The problem of several delta-shell potentials in the Lippmann–Schwinger formulation

Abstract

The quantum-mechanical problem of the bound states corresponding to a sum of several delta-shell potentials is solved exactly as an illustration of the applications of the Lippmann-Schwinger formulation. We obtain the bound-state energies and illustrate their relation to the poles of the corresponding S matrix. In addition, the several-delta-shell-potentials problem also gives a good example for the study of localization of states in an elementary level, and we do this in some detail. It is found that for two well-separated delta-shell potentials, the ground state is localized in the stronger well whereas the excited state is localized in the weaker well. This result is generalized for n delta-shell potentials. A simple proof that n attractive delta-shell potentials cannot have more than n S-wave bound states is also given. Like other classical problems of quantum mechanics, namely the harmonic oscillator and the hydrogen atom, this problem of several delta-shell potentials serves as a good illustration of various physical and mathematical aspects of quantum mechanics.