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Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains

  • Quanyu Tang
  • Zuoqin Wang
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Abstract

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

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Cite

BibTeXRIS
BibTeX
@misc{tang2026strict,
  title = {{Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains}},
  author = {Quanyu Tang and Zuoqin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13139},
  eprint = {2610.13139},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13139},
}
RIS
TY  - UNPB
TI  - Strict Dirichlet Pólya Inequality for Simply Connected Planar Domains
AU  - Quanyu Tang
AU  - Zuoqin Wang
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13139
UR  - https://arxiv.org/abs/2610.13139
AB  - We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.
ER  -