On isospectral deformations of riemannian metrics. II
On lens spaces which are isospectral but not isometric
On -accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry.
On -sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry
We consider a Schrödinger-type differential expression , where is a -bounded Hermitian connection on a Hermitian vector bundle of bounded geometry over a manifold of bounded geometry with metric and positive -bounded measure , and is a locally integrable section of the bundle of endomorphisms of . We give a sufficient condition for -sectoriality of a realization of in . In the proof we use generalized Kato’s inequality as well as a result on the positivity of satisfying the...
On nodal sets and nodal domains on and
We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on . We also construct a solution of the equation in that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
On quantum limits on flat tori.
On Riesz means of eigenfunction expansions for the Kohn-Laplacian.
On spectra of geometric operators on open manifolds and differentiable groupoids.
On spherical space forms with meta-cyclic fundamental group which are isospectral but not equivariant cobordant
On the Analytic Continuation of Rank One Eisenstein Series.
On the characterizations of flat metrics by the spectrum.
On the differential form spectrum of hyperbolic manifolds
We give a lower bound for the bottom of the differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds.
On the distribution of resonances for some asymptotically hyperbolic manifolds
We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute -matrix that is unitary for real values of the energy. This paramatrix is the -matrix for a model laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance...
On the error term in Weyl's law for Heisenberg manifolds
On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space.
On the generic spectrum of a riemannian cover
Let be a compact manifold let be a finite group acting freely on , and let be the (Fréchet) space of -invariant metric on . A natural conjecture is that, for a generic metric in , all eigenspaces of the Laplacian are irreducible (as orthogonal representations of ). In physics terminology, no “accidental degeneracies” occur generically. We will prove this conjecture when dim dim for all irreducibles of . As an application, we construct isospectral manifolds with simple eigenvalue...
On the multiplicity of eigenvalues of conformally covariant operators
Let be a compact Riemannian manifold and an elliptic, formally self-adjoint, conformally covariant operator of order acting on smooth sections of a bundle over . We prove that if has no rigid eigenspaces (see Definition 2.2), the set of functions for which has only simple non-zero eigenvalues is a residual set in . As a consequence we prove that if has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the -topology....
On the Sobolev constant and the -spectrum of a compact riemannian manifold
On the spectral theory and dynamics of asymptotically hyperbolic manifolds
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
On the spectral theory of the laplacian on noncompact hyperbolic manifolds