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Spectrum of the Laplace–Beltrami Operator on Antisymmetric Functions

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We study properties of the eigenvalues of the Laplace–Beltrami operator on L2\(\left({\mathbb{S}}^{dN-1}\right)\), 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.

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Authors and Affiliations

  1. Sirius Mathematics Center, Sirius University of Science and Technology, 1, Olympic Av, Sochi, 354340, Russia

    I. A. Shcherbakov

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  1. I. A. Shcherbakov
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Correspondence to I. A. Shcherbakov.

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Translated from Problemy Matematicheskogo Analiza 126, 2024, pp. 99-106.

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Shcherbakov, I.A. Spectrum of the Laplace–Beltrami Operator on Antisymmetric Functions. J Math Sci 279, 563–572 (2024). https://doi.org/10.1007/s10958-024-07032-0

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  • DOI: https://doi.org/10.1007/s10958-024-07032-0

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