I am calculating the eigenvalues of Laplacian on sphere. The Laplace's problem(with the Dirichlet's boundary condition) on sphere is as follows: $$L u + \lambda u = 0$$ where $L$ has the following meaning.
Now I am interested in computing the Weyl's law for the sphere. Weyl's law for bounded domain is as follows.
where $N(\lambda)$ is the eigenvalue counting function, that is, $N(\lambda)$ is the number of eigenvalues less than or equal to $\lambda$ for a given $\lambda$. Now , my question is what is Weyl's for sphere, more precisely, how to verify Weyl's law for the sphere?
Please help me. Thanking in advanced.
See theorem 22.1 here - there is a complete derivation of the eigenfunctions (and eigenvalues) of the Laplace operator on the sphere.