Low energy spectral and scattering theory for relativistic Schroedinger operators

TL;DR

This paper investigates the low energy spectral and scattering properties of relativistic Schrödinger operators, revealing unique threshold behaviors, the absence of zero-energy resonance, and deriving explicit formulas for the free evolution group.

Contribution

It provides new insights into the low energy spectral and scattering theory for relativistic Schrödinger operators, including explicit formulas and threshold property analysis.

Findings

Absence of zero-energy resonance at thresholds

Explicit formula for the free evolution group derived

Absolute continuity of the spectrum established under certain conditions

Abstract

Spectral and scattering theory at low energy for the relativistic Schroedinger operator are investigated. Some striking properties at thresholds of this operator are exhibited, as for example the absence of 0-energy resonance. Low energy behavior of the wave operators and of the scattering operator are studied, and stationary expressions in terms of generalized eigenfunctions are proved for the former operators. Under slightly stronger conditions on the perturbation the absolute continuity of the spectrum on the positive semi axis is demonstrated. Finally, an explicit formula for the action of the free evolution group is derived. Such a formula, which is well known in the usual Schroedinger case, was apparently not available in the relativistic setting.