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

Using the double covering approach introduced by B. Helffer, M. and T. HoffmannOstenhof 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.