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 S̃ such that the spectral parts S̃J and RJ are unitarily equivalent. The proofs relies on the technique of boundary triples and associated Weyl functions which allows in addition, to investigate the spectral properties of S̃ within the spectrum of S0. So it is shown that for any extension S̃ of S the absolutely continuous spectrum of S0 is contained in that one of S̃. Moreover, for a wide class of extensions the absolutely continuous parts of S̃ and S are even unitarily equivalent.