# 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

## 2 Answers

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.

• The higher-dimensional spherical harmonics factorise into functions of each angular variable separately. So the vanishing on the circle $x_N$ only constrains the factor that depends on the "last" angular variable, right? If so, the spectrum is the same as that of the full sphere ($\ell(\ell+N-1)$), but with less degeneracy. – AccidentalFourierTransform Oct 18 '18 at 19:01
• ya i figured one could extend to full sphere and play around...but I thought there might be known explicit formula. So without any multiplicity concerns it appears the first two eigenvalues would be $N-1$ and $2N$ ? does this appear correct? – Math604 Oct 18 '18 at 19:08
• i confused on a few things...but... @Gabe K. This $N$ and $2N+1$ is off by one from my $N-1$ and $2N$. So for $S^{N-1}$ the eigenvalues of $\Delta_\theta$ are $\lambda_0=0$, $\lambda_1=N-1$ and $\lambda_2=2N$. ? – Math604 Oct 19 '18 at 1:46
• I think I'm making a bunch of off-by-one mistakes. The correct formula is given on page 6 of the following pdf: macs.hw.ac.uk/~hg94/pdst11/pdst11_sphere.pdf – Gabe K Oct 19 '18 at 3:16

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.