Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions

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 S0 of S such that J is contained in the resolvent set of S0 and the associated Weyl function of the pair {S,S0} is monotone with respect to J, then for any self-adjoint operator R there exists a self-adjoint extension S˜ such that the spectral parts S˜J and RJ are unitarily equivalent. It is shown that for any extension S˜ of S the absolutely continuous spectrum of S0 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.