Related papers: Distribution of Eigenvalues of the Kohn Laplacian …

We obtain an analog of Weyl's law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are…

We present an explicit formula for the leading coefficient in the asymptotic expansion of the eigenvalue counting function of the Kohn Laplacian on the unit sphere $\mathbb{S}^{2n-1}$.

In this article we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for…

In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory…

We prove that the spectrum of the Kohn Laplacian does not determine the equivalence classes of CR manifolds. We construct pairs of odd-dimensional elliptic manifolds that are not equivalent as CR manifolds but whose Kohn Laplacians have the…

Motivated by a sharp eigenvalue estimate for the Kohn Laplacian, we prove a theorem that characterizes the CR sphere in terms of the existence of a non-trivial complex-valued function satisfying a certain overdetermined system.

We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features…

We look at the action of finite subgroups of $\operatorname{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in $\mathbb{C}^2$ and with a perturbed CR structure known as the Rossi sphere. We…

This paper is devoted to establish semiclassical Weyl formulae for the Robin Laplacian on smooth domains in any dimension. Theirs proofs are reminiscent of the Born-Oppenheimer method.

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete…

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that…

We construct a general quantization procedure for square integrable functions on well-behaved connected exponential Lie groups. The Lie groups in question should admit at least one co-adjoint orbit of maximal possible dimension. The…

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of…

This paper is the fourth of a multi-part series in which we study the geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times\mathbb{C}^m$. The goal of this article is…

In this note, we study the spectrum of the Kohn Laplacian on the unit spheres in $\mathbb{C}^n$ and revisit Folland's classical eigenvalue computation. We also look at the growth rate of the eigenvalue counting function in this context.…

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of…

We study spectral asymptotics for a large class of differential operators on an open subset of $\R^d$ with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with…

Given a polynomial P of partial derivatives of the Kahler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for $P$ to be an invariant polynomial, i.e. invariant…

We extend the Siu--Beauville theorem to a certain class of compact Kaehler--Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As…

By comparing the Laplace spectrum of the sphere $\mathbb{S}^n$ to its Weyl function $w(x) = \frac{\omega_n}{(2\pi)^n}|\mathbb{S}^n|x^{n/2}$, we show that no analogue of P\'olya's eigenvalue conjecture holds in general for Riemannian…