×

Localization for a continuum Cantor-Anderson Hamiltonian. (English) Zbl 1137.81011

Germinet, François (ed.) et al., Adventures in mathematical physics. International conference in honor of Jean-Michel Combes on transport and spectral problems in quantum mechanics, September 4–6, 2006, Cergy-Pontoise, France. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4241-6/pbk). Contemporary Mathematics 447, 103-112 (2007).
Summary: We prove localization at the bottom of the spectrum for a random Schrödinger operator in the continuum with a single-site potential probability distribution supported by a Cantor set of zero Lebesgue measure. This distribution is too singular to be treated by the usual methods. In particular, an “a priori” Wegner estimate is not available. To prove the result we perform a multiscale analysis following the work of Bourgain and Kenig for the Bernoulli-Anderson Hamiltonian, and obtain the required Wegner estimate scale by scale. To do so, we generalize their argument based on Sperner’s Lemma by resorting to the LYM inequality for multisets, and combine it with the concept of scale dependent equivalent classes of configurations introduced by Germinet, Hislop and Klein for the study of Poisson Hamiltonians.
For the entire collection see [Zbl 1130.81005].

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35R60 PDEs with randomness, stochastic partial differential equations
PDFBibTeX XMLCite
Full Text: arXiv