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 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.

Selfadjoint Extensions with Several Gaps: Finite Deficiency Indices

“…The assumption that A admits a scalar-type Weyl function has far going spectral implications beyond the gap. In [3] it was conjectured that already the monotonicity assumption is sufficient to solve the Problem 1.2. In the following we make a first step to verify this conjecture for the special case that the deficiency indices are finite.…”

Dirac operator coupled to bosons

“…holds where I H is the identity operator in H, see [5]. Obviously, M (·) is of a scalar type if n ± (A) = 1.…”

Boundary Data Maps for Schrödinger Operators on a Compact Interval

“…), we refer, for instance, to [2], [3], [4], [11], [12], [13]- [19], [24]- [28], [33]- [39], [42], [44,Ch. 3], [45], [46,Ch.…”

“…That H θ 0 ,θ R is indeed a closed operator follows, for instance, from [29,Sect. XII.4], especially, by combining Lemma 5 (c) and the first part of the proof of Lemma 26 and noting that g(0), g (0) (resp., g(R), g (R)) are a complete set of boundary values for the minimal operator H min associated with the differential expression −d 2 …”