Werner Kirsch, Bernd Metzger
The Integrated Density of States for Random Schr dinger Operators
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ABSTRACT. We survey some aspects of the theory of the integrated density of
states (IDS) of random Schr dinger operators. The first part motivates the
problem and introduces the relevant models as well as quantities of interest.
The proof of the existence of this interesting quantity, the IDS, is discussed in
the second section. One central topic of this survey is the asymptotic behavior
of the integrated density of states at the boundary of the spectrum. In particular,
we are interested in Lifshitz tails and the occurrence of a classical and
a quantum regime. In the last section we discuss regularity properties of the
IDS. Our emphasis is on the discussion of fundamental problems and central
ideas to handle them. Finally, we discuss further developments and problems
of current research.