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… Show more
“…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.…”
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.
“…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.…”
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.
“…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.…”
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.
“…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.…”
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. Relying on a basis of pseudodifferential projections commuting with A, we construct an almost-unitary pseudodifferential operator that diagonalizes A modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of A and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.
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