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On a Riemann surface $\Sigma$, consider the space $H_\lambda$ spanned by eiigenfunctions corresponding to eigenvalues $\leq \lambda$. By Weyl's asymptotic formula we know that $$ \dim H_\lambda \s …

Yes there is. Here is how you do it. First find an orthogonal change in variables $$ x_j=\sum_{jk} s_{jk}y_k $$ $(s_{jk})$ orthogonal matrix, so that in the new coordinates we have $$ \sum_{i, …

Suppose that $(M,g)$ is a compact Riemann manifold and for any positive integer $N$ there exists an eigenvalue of the Laplacian that has multiplicity $>N$. …

The Laplacian defines an isomorphism $\Delta E\to H^{-1}$ with inverse $\Delta^{-1}$. …

In principle, one can compute the spectrum of any homogeneous compact Riemannian manifold because in this case the problem is essentially representation theoretic. However, performing this computation …

The most common boundary conditions in the case of Laplacian are the Dirichlet and the Neumann conditions. The only tricky part in the case with boundary is regularity along the boundary. …

If $A$ is a symmetric partial differential operator of order $2k$ on a compact manifold whose principal symbol is positive definite, then for $\lambda\gg 0$ the operator $A+\lambda$ is positive d …

For a generic metric on an $m$-dimensional the manifold the eigenvalues of the Laplacian are all simple. …