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Consider a differential operators acting on differentiable functions of two real variables. Under what conditions does such an operator admit a basis of separable eigenfunctions, i.e. eigenfunctions of the form $f_n(x,y)= a_n(x) b_n(y)$?

I've heard this is true when the operator is itself separable, i.e. $\hat{D} = \hat{A}_x \hat{B}_y$, where $A_x$ is a differential operator acting on the $x$ variable, and $B_y$ is a differential operator acting on the variable $y$. Is it true even if the operator to be hermitian?

And how about the case $\hat{D} = \hat{A}_x \hat{B}_y + \hat{F}_x \hat{G}_y$, where $\hat{F}$ and $\hat{G}$ are two more differential operators acting each on a single variable?

Thanks for sharing any tips or references you may think of.

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