Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

TL;DR

This paper develops a generalized Mehler formula for time-dependent non-selfadjoint quadratic operators, providing a detailed analysis of their evolution equations and the propagation of Gabor singularities.

Contribution

It introduces a new generalized Mehler formula for non-autonomous quadratic operators and explores their impact on the propagation of singularities in solutions.

Findings

Solution operators are Fourier integral operators with Gaussian kernels.

Derived a generalized Mehler formula for Weyl symbols.

Applied results to study Gabor singularity propagation.

Abstract

We study evolution equations associated to time-dependent dissipative non-selfadjoint quadratic operators. We prove that the solution operators to these non-autonomous evolution equations are given by Fourier integral operators whose kernels are Gaussian tempered distributions associated to non-negative complex symplectic linear transformations, and we derive a generalized Mehler formula for their Weyl symbols. Some applications to the study of the propagation of Gabor singularities (characterizing the lack of Schwartz regularity) for the solutions to non-autonomous quadratic evolution equations are given.