Eigenvalue problem with operators as matrix elements

  • Thread starter wil3
  • Start date
  • #1
179
1
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:

[tex]

\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}
[/tex]

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

Answers and Replies

  • #2
611
24
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:

[tex]

\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}
[/tex]

where all x dependencies have been declared. I know the definition of the function [itex] u(x)[/itex], 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.
 
  • #3
179
1
Typo in question: [itex] \omega [/itex] should not depend on x. I want to solve for [itex] \omega [/itex], 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.

[itex] \partial_x [/itex] 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).
 

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