Negative-energy band of an attractive delta comb

With well strength $\lambda$ and spacing $a$, negative-energy Bloch states obey $\tanh y\le 2y/(\lambda a)\le \coth y$, where $y=\kappa a/2$. There is one negative band: it reaches zero when $\lambda a\le 2$ and is separated from zero when $\lambda a>2$.