Eigenvalues of Laplace-Beltrami on half sphere 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
 A: Tools for computing eigenvalues on disks in constant-curvature space forms are worked out in Chapter II, section 5 of Chavel's book Eigenvalues in Riemannian Geometry although skimming it I do not see the spectrum itself explicitly written out. Basic idea is separation of variables in polar coordinates.
A: 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.
