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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