SERGIO ALBEVERIO, JOHANNES F. BRASCHE, MARK MALAMUD, HAGEN NEIDHARDT
INVERSE SPECTRAL THEORY FOR SYMMETRIC OPERATORS WITH SEVERAL GAPS:
SCALAR-TYPE WEYL FUNCTIONS
(449K, pdf)
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 $\wt{S}$
such that the spectral parts $\wt{S}_J$ and $R_J$ 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
$\wt{S}$ within the spectrum of $S_0$. So it
is shown that for any extension $\wt{S}$ of $S$ the absolutely
continuous spectrum of $S_0$ is contained in that one of
$\wt{S}$. Moreover, for a wide class of extensions the absolutely
continuous parts of $\wt{S}$ and $S$ are even unitarily equivalent.