Shubin type Fourier integral operators and evolution equations

Abstract

We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin's class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.2010 Mathematics Subject Classification. Primary: 53S30. Secondary: 35S10, 35A22. Key words and phrases. Fourier integral operator, Schrödinger equation, semigroup, perturbation. R. Schulz gratefully acknowledges support of the project "Fourier Integral Operators, symplectic geometry and analysis on noncompact manifolds" received by the University of Turin in form of an "I@Unito" fellowship as well as institutional support by the University of Hannover.