Bound quantum states with no classical turning points.

Abstract

For a two-parameter class of local potentials which support a bound state with energy ${\mathit{E}}_{1}$ embedded in the continuous spectrum, we determine the conditions that the potential has m wells with barriers of height greater than ${\mathit{E}}_{1}$. Although ${\mathit{E}}_{1}$\ensuremath{\gtrsim}V(\ensuremath{\infty})=0, such wells act as traps for a classical particle, with classical turning points wherever a barrier height reaches the energy of the particle. However, the parameters of the potential may easily be chosen so that there are no classical turning points, even though the potential supports a bound quantum state.