I am trying to solve directly (numerically if direct methods are not possible) for the spectrum of the differential operator \begin{align*} & \mathscr{L}v \equiv (D\partial_{\xi\xi} -c \partial_{\xi} + J(\xi))v \newline & v(0) = v(1) \end{align*}
where $\xi \in [0, 1]$, $v:[0, 1] \mapsto \mathbb{R}^n$ is a vector function, $D \in \mathbb{R}^{n \times n}$ is a diagonal matrix with non-negative entries (i.e. it's non necessarily invertible), $c \in \mathbb{R}$, and $J: [0, 1] \mapsto \mathbb{R}^{n \times n}$ is smooth on $[0, 1]$.
Is there an elegant way of solving for the spectrum? I'm able to estimate it using finite differencing, however for certain $J(\xi)$, the eigenvalues $\lambda$'s are either too inaccurate, or certain (important for my problem) eigenvalues are missing from the approximate spectrum.
Any references/solutions would be greatly appreciated.