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Large eigenvalues of the Connes-Moscovici operator

  • Kouichi Taira
  • Max Willems
  • Michał Wrochna
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Abstract

We prove the logarithmic Weyl law predicted by Connes and Moscovici for large positive eigenvalues of the distinguished self-adjoint extension of the operator P=−∂x(x2−1)∂x−4π2x2 on the real line. More precisely, we use the WKB method and show a Bohr-Sommerfeld quantization rule characterizing sufficiently large eigenvalues. A particular difficulty is that the associated h-dependent Lagrangian submanifolds are non-compact, which requires a renormalization of the action integral. Furthermore, to deal with the degeneracy of the coefficients at x =± 1, we use a quasimode matching argument based on radial estimates and Lagrangian regularity.

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Cite

BibTeXRIS
BibTeX
@misc{taira2026large,
  title = {{Large eigenvalues of the Connes-Moscovici operator}},
  author = {Kouichi Taira and Max Willems and Michał Wrochna},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32639},
  eprint = {2609.32639},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32639},
}
RIS
TY  - UNPB
TI  - Large eigenvalues of the Connes-Moscovici operator
AU  - Kouichi Taira
AU  - Max Willems
AU  - Michał Wrochna
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32639
UR  - https://arxiv.org/abs/2609.32639
AB  - We prove the logarithmic Weyl law predicted by Connes and Moscovici for large positive eigenvalues of the distinguished self-adjoint extension of the operator P=−∂_x(x^2−1)∂_x−4π^2x^2 on the real line. More precisely, we use the WKB method and show a Bohr-Sommerfeld quantization rule characterizing sufficiently large eigenvalues. A particular difficulty is that the associated h-dependent Lagrangian submanifolds are non-compact, which requires a renormalization of the action integral. Furthermore, to deal with the degeneracy of the coefficients at x =± 1, we use a quasimode matching argument based on radial estimates and Lagrangian regularity.
ER  -