Operator Group Generated by a One-Dimensional Dirac System

Abstract

In this paper, we construct a strongly continuous operator group generated by a one-dimensional Dirac operator acting in the space \(\mathbb{H} = {{\left( {{{L}_{2}}[0,\pi ]} \right)}^{2}}\). The potential is assumed to be summable. It is proved that this group is well-defined in the space \(\mathbb{H}\) and in the Sobolev spaces \(\mathbb{H}_{U}^{\theta }\), \(\theta > 0\), with a fractional index of smoothness θ and boundary conditions U. Similar results are proved in the spaces \({{\left( {{{L}_{\mu }}[0,\pi ]} \right)}^{2}}\), \(\mu \in (1,\infty )\). In addition, we obtain estimates for the growth of the group as \(t \to \infty \).