Invariant subspaces of elliptic systems I: pseudodifferential projections

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 existence and uniqueness of m orthonormal pseudodifferential projections commuting with the operator A and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of L 2 (M ) into invariant subspaces under the action of A modulo C ∞ (M ). Furthermore, they allow us to decompose A into m distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator A in terms of pseudodifferential projections and discuss physically meaningful examples.

Invariant subspaces of elliptic systems II: spectral theory

“…Another important special case is that of nonnegative second order operators. For example, the operator of linear elasticity (Lamé operator) falls into this category, see [10,Subsection 8.2] for details. For such operators we have m = m + and the propagator is defined as…”

Diagonalization of elliptic systems via pseudodifferential projections

“…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).…”

“…Let us begin by computing (B j ) sub , j = 1, 2. It was shown in [12] that (P j ) sub = 0, j = 1, 2. (6.14) Furthermore, it is not hard to see that {P (1) , v (1) } = ǫ{P (2) , v (2) } = 0.…”

“…In this paper we will approach diagonalization of elliptic pseudodifferential systems by a different route, one that is based on the theory of pseudodifferential projections, as developed by Vassiliev and the author in [12]. Our global invariant construction is different from those listed above, and will lead to an explicit algorithm for the determination of the full symbol of the almost-unitary diagonalizing operator.…”