Resolvents of selfadjoint extensions of the Laplace operator on the solenoidal subspace

On the space of solenoidal vector-valued functions vanishing at the origin with their derivatives, the Laplace operator is symmetric and has defect indices $(3,3)$. With the help of the Krein formula, an expression for the kernel of the resolvent for selfadjoint extensions of this operator is found as the sum of the Green function for the Laplace operator on the space of all vector-valued functions and a certain finite rank addendum.