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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 {\it J.-C. Guillot}, {\it J. Ralston} and {\it 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 $A(x)$ is the associated linear magnetic potential, $V(x)$ is the periodic electric potential and $\varepsilon$ is a small parameter, the author considers the evolution equation $$i\varepsilon \partial u/ \partial t= (i\partial/ \partial x+ \varepsilon A(x))\sp 2 u+Vu$$ and looks for an “ansatz” of the form $u(x)= \exp(-i\varphi(y,t))m(x,y,t,\varepsilon)$ (with $y=\varepsilon x$ and $m$ periodic with respect to the $x$ variable) in the asymptotic situation $\varepsilon\to 0$. In this case (which is reminiscent of the homogenization theory) one has to analyze the spectral properties of the family of operators $L(k)= (i\partial/ \partial x+k)\sp 2+ V(x)$ (where $k\in \bbfR\sp 3$) with the lattice conditions. The construction of the ansatz in the case when one eigenvalue $E\sb n(k)$ of $L(k)$ is simple was studied in the above mentioned reference. The author considers here a generic situation where two eigenvalues of $L(k)$ are crossing for some $k$. Connected results have been obtained by {\it V. Buslaev} [Sémin. Equ. Dériv. Partielles, Ec. Polytech., Cent. Math., Palaiseau 1990-1991, No. XXIII (1991; Zbl 0739.35053)], {\it B. Helffer} and {\it J. Sjöstrand} [Ann. Inst. Henri Poincaré, Phys. Théor. 52, 303-375 (1990; Zbl 0715.35070)] and {\it J. Sjöstrand} [Proc. Int. Conf., Birmingham/AL (USA) (1990), Math. Sci. Eng. 186, 295-345 (1992; Zbl 0778.35089)]. For the entire collection see [Zbl 0778.00035].

35Q55NLS-like (nonlinear Schrödinger) equations
81Q05Closed and approximate solutions to quantum-mechanical equations
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