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:

<br /> <br /> \begin{bmatrix}<br /> \partial_x^2 + \mu + u(x) &amp; u(x)^2 \\<br /> \bar{u(x)}^2 &amp; \partial_x^2 + \mu + u(x)<br /> \end{bmatrix}<br /> \begin{bmatrix}<br /> a(x)\\<br /> b(x)<br /> \end{bmatrix}<br /> =<br /> \omega(x)<br /> \begin{bmatrix}<br /> a(x)\\<br /> b(x)<br /> \end{bmatrix}<br />

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:

<br /> <br /> \begin{bmatrix}<br /> \partial_x^2 + \mu + u(x) &amp; u(x)^2 \\<br /> \bar{u(x)}^2 &amp; \partial_x^2 + \mu + u(x)<br /> \end{bmatrix}<br /> \begin{bmatrix}<br /> a(x)\\<br /> b(x)<br /> \end{bmatrix}<br /> =<br /> \omega(x)<br /> \begin{bmatrix}<br /> a(x)\\<br /> b(x)<br /> \end{bmatrix}<br />

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).