Editing Apps/ImamuraHadleyCollaboration (section)

=====Trial Eigenfunction=====
Following [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; see our [[Apps/Blaes85SlimLimit#Oscillations_of_PP_Tori_in_the_Slim_Torus_Limit|accompanying discussion]] for details &#8212; If we assume that the eigenfunction is of the form,
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<math>~\delta W^{(0)}</math>
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<math>~=</math>
  </td>
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<math>~\biggl[\eta^{|k|} \exp(ik\theta) \biggr] \Upsilon(\eta) \, ,</math>
  </td>
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we find that the function, <math>~\Upsilon(\eta)</math>, must satisfy the one-dimensional, 2<sup>nd</sup>-order ODE,

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<math>~(1-\eta^2)
\frac{d^2\Upsilon}{d\eta^2} + \frac{1}{\eta}\biggl[(2|k|+1) - (2|k|+1+2n)\eta^2\biggr]\frac{d\Upsilon}{d\eta} 
</math>
  </td>
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<math>~=</math>
  </td>
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<math>~
- 2n \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon  \, ,
</math>
  </td>
</tr>
</table>
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which is a fairly standard looking eigenvalue problem.  If, furthermore, we adopt an independent variable given by the expression,
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<math>\Lambda \equiv 2\eta^2 - 1 \, ,</math>
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in which case, 
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<math>\frac{d}{d\eta} \rightarrow 2^{3/2} (\Lambda+1 )^{1/2}\frac{d}{d\Lambda}</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>\frac{d^2}{d\eta^2} \rightarrow 
\biggl[ 4\frac{d}{d\Lambda} + 8(\Lambda+1)\frac{d^2}{d\Lambda^2} \biggr] \, ,
</math>
</div>
<!-- DETAILS
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<math>\frac{d}{d\eta} \rightarrow \frac{d\Lambda}{d\eta} \cdot \frac{d}{d\Lambda} = 4\eta \frac{d}{d\Lambda} 
= 2^{3/2}(\Lambda+1)^{1/2}\frac{d}{d\Lambda}</math>
</div>
and, 
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<math>\frac{d^2}{d\eta^2} \rightarrow 2^{3/2}\biggl[\Lambda+1\biggr]^{1/2}\cdot 
\frac{d}{d\Lambda} \biggl\{ 2^{3/2}\biggl[\Lambda+1\biggr]^{1/2}\frac{d}{d\Lambda}\biggr\}
=2^{3}\biggl[\Lambda+1\biggr]^{1/2} \biggl\{ 
\frac{1}{2}\biggl[\Lambda+1\biggr]^{-1/2}\frac{d}{d\Lambda} + \biggl[\Lambda+1\biggr]^{1/2}\frac{d^2}{d\Lambda^2}\biggr\}
=
4\frac{d}{d\Lambda} + 8 (\Lambda+1 )\frac{d^2}{d\Lambda^2}
</math>
</div>
END DETAILS -->
the statement of the eigenvalue problem becomes,

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<math>~
- 2n \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1-\tfrac{1}{2}(\Lambda+1) \biggr]
\biggl[ 4\frac{d\Upsilon}{d\Lambda} + 8(\Lambda+1)\frac{d^2\Upsilon}{d\Lambda^2} \biggr]
+ 2^{1/2}(\Lambda+1)^{-1/2} \biggl[(2|k|+1) - (2|k|+1+2n)\tfrac{1}{2}(\Lambda+1)\biggr] 2^{3/2} (\Lambda+1 )^{1/2}\frac{d\Upsilon}{d\Lambda} 
</math>
  </td>
</tr>

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&nbsp;
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<math>~=</math>
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<math>~2(1-\Lambda)
\biggl[ \frac{d\Upsilon}{d\Lambda} + 2(\Lambda+1)\frac{d^2\Upsilon}{d\Lambda^2} \biggr]
+ 2\biggl[(4|k|+2) - (2|k|+1+2n)(\Lambda+1)\biggr] \frac{d\Upsilon}{d\Lambda} 
</math>
  </td>
</tr>
-->
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  <td align="right">
<math>~
\frac{n}{2} \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon
</math>
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<math>~=</math>
  </td>
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<math>~
(\Lambda^2-1)\frac{d^2\Upsilon}{d\Lambda^2} 
+ \biggl[(n - |k|-1 ) + (|k|+1+n)\Lambda\biggr] \frac{d\Upsilon}{d\Lambda} \, .
</math>
  </td>
</tr>
</table>
</div>

This definition of the eigenvalue problem is exactly of the [[#SingularSturmLiouville|singular Sturm-Liouville form, as described above]], from which we draw the following coefficient associations:


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<math>~(\alpha - \beta) \leftrightarrow (n-|k|-1)</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~(\alpha+\beta+2) \leftrightarrow (|k|+1+n) \, ,</math>
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that is,

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<math>~\alpha \leftrightarrow (n-1)</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~\beta\leftrightarrow |k|\, .</math>
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Hence the j<sup>th</sup> solution to this eigenvalue problem is,
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<math>~\Upsilon_j(\Lambda)</math>
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<math>~=</math>
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<math>~J_j^{n-1,|k|}(\Lambda) \, ,</math>
  </td>
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</table>
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with the associated eigenvalue,
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<math>~\frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2 -|k|\biggr] </math>
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<math>~=</math>
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<math>~j(j+n + |k|)</math>
  </td>
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<math>~\Rightarrow~~~ \biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2  </math>
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<math>~=</math>
  </td>
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<math>~\frac{1}{n}\biggl[2j^2+2jn + 2j|k| + n|k|\biggr] \, .</math>
  </td>
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</table>
</div>
Table 4 presents the polynomial expressions that are relevant to the first three (j = 0, 1, 2) eigenfunctions of "slender" PP-tori, along with the associated eigenvalues.

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<tr>
  <th align="center" colspan="3"><font size="+1">Table 4:  Example Jacobi Polynomials Relevant to "Slender" PP-Tori Eigenvectors</font></th>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~J_j^{n-1,|k|}(2\eta^2-1)</math></td>
  <td align="center"><math>~\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2</math></td>
</tr>
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<math>~0</math>
  </td>
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<math>~1</math>
  </td>
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<math>~0</math>
  </td>
</tr>
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<math>~1</math>
  </td>
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<math>~(n + 1 + |k|)\eta^2 - (1 + |k|)</math>
  </td>
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<math>~\tfrac{1}{n}[2+2n+(2+n)|k|]</math>
  </td>
</tr>
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  <td align="center">
<math>~2</math>
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<math>~
\tfrac{1}{2}[(6 + 5|k|+k^2) +n(5+2|k|) +n^2 ]\eta^4
- [4 + 4|k|+k^2 +n(2+|k|)]\eta^2 
+  \tfrac{1}{2} [2 + 3|k| + k^2] 
</math>
  </td>
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<math>~\tfrac{1}{n}[8+4n + (4+n)|k|] </math>
  </td>
</tr>
</table>
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