CRITICAL-BEHAVIOR OF WEAKLY-BOUND SYSTEMS

We consider the class of three-dimensional finite-range, or similar, p otentials lambda W(r), depending on a strength constant lambda. We stu dy the behaviour of the eigenvalue E as a function of lambda - lambda( c), where lambda(c) is the critical value at the transition from 0 --> 1 bound state. For the l = 0 case, we find E proportional to (lambda - lambda(c))(2), whereas the relationship is linear for l greater than or equal to 1. Treating l as a continuous parameter in the radial Sch rodinger equation, we give the evolution of the power law between l = 0 and l = 1. Besides spherically symmetric scalar potentials, we also discuss the case of a repulsive scalar potential combined with a spin- orbit component of the Thomas form.