R. del Rio,S.Fuentes, A. Poltoratski
Families of spectral measures with mixed types
(26K, LaTex)

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.