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The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneous magnetic field. (English) Zbl 0824.35151

Starting from a semiclassical Schrödinger operator \(H^ h_ 0\) \((h > 0\) is the Planck constant) with a constant magnetic field of intensity \(B\) defined on \(\mathbb{R}^ 3\), the author wants to study the spectral properties of the perturbed operator \(H = H^ h_ 0 - Bh + V\) where \(V\) is a potential tending to 0 as \(| x | \to \infty\). More precisely, the following spectral quantity is analyzed. If, for \(s \geq 0\), \(g_ s\) is the function \[ g_ s (\lambda) = \bigl( \sup (- \lambda, 0) \bigr)^ s \] and, if \(\psi\) is a compactly supported \(C^ \infty\)- function, then the author considers asymptotics as \(h \to 0\) and \(B \to \infty\) for the Riesz means localized by \(\psi\) that is: \[ \text{Tr} \bigl\{ \psi g_ s (H) \bigr\}. \] The remainders are carefully analyzed. The author applies techniques introduced by V. Ivrii (as used for example in the paper of V. Ivrii and I. M. Sigal [Ann. Math. II. Ser. 138, No. 2, 243-335 (1993; Zbl 0789.35135)] and these results are connected with the recent study of E. H. Lieb, J. P. Solovej and J. Yngason [Phys. Rev. Lett. 69, 749-752 (1992)].
Reviewer: B.Helffer (Orsay)

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P20 Asymptotic distributions of eigenvalues in context of PDEs

Citations:

Zbl 0789.35135
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