I have a Schrodinger equation
$$-\frac{1}{m}\frac{\partial^{2} \psi}{\partial^{2} x} + V(x) \psi = E \psi$$
where,
$$V(x) = \frac{1}{x^{6} - 1} $$
I want to analyze this differential equation near the singular point x = 1. Actually, I need to find the discontinuity of the derivative of the wave function across the singularity.
It is given that the above differntial equation can be approximated to the following form,
$$\frac{1}{U^{2}}\frac{\partial^{2} \psi}{\partial^{2} x} = \frac{1}{x-1}\psi$$
where,
$$U \approx \frac{1}{\sqrt{6}}(const.)$$
If shows that potential function follows $1/(x-1)$ near the singularity.
What are the approximation methods to come up with this? Is there any series expansion I can do for this differential equation? (I guess there is a series expansion as there is $1/\sqrt{6}$ term in $U$).
$$-\frac{1}{m}\frac{\partial^{2} \psi}{\partial^{2} x} = -\frac{1}{x^{6} - 1} \psi + E \psi$$ Change of variable $x=1+\epsilon \quad\to\quad \frac{1}{x^{6} - 1}=\frac{1}{(1+\epsilon)^{6} - 1} \simeq \frac{1}{6\epsilon}$ $$-\frac{1}{m}\frac{\partial^{2} \psi}{\partial^{2} \epsilon} \simeq -\frac{1}{6\epsilon} \psi + E \psi$$ $\frac{1}{6\epsilon}$ is large compared to $E \quad\to\quad -\frac{1}{m}\frac{\partial^{2} \psi}{\partial^{2} \epsilon} \simeq -\frac{1}{6\epsilon} \psi = -\frac{1}{6(x-1)}\psi$. $$\frac{1}{U^{2}}\frac{\partial^{2} \psi}{\partial^{2} x} \simeq \frac{1}{x-1} \psi \qquad \text{where}\quad U=\sqrt{\frac{m}{6}}$$