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Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential. (English) Zbl 0851.35145
The author presents a study of the asymptotics for the density of states of the magnetic Schrödinger operator with a random potential defined on \(L^2(\mathbb{R}^2)\) \[ P^\omega_{B, V}= (D_x+ By)^2+ D^2_y+ V^\omega(x, y), \] where \(D_x= {1\over i} \partial_x\), \(D_y= {1\over i} \partial_y\) and \(B> 0\) is a constant. If \(v\) is a function in \(C^\infty_0(\mathbb{R}^2)\), then the potential \(V^\omega\) is defined as \[ V^\omega(x, y)= \sum_{(i, j)\in \mathbb{Z}^2} \alpha_{i, j} v(x- i, y- j), \] where \(\alpha= \{\alpha_{i, j}\}_{(i, j)\in \mathbb{Z}^2}\) is a family of random variables on a probability space \((\Omega, P)\).
By using the method of effective Hamiltonian [see for example the reviewer and J. Sjöstrand, Lect. Notes Phys. 345, 118-197 (1989; Zbl 0699.35189)], complex dilation and complex translation, the author obtains in the large magnetic field limit, the asymptotic expansion for the density of states measure considered as distribution. This justifies the Wegner approximation in some weak sense [cf. F. Wegner, Z. Phys. B. Condensed Matter 51, No. 4, 279-285 (1983)].
Reviewer: B.Helffer (Orsay)

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35R60 PDEs with randomness, stochastic partial differential equations
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