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

“…. , −m − , imply a result first proved in [13], namely that the spectrum of A partitions, up to an error with superpolynomial decay, into m distinct series, one for each eigenvalue of A prin . The advantage of the approach adopted in the current paper is that these series are characterised as the spectra of m semibounded scalar (as opposed to non-semibounded matrix) elliptic operators, the operators a j .…”

“…We believe that the self-contained explicit construction presented here will prove useful in applications -see, for example, [4,15,5] and [24,25,26,43] -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 [12,13], 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 [12]. Pseudodifferential projections were exploited in [13] 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 2.3 tells us that pseudodifferential projections allow one to decompose A into m sign-semidefinite operators P * j AP j ∈ Ψ s m .…”

“…Theorem 2.3 tells us that pseudodifferential projections allow one to decompose A into m sign-semidefinite operators P * j AP j ∈ Ψ s m . However, these operators are not elliptic, and this was one of the major obstacles one had to overcome in [13].…”

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

“…For later convenience we shall denote by ρ t : Γ(E) → Γ(E| Σt ) the restriction map for smooth sections: Notice that, ρ t is a right-inverse for U S,t . As shown in [16][17][18][19], on globally hyperbolic manifolds with empty boundary and compact Cauchy hypersurfaces, the evolution operator can be realized as a Fourier integral operator. As a matter of fact, the Fourier integral representation of the propagator contains the information on how singularities propagates in the manifold.…”