On weak and strong solution operators for evolution equations coming from quadratic operators

TL;DR

This paper studies solution operators for evolution equations generated by quadratic operators, revealing their boundedness, decay, and regularizing properties, and connecting short- and long-term behaviors through spectral analysis.

Contribution

It introduces a change of variables approach to analyze solution operators for quadratic evolution equations, unifying their properties across different operator classes.

Findings

Sharp results on boundedness and decay of solution operators

Connection between short-time behavior and symbol range

Existence of compact, regularizing solution operators for certain spectra

Abstract

We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to a broad class of supersymmetric quadratic multiplication-differentiation operators acting on $L^2(\Bbb{R}^n)$ which includes the elliptic and weakly elliptic quadratic operators. We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the symbol and the long-time behavior with the eigenvalues of their generators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.