Threshold energy dependence as a function of potential strength and the nonexistence of bound states

The difficulty in attempting to prove that a given set of particles cannot form a bound state is the absence of a margin of error; the possibility of a bound state of arbitrarily small binding energy must be ruled out. At the sacrifice of rigor, one can hope to bypass the difficulty by studying the ground-state energy $E\left(\lambda \right)$ associated with $H\left(\lambda \right)\equiv H_{\mathrm{true}}+{\lambda }_v$, where $H_{\mathrm{true}}$ is the true Hamiltonian, $v$ is an artificial attractive potential, and $\lambda >0\mathrm{}$. $E\left(\lambda \right)$ can be estimated via a Rayleigh-Ritz calculation. If $H_{\mathrm{true}}$ falls just short of being able to support a bound state, $H\left(\lambda \right)$ for $\lambda$ "not too small" will support a bound state of some significant binding. A margin of error is thereby created; the inability to find a bound state for $\lambda$ "not too small" suggests not only that $H\left(\lambda \right)$ can support at best a weakly bound state but that $H_{\mathrm{true}}$ cannot support a bound state at all. To give the argument real substance, we study $E\left(\lambda \right)$ in the neighborhood of $\lambda ={\lambda }_0$, the (unknown) smallest value of $\lambda$ for which $H\left(\lambda \right)$ can support a bound state. A comparison of $E\left(\lambda \right)$ determined numerically with the form of $E\left(\lambda \right)$ obtained with the use of a crude bound-state wave function in the Feynman theorem gives a rough self-consistency check. One thereby obtains a believable lower bound on the energy of a possible bound state of $H_{\mathrm{true}}$ or a believable argument that no such bound state exists. The method is applied to the triplet state of ${\mathrm{H}}^{-}$.