For the one-dimensional potential

$$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{n} \delta(x-n a)$$

solve the Schrödinger equation for negative energy and obtain an equation that determines possible energy bands. Show that the results agree with the tight-binding model in appropriate limits.

[It may be useful to note that $\left.V(x)=-\frac{\hbar^{2} \lambda}{m a} \sum_{n} e^{2 \pi i n x / a} .\right]$