Finding Band Gaps for Dirac Comb Potential

[MisterX]

Homework Statement

Find band gaps for Dirac Comb potential
$$V = \sum_n aV_0(x-na) $$

Homework Equations

Bloch Theorem
$$\psi(x+a) = e^{ika}\psi(x)$$

The Attempt at a Solution

I can solve exactly up to
$$\cos(k a) = \cos(\kappa a) + \frac{2ma^2V_0}{\hbar^2}\frac{\sin(\kappa a)}{\kappa a} = f(\kappa a)$$
Where $E = \frac{\hbar^2 \kappa^2}{2m}$ and $k$ is the Bloch wavenumber. The goal is to solve for the gaps between the energy bands. Due to the transcendental nature we don't expect an exact solution, but can we get an approximate solution for the first few gaps without resorting to numerical methods?
A gap will be either

• start at $\kappa_1$: $f(\kappa_1 a) =1$ and $f'(\kappa_2 a) > 0$ and end at the next $\kappa$ where $f(\kappa_2 a) =1$ and $f'(\kappa_2 a) < 0$
• start at $\kappa_1$: $f(\kappa_1 a) =-1$ and $f'(\kappa_2 a) < 0$ and end at the next $\kappa$ where $f(\kappa_2 a) =-1$ and $f'(\kappa_2 a) > 0$

So maybe there is a way to approximate $f(\kappa a)$ for example that will enable estimating the size of the first couple band gaps? I tried doing a power series expansion to forth order in $\kappa$ around $\kappa=0$ but even that seemed too complicated.

 

[MisterX]

I came up with a way to estimate the bands. I noticed that $f(\kappa a) = n\pi$ is always an exact solution for the start of a forbidden band. I then made an expansion about those points to estimate where the forbidden band ended.