= Supersymmetric factorization and zero-mode normalizability
{title2=$A=\partial_x+w(x),\quad H_-=A^\dagger A/2,\quad H_+=AA^\dagger/2$}
The candidate zero modes are $e^{-\int^xw}$ in the minus sector and $e^{+\int^xw}$ in the plus sector; this follows from the positive squared-norm expressions for their energy. Their membership in the <Hilbert space> determines whether <supersymmetry> is unbroken. For $w=x$ only the first is normalizable. For $w=x^2+a$, $a>0$, neither is normalizable, while both partner potentials grow quartically and have discrete spectra. Thus their strictly positive lowest energy is paired and <supersymmetry> is broken, with zero <Witten index>.
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