The Integrated Density of States for Random Schroedinger Operators

Abstract

We survey some aspects of the theory of the integrated density of states (IDS) of random Schroedinger 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.

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Cited by in corpus (7)

• An Invitation to Random Schroedinger operators
• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
• A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states
• Continuity of the integrated density of states on random length metric graphs
• The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs
• Lifshitz asymptotics for Hamiltonians monotone in the randomness
• Regularity of the density of states for random Dirac operators