Boundary spectral estimates for semiclassical Gevrey operators

TL;DR

This paper establishes spectral and resolvent estimates for semiclassical Gevrey operators, revealing a boundary spectrum free region of size proportional to a fractional power of the semiclassical parameter, extending known results to Gevrey regularity.

Contribution

It provides the first spectral boundary estimates for semiclassical pseudodifferential operators with Gevrey regular symbols, generalizing previous smooth and analytic results.

Findings

Boundary spectrum free region size is ${ m O}(h^{1-rac{1}{s}})$.

Resolvent is at most fractional exponentially large in $h$.

Results extend spectral estimates to Gevrey regular operators.

Abstract

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size ${\mathcal O}(h^{1-\frac{1}{s}})$ where the resolvent is at most fractional exponentially large in $h$, as the semiclassical parameter $h\to 0^+$. This is a natural Gevrey analogue of a result by N. Dencker, J. Sj{\"o}strand, and M. Zworski in the $C^{\infty}$ and analytic cases.