Spectrum of the Laplace–Beltrami Operator on Antisymmetric Functions

We study properties of the eigenvalues of the Laplace–Beltrami operator on L² \(\left({\mathbb{S}}^{dN-1}\right)\) $\left({\mathbb{S}}^{dN-1}\right)$ , restricted to the class of antisymmetric functions. We prove that Vandermonde type determinants generate a basis for the space of antisymmetric homogeneous harmonic polynomials. Based on this fact, we calculate the multiplicity of each eigenvalue for d = 1 and present an algorithm for calculating eigenvalues for d ⩾ 2.