The quantum-mechanical box effect

A particle moves in one spatial dimension in an attractive symmetric negative potential upsilon f(x), and obeys non-relativistic quantum mechanics. If the system is enclosed in a finite box [-l, l], then, unlike the situation in R the ground-state energy E(0) is negative only when the coupling upsilon is sufficiently large. General upper- and lower-bound formulae are derived for the critical coupling upsilon(c) which corresponds to E(0) = 0. Some generalizations are derived for potentials which change sign and for unsymmetric unimodal potentials.