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

Abstract

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., Nodal Domains and Spectral Minimal Partitions. 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., and Bonnaillie-Noël et al. Aharonov–Bohm Hamiltonians, Isospectrality and Minimal Partitions.”, 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. This illustrates also recent results of B. Noris and S. Terracini; see Noris and Terracini 10, Nodal Sets of Magnetic Schrödinger Operators of Aharonov–Bohm Type and Energy Minimizing Partitions”. This finally supports or disproves conjectures for the minimal 3- and 5-partitions on the square.