Internal Lifshitz tails for discrete Schrödinger operators
The main object under consideration in the paper is the following second-order difference random operator which acts in \(\ell^2 (\mathbb Z^d)\) \[ (H_\omega\psi )(n) = 2d\psi (n) - \sum_{|m-n|=1} \psi (m) + (W_{per} (n) + V_{\omega} (n))\psi (n). \] Hamiltonians with a potential energy, which is the sum of periodic and random ones, give the examples of models with gaps in the spectrum for which the investigation of the internal Lifshitz tails is a natural problem. For such operators the integrated density of states (IDS) exists and its topological support is equal to the almost sure spectrum. The main goal of the paper is to determine the behavior of the IDS near the gap edges and to prove that is has a Lifshitz behavior if and only if the IDS of a well-chosen periodic operator is nondegenerate at the same edges. An application to the Anderson localization is given.
scientific article; zbMATH DE number 5304379
scientific article