Diagonalization of a matrix with operators as elements How to diagonalize a hamiltonian matrix that has differential operators as elements? My matrix looks something like:
\begin{bmatrix}
A \frac{d^{2}}{{d\theta}^{2}}+ B_{1} & a\cos{(b\theta +c)}\\
 a\cos{(b\theta +c)} & A \frac{d^{2}}{{d\theta}^{2}}+ B_{2}
\end{bmatrix}
where A, $B_{1}$, $B_{2}$, a,b and c are all constants.
How to do I find the spectrum of eigen values for such systems. Are there any standard methods available.
 A: Let your matrix operate on a vector of functions (of $\theta$, in this case), and look for solutions where it produces a constant multiple of that vector. In other words, solve
\begin{equation}
\begin{bmatrix}
A \dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}} + B_{1} & & a\cos(b\theta +c) \\
 a\cos{(b\theta +c)} && A \dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}}+ B_{2}
\end{bmatrix}
\begin{bmatrix}
f(\theta)\vphantom{\dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}}} \\
g(\theta)\vphantom{\dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}}}
\end{bmatrix}
=\lambda
\begin{bmatrix}
f(\theta)\vphantom{\dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}}} \\
g(\theta)\vphantom{\dfrac{\mathrm d^{2}}{{\mathrm d\theta}^{2}}}
\end{bmatrix}
\tag{01}\label{eq01}       
\end{equation}
which is a coupled set of second-order differential equations for $f(\theta)$ and $g(\theta)$. To help solve this system and find values of $\lambda$ which produce periodic solutions so that there is no discontinuity in $\theta$, read about Mathieu functions.
