Supersymmetric partner potentials arising from nodeless half bound states

TL;DR

This paper demonstrates how nodeless half bound states can be used to generate supersymmetric partner potentials that have no bound states, while their scaled negative counterparts do have bound states, revealing new insights into potential construction.

Contribution

It introduces a method to construct supersymmetric partner potentials from nodeless half bound states, highlighting their bound state properties and conditions for bound state existence.

Findings

Nodeless HBS can generate partner potentials with no bound states.

Negative scaled potentials of these partners always have at least one bound state.

Scaling factors influence the bound state conditions of the potentials.

Abstract

A Half Bound State (HBS) $\psi_*(x)$ can be defined as a single, conditional, zero-energy, continuous solution of the one dimensional Schr{\"o}dinger equation for a scattering potential well $V(x)$ ($s.t ~ V(\pm \infty)=0$). The non-normalizable and solitary HBS of a potential satisfies Neumann boundary condition that $\psi'_*(\pm \infty)=0$ 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 $(-cV_{\pm}(x)),c>0$ 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 $c>>1$.