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Robin counting functions on cuboids in the critical scaling regime

  • Matthias Baur
  • Simon Larson
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Abstract

We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off λ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to √λ. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a Pólya-type inequality for the counting function whenever the ratio of the Robin parameter and √λ exceeds a dimension-dependent threshold.

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Cite

BibTeXRIS
BibTeX
@misc{baur2026robin,
  title = {{Robin counting functions on cuboids in the critical scaling regime}},
  author = {Matthias Baur and Simon Larson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02297},
  eprint = {2609.02297},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02297},
}
RIS
TY  - UNPB
TI  - Robin counting functions on cuboids in the critical scaling regime
AU  - Matthias Baur
AU  - Simon Larson
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02297
UR  - https://arxiv.org/abs/2609.02297
AB  - We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off λ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to √λ. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a Pólya-type inequality for the counting function whenever the ratio of the Robin parameter and √λ exceeds a dimension-dependent threshold.
ER  -