No abstract

We characterize the spectra of self-adjoint extensions of a symmetric operator with equal deficiency indices in terms of boundary values of their Weyl functions. A complete description is obtained for the point and absolutely continuous spectrum while for the singular continuous spectrum additional assumptions are needed. The results are illustrated by examples.

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S 0 of S such that J is contained in the resolvent set of S 0 and the associated Weyl function of the pair {S, S 0 } is monotone with respect to J, then for any selfadjoint operator R there exists a self-adjoint extension S such that the spectral parts S J and R J are unitarily equivalent. It is shown that for any extension S of S the absolutely continuous spectrum of S 0 is contained in that one of S. Moreover, for a wide class of extensions the absolutely continuous parts of S and S are even unitarily equivalent.