Contraction semigroups of elliptic quadratic differential operators

TL;DR

This paper investigates the behavior of contraction semigroups generated by elliptic quadratic differential operators, demonstrating exponential decay under specific ellipticity and symbol conditions.

Contribution

It establishes that elliptic quadratic differential operators with non-zero non-positive real parts of their symbols generate contraction semigroups with exponential decay.

Findings

Semigroups decay exponentially over time

Decay depends on ellipticity and symbol properties

Results apply to non-selfadjoint operators

Abstract

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.