00-489 R. del Rio,S.Fuentes, A. Poltoratski

Families of spectral measures with mixed types (26K, LaTex) Dec 7, 00

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Abstract. Consider a family of Sturm-Liouville 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 half-plan. We also discuss applications to the Krein spectral shift.

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