspect…

Assuming the negative part of the potential is uniformly locally $L^1$, we prove a pointwise $L^p$ estimate on derivatives of eigenfunctions of one-dimensional Schrodinger operators. In particular, if an eigenfunction is in $L^p$, then so…

In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent…

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4:…

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have…

We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(\mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special…

Consider an n x n Hermitian random matrix with, above the diagonal, independent entries with alpha-stable symmetric distribution and 0 < alpha < 2. We establish new bounds on the rate of convergence of the empirical spectral distribution of…

Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional…

In this paper we study an eigenvalue problem for the so called $(p,2)$-Laplace operator on a smooth bounded domain under a nonlinear Steklov type boundary condition, namely \begin{equation} \left\{ \begin{aligned} -\Delta_pu-\Delta u &…

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including…

We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can…

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging…