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

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_3/2,1² 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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BibTeX

@misc{sundqvist2026ypartition,
  title = {{The Y-partition is the optimal three-partition for the disc and the harmonic oscillator}},
  author = {Mikael Sundqvist},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21964},
  eprint = {2609.21964},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21964},
}

RIS

TY  - UNPB
TI  - The Y-partition is the optimal three-partition for the disc and the harmonic oscillator
AU  - Mikael Sundqvist
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21964
UR  - https://arxiv.org/abs/2609.21964
AB  - 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.
ER  -

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