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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.

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  • $\begingroup$ The multiplicity is $1$ or $2$, but you have to analyze this case-by-case, there's nothing you can say in this generality. $\endgroup$ – Christian Remling Aug 7 '15 at 17:58
  • $\begingroup$ Thanks for replying. I understand the multiplicity is either 1 or 2, but how does one go about actually determining whether it is 1 or 2? Do you know of any non-trivial examples where they determine the multiplicity? If so, what do they do? $\endgroup$ – k3thomps Aug 7 '15 at 18:13
  • $\begingroup$ What do you know about the Jordan structure of $V_\infty$ in your problem? $\endgroup$ – Igor Khavkine Aug 7 '15 at 21:32
  • $\begingroup$ Suppose that $V_\infty$ has 4 distinct eigenvalues: $\pm \mu \neq 0$ and $\pm \nu \neq 0$. $\endgroup$ – k3thomps Aug 7 '15 at 22:27
  • $\begingroup$ @k3thomps: It all depends on the details (for example think of the case $V$ diagonal), your question is really much too general in this form to admit specific answers (even in the case $V_{\infty}=0$). $\endgroup$ – Christian Remling Aug 8 '15 at 0:12

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