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Q:
Nonoverlapping groups of stationary points
I read something in the book “Mathematical and computational problems from applied mathematics” that I’m not able to understand. The title of the problem is:
If we have $\frac{d^2}{dx^2}-V(x)$ we know that there exist infinitely many stationary points (or is it only linearly independent stationary points?). However, we can exclude these stationary points from being the only linearly independent stationary points.
Here, a stationary point is a point $x_{0}$ where the function is stationary ($f'(x_{0})=0$). I want to know why we can exclude the stationary points from being the only linearly independent stationary points.
A:
Note that if there are linearly independent stationary points for the differential operator $D$, then there exist linearly independent solutions of the form
$$y_1(x),\;y_2(x),\ldots,\;y_n(x)$$
where each $y_i(x)$ is a linear combination of the $y_i(x_0)$ for the stationary points $x_0$ of $D$. But I think this result is well-known anyway.
Now it’s possible that the $x_0$’s form nonlinear relations (i.e. $x_0
e x_0$). If that happens we can take the $\cos(2\pi x)$ and $\sin(
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