Fourier integral operators on non-compact manifolds

Abstract

We consider Fourier integral operators on non-compact manifolds and their applications, in particular in spectral theory. Fourier integral operators appear naturally as the solution operators of certain pseudodifferential evolution equations, such as the Schrödinger equation or the wave equation. For Euclidean space there are two important global pseudodifferential calculi: First there is the isotropic calculus, which contains the quantum harmonic oscillator, its inverse, and similar operators. We consider the solution operator to the dynamical Schrödinger equation with an isotropic pseudodifferential operator of order two and show how singularities and growth evolve with time. Moreover we show that for generic lower order perturbations of the harmonic oscillator the eigenvalues are more equally distributed then in the case of the unperturbed operator. The second important calculus, the scattering calculus, contains the Laplacian plus a bounded potential on asymptotically Euclidean manifolds. We define a class of geometric distributions that are related to the solution operators of the Klein-Gordon equation of quantum field theory and contain certain distributions that are appear in the scattering theory of the Laplacian. We show that these distributions have a symbol structure that admits an invariantly defined order and the existence of a principal symbol.