What process can be used to solve for the eigenvalues and eigenfunctions of the following differential operator? $$H=A\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\cos(x-y)$$
Where the solution is periodic for $x$ and $y$ on $(0,2\pi)$ and $A$ is an arbitrary constant. The eigenvalues of a 2D Laplacian operator with such periodic boundary conditions are $\lambda=m^2+n^2$ with eigenfunctions $$F(x,y)=\sum_{m,n=-\infty}^{\infty}a_{m,n}\cos(mx+ny)+b_{m,n}\sin(mx+ny)$$ but I am not sure if this will help me find a solution.
Substitute $2u=x+y$, $2v=x-y$, then
$$ H(u,v) = \frac{A}{2}\left(\frac{\partial^2}{\partial u^2} + \frac{\partial^2}{\partial v^2}\right) + \cos(2v) $$
Let $f(u,v) = U(u)V(v)$, then the problem becomes
$$ \frac{A}{2} \left(\frac{U''}{U} + \frac{V''}{V}\right) + \cos(2v) = \lambda $$
which we can separate to get
\begin{align} U'' + \mu U &= 0 \\ V'' + \left(\mu - \frac{2\lambda}{A} + \frac{2}{A}\cos(2v)\right)V &= 0 \end{align}
You can look for solutions that are $\pi$-periodic in $u,v$, which will then be $2\pi$-periodic in $x,y$. The $V$ components are Mathieu functions, of which periodic solutions do exist for specific values of $\lambda$.