Spectra of Anderson type models with decaying randomness (Q5947473)

The author considers Hamiltonians on the Hilbert spaces \(\ell(Z^d)\) and \(\ell(Z_+^d)\). The Hamiltonians are of the form \(H=H_0 + V\). \(H_0\) is a Toeplitz matrix (i.e. a multiplication operator in Fourier space), \(V\) is a random potential on \(Z^d\) of the form \(V(n)=a_n q_n(\omega)\). The random variables \(q_n\) are independent and identically distributed and \(a_n\) is a deterministic decaying sequence. An example of particular interest for \(H_0\) is the discrete Laplacian but more general free operators are allowed. Under suitable assumptions the authors proof that on the spectrum of the operator \(H_0\) the spectrum of \(H\) is absolutely continuous. At the same time there may be singular essential spectrum for \(H\) outside the spectrum of \(H_0\). For this the random variables \(q_n\) must be unbounded so that \(V\) is not going to zero at infinity. In particular examples it is proven that the dense pure point spectrum of \(H\) consists of \(R \setminus \sigma(H_0)\). The proof is scattering theoretic in nature and develops the scattering theory for this kind of free oparators. This is combined with results by Jaksic and Last. To prove pure point spectrum the authors use (their variant of) the fractional moment method by Aizenman and Molchanov.