Related papers: Eigenfunction maxima and spherical means

We study eigenvalues and eigenfunctions of the Laplacian on the surfaces of four of the regular polyhedrons: tetrahedron, octahedron, icosahedron and cube. We show two types of eigenfunctions: nonsingular ones that are smooth at vertices,…

In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragm\'{e}n-Lindel\"{o}f principle we prove…

We consider a one parameter family of Laplacians on a closed manifold and study the semi-classical limit of its analytically parametrized eigenvalues. Our results are analogous to a theorem for scalar Schr\"odinger operators on Euclidean…

In this short survey, we derive some weyl-type universal inequalities of eigenvalues of the Laplacian on a closed Riemannian manifold of nonnegative Ricci curvature. We also give upper bounds for the $L_{\infty}$ norm of eigenfunctions of…

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

This paper explores the existence and properties of \emph{basic} eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie…

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary…

Let $\Omega$ be an $n$-dimensional compact Riemannian manifold $(n \geq 3)$ with $C^\infty$ boundary, and consider $L^2$-normalized eigenfunctions $ - \Delta \phi_{\lambda} = \lambda^2 \phi_\lambda$ with Dirichlet or Neumann boundary…

The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement $\Omega := \{(z,w) \in \widehat{\mathbb{C}}^2 \colon z \cdot w \neq 1\}$ of the "complexified unit circle" $\{(z,w) \in…

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is…

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_\Omega$ on a domain $\Omega$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain…

We establish inequalities for the eigenvalues of Schr\"{o}dinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related…

Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds,…

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and we provide a number of geometric…

By methods of stochastic analysis on Riemannian manifolds, we develop two approaches to determine an explicit constant $c(D)$ for an $n$-dimensional compact manifold $D$ with boundary such that $\frac{\lambda}{n}\,\|\phi\|_{\infty} \leq…

Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined…

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hd\sigma_H\Big| $$ where $H\subset M$ is a submanifold and $\sigma_H$ the induced by the Riemannian…

In this note we analyse $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular we use previously determined estimates to produce a new set of estimates for restriction to thickened…

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For…