Eigenvalues variations for Aharonov-Bohm operators

Corentin Léna (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We study how the eigenvalues of a magnetic Schr{ö}dinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential. We consider a magnetic potential defined everywhere in $\mathbb{R}^2$ except at a finite number of singularities, so that the associated magnetic field is zero. On a fixed planar domain, we define the corresponding magnetic Hamiltonian with Dirichlet boundary conditions, and study its eigenvalues as functions of the singularities. We prove that these functions are continuous, and in some cases even analytic. We sketch the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain.

Mots clés: spectral theory; partial differential equations; Aharonov-Bohm Hamiltonians; eigenvalues; perturbation theory; minimal partitions; nodal domains

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Corentin Léna. Eigenvalues variations for Aharonov-Bohm operators. Journal of Mathematical Physics, 2015, 56 (1), pp.011502. ⟨10.1063/1.4905647⟩. ⟨hal-00959975⟩