Harmonic Approximation and Resolvent Estimates for Non-Self-Adjoint Operators

TL;DR

This paper establishes a semiclassical resolvent estimate for a wide class of non-self-adjoint pseudodifferential operators, utilizing dynamical conditions on the symbol to improve ellipticity and analyze spectral properties.

Contribution

It introduces a novel approach to obtain resolvent estimates for non-self-adjoint operators by imposing dynamical conditions on the symbol's average along Hamiltonian flow.

Findings

Resolved spectral localization near the origin for complex potential Schrödinger operators

Developed a method to improve ellipticity via dynamical conditions

Provided resolvent bounds in the low-lying spectral regime

Abstract

We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the operator, which we obtain by imposing a dynamical condition on the average of the real part of the symbol along the Hamiltonian flow generated by its imaginary part. An application of the resolvent estimate to a family of semiclassical Schr\"{o}dinger operators with complex potentials allows us to localize the spectral problem to an $O(h)$-sized neighborhood of the origin.