The absolutely continuous spectrum of Jacobi matrices

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We study general structural properties of Jacobi matrices with some absolutely continuous spectrum. The starting point of this analysis is the recently discovered fact that there are universal building blocks of a rather special type, and with peculiar poperties that have to be used to produce any kind of absolutely continuous spectrum. More precisely, this role is played by reflectionless operators, and these will take center stage in our investigations. We would like to analyze their properties and their dynamical properties associated with the shift map.Spectral theory is at the heart of the mathematical theory of quantum mechanics and our understanding of matter. The plan outlined above concerns mathematical issues of these theories. While these belong to the domain of fundamental research, there are also close connections to physics and potential long term applications in materials science.