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L’asymptotique de Weyl pour les bouteilles magnétiques. (The Weyl asymptotic formula for magnetic bottles). (French) Zbl 0612.35102
The Schrödinger operator \(H=(i\vec V-\vec a)^ 2\) with magnetic field and without electric one is considered. Let \(B=(\partial_ ja_ k- \partial_ ka_ j)_{jk}\) be a matrix of the intensity of the magnetic field and let the following conditions be fulfilled: \(\| B(x)\| \leq C\| B(x')\|\) for \(| x-x'| \leq 1\) and \(\| B(x)\| \to \infty\) and \(| D^{\beta}a(x)| =o(\| B(x)\|^{3/2})\) as \(| \beta | =2\) and \(| x| \to \infty\). Then for every \(\epsilon >0\) the eigenvalue counting function N(\(\lambda)\) of H satisfies the following asymptotical estimates: \[ N_{as}(\lambda (1- \epsilon))\lesssim N(\lambda)\lesssim N_{as}(\lambda (1+\epsilon))\quad as\quad \lambda \to +\infty \] where \[ N_{as}(\lambda)=\int_{{\mathbb{R}}^ d}\nu_{B(x)}(\lambda)dx,\quad \nu_ B(\lambda)=c_{d,r}b_ 1...b_ r\sum_{n_ i\geq 0}(\lambda - \sum^{r}_{i=1}(2n_ i+1))_+^{d/2-r} \] \(b_ j>0\) and \(\pm ib_ j\) are all the non-zero eigenvalues of B.
These estimates are proved by the variational method. The two-dimensional case was investigated in the previous papers of the author and three- dimensional one in preprint of H. Tamura (1985). Moreover, the precise remainder estimates in the three-dimensional case in the presence of electric field are obtained by the reviewer (to appear in Proc. Intern. Congress of Mathematicians, Berkeley-1986 and in Soviet Math. Dokl.).
Reviewer: V.Ivrii

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J10 Schrödinger operator, Schrödinger equation
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