 04177 SERGIO ALBEVERIO, JOHANNES F. BRASCHE, MARK MALAMUD, HAGEN NEIDHARDT
 INVERSE SPECTRAL THEORY FOR SYMMETRIC OPERATORS WITH SEVERAL GAPS:
SCALARTYPE WEYL FUNCTIONS
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Jun 4, 04

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Abstract. Let $S$ be the orthogonal sum of infinitely many pairwise unitarily
equivalent symmetric operators with nonzero deficiency
indices. Let $J$ be an open subset of $\R$. If
there exists a selfadjoint 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 selfadjoint 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.
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