Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov-Bohm Hamiltonians on the Square with Application to Minimal Partitions

This paper is devoted to presenting numerical simulations and a theoretical interpretation of results for determining the minimal k-partitions of a domain Omega as considered in [Helffer et al. 09]. More precisely, 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 and S. Terracini in [Helffer et al. 09, Bonnaillie-Noel et al. 09], 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 and without any symmetric assumption, in contrast to our previous works [Bonnaillie-Noel et al. 10, Bonnaillie-Noel et al. 09]. This illustrates also recent results of B. Noris and S. Terracini; see [Noris and Terracini 10]. This finally supports or disproves conjectures for the minimal 3- and 5-partitions on the square.