Supersymmetric partner potentials arising from nodeless half bound states

A Half Bound State (HBS) ${\psi }_{\ast }(x)$ can be defined as a single, conditional, zero-energy, continuous solution of the one dimensional Schr{"o}dinger equation for a scattering potential well $V(x)$ ($s\mathrm{.}tV(\pm \mathrm{\infty })=0$). The non-normalizable and solitary HBS of a potential satisfies Neumann boundary condition that ${\psi }_{\ast }^{\mathrm{\prime }}(\pm \mathrm{\infty })=0$ and it can have $n$ (= 0,1,2,…) number of nodes indicating $n$ number of bound states in $V(x)$ below $E=0$. Here we show that starting with a nodeless HBS, we can construct a (supersymmetric) pair of finite potentials (well, double wells, well-barrier): $V_{\pm }(x)$ having no bound state and they enclose positive area on $x$-axis. On the contrary their negative counterparts $(-c{V}_{\pm }(x)),c>0$ do have at least one bound state for any arbitrary positive value of $c$. Furthermore, $c V_{\pm}(x),~ c >0$ which binds positive area on x-axis in conformity with Simon’s theorem can have at least one bound state only conditionally for instance when $c>1$ or $c>>1$.