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arXiv · 2609.21964

The Y-partition is the optimal three-partition for the disc and the harmonic oscillator

Mikael Sundqvist
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Abstract

We prove that the Y-partition into three equal sectors is the minimal spectral three-partition both for the Dirichlet Laplacian on the unit disc and for the planar harmonic oscillator $-Δ+|x|^2$, with minimal energies $j_{3/2,1}^{2}$ and $5$; for the disc, this confirms a conjecture of Helffer and Hoffmann-Ostenhof. The minimizing regular strong partition is unique up to rotation, and every open minimizing partition has cells with the Dirichlet form domains of the sectors. The proof is a positive radial transplantation to the sphere that preserves segregation and matches the angular-energy measures of the separated model states; the three-lune theorem of Helffer, Hoffmann-Ostenhof, and Terracini then gives the lower bound. The transplantation lowers the shifted quadratic form by a nonnegative defect with strictly positive radial weight; in the equality case, spherical equipartition makes the defects vanish, which separates variables, and a Poincaré inequality on the circle identifies the sectors.

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Mikael Sundqvist. 2026-09-18. The Y-partition is the optimal three-partition for the disc and the harmonic oscillator. https://arxiv.org/abs/2609.21964

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