Common Eigenvalue Problem and Periodic Schrِdinger Operators

Let A be a subset of the family of all self-adjoint extensions of a symmetric operator A0 with equal deficiency indices in a Hilbert space. Assuming that A0 has a purely residual spectrum we describe the set of eigenvalues common to all self-adjoint extensions from A. This abstract result is used to show that the one-dimensional periodic Schrِdinger operator with local point interactions is absolutely continuous.