Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions

Virginie Bonnaillie-Noël (1) , Bernard Helffer (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

Using the double covering approach introduced by B. Helffer, M. and T. Hoffmann-Ostenhof and M. Owen and further developed for questions of isospectrality by the authors in collaboration with T. Hoffmann-Ostenhof, we analyze the variation of the eigenvalues of the one pole Aharonov-Bohm Hamiltonian on the square and the nodal picture of the associated eigenfunctions as a function of the pole. This leads us to discover new candidates for minimal $k$-partitions of the square with a specific topological type. This illustrates also recent results of B. Noris and S. Terracini. This finally supports or disproves conjectures for the minimal $3$ and $5$-partitions on the square.

Mots clés: Numerical simulations; Spectral theory; minimal partitions; Nodal domains; Aharonov-Bohm Hamiltonian

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Virginie Bonnaillie-Noël, Bernard Helffer. Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square and application to minimal partitions. Experimental Mathematics, 2011, 20 (3), pp.304-322. ⟨10.1080/10586458.2011.565240⟩. ⟨hal-00429551⟩