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Magnetic breakdown. (English) Zbl 0789.35151
Robert, D. (ed.), Méthodes semi-classiques, Volume 2. Colloque international (Nantes, juin 1991). Paris: Société Mathématique de France, Astérisque. 210, 263-282 (1992).

This paper is the continuation of J.-C. Guillot, J. Ralston and E. Trubowitz [Commun. Math. Phys. 116, No. 3, 401-415 (1988; Zbl 0672.35014)]. The author considers the Schrödinger equation for a single electron in a crystal lattice of ions in a constant magnetic field $B$. More precisely, if $A\left(x\right)$ is the associated linear magnetic potential, $V\left(x\right)$ is the periodic electric potential and $\epsilon$ is a small parameter, the author considers the evolution equation

$i\epsilon \partial u/\partial t={\left(i\partial /\partial x+\epsilon A\left(x\right)\right)}^{2}u+Vu$

and looks for an “ansatz” of the form $u\left(x\right)=exp\left(-i\phi \left(y,t\right)\right)m\left(x,y,t,\epsilon \right)$ (with $y=\epsilon x$ and $m$ periodic with respect to the $x$ variable) in the asymptotic situation $\epsilon \to 0$. In this case (which is reminiscent of the homogenization theory) one has to analyze the spectral properties of the family of operators $L\left(k\right)={\left(i\partial /\partial x+k\right)}^{2}+V\left(x\right)$ (where $k\in {ℝ}^{3}$) with the lattice conditions.

The construction of the ansatz in the case when one eigenvalue ${E}_{n}\left(k\right)$ of $L\left(k\right)$ is simple was studied in the above mentioned reference. The author considers here a generic situation where two eigenvalues of $L\left(k\right)$ are crossing for some $k$.

Connected results have been obtained by V. Buslaev [Sémin. Equ. Dériv. Partielles, Ec. Polytech., Cent. Math., Palaiseau 1990-1991, No. XXIII (1991; Zbl 0739.35053)], B. Helffer and J. Sjöstrand [Ann. Inst. Henri Poincaré, Phys. Théor. 52, 303-375 (1990; Zbl 0715.35070)] and J. Sjöstrand [Proc. Int. Conf., Birmingham/AL (USA) (1990), Math. Sci. Eng. 186, 295-345 (1992; Zbl 0778.35089)].

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 81Q05 Closed and approximate solutions to quantum-mechanical equations