Spectra and semigroup smoothing for non-elliptic quadratic operators

TL;DR

This paper investigates the smoothing properties and spectral characteristics of non-elliptic quadratic differential operators, revealing conditions under which their associated semigroups exhibit smoothing effects and their spectra are discrete.

Contribution

It introduces the concept of a singular space for these operators and characterizes the spectral and smoothing behavior based on the symplectic structure of this space.

Findings

The heat semigroup is smoothing in directions orthogonal to the singular space when it has a symplectic structure.

The spectrum of elliptic operators on the singular space is discrete and explicitly describable.

Large time behavior of contraction semigroups is characterized.

Abstract

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space, intrinsically associated to the Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal complement. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the globally elliptic case. We also describe the large time behavior of contraction semigroups generated by these operators.