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

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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 $E(\ensuremath{\lambda})$ associated with $H(\ensuremath{\lambda})\ensuremath{\equiv}{H}_{\mathrm{true}}+{\ensuremath{\lambda}}_{v}$, where ${H}_{\mathrm{true}}$ is the true Hamiltonian, $v$ is an artificial attractive potential, and $\ensuremath{\lambda}>0$. $E(\ensuremath{\lambda})$ can be estimated via a Rayleigh-Ritz calculation. If ${H}_{\mathrm{true}}$ falls just short of being able to support a bound state, $H(\ensuremath{\lambda})$ for $\ensuremath{\lambda}$ "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 $\ensuremath{\lambda}$ "not too small" suggests not only that $H(\ensuremath{\lambda})$ 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 $E(\ensuremath{\lambda})$ in the neighborhood of $\ensuremath{\lambda}={\ensuremath{\lambda}}_{0}$, the (unknown) smallest value of $\ensuremath{\lambda}$ for which $H(\ensuremath{\lambda})$ can support a bound state. A comparison of $E(\ensuremath{\lambda})$ determined numerically with the form of $E(\ensuremath{\lambda})$ 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 ${\mathrm{H}}^{\ensuremath{-}}$.