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

up vote 14 down vote accepted

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.

  • 2
    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 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 at 19:08
  • 1
    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 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 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.

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.