Calculations with supersymmetric potentials

We present numerical solutions for two problems in one-dimensional supersymmetric quantum mechanics. The first case deals with the superpotential W(x)=$x^3$, which is about the simplest case with no known analytical solution. We compute the eigenvalues for the states above the supersymmetric E=0 state of the corresponding ordinary potential $V_{\mathrm{-}}$(x)=$x^6$-α$x^2$ for α=3; those states are also the bound states of the partner convex potential $V_{+}$(x)=$x^6$+3$x^2$. We discuss in which sense the double-well potential $V_{\mathrm{-}}$(x) is a critical potential; increasing α≥3 we obtain a well-defined grouping of positive- and negative-parity levels, corresponding to an increasing barrier between the two wells. For the second case we start with a ground-state wave function of Lorentzian shape, namely, $u_0$=(1+$x^2$${)}^{\mathrm{-}1}$, which gives rise to the superpotential W(x)=2x/(1+$x^2$) and the ordinary potentials $V_{+}$(x)=2/(1+$x^2$) and $V_{\mathrm{-}}$(x) =(6$x^2$-2)/(1+$x^2$${)}^2$.

There is only a bound state at zero energy for $V_{\mathrm{-}}$(x), the partner potential $V_{+}$(x) being a repulsive barrier. The potentials $V_{+}$(x) and $V_{\mathrm{-}}$(x) decay like l(l+1)/$x^2$ for l=1 and 2, respectively, at large distances (‘‘intermediate range’’ potentials) and this produces in particular anomalous phase shifts ${\delta }_{\mathrm{even}}$ and ${\delta }_{\mathrm{odd}}$ at very low energies. We calculate these phase shifts for $V_{+}$, those of the partner potential $V_{\mathrm{-}}$(x) being fixed by supersymmetry. We also show the peculiar character of these potentials by changing the parameters. In particular for the ‘‘craterlike’’ potential $V_{\mathrm{-}}$(x) the replacement of 6$x^2$-2 by 6$x^2$-1 gives rise to a distinctive resonance in the even phase shift. Some considerations regarding $x^{\mathrm{-}2}$ potentials, factorization, and scale invariance are relegated to an appendix.