Wegner-Stollmann type estimates for some quantum lattice systems. (English) Zbl 1145.47061

Chulaevsky, Victor

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, 17-28 (2007).

The author discusses two functional analytic techniques allowing to obtain Wegner–Stollmann type estimates of the density of states for multi-particle quantum lattice systems and for single- or multi-particle systems with generic deterministic potentials. The first approach is based on a mathematically rigorous representation of unitary propagators via Molchanov’s path integral. The second approach uses wavelet-like expansions with random coefficients and provides valuable information about the regularity of a density of states of generic deterministic random operators, otherwise difficult to obtain.
For the entire collection see [Zbl 1130.81005].

Reviewer: Kun Soo Chang (Seoul)

MSC:

47N55	Applications of operator theory in statistical physics (MSC2000)
47E05	General theory of ordinary differential operators
35P20	Asymptotic distributions of eigenvalues in context of PDEs
60J75	Jump processes (MSC2010)
82B10	Quantum equilibrium statistical mechanics (general)

Keywords:

path integrals; wavelets; density of states; Anderson localization