Confining quantum particles with a purely magnetic field

Colin de Verdière, Yves; Truc, Françoise

Annales de l'Institut Fourier, Tome 60 (2010) no. 7, pp. 2333-2356.

Nous considérons un ouvert à bord compact d’un espace euclidien et un opérateur de Schrödinger avec champ magnétique dans cet ouvert. Nous donnons des conditions suffisantes sur la croissance du champ magnétique près du bord qui assurent que l’opérateur de Schrödinger est essentiellement auto-adjoint. Du point de vue de la physique, cela signifie que la particule quantique est confinée dans l’ouvert par le champ magnétique. Nous construisons des exemples dans les polytopes et dans des ouverts à frontières lisses  ; ces exemples de “bouteilles magnétiques” sont des modèles extrêmement simplifiés de ce qui est nécessaire pour la fusion nucléaire dans les tokamacs. Nous présentons aussi des problèmes ouverts.

We consider a Schrödinger operator with a magnetic field (and no electric field) on a domain in the Euclidean space with a compact boundary. We give sufficient conditions on the behaviour of the magnetic field near the boundary which guarantees essential self-adjointness of this operator. From the physical point of view, it means that the quantum particle is confined in the domain by the magnetic field. We construct examples in the case where the boundary is smooth as well as for polytopes; These examples are highly simplified models of what is done for nuclear fusion in tokamacs. We also present some open problems.

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