 00489 R. del Rio,S.Fuentes, A. Poltoratski
 Families of spectral measures with mixed types
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Dec 7, 00

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Abstract. Consider a family of SturmLiouville operators $H_\theta$ on the half
axis defined as $$H_\theta u=u^{\prime\prime}+q(x)u\qquad 0\leq x<\infty
$$ with boundary condition $$u(0)\cos\theta +u^\prime(0)\sin\theta=0\
qquad 0\leq theta <\pi$$ and the limit point case at infinity. We show
that it is possible for all $H_\theta$ to have dense absolutely
continuous and dense singular spectrum. The construction is based on
integral representations of Pick functions in the upper halfplan. We
also discuss applications to the Krein spectral shift.
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