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Champs magnétiques, groupes discrets et asymptotique semi-classique. (Magnetic fields, discrete groups and semiclassical asymptotics). (French) Zbl 0932.58030

The results in the first part of the paper of M. A. Shubin [Geom. Funct. Anal. 6, 370-409 (1996; Zbl 0861.58039)] are extended in the paper under review to the case of magnetic Schrödinger operators. This is done by adapting the methods in the paper of Shubin to operators which commute with magnetic translations. The methods apply in particular to magnetic Schrödinger operators in \({\mathbb R}^n\): \(H_{(h)}=h(-i\nabla+h^{-1}A)^2+h^{-1}V\) with magnetic field \(dA\) and electric potential \(V\;\Gamma\) invariant. Here \(\Gamma\) is a lattice in \({\mathbb R}^n\). It is also supposed that \(V\) is non-negative and the Hessian of \(V\) is positive definite in the points where \(V\) is zero. Then a pure point spectrum model operator \(K\) is constructed such that: \[ N(\lambda-Ch^{1/2},K)\leq N_{\Gamma}(\lambda,H_{(h)})\leq N(\lambda+Ch^{1/5},K) \] for \(\lambda\in [-R,R]\) and \(h\leq h_0\). \(N_{\Gamma}(\lambda,H_{(h)})\) is the usual spectral distribution function of \(H_{(h)}\), \(N_{\Gamma}(\lambda,K)\) is the von Neumann spectrum distribution function of \(K\) and the constants \(C\) and \(h_0\) depend on \(R\).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators

Citations:

Zbl 0861.58039
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