We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or unitary operators. In the n-dimensional case, we prove that for any cyclic self-adjoint operator $A$, operator $A_��= A + ��_{k=1}^n ��_k(\cdot,��_k)��_k$ is pure point for a. e. $��=(��_1,��_2,...,��_n) \in\Bbb R^n$ iff operator $A_��=A+��(\cdot,��_k)��_k$ is pure point for a.e.\ $��\in\Bbb R$ for $k=1,2,...,n$. We also show that if $A_��$ is pure point for a.e.\ $��\in \Bbb R^n$ then $A_��$ is pure point for a.e.\ $��\in ��$ for any analytic curve $��\in\Bbb R^n$.