Anti-symmetric square well and barrier potential between two rigid walls

TL;DR

This paper investigates a unique anti-symmetric square well and barrier potential between rigid walls, revealing special eigenstates at specific potential values where energies like 0 or V_0 become eigenvalues, differing from typical double-well systems.

Contribution

It introduces a novel analysis of anti-symmetric potentials with special eigenstates at calculable potential values, expanding understanding of quantum well spectra.

Findings

Certain potential values yield eigenstates with energies 0 or V_0.

Special eigenstates emerge only with linear solutions in specific regions.

The spectrum differs from conventional double-well systems.

Abstract

We study an anti-symmetric (square) well and barrier potential of depth/height $(V_0)$ placed between two rigid walls. Unlike the usual double-well, here the closely lying sub-barrier doublets need not be the lowest ones in the spectrum. When $V_0$ admits certain calculable values, $E=0$ or $E=V_0$ or both could become energy eigenvalues of the special eigenstate which emerge only if one seeks a linear solution of Schr{\"o}dinger equation in the appropriate regions.