Harmonic approximation near a potential minimum
= Harmonic approximation near a potential minimum
Near a nondegenerate minimum $x_0$, a smooth potential satisfies
$$
V(x)=V(x_0)+\frac12V''(x_0)(x-x_0)^2+\cdots.
$$
The localized levels are therefore approximately those of a harmonic oscillator of frequency $\sqrt{V''(x_0)/m}$; tunnelling broadens each level into a narrow band.