Bound states in bottomless potentials

TL;DR

This paper investigates classical and quantum behaviors in bottomless potentials, revealing that quantum bound states can exist and most classical trajectories become trapped, leading to unique dynamical properties.

Contribution

It introduces explicit constructions of bottomless potentials supporting quantum bound states and characterizes the classical trapping phenomena in these potentials.

Findings

Quantum bound states exist in certain bottomless potentials.

Most classical trajectories are trapped and do not escape to infinity.

Classical dynamics in these potentials is characterized by a zero measure set of escaping trajectories.

Abstract

We consider classical and quantum dynamics on potentials that are asymptotically unbounded from below. By explicit construction we find that quantum bound states can exist in certain bottomless potentials. The classical dynamics in these potentials is novel. Only a set of zero measure of classical trajectories can escape to infinity. All other trajectories get trapped as they get further out into the asymptotic region.