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I am computing eigenvalues and eigenfunctions of Laplacian on a unit square $[0,1]^{2}$ numerically.

Consider the eigenvalue problem with the Dirichlet boundary condition that is, $$L u(x, y) = \lambda u(x, y)$$ where $$L = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}.$$

The boundary condition is that the one the boundary of the square $ u = 0$.

I have computed the eigenvalues and eigenfunctions:

$\textbf{Eigenvalues:}$ $\lambda _{mn} = (m^2 +n^2) \pi ^2$ for $m, n = 1, 2, 3\dotsc$

$\textbf{Eigenfunctions:}$ $u_{mn} (x, y) = \sin(m\pi x) \sin (n \pi y)$ for $m, n = 1, 2, 3\dotsc$

$\textbf{QUESTION:}$ How do I calculate the dimension of the eigenspace/the number of linearly independent eigenvectors corresponding to a given eigenvalue?

EDIT: This is not entirely trivial, for example we have $1^2 + 7^2 = 50 = 5^2 + 5^2$ which indicates that the multiplicity of the eigenvalue $50\pi^2$ is at least $3$ corresponding to $$(m,n) \in \{(1,7),(7,1),(5,5)\}.$$

Carl Christian
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  • I replaced two of your tags with others that are closer fit and did minor formatting. I added a paragraph demonstrating that your question is not trivial. Your question is highly relevant for me because I am interested in the case of distinct eigenvalues with tightly packed clusters. – Carl Christian Feb 11 '21 at 23:04
  • Upon reflection, this is a question that should be cast in terms of number theory. Before you do so, you should have a very close look at this link https://math.stackexchange.com/questions/787321/how-to-determine-whether-a-number-can-be-written-as-a-sum-of-two-squares (question and answers) – Carl Christian Feb 13 '21 at 10:04
  • Linear independence comes from the fact that the relevant functions are in fact orthogonal to each other with respect to the $L^2$ inner product. – Carl Christian Feb 13 '21 at 12:49

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