A simple separable Hamiltonian having bound states in the continuum

Abstract

The author discusses a simple separable two-freedom Hamiltonian H describing the motion of a point particle in a plane region (V=0) between two infinite parallel hard walls (V= infinity ) and with a rectangular finite potential well (V=-V0). The quantal spectrum of H is discrete below the continuum limit (ionisation energy) different from the classical escape energy (E=0) by the zero point energy of the transverse motion and has infinitely many bound states embedded in the continuum. The author shows that this spectrum is fragile against perturbations of H: if the hard walls are made diverging, the bound states in the continuum disappear, while if they are converging, the continuum disappears and the spectrum is pure point.