Minimal velocity bound for Schroedinger-type operator with fractional powers
Atsuhide Ishida
TL;DR
The paper proves a minimal velocity bound for Schrödinger-type operators with fractional powers by developing a comprehensive propagation-estimate framework. It constructs and analyzes Mourre-type conjugate operators, employs almost analytic extensions and Helffer–Sjöstrand formulas to handle commutators, and derives maximal, middle, and minimal velocity bounds under broad potential classes including long-range and Coulomb-type singularities. The results yield information on the spectral properties via a Mourre estimate, including isolation and finite multiplicity of eigenvalues away from zero, and provide a pathway to asymptotic completeness for fractional and relativistic Schrödinger operators in short-range, long-range, and N-body contexts. These propagation bounds are crucial for understanding scattering and asymptotic behavior in fractional quantum dynamics.
Abstract
It is known in scattering theory that the minimal velocity bound plays a conclusive role in proving the asymptotic completeness of the wave operators. In this study, we prove the minimal velocity bound and other important estimates for the two-body Schroedinger-type operator with fractional powers. We assume that the pairwise potential functions belong to broad classes that include long-range decay and Coulomb-type local singularities. Our estimates are expected to be applied to prove the asymptotic completeness for the fractional Schroedinger-type operators in various (not only short-range but also long-range and N-body) situations.