Brief description: Model theory for rank one perturbations with mixed spectra
Aleksandrov-Clark theory connects families of (unitary) rank one perturbations and model theory and hence with many other deep questions of complex analysis. If those families have purely singular spectral measures (the absolutely continuous parts preserved under rank one perturbations by the Kato--Rosenblum theorem; for an easy proof and more theory of rank one perturbations), then the adjoint of the Clark operator is represented by the normalized Cauchy transform, probably the most important object in the field. The corresponding model spaces are (complex) single-valued.
In applications, e.g. to differential equations, we are frequently confronted with rank one perturbations of so-called mixed spectral type. In particular, the spectra with singular continuous parts embedded in the absolutely continuous spectrum are mathematically challenging. The functional models arising in the mixed case are vector-valued.
This paper concerns the generalization of Aleksandrov-Clark theory to rank one perturbations of mixed spectral type. The main result is a concrete representation for the adjoint of Clark's operator. In other words, for general spectral measures we find an object that plays the same role as the normalized Cauchy transform for the case of purely singular spectral families.