Spectral Theory of Riemannian Manifolds

Purdue University, West Lafayette IN

Investigators

Abstract

ABSTRACT DMS - 0203070. The principal investigator plans to study three separate topics: i) Eigenvalue estimates for the Laplacian on Hermitian holomorphic line bundles, ii) Quantum Unique Ergodicity, and iii) Behavior of eigenfunctions near the ideal boundary of hyperbolic space. Consider the tensor powers of a Hermitian holomorphic line bundle, over a compact complex manifold. If the curvature form of the line bundle is strictly positive, then the first non--zero eigenvalue, of the Laplacian, acting on sections of the $k$th power, is bounded below uniformly in $k$. The proposal is to prove that the first non--zero eigenvalue is uniformly bounded below when the curvature is semipositive everywhere and positive for at least one point. The problem can be reformulated in terms of CR geometry. Microlocal analysis will be applied to the reformulated problem. The problem of quantum unique ergodicity concerns concentration of eigenfunctions on manifolds with ergodic geodesic flow. We propose to construct examples where sequences of eigenfunctions concentrate along isolated closed geodesics or on one parameter families of closed geodesics. The first step is to find quasimodes (approximate eigenfunctions). Next one must show that these quasimodes correspond to individual eigenfunctions rather than sums of several eigenfunctions. The hyperbolic space has essential spectrum which is a proper subset of the positive real line. There do exist eigenfunctions, defined on the complements of compact sets, whose eigenvalue lies below the start of the essential spectrum. The behavior of these eigenfunctions will be studied near the ideal boundary at infinity. The goal is to understand the nodal set by means of a perturbation expansion. It appears that the case of surfaces is much more tractable than the higher dimensional cases. To develop our understanding of the quantum phenomena, mathematicians are often inspired by the analogy with classical mechanics. The passage from the classical to the quantum level is called the semiclassical limit. If the classical motion is chaotic, one expects the probability distribution of the quantum particle to be dispersed. Exceptions to this pattern, where the particle concentrates, are of particular interest. Similarly, if an energy estimate holds under strict positivity conditions of a classical curvature form, one naturally investigates the borderline case where the curvature form is non--negative. One hopes that regularity properties will persevere in the more general situation. This type of question is interesting because non--negative objects often occur as the limits of positive objects. The nodal set of a quantum particle is the stationary set for the associated wave motion. Its distribution and shape are of fundamental interest, but poorly understood, especially in dimensions larger than two.