Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics

TL;DR

This paper develops semiclassical hypoelliptic estimates for a class of non-selfadjoint pseudodifferential operators with double characteristics, providing bounds on resolvents and eigenvalues under partial ellipticity assumptions.

Contribution

It introduces new semiclassical hypoelliptic estimates for operators with double characteristics assuming partial ellipticity along the singular space.

Findings

Established resolvent bounds for the operators.

Derived estimates for low-lying eigenvalues.

Provided spectral localization results.

Abstract

For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we establish semiclassical hypoelliptic a priori estimates with a loss of the full power of the semiclassical parameter, giving a localization for the low lying spectral values of the operator.