Invariant subspaces of elliptic systems II: spectral theory

TL;DR

This paper analyzes the spectral decomposition of elliptic self-adjoint pseudodifferential operators with simple eigenvalues, revealing a near-exact division of the spectrum into distinct series and applying these results to hyperbolic systems.

Contribution

It introduces a spectral decomposition framework for elliptic operators with simple eigenvalues and applies pseudodifferential projections to study singularity propagation in hyperbolic systems.

Findings

Spectrum decomposes into $m$ distinct series with superpolynomial accuracy.

Pseudodifferential projections create almost-invariant subspaces under operator and evolution.

Results enhance understanding of singularity propagation in hyperbolic PDEs.

Abstract

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of $A$ decomposes, up to an error with superpolynomial decay, into $m$ distinct series, each associated with one of the eigenvalues of the principal symbol of $A$. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose $L^2(M)$ into almost-orthogonal almost-invariant subspaces under the action of both $A$ and the hyperbolic evolution.