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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
##### Citations:
Zbl 0719.47031; Zbl 0652.35091; Zbl 0623.35048
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##### References:
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