# Notation in One-dimensional Schrodinger equation [closed]

I am reading a book where it write the one-dimensional stationary Schrodinger equation as $$[-\frac{\hbar^2}{2m}\frac{d^2}{dζ^2}-Γ{\rm sech}^2(bζ)]ψ(ζ)=Eψ(ζ).$$ It is known that the equation is usually written as $$[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)]ψ(x)=Eψ(x).$$ I speculate that the author may have $$bζ$$ instead of $$x$$. But I am confused with the component of potential energy $$-Γ{\rm sech}^2(bζ)$$. My guess is that we may have $$2mΓb^2(2π)^2/h^2$$ instead of the $$V$$. Isn't it?

It seems most likely that the potential energy function given in your equation was a deliberate choice, as it is a famous potential: the Pöschl-Teller potential. It can be solved exactly, giving a finite number of bound states. Also, for suitable values of the strength parameter $$\Gamma$$, it is reflectionless, meaning that incident waves transmit perfectly through it.
The value of $$b$$ establishes a length scale for the potential, and your speculation about the combination of $$\Gamma$$ and factors of $$\hbar^2/2m$$ (where $$\hbar=h/2\pi$$) probably relates to making a convenient choice of energy scale so as to reduce the equation to a dimensionless form (see the pages I referenced, as well as this one).