Let $ \Delta\_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-1}\_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?  

thanks
Craig



Eigenvalues of Laplace-Beltrami on half sphere
Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $\mathbb{S}^{N-1}$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x\_N \equiv 0$. The reference that I've seen that explicitly constructs the $N-1$ dimensional spherical harmonics is the following paper of Frye and Efthimiou: <https://arxiv.org/pdf/1205.3548.pdf>


In theory, this reduces your question to a combinatorial problem involving Legendre polynomials, though I haven't solved out the combinatorics explicitly. For the 2-sphere, it seems like the eigenfunctions (and their eigenvalues) you are looking for are the $Y\_l^m$ where $m+l$ is odd. From this, you can see the that spectrum is $l(l+1)$ but with less degeneracy than with the full sphere.


