# 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.

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.