Related papers: Theorems of Chernoff and Ingham for certain eigenf…

We prove an exact analogue of Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose…

This paper studies the uncertainty principle for spherical $h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the…

Let X be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on X, by techniques from the theory of partial differential equations.

In 1975, P.R. Chernoff used iterates of the Laplacian on $\mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $\mathbb{R}^n$ to be quasi-analytic. In this…

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…

We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem…

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator.…

Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by…

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group $G$, criteria for the existence of $G$-invariant self-adjoint extensions of…

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with…

Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami…

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we…

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schr\"odinger operator has a finite or infinite number of the discrete pectrum. In…

An $L^2$ version of the classical Denjoy-Carleman theorem regarding quasi-analytic functions was proved by P. Chernoff on $\mathbb R^n$ using iterates of the Laplacian. We give a simple proof of this theorem which generalizes the result on…

We prove an analogue of Chernoff's theorem for the Laplacian $ \Delta_{\mathbb{H}} $ on the Heisenberg group $ \mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ \mathbb{H}^n $ and also for…

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…

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 extend the uncertainty principle, the Cowling--Price theorem, on non-compact Riemannian symmetric spaces $X$. We establish a characterization of the heat kernel of the Laplace--Beltrami operator on $X$ from integral estimates of the…

We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed…

An $L^2$ version of the celebrated Denjoy-Carleman theorem regarding quasi-analytic functions was proved by Chernoff \cite{CR} on $\mathbb R^d$ using iterates of the Laplacian. In $1934$ Ingham \cite{I} used the classical Denjoy-Carleman…