Can degenerate bound states occur in one-dimensional quantum mechanics?

¹  Department of Physics and Centre for Theoretical Studies, Indian Institute of Technology - Kharagpur 721 302, WB, India
²  Department of Physics and University Scholars Programme, National University of Singapore - Kent Ridge, Singapore

Corresponding authors: sayan@cts.iitkgp.ernet.in parwani@nus.edu.sg

Received: 13 July 2007
Accepted: 13 September 2007

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.