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PreprintarXiv licencearXiv

Characterization of the Tau Spectrum for Eigenvalue Problems

  • Keaton J. Burns
  • Ilaria Fontana
  • Daniel Lecoanet
  • Geoffrey M. Vasil
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Abstract

We consider tau approximations of a general eigenvalue problem for a K-th-order linear differential operator in one dimension, and we show that the approximate spectrum satisfies a resolvent-based spectral determinant property, which we use to investigate it. More specifically, we analyze the spectral determinant property for a second-order Sturm-Liouville eigenvalue problem approximated using Gegenbauer tau methods. Notably, a quantitative error analysis identifies two values of the Gegenbauer parameter γ leading to better approximation properties. These values correspond to Legendre polynomials and their derivatives. Moreover, we discuss the behavior of the largest eigenvalue of the tau approximation. The results are validated with a numerical implementation.

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Cite

BibTeXRIS
BibTeX
@misc{burns2026characterization,
  title = {{Characterization of the Tau Spectrum for Eigenvalue Problems}},
  author = {Keaton J. Burns and Ilaria Fontana and Daniel Lecoanet and Geoffrey M. Vasil},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40307},
  eprint = {2610.40307},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40307},
}
RIS
TY  - UNPB
TI  - Characterization of the Tau Spectrum for Eigenvalue Problems
AU  - Keaton J. Burns
AU  - Ilaria Fontana
AU  - Daniel Lecoanet
AU  - Geoffrey M. Vasil
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40307
UR  - https://arxiv.org/abs/2610.40307
AB  - We consider tau approximations of a general eigenvalue problem for a K-th-order linear differential operator in one dimension, and we show that the approximate spectrum satisfies a resolvent-based spectral determinant property, which we use to investigate it. More specifically, we analyze the spectral determinant property for a second-order Sturm-Liouville eigenvalue problem approximated using Gegenbauer tau methods. Notably, a quantitative error analysis identifies two values of the Gegenbauer parameter γ leading to better approximation properties. These values correspond to Legendre polynomials and their derivatives. Moreover, we discuss the behavior of the largest eigenvalue of the tau approximation. The results are validated with a numerical implementation.
ER  -