Notes
2026-10-30
So I am starting to further the the study of change of the topology and geometry of the attractive case of the model, I will describe this in ‘Renormalisation’
So I am considering the case that the displayed, relative volume should be a constant value \(c\). I am looking for \(\epsilon\), to find how the width should be displayed.
\[\frac{\int_{-\epsilon}^{\epsilon} x(p)dp}{\int_{-\infty}^{\infty} x(p)dp}=c\]
The Lorenzian function \(x(p)\propto \tfrac{1}{2m_2E-p^2}\) is obtained by Looking at the boundary of the surface of section. If the boundary is considered, we obtain
\[x(p)= \frac{2m_2 \mathcal{C}}{L} \frac{1}{2m_2E-p^2}-\partial\mathcal{B}_{1\text{R}}\]
The integral of the lorenzian term is
\[\int_{-\infty}^{\infty}\frac{dp}{p^2+\kappa^2} = \frac{\pi}{\kappa}, \qquad \int_{-\varepsilon}^{\varepsilon}\frac{dp}{p^2+\kappa^2} = \frac{2}{\kappa}\arctan\frac{\varepsilon}{\kappa}\] \[\frac{2}{\pi}\arctan\frac{\varepsilon}{\kappa} = c \quad\Longrightarrow\quad \boxed{\varepsilon = \sqrt{-2mE}\;\tan\!\left(\frac{\pi c}{2}\right)}, \qquad 0<c<1.\]
As the boundary only funcitons as an offset, and only cases \(E<0\) shall be considered, this consideres the problem to be solved.
When discussing with Awais, he suggested, attractive and repulsive should behave the same, with the difference of changing the order of the boxes?