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)


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