Eigenvalue problem with operators as matrix elements

[wil3]

Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem:

$$\begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu + u(x) \end{bmatrix} \begin{bmatrix} a(x)\\ b(x) \end{bmatrix} = \omega(x) \begin{bmatrix} a(x)\\ b(x) \end{bmatrix}$$

where all x dependencies have been declared. I know the definition of the function $u(x)$, but I need to solve for the eigenfrequency and eigenvectors.

 

[Mandelbroth]

Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem:

$$\begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu + u(x) \end{bmatrix} \begin{bmatrix} a(x)\\ b(x) \end{bmatrix} = \omega(x) \begin{bmatrix} a(x)\\ b(x) \end{bmatrix}$$

where all x dependencies have been declared. I know the definition of the function $u(x)$, but I need to solve for the eigenfrequency and eigenvectors.

I assume that you mean to find the function omega of x? You need to specify, because it's unclear what you want. Is your use of del indicating a partial derivative or a directional derivative. Be specific.

 

[wil3]

Typo in question: $\omega$ should not depend on x. I want to solve for $\omega$, hence why I am calling this an eigenvalue problem. I'm not sure if getting w will also give a and b, like in a standard linear system.

$\partial_x$ indicates a partial derivative, which is standard notation in physics for this sort of problem. But the problem would be identical if you wanted to interpret that as a directional derivative (both because the problem is one dimensional and because the subscript x would suggest a derivative in the x direction).