Diagonalization of a matrix with operators as elements

Question

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.

Answer 1

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.