OPERATOR GROUP GENERATED BY A ONE-DIMENSIONAL DIRAC SYSTEM

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 fractional index of smoothness $\theta$ and under boundary conditions $U$. Similar results are proved in the spaces ${\left(L_{\mu }[0,\pi ]\right)}^2$, $\mu \in (1,\mathrm{\infty })$. In addition we obtain estimates for the growth of the group as $t\to \mathrm{\infty }$.