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2011 Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitions
V. Bonnaillie-Noël, B. Helffer
Experiment. Math. 20(3): 304-322 (2011).

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.

Citation

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V. Bonnaillie-Noël. B. Helffer. "Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitions." Experiment. Math. 20 (3) 304 - 322, 2011.

Information

Published: 2011
First available in Project Euclid: 6 October 2011

zbMATH: 1270.35025
MathSciNet: MR2836255

Subjects:
Primary: 35B05 , 35J05 , 35P15 , 65F15 , 65N25

Keywords: Aharonov-Bohm Hamiltonian , minimal partitions , nodal domains , numerical simulations , Spectral theory

Rights: Copyright © 2011 A K Peters, Ltd.

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Vol.20 • No. 3 • 2011
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