- #1
robousy
- 332
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Hey, this is rather involved but I hope someone can help me out.
I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :
g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0
to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:
\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )
I'll show you my attempts and if anyone has the patience can maybe help me.
First express field \Phi as:
\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)
Plug this into the original differential equation
(g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0
Divide by \phi\chi to obtain
\frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+ <br /> <br /> <br /> \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0
As the first term only depends on x_\mu and the second on y, each term in the equation must be a constant, so for the \chi(y) term we can write:
\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2
where n^2 is some constant.
Tidying this up a bit we get:
e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0
Make this easier to compare to Bessels equation by writing
e^{2ky} \rightarrow x^2, \chi\rightarrow y
My final equation now looks like:
x^2y''-4kx^2y'-n^2y=0
Whereas Bessels equation is
x^2y''+xy'+(x^2-n^2)y=0
So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??
Thank you so much.
I am reading http://arxiv.org/abs/0704.3626 ( the casimir force in randall sundrum models) and am trying to get from equation 2.1 :
g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0
to equation 2.2 ...'The general solution for the non-zero modes can be eppressed in terms of Bessel functions of the 1st and 2nd kind as:
\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )
I'll show you my attempts and if anyone has the patience can maybe help me.
First express field \Phi as:
\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)
Plug this into the original differential equation
(g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0
Divide by \phi\chi to obtain
\frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+ <br /> <br /> <br /> \frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0
As the first term only depends on x_\mu and the second on y, each term in the equation must be a constant, so for the \chi(y) term we can write:
\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2
where n^2 is some constant.
Tidying this up a bit we get:
e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0
Make this easier to compare to Bessels equation by writing
e^{2ky} \rightarrow x^2, \chi\rightarrow y
My final equation now looks like:
x^2y''-4kx^2y'-n^2y=0
Whereas Bessels equation is
x^2y''+xy'+(x^2-n^2)y=0
So this is my problem, I obtain an expression that is not quite bessels equation. Can anyone see the error of my logic??
Thank you so much.
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