Can degenerate bound states occur in one dimensional quantum mechanics?

TL;DR

This paper demonstrates that one-dimensional quantum systems with certain unbounded potentials can host degenerate bound states of different parity, challenging traditional assumptions and suggesting potential experimental observations.

Contribution

It introduces a class of non-singular, unbounded potentials in 1D quantum mechanics that support degenerate bound states with different parity, providing explicit constructions and analytic expressions.

Findings

Degenerate bound states can occur in 1D systems with unbounded-from-below potentials.

Such states can exist both above and below the potential maximum.

Potential for observing these states in mesoscopic systems.

Abstract

We point out that bound states, degenerate in energy but differing in parity, may form in one dimensional quantum systems even if the potential is non-singular in any finite domain. Such potentials are necessarily unbounded from below at infinity and occur in several different contexts, such as in the study of localised states in brane-world scenarios. We describe how to construct large classes of such potentials and give explicit analytic expressions for the degenerate bound states. Some of these bound states occur above the potential maximum while some are below. Various unusual features of the bound states are described and after highlighting those that are ansatz independent, we suggest that it might be possible to observe such parity-paired degenerate bound states in specific mesoscopic systems.