Local and global continuity of the integrated density of states. (English) Zbl 1060.47042

Combes, J.-M.; Hislop, P. D.; Klopp, F.

Karpeshina, Yulia (ed.) et al., Advances in differential equations and mathematical physics. Proceedings of the 9th UAB international conference, University of Alabama, Birmingham, AL, USA, March 26–30, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3296-4). Contemp. Math. 327, 61-74 (2003).

In their paper [Int. Math. Res. Not. 2003, No. 4, 179–209 (2003; Zbl 1022.47028)], the authors studied the continuity property of the integrated density of states at all energies for a class of random Schrödinger operators \(H_{\omega}(\lambda) = H_0 + \lambda V_{\omega}\) on \(L^2({\mathbb R}^d)\). Here \(H_0 = (-i\nabla - A_0)^2 + V_0\) is a deterministic background operator and \(V_{\omega}\) an Anderson-type random potential constructed from nonnegative single-site potentials together with random coupling constants having essentially bounded probability densities with bounded support.
The paper under review is a rather expository note based on it, but the authors also give, localizing their previous methods to energy intervals, some new results on local continuity property of the integrated density of states and regularity property of the density of states measure for such random Schrödinger operators, especially with constant magnetic fields. They give local and global-in energy Wegner estimates, which are strong enough to yield the density of states measure is absolutely continuous with a density in \(L^q_{\text{loc}}({\mathbb R})\) for any \(1\leq q <\infty\). The integrated density of states is also shown to be locally or globally Hölder continuous with exponent \(1/q\) for any \(q>1\).
For the entire collection see [Zbl 1015.00019].

Reviewer: Takashi Ichinose (Kanazawa)

MSC:

47B80	Random linear operators
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81Q15	Perturbation theories for operators and differential equations in quantum theory
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60H25	Random operators and equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
35Q40	PDEs in connection with quantum mechanics

Keywords:

random Schrödinger operator; integrated density of states; Wegner estimate; localization

Citations:

Zbl 1022.47028