Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators

Abstract

This survey is a short version of a chapter written by the first two authors in the book [66] (where more details and references are given) but we have decided here to emphasize more on the role of the Aharonov-Bohm operators which appear to be a useful tool coming from physics for understanding a problem motivated either by spectral geometry or dynamics of population. Similar questions appear also in Bose-Einstein theory. Finally some open problems which might be of interest are mentioned.

On Exact Pleijel's Constant for Some Domains

“…Note that the "irrational" rectangles discussed above are the most simple domains whose Pleijel's constant can be worked out explicitly. However, to the best of our knowledge, the exact value of the Pleijel constant has not been known for any other domains 2 , and the question of finding of such domains was proposed by Bonnaillie-Noël et al [5,Section 6.1]. Furthermore, numerical experiments (see, e.g., [4,1,14]) demonstrate relatively slow convergence of the ratio µ(ϕn) n as n → ∞.…”

On exact Pleijel's constant for some domains

“…Note that the "irrational" rectangles discussed above are the most simple domains whose Pleijel's constant can be worked out explicitly. However, to the best of our knowledge, the exact value of the Pleijel constant has not been known for any other domains, 2 and the question of finding of such domains was proposed by Bonnaillie-Noël et al [4,Section 6.1]. Furthermore, numerical experiments even for the planar disk and square (see, e.g., Blum et al [3]) demonstrate relatively slow convergence of ratio µ(ϕn) n as n → ∞.…”

Asymptotic relation for zeros of cross-product of Bessel functions and applications

“…Here, 𝑃 𝑙(Ω) is called Pleijel constant of Ω. We refer the reader to the surveys [4,10] for the overview of results in this direction.…”

“…In connection with the conjecture of Polterovich [18, Remark 2.2], there is an interesting question to determine the exact value of the Pleijel constant 𝑃 𝑙(Ω) for particular domains, see [4,Section 6.1]. In the article [2], we investigated the values and expressions of 𝑃 𝑙(Ω) for some symmetric domains like a disk, annuli (rings), and their sectors.…”