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

Abstract

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 self-adjoint 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.