Connection between the spectra of a family of matrices and a modelization of particles' scattering?

I'm don't think the curves would be hyoerbolæ anyway if it were a model for classical scattering.

The polar equation of the particle being deflected by a point charge is

$$r=\frac{2a^2}{\sqrt{b^2+4a^2}\sin\theta -b}$$

where $a$ is the impact parameter which is the closest approach to the nucleus were the path undeviated; $b$ is the closest approach of a head-on ( $a=0$) particle with repulsion operating, and $\theta$ is the angle between the radius vector of the particle (nucleus at origin) and the line joining the nucleus to point of closest approach with charge in place (not the axis of approach - ie the line through the nucleus that particle is moving parallel to & distance $a$ from when it is yet infinite distance away).

Is that a polar coördinate representation of a hyperbola? I suppose it ought to be, as it is an inverse-square force!

Anyway - it's likely I'll look into it will-I-nill-I, now.

Oh ... the diæræsis in "coördinate" & the ligature in "diæræsis" ... I'm hoping you don't mind that archaïck stuff too much - I love it ... but many would probably go ballistic if I used that kind of idiom more generally on here. They even complain about my italics & emphasis! The diæræsis over the second vowel in a conjunction of vowels was a device used to denote that the conjunction does not form a diphthong. It never was actually a very widely-used convention!