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To find the eigenfunctions of the Laplacian that are invariant under
horizontal and vertical translations, we seek eigenfunctions
Of course, one way to solve a problem in PDEs is to guess the solution correctly. It's very simple, by means of another application of the technique of separation of variables , to find that the totality of translation invariant eigenfunctions is the set consisting of
where m and n are integers. For a given pair
(n,m), we have a four-dimensional eigenspace with eigenvalue
The values of K and L may be adjusted to give the rectangular cell any desired dimensions.