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robousy

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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 :

[tex]g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0[/tex]

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:

[tex]\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )[/tex]

I'll show you my attempts and if anyone has the patience can maybe help me.

First express field [tex]\Phi[/tex] as:

[tex]\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)[/tex]

Plug this into the original differential equation

[tex](g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0[/tex]

Divide by [tex]\phi\chi[/tex] to obtain

[tex] \frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+

\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0[/tex]

As the first term only depends on [tex]x_\mu[/tex] and the second on y, each term in the equation must be a constant, so for the [tex]\chi(y)[/tex] term we can write:

[tex]\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2[/tex]

where [tex]n^2[/tex] is some constant.

Tidying this up a bit we get:

[tex] e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0[/tex]

Make this easier to compare to Bessels equation by writing

[tex]e^{2ky} \rightarrow x^2, \chi\rightarrow y[/tex]

My final equation now looks like:

[tex]x^2y''-4kx^2y'-n^2y=0[/tex]

Whereas Bessels equation is

[tex]x^2y''+xy'+(x^2-n^2)y=0[/tex]

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 :

[tex]g^{\mu\nu}\partial_\mu\partial_\nu\Phi+e^{2ky}\partial_y(e^{-4ky}\partial_y\Phi)=0[/tex]

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:

[tex]\chi(y)=e^{2ky}(a_1 J_2(\frac{m_Ne^{ky}}{k}) +a_2Y_2(\frac{m_Ne^{ky}}{k}) )[/tex]

I'll show you my attempts and if anyone has the patience can maybe help me.

First express field [tex]\Phi[/tex] as:

[tex]\Phi(x_\mu,y)=\phi(x_\mu)\chi(y)[/tex]

Plug this into the original differential equation

[tex](g^{\mu\nu}\partial_\mu\partial_\nu\phi)\chi+e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)\phi=0[/tex]

Divide by [tex]\phi\chi[/tex] to obtain

[tex] \frac{ (g^{\mu\nu}\partial_\mu\partial_\nu\phi) }{ \phi}+

\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=0[/tex]

As the first term only depends on [tex]x_\mu[/tex] and the second on y, each term in the equation must be a constant, so for the [tex]\chi(y)[/tex] term we can write:

[tex]\frac{e^{2ky}(-4ke^{-4ky}\partial_y\chi +e^{-4ky}\partial^2_y\chi)}{\chi}=n^2[/tex]

where [tex]n^2[/tex] is some constant.

Tidying this up a bit we get:

[tex] e^{-2ky}\partial^2_y\chi} -4ke^{-2ky}\partial_y\chi-n^2\chi=0[/tex]

Make this easier to compare to Bessels equation by writing

[tex]e^{2ky} \rightarrow x^2, \chi\rightarrow y[/tex]

My final equation now looks like:

[tex]x^2y''-4kx^2y'-n^2y=0[/tex]

Whereas Bessels equation is

[tex]x^2y''+xy'+(x^2-n^2)y=0[/tex]

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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