Critical domain of coupling constants of short-range potentials

In this paper, we consider three-dimensional short-range potentials of several components, i.e. depending on several coupling constants. For N components, the critical domain is defined as the (N-1)-dimensional surface separating the regions with and without a bound state in the space of the coupling constants. It is shown that the problem of finding the critical domain can be solved by generalizing the methods established for the determination of the critical coupling constant in the case of a single-component potential. Applications are made for the truncated harmonic oscillator and the square-well potentials with a spin-orbit interaction of the Thomas form.