Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

TL;DR

This paper investigates resolvent bounds and spectral projections for partially elliptic quadratic differential operators, providing exponential estimates, asymptotic expansions, and characterizations of orthogonal projections, especially for Kramers-Fokker-Planck operators.

Contribution

It introduces new exponential resolvent bounds and asymptotic spectral projection expansions for partially elliptic quadratic operators, including a full characterization of orthogonal spectral projections.

Findings

Exponential resolvent bounds established for these operators.

Complete asymptotic expansions for spectral projections in one dimension.

Exponential upper bounds and growth rates in higher dimensions.

Abstract

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.