On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current

Vincent Bruneau (1) , Nicolas Popoff (2, 3, 4)

1 IMB - Institut de Mathématiques de Bordeaux
2 CPT - E8 Dynamique quantique et analyse spectrale
3 CPT - Centre de Physique Théorique - UMR 7332
4 Équipe EDP et Physique Mathématique

We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band function near the ground energy and we exhibit a semi-classical behavior. We perturb the hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential, an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. Our results show different borderline type conditions that in the case where there is no magnetic field.

Mots clés: Perturbation électrique; Laplacien magnétique; Spectral Theory; Analyse asymptotique

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Vincent Bruneau, Nicolas Popoff. On the ground state energy of the Laplacian with a magnetic field created by a rectilinear current. Journal of Functional Analysis, 2015, 268 (5), pp.1277-1307. ⟨10.1016/j.jfa.2014.11.015⟩. ⟨hal-00911208v3⟩