Consider the following operator acting on $H^1(\mathbb{R})$ $$ \mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \left(\begin{array}{c} \phi \\ \psi \end{array}\right) $$ where $V(x)\rightarrow V_\infty$ ($V_\infty$ a constant) exponentially fast, $V(x)$ is a real 2x2 matrix for all $x$, and $V(x)$ is smooth.

If I know that $\lambda$ is an eigenvalue of $\mathcal{L}$, how does one go about determining the geometric multiplicity of $\lambda$?

References would be much appreciated.