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Grushin spherical harmonics in non-degenerate dimension one
Abstract
The Grushin Laplacian $Δ_α= Δ_x +|x|^{2α}Δ_y $ is a degenerate elliptic operator in ℝh+k that degenerates on {0} × ℝk. We explicitly compute the Grushin spherical harmonics on the gauge sphere {(x,y) ∈ ℝh+k : |x|2(α+1)+(α+1)2|y|2=1} and the corresponding spectrum when the non-degenerate dimension h=1, a case not covered by the existing literature. The eigenfunctions consist of homogeneous Grushin-harmonic polynomials of degrees n ≡ 0,1 pmodα+1, even and odd in the x-variable respectively, which can be described in terms of Jacobi polynomials.
Related
- Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type
- A class of degenerate elliptic problems involving the Grushin operator: critical mass and ground states
- Spherical harmonics with maximal Lp norm growth
- Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations
- Spherical harmonics and infinitesimal projective transformations
BibTeX
@misc{bramati2026grushin,
title = {{Grushin spherical harmonics in non-degenerate dimension one}},
author = {Roberto Bramati and Paolo Luzzini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31137},
eprint = {2610.31137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31137},
}
RIS
TY - UNPB
TI - Grushin spherical harmonics in non-degenerate dimension one
AU - Roberto Bramati
AU - Paolo Luzzini
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31137
UR - https://arxiv.org/abs/2610.31137
AB - The Grushin Laplacian $Δ_α= Δ_x +|x|^{2α}Δ_y $ is a degenerate elliptic operator in ℝ^h+k that degenerates on {0} × ℝ^k. We explicitly compute the Grushin spherical harmonics on the gauge sphere {(x,y) ∈ ℝ^h+k : |x|^2(α+1)+(α+1)^2|y|^2=1} and the corresponding spectrum when the non-degenerate dimension h=1, a case not covered by the existing literature. The eigenfunctions consist of homogeneous Grushin-harmonic polynomials of degrees n ≡ 0,1 pmodα+1, even and odd in the x-variable respectively, which can be described in terms of Jacobi polynomials.
ER -