Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 23 min ago
PreprintCC BY 4.0arXiv

The para--Krawtchouk oscillator and its continuum limit

  • Stéphane Z. Beaulac
  • Nicolas Crampé
  • Quentin Labriet
  • Lucia Morey
  • Marlon Josue Rivera Valladares
  • Luc Vinet
Open at sourcePDFDOICite

Abstract

The para-Krawtchouk polynomials define, through their recurrence relation, a finite oscillator model that exhibits fractional revival. We determine the continuum limit of this Hamiltonian, both from its recurrence relation and directly on the polynomials. The limit is the singular (isotonic) oscillator −∂η2+η2+(γ2−1)/4η2. The deformation parameter γ survives as a central impurity of the couplings, which decays as the inverse square of the distance. This is the marginal rate, which the continuum scaling turns into a finite sfrac1η2 term. Using the decomposition of the para-Krawtchouk polynomials into two Hahn families, we prove the confluence, for every degree, of the eigenvectors on the two sub-lattices to generalized Hermite functions. The bispectral pair of the discrete model, which generates the Hahn algebra, becomes in the limit a pair of generators of su, the dynamical algebra of the singular oscillator.

Related

  1. The Fixed-q Edge Limit of the q-Krawtchouk Ensemble
    Pavel Nikitin
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  2. A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
    Anier Soria Lorente, Junior Michel
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. Mean-Field Oscillator Ising Machines: Gradient Flows and Classification of Limit Solutions
    Arvind R. Venkatakrishnan, Max Emerick, Bassam Bamieh +1
    arXivPreprintarXiv licenceFull textDynamical systemsOptimisation & controlarXiv
  4. The Continuum Limit Analysis of Causal Fermion Systems for Curved Spacetimes
    Felix Finster, Patrick Fischer
    arXivPreprintarXiv licenceFull textMathematical physicsGravitation & general relativityarXiv
  5. Continuum Limit for One-Dimensional Moiré Relaxation
    Hao Lan, Michael Hott, Jeff Calder +1
    arXivPreprintarXiv licenceFull textMathematical physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{beaulac2026parakrawtchouk,
  title = {{The para--Krawtchouk oscillator and its continuum limit}},
  author = {Stéphane Z. Beaulac and Nicolas Crampé and Quentin Labriet and Lucia Morey and Marlon Josue Rivera Valladares and Luc Vinet},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30619},
  eprint = {2609.30619},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30619},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The para--Krawtchouk oscillator and its continuum limit
AU  - Stéphane Z. Beaulac
AU  - Nicolas Crampé
AU  - Quentin Labriet
AU  - Lucia Morey
AU  - Marlon Josue Rivera Valladares
AU  - Luc Vinet
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.30619
UR  - https://arxiv.org/abs/2609.30619
AB  - The para-Krawtchouk polynomials define, through their recurrence relation, a finite oscillator model that exhibits fractional revival. We determine the continuum limit of this Hamiltonian, both from its recurrence relation and directly on the polynomials. The limit is the singular (isotonic) oscillator −∂_η^2+η^2+(γ^2−1)/4η^2. The deformation parameter γ survives as a central impurity of the couplings, which decays as the inverse square of the distance. This is the marginal rate, which the continuum scaling turns into a finite sfrac1η^2 term. Using the decomposition of the para-Krawtchouk polynomials into two Hahn families, we prove the confluence, for every degree, of the eigenvectors on the two sub-lattices to generalized Hermite functions. The bispectral pair of the discrete model, which generates the Hahn algebra, becomes in the limit a pair of generators of su, the dynamical algebra of the singular oscillator.
ER  -