## 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 $$A(x)$$ 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 $i\varepsilon \partial u/ \partial t= (i\partial/ \partial x+ \varepsilon A(x))^ 2 u+Vu$ 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 $$L(k)= (i\partial/ \partial x+k)^ 2+ V(x)$$ (where $$k\in \mathbb{R}^ 3$$) with the lattice conditions.
The construction of the ansatz in the case when one eigenvalue $$E_ n(k)$$ 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 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)].
For the entire collection see [Zbl 0778.00035].
Reviewer: B.Helffer (Paris)

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

### Citations:

Zbl 0672.35014; Zbl 0739.35053; Zbl 0715.35070; Zbl 0778.35089