Shubin type Fourier integral operators and evolution equations

TL;DR

This paper investigates the Schrödinger-type evolution equations with quadratic Hamiltonians plus perturbations, proving the propagator is a Shubin type Fourier integral operator and deriving phase space estimates for solutions.

Contribution

It establishes that the propagator for such evolution equations is a Shubin type Fourier integral operator of order zero, extending the understanding of their structure and properties.

Findings

The propagator is a Fourier integral operator of Shubin type of order zero.

Phase space estimates for the propagator and solutions are derived.

The results connect the evolution equations with advanced Fourier integral operator theory.

Abstract

We study the Cauchy problem for an evolution equation of Schr\"odinger 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.