Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions.(English)Zbl 1270.35025

Summary: This paper is devoted to presenting numerical simulations and a theoretical interpretation of results for determining the minimal $$k$$-partitions of a domain $$\Omega$$. 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, 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 [Indiana Univ. Math. J. 59, No. 4, 1361–1404 (2010; Zbl 1219.35054)]. This finally supports or disproves conjectures for the minimal 3- and 5-partitions on the square.

MSC:

 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P15 Estimates of eigenvalues in context of PDEs 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

Zbl 1219.35054

Triangle
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References:

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