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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(x) is the associated linear magnetic potential, V(x) is the periodic electric potential and ε is a small parameter, the author considers the evolution equation

iεu/t=(i/x+εA(x)) 2 u+Vu

and looks for an “ansatz” of the form u(x)=exp(-iϕ(y,t))m(x,y,t,ε) (with y=εx and m periodic with respect to the x variable) in the asymptotic situation ε0. 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/x+k) 2 +V(x) (where k 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)].

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