Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators

TL;DR

This paper establishes semiclassical resolvent estimates for a class of pseudodifferential operators with non-elliptic quadratic symbols, under averaging assumptions, allowing spectral parameters to grow faster than the semiclassical parameter.

Contribution

It introduces new resolvent estimates for non-selfadjoint operators with non-elliptic quadratic parts under averaging conditions, expanding semiclassical analysis techniques.

Findings

Semiclassical resolvent estimates are proven for operators with non-elliptic quadratic symbols.

The spectral parameter can grow slightly faster than the semiclassical parameter.

Results apply to a class of pseudodifferential operators with doubly characteristic points.

Abstract

We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter.