The first eigenvalues of \(-\Delta_{\mathbb{S}^d_R}\) are

  • \(\lambda_0 = 0\)
  • \(\lambda_1 = \frac{d}{R^2}\)
  • \(\lambda_2 = \frac{2d+2}{R^2}\)
  • \(\lambda_3 = \frac{3d+6}{R^2}\)

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