... problem

Strictly speaking, this is not a Sturm-Liouville problem because the dependence of the eigenvalue with the equilibrium model is rather complicated. However, the main properties of this oscillatory system are quite similar to the classical Sturm-Liouville problem. This has been put in evidence by Ledoux & Walraven (1958) for the high order gravity modes and the high order acoustic modes, where a second-order system of stellar oscillations has been obtained under the Cowling approximation (see Sect. 3.4).

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...$\Psi_r$

Hereafter we will refer to $\Psi$ as $\Psi_r$, if not stated otherwise.

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... conjugates

We observe that if $k_{\rm r}^2$ is real, then $\omega_{-}^2$ and $\omega_{+}^2$ are both real or complex conjugates. Consequently $f_{\rm p}$ and $f_{\rm g}$ are also complex conjugate functions.

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... invariant

In this context, invariant means that the different differential equations, that can be obtained by any type of transformation (of the independent and dependent variables or both), has $k_{\rm r}^2$ as a characteristic quantity, independent of the transformation considered.

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... simple

A star is considered simple in this context: when its propagation diagram can be obtained by a topological deformation of a simple polytrope such as polytrope $n_{\rm e}=3$.

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... phase

For some modes the influence of the boundary conditions is negligible. In those cases only the layers of the evanescent region, very near the turning point, contribute to the phase. Such cases, correspond to consider that the endpoints of the problem are: $-\infty$ and $+\infty$.

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... sequence

by "well-ordered sequence'' we mean that two modes of different frequencies never cross.

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