In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry in Non-standard Domains page $39$, they use that concept, but I don't know why it works. Could someone be able to explain to me why the point $7$ of page $39$ is sufficient? In other words, why it is sufficient the use the spherical harmonics to find the spectrum of the sphere? A simple example could be appreciate to understand why it works.

Thanks for your help!