Can increasing the potential width α decrease the coupling constant λ?

[intervoxel]

Hi,

I'm studying a theorem that is valid for a small coupling constant λ in the Schrödinger operator below

$-\frac{d^2}{dx^2}+\lambda V$

The potential has a parameter α which defines its width.

My question is: Starting with λ=1 and α=1, can I claim that if I increase the value of α (α >> 1) is equivalent to decrease λ (λ << 1)?

 

[Ken G]

Actually, I would have said that you can prove that an increase in a is equivalent to an increase in lambda. The way to do it is define a new energy variable, call it F = a² E, and a new position variable, call it y = x/a. Then in terms of the new energy F, the Shroedinger operator is
$-\frac{d^2}{dy^2}+\lambda a^2 V(y)$,
and we see that the equivalence class is defined by the value of $\lambda a^2 V(x=a)$, which characterizes the strength of the potential term relative to the kinetic energy term.

 

[intervoxel]

Thank you for the answer