04-177 SERGIO ALBEVERIO, JOHANNES F. BRASCHE, MARK MALAMUD, HAGEN NEIDHARDT

INVERSE SPECTRAL THEORY FOR SYMMETRIC OPERATORS WITH SEVERAL GAPS: SCALAR-TYPE WEYL FUNCTIONS (449K, pdf) Jun 4, 04

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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.

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