Krdytkyu

In the excellent book "Numerical Computing with MATLAB" by Cleve B. Moler (SIAM 2004), [Moler is the "father" of Matlab], one finds, on pages 298-299, the following graphical representation (I have reconstructed it making minor changes : Matlab program below) with the following explanations (personal adaptation of the text):

Let $n$ be a fixed integer $>1$. Let $tin (0,1)$, and let $A_t$ be the matrix with entries

$$A_{t,i,j}=dfrac{1}{i-j+t}$$

Fig. 1 gathers the spectra of matrices $A_t$, for $n=11$, and $0.1 leq t leq 0.9$ with values of $t$ increased by steps $Delta t=0.005$.

Figure 1. Interpretation : the 11 rightmost points correspond to the spectrum of matrix $A_t$ for $t=0.1$.

The similarity with particles' scattering by a nucleus situated at the origin and their hyperbolic trajectories is striking ; see for example this

Thus, my question is this one : can somebody indicate a model of particle scattering using this type of matrices ?

My thoughts : could these (Toeplitz) matrices be associated with a quadratic form defined by its moment's matrix $int_C z^{i}bar{z}^{j}z^{t-1}dz$ but for which curve $C$, $z^{t-1}$ playing the rôle of a weight function, and what can be done afterwards ? I have also attempted to prove without success that the points are situated on curves likely to be hyperbolas...

Matlab program for Fig. 1 :

 clear all;close all;hold on;axis equal

 n=11;

 [I,J]=meshgrid(1:n,1:n);

 E=;

 for t=0.1:0.005:0.9

   A=1./(I-J+t);

   E=[E,eig(A)];

 end;

 plot(E,'.k');

Addendum (November 22, 2018) : Here is a second figure that provides a "bigger" view, with $n=20$ and $0.005 leq t leq 0.995$. The eigenvalues corresponding

• to $t=0.005$ are grouped into the rightmost big red blob on the right,




• to $t=0.995$ are blue filled dots (quasi-circle with radius $approx 110$).

_Figure 2 : [enlarged version of Fig. 1 ; case $n=15$] Everything happens as if a planar wave enters at $t=0$ from the right, is slowed down by the repulsive effect of the nucleus, then scattered by it as a circular wave..._

Fig. 3 : Two cases are gathered here : $n=51$ (empty circles) and $n=100$ (stars), both for $t=0.995$. One can note that the eigenvalues are very close from being $(n+1)$-th roots of unity except one. For this value of $t$, the radii are given by experimental formula : $R_n=200(1-12.4/n+212/n^2-3110/n^3)$.

Following the precious remarks of @AmbretteOrrisey who gave the following polar representation for the trajectory of particles : [citation follows ; more details can be found in his/her answer]
"The polar equation of the trajectory of a particle being deflected by a point charge is

$$r=frac{2a^2}{sqrt{b^2+4a^2}sintheta -b}$$

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."

Figure 4 displays a reconstruction result that takes into account the fact that (n+1)th roots of unity give asymptotic directions.

Figure 4 : Case $n=201$ : An approximate reconstruction of 21 trajectories among the $n$ trajectories (Matlab program below ; polar equation can be seen on line 5). Note the "ad hoc" coefficient $3.0592$...

 n=201;

 for k=1:20:n

    d=pi*k/(2*(n+1));c=cos(d);

    t=-d+0.01:0.01:d-0.01;

    r=3.0592*(1-c)*exp(i*(t-d))./(cos(t)-c);

    plot(r);plot(conj(r));

 end;

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

$$r=frac{2a^2}{sqrt{b^2+4a^2}sintheta -b}$$

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!