= Bound-state count of an attractive symmetric double-delta potential
{title2=$g_c=-\hbar^2/(2ma)$}
For $V(x)=g[\delta(x-a)+\delta(x+a)]$ with $a>0$ and $\nu=-mg/\hbar^2>0$, an even <bound state> obeys $\kappa=\nu(1+e^{-2a\kappa})$, and an odd <bound state> obeys $\kappa=\nu(1-e^{-2a\kappa})$, with <energy> $E=-\hbar^2\kappa^2/(2m)$ and $\kappa>0$. The even right-hand side is strictly decreasing from $2\nu$, so it meets $\kappa$ once. For the odd equation, $h(\kappa)=\nu(1-e^{-2a\kappa})-\kappa$ is strictly <concave>, starts at zero with slope $2a\nu-1$, and tends to negative infinity. It has one positive <equation root> precisely when $2a\nu>1$. Thus there is one <bound state> for $-\hbar^2/(2ma)\leq g<0$, and two for $g<-\hbar^2/(2ma)$. For $g\geq0$ positivity of the <Hamiltonian operator> excludes negative <energy>, and square integrability excludes nonnegative-<energy> exterior solutions. At $g=g_c$ the odd zero-<energy> solution is bounded but not square integrable: it is a <zero-energy scattering resonance>, not a second <bound state>.
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