Grushin spherical harmonics in non-degenerate dimension one

Abstract

The Grushin Laplacian ${\Delta }_{\alpha }={\Delta }_x+|x{|}^{2\alpha }{\Delta }_y$ is a degenerate elliptic operator in ${\mathbb{R}}^{h+k}$ that degenerates on $\{0\}\times {\mathbb{R}}^k$. We explicitly compute the Grushin spherical harmonics on the gauge sphere $\{(x,y)\in {\mathbb{R}}^{h+k}:|x{|}^{2(\alpha +1)}+(\alpha +1{)}^2|y{|}^2=1\}$ 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 $n\equiv 0,1(\mathrm{mod}\alpha +1)$, even and odd in the $x$-variable respectively, which can be described in terms of Jacobi polynomials.