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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

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2 Answers 2

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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.

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    $\begingroup$ 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. $\endgroup$ Oct 18, 2018 at 19:01
  • $\begingroup$ 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? $\endgroup$
    – Math604
    Oct 18, 2018 at 19:08
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    $\begingroup$ 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$. ? $\endgroup$
    – Math604
    Oct 19, 2018 at 1:46
  • $\begingroup$ 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 $\endgroup$
    – Gabe K
    Oct 19, 2018 at 3:16
  • $\begingroup$ Math604, You mentioned that the eigenvalues are well known, can you please share the reference where I can find this. $\endgroup$
    – Adi
    Mar 9 at 19:30
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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.

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