Magnetic WKB Constructions

Virginie Bonnaillie-Noël (1) , Nicolas Raymond (2) , Frédéric Hérau (3)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Equations aux dérivées partielles

This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman-Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions. We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal estimates of Agmon and upper bound of the tunelling effect (in symmetric cases). We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades.

Mots clés: magnetic laplacian; tunnel effect; Born-Oppenheimer approximation; Agmon estimates; coherent states; WKB expansion

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

Virginie Bonnaillie-Noël, Nicolas Raymond, Frédéric Hérau. Magnetic WKB Constructions. Archive for Rational Mechanics and Analysis, 2016, 221 (2), pp.817-891. ⟨10.1007/s00205-016-0987-x⟩. ⟨hal-00966003v3⟩