Editing SSC/Perturbations (section)

===What Makes This an Eigenvalue Problem?===
Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has fostered the following appreciation.  Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center &#8212; see, for example, the expansion that is appropriate for [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement|polytropic]] or for [[Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement|isothermal]] spheres &#8212; the [[#2ndOrderODE|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (or at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>.  As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition.  This is illustrated, for example, by Figure 2 in our [[SSC/Stability/n3PolytropeLAWE#Figure2|discussion of oscillating n = 3 polytropes]].  Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.

The continuous spectrum is transformed into a discrete spectrum of oscillation ''modes'' when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' matches a specific surface boundary condition.  For example, as [[#Implications|described above]], if the aim is to ensure that there are no pressure fluctuations at the surface throughout an oscillation, then the only physically relevant displacement functions are the ones whose logarithmic radial derivative at the surface is negative three.  And each of these relevant displacement functions &#8212; now, an ''eigenfunction'' &#8212; will be associated with a ''discrete'' value the oscillation frequency &#8212; the associated ''eigenfrequency.''  Understanding this interplay between a derived solution to the LAWE and the imposed boundary condition helps clarify why this is an eigenvalue problem.