Geometric wave propagator on Riemannian manifolds

Abstract

We study the propagator of the wave equation on a closed Riemannian manifold M . We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant de nition of the full symbol of the propagator -a scalar function on the cotangent bundle -and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise topological obstructions and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.

Invariant subspaces of elliptic systems II: spectral theory

“…Before addressing the proof of Theorem 2.4, let us recall, in an abridged manner and for the convenience of the reader, the propagator construction from [9], which builds upon [18,26,11] and is an extension to first order systems of earlier results for scalar operators [8,7]. Let A ∈ Ψ 1 be an operator as in Section 1.…”

Invariant subspaces of elliptic systems I: pseudodifferential projections

“…In this section we discuss some applications of the above results. The most important application -the partition of the spectrum of a positive order pseudodifferential system -will be the subject of a separate paper [15], where, among other things, results from [16,12,11,14] will be refined and improved. Throughout this section we adopt Einstein's summation convention over repeated indices.…”

Diagonalization of elliptic systems via pseudodifferential projections

“…7]. We refer the reader to [14,8,11,7] for additional details on U(t), U (j) (t) and their explicit construction. Properties (5.44) and (5.45) allowed us to use Levitan's wave method to compute local asymptotics for the spectral density 'along invariant subspaces' and express the second local Weyl coefficients of A as the sum of the second local Weyl coefficients of the A j 's.…”