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The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis of objects. I wonder, is my intuition true? What is the physical meaning of Laplace-Beltrami eigenfunctions?

For now, I only know that the eigenfunctions of the Laplace-Beltrami operator are real and orthogonal, thus they could be used as the basis of functions on the manifold where the functions are defined.

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Cross-posted to – Qmechanic Dec 24 '12 at 9:25
up vote 5 down vote accepted

Eigenfunctions of the Laplacian can be used to construct solutions to differential equations involving the Laplacian, most notably the wave equation, the heat equation, and the Schrödinger equation. The wave equation gives the collection to oscillation; Laplacian eigenfunctions describe standing wave solutions and their eigenvalues describe the period of the oscillation.

See also hearing the shape of a drum.

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