# Cauchy problem for Dirac operator

Jorge Vargas
We consider the unit ball $\mathcal D^n$ in $\mathbb R^n$ together with its non-Euclidean metric. Hence, the connected isometry group for $\mathcal D^n$ is isomorphic to the group $Spin(n,1).$ For each irreducible representation $(\tau, V)$ of $Spin(n),$ let $D_V$ denote the twisted Dirac operator on $\Gamma^\infty( G\times_K V).$ In this talk we describe by means of representation theory the $L^2-$kernel of $D_V$ and the discrete spectrum of $D_V.$ Next, we fix a closed reductive subgroup $H$ of $Spin(n,1)$ (for example: $H=Spin(r) \times Spin(n-r,1)$), hence $H/H\cap Spin(n)$ ($\mathcal D^{n-r}$ for the example) may be thought as a closed sub-manifold of $\mathcal D^n.$ We apply the above results to the Cauchy problem for domain $\mathcal D^n,$ boundary $\mathcal D^{n-r},$ and differential operator $D_V.$ Certainly, the previous setting is example of a more general theory. Part of the work is joint with Bent Orsted and Esther Galina.