= Band edges of a repulsive delta-comb potential
For $V(x)=\alpha\sum_j\delta(x-ja)$ with $\alpha>0$, put $P=m\alpha a/\hbar^2$ and $x=ka$. The <Floquet discriminant of the delta-comb Kronig-Penney model> is $f(x)=\cos x+P\sin x/x$. Extended <Bloch states> require $|f(x)|\leq1$. Since $f(0)=1+P>1$, there is a positive lower band edge; a repulsive potential does not support negative-energy states in this energy convention.
The $j$th positive-energy band ends at $x=j\pi$. Its lower edge lies in $((j-1)\pi,j\pi)$ and solves $f(x)=(-1)^{j-1}$. A finite periodic ring selects only the discrete allowed values of <crystal momentum> within these bands. Nonzero <band gaps> do not by themselves imply an insulator: the <band filling> must also place the <Fermi energy> in a gap.
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