Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

TL;DR

This paper establishes dispersive estimates for certain non-selfadjoint Schrödinger operators in three dimensions, especially when zero energy eigenvalues or resonances are present, without relying on specific NLS properties.

Contribution

It provides a general, axiomatic framework for dispersive bounds applicable to a broad class of operators, extending previous results to cases with zero energy eigenvalues or resonances.

Findings

Dispersive bounds are proven without assuming regular edges of the spectrum.

The approach is independent of specific NLS properties, making it broadly applicable.

Results are obtained under four general assumptions, applicable in three dimensions.

Abstract

We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.