Pseudodifferential Projections Research Articles

Invariant subspaces of elliptic systems II: Spectral theory

Consider an elliptic self-adjoint pseudodifferential operator A acting on m -columns of half-densities on a closed manifold M 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.

Invariant subspaces of elliptic systems I: Pseudodifferential projections

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 L2(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.

Diagonalization of elliptic systems via pseudodifferential projections

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.

Elliptic Complexes on Manifolds with Boundary

We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.

An Algebra of Boundary Value Problems Not Requiring Shapiro–Lopatinskij Conditions

We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro–Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.

Elliptic operators in odd subspaces

An elliptic theory is constructed for operators acting in subspaces defined by means of odd pseudodifferential projections. Index formulae for operators on compact manifolds without boundary and for general boundary-value problems are obtained. A connection with Gilkey's theory of η-invariants is established.

Elliptic operators in even subspaces

An elliptic theory is constructed for operators acting in subspaces defined in terms of even pseudodifferential projections. Index formulae are obtained for operators on compact manifolds without boundary and for general boundary-value problems. A connection with Gilkey's theory of η-invariants is established.

Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems

We calculate the homotopy groups of the space of elliptic boundary value problems for an elliptic differential operatorA of a first order and of the space of elliptic self-adjoint boundary value problems whenA is a formally self-adjoint. In particular we show that the spectral flow of anS1 family of self-adjoint elliptic boundary value problems is well defined. This provides some information on spectral properties along the lines of the Vafa-Witten approach to spectral inequalities.