CONTEXT
I'm working with a problem tangentially like the Kepler problem [1].
I'm trying to make sense of eccentricities $e$, where $e = -1$. There are some thoughts about such eccentricities [2]. For example, from what I gather from [2] the trajectories are scattered orbits resulting from repulsive central potentials.
Nonetheless, I'd like to work out the following boundary value problem. Starting with the solution of the Kepler problem in [1], I eventually arrive at $$ \frac{d^2 u}{d\theta^2} + u = -\frac{m}{L^2} \frac{d}{du} V\left(\frac 1 u\right). $$
I'd like to consider an infinite radial step function. By $R$, I denote a positive real number. $H(r-R)$, I denote a radial Heaviside step function I write as $$H(r-R) = \begin{cases} 0,& \textrm{for}~r < R;~\textrm{and} \\ 1,& \textrm{for}~r \geq R. \end{cases}$$ So, I write $$V({\bf{r}}) = \lim _{k\rightarrow \infty} k H(r-R). $$ Further \begin{align} \dfrac{dV({\bf{r}})}{dr} &= \lim _{k\rightarrow \infty} k \dfrac{dH(r-R)}{dr} \\ &= \lim _{k\rightarrow \infty} k \delta(r-R). \end{align} where $\delta$ is the Dirac delta function.
I now substitute $l$ in place of $\frac{mk}{L^2} $so collecting all the constants $$ \frac{d^2 u}{d\theta^2} + u = \lim_{l\rightarrow \infty} \,l\,\delta\left(\frac{1}{u} - R\right). $$
It seems to me that, owing to the nature of the Dirac delta function, I can arrive at the following differential equation. $$ \frac{d^2 u}{d\theta^2} + u = \delta\left(\frac{1}{u} - R\right). $$
MY SOLUTION
I'm not in a position to go any further at this moment. I reckon I would do something similar to [3], which is to take the Laplace transform, use partial-fraction expansion, and then take the inverse Laplace transform.
I don't trust my own work herein in the slightest.
QUESTIONS
What is the equation of motion for the radius $r$ of a particle of mass $m$ in a radial Heaviside (infinite) potential?
Constructive comments are welcome as are meaningful discussion.
Bibliography
[1] https://en.wikipedia.org/wiki/Kepler_problem
[2] http://bohr.physics.berkeley.edu/reinsch/phys105spr2014/lectures/Lecture10_Charman__notes_two_body_16.pdf, Holland, Peter R, "The quantum theory of motion : an account of the de Broglie-Bohm causal interpretation of quantum mechanics," Cambridge University Press, 1993.
[3] http://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx