1
$\begingroup$

I have the following system of coupled differential equations (the Bogoliubov-de Gennes equations for a certain geometry):

$$ \begin{cases} -\frac{1}{2}u''(r) + \big[\frac{1}{2r^2} + f_1(r)\big]u(r) + f_2(r)v(r)=\omega u(r)\\ -\frac{1}{2}v''(r) + \big[\frac{1}{2r^2} + f_1(r)\big]v(r) + f_2(r)u(r)=-\omega v(r), \end{cases} $$ where $f_i(r)$ are known functions which go to zero for large $r$ and are finite for small $r$. This is in fact an eigenvalue problem, because I need to find the value of $\omega$ for which this system has (normalizable) solution.

Now I would like to solve this system numerically: hence find the value of $\omega$ such that the system has a solution and then also the functions $u(r),v(r)$. I am familiar with numerical recipes to solve coupled differential equations, but now that this is an eigenvalue problem as well, I am at a loss. Could anyone give me some pointers on how to tackle such a problem? Some references where this is described would be appreciated as well!

$\endgroup$
  • 2
    $\begingroup$ For $r\to\infty$ the system becomes $u''(r)=-2\omega u(r)$ and $v''(r)=2\omega v(r)$. So either of the solutions will become unstable, or would you still consider this a (normalizable) solution? $\endgroup$ – fibonatic May 12 '16 at 16:13

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.