Let $A,B,C\in\mathbb{C}[\partial_x,\partial_y]$ be differential operators. Diagonalize the matrix $$M=\begin{pmatrix}A+B&C\\-C&A-B\end{pmatrix}.$$
If $B,C$ were operators with purely complex image and $A$ an operator with real image I could've imagined that one should think on $M$ as a matrix in some version of $SU(1,1)$. Another idea I thought about is to try and write the operators in form of continued fractions but inverting the operators explicitly can be hard. How would you proceed?