Related papers: Eigenfunctions of the Laplace-Beltrami Operator on…

We show that for any hyperbolic surface of genus g, the eigenvalue $\lambda _{2g-2}$ of the Laplace operator is > 1/4.

Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions while solutions to this…

The inner plethysm of symmetric functions corresponds to the $\lambda$-ring operations of the representation ring $R({\mathfrak S}_n)$ of the symmetric group. It is known since the work of Littlewood that this operation possesses stability…

The Neumann-Poincar\'e operator is a boundary-integral operator associated with harmonic layer potentials. This article proves the existence of eigenvalues within the essential spectrum for the Neumann-Poincar\'e operator for certain…

We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M, g)$. We conclude under which conditions on…

We consider a compact Riemannian manifold with boundary with a certain class of critical singular Riemannian metrics that are singular at the boundary. The corresponding Laplace-Beltrami operator can be seen as a Grushin-type operator plus…

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part,…

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…

We use the sum-of-squares theorem from number theory to construct eigenfunctions of the Laplacian on the $d$-dimensional torus, $d \geq 2$, which vanish to any prescribed order at some point. These functions are then applied to provide a…

We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of…

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds of arithmetic type. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras…

We examine spectra of Dirac operators on compact hyperbolic surfaces. Particular attention is devoted to symmetry considerations, leading to non-trivial multiplicities of eigenvalues. The relation to spectra of Maass-Laplace operators is…

This paper deals with the problem of factorizing integer powers of the Laplace operator acting on functions taking values in higher spin representations. This is a far-reaching generalization of the well-known fact that the square of the…

We discuss representation of certain functions of the Laplace operator $\Delta$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-\Delta)^{1/2}$, the square root of the…

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $\Delta_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalized eigenfunctions…

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace-Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in…

Let $ X = G/K $ be a rank one Riemannian symmetric space of noncompact type. In view of the Iwasawa decomposition $ G = NAK $ of the underlying semisimple Lie group, we can also view $ X $ as the solvable extension $ S = NA $ of the Iwasawa…

We derive a numerical approximation of the Laplace-Beltrami operator on compact surfaces embedded in $\mathbb{R}^3$ with an axial symmetry. To do so we use a noncommutative Laplace operator defined on the space of finite dimensional…

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized…