Long distance limit BdG-equations
Consider the dimensionless Bogoliubov-de Gennes equations for the excitations in a BEC: $$ \begin{cases} \big(-\frac{1}{2}\nabla^2+2gn({\bf r}) - \mu -\omega\big) u({\bf r})-n({\bf r})g v({\bf r}) = 0\ \big(-\frac{1}{2}\nabla^2+2gn({\bf r}) - \mu +\omega\big) v({\bf r})-n({\bf r})g u({\bf r}) = 0, \end{cases} $$ where $g$ is the value of the contact interaction between the atoms, $n({\bf r})$ is the condensate density, $\mu$ is the chemical potential and $\omega$ is the eigenvalue of the system which should describe the frequency of the oscillating excitation wave functions $u({\bf r})$ and $v({\bf r})$.
First suppose we consider a homogeneous system. Then the condensate density is a constant: $n({\bf r}) = n_0$ (and from the Gross-Pitaevski equation one has $\mu = n_0 g$) . Going to Fourier space $u({\bf r}) \propto e^{i {\bf k\cdot r}}$ (and similarly for $v({\bf r})$, one easily solves for $\omega$ by demanding that the determinant of the system vanishes and one finds $4\omega^2 = k^2(k^2 + 4n_0g) $.
Now suppose we consider a `realistic' radially symmetric system where $\lim_{r\rightarrow\infty}n(r) = 0$, i.e., the condensate density is only finite in a finite region of the space. In this case the coupling between the two equations vanishes and one has $u''(r) = -2\omega u(r)$ and $v''(r) = 2\omega v(r)$. 649
Making a similar ansatz for u and v, you find $k^2 = \pm 2\omega$. Hence, one solution has an imaginary part and is unstable.
I do not know how to interpret this result. I have the feeling that something is wrong in my reasoning, because I would expect similar behavior for large $r$ in the two cases.
Note: the reason I am asking this is because I want to numerically solve a somewhat more complicated form of the BdG-equations for a radially symmetric system. I have already obtained a numerical solution for $n(r)$ from the Gross-Pitaevski equation, but in order to solve for $u(r)$ and $v(r)$ I need proper boundary conditions on $r=0$ and $r\rightarrow \infty$. That is why I was thinking about this limit. 521