Invariant subspaces of elliptic systems II: Spectral theory

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.

Diagonalization of elliptic systems via pseudodifferential projections

“…We believe that the self-contained explicit construction presented here will prove useful in applications -for example, in the context of quantum field theory in curved spacetime, compare [4,16,5] and [25,26,27,49] -and be more accessible for the wider mathematical physics community. From a theoretical perspective, this paper complements the analysis of invariant subspaces of elliptic systems carried out in [13,14], and the spectral results therein.…”

“…The operators P j from the above theorem form an orthonormal basis of pseudodifferential projections commuting with A, and an explicit algorithm for the construction of their full symbols was given in [13]. Pseudodifferential projections were exploited in [14] to partition the spectrum of A into precisely m infinite series of eigenvalues, singling out the contribution to the spectrum of A of individual eigenvalues of A prin (more details about this will be recalled later on). Theorem 2.4 tells us that pseudodifferential projections allow one to decompose A into m sign-semidefinite operators P * j AP j ∈ Ψ s m .…”

Topological obstructions to the diagonalisation of pseudodifferential systems

“…However, diagonalising a system may not be possible due to topological obstructions, as is the case for some important physically meaningful operators; see Sections 3-5. For this reason, other approaches to the study of the spectrum of systems, such as the use of pseudodifferential projections [8,9], are perhaps more natural, in that they always work and circumvent topological obstructions altogether. Let {e j } 3 j=1 be a positively oriented global framing of M , namely, a set of three orthonormal smooth vector fields on M , whose orientation agrees with that of M .…”

Topological obstructions to the diagonalisation of pseudodifferential systems

“…However, diagonalising a system may not be possible due to topological obstructions, as is the case for some important physically meaningful operators; see Section 3. For this reason, other approaches to study the spectrum of systems, such as the use of pseudodifferential projections [7,8], are perhaps more natural, in that they always work and circumvent topological obstructions altogether.…”