Minimal velocity bound for Schroedinger-type operator with fractional powers

TL;DR

This paper establishes a minimal velocity bound for fractional Schroedinger-type operators, aiding in proving asymptotic completeness in complex quantum scattering scenarios with diverse potentials.

Contribution

It proves the minimal velocity bound for two-body fractional Schroedinger operators with broad potential classes, extending scattering theory results.

Findings

Proved minimal velocity bound for fractional Schroedinger operators.

Derived key estimates applicable to long-range and Coulomb-type potentials.

Facilitated future proofs of asymptotic completeness in various quantum systems.

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.