Abstract:
In this paper, we construct a strongly continuous operator group generated by a one-dimensional Dirac operator acting in the space $\mathbb{H}=(L_2[0,\pi])^2$. 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 $\mathbb{H}^\theta_U$, $\theta>0$, with fractional index of smoothness $\theta$ and under boundary conditions $U$. Similar results are proved in the spaces $(L_\mu[0,\pi])^2$, $\mu\in(1,\infty)$. In addition we obtain estimates for the growth of the group as $t\to\infty$.
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