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What I am missing in your question, is the dimension of $\lambda$. You can see from your second formula, that the dimension of $\lambda$ is $\text{m}^{-2}$ (because of the Laplacian). Further, to add a bit more information, Normally we would write your Eigenvalue problem as $$\Delta v = - k^2 v.$$ We use $k^2$ here mostly because it will make the answer ...

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