Transparency Resonances and Bound States of the $\delta^\prime$ Junction

TL;DR

This paper provides exact solutions for a Schrödinger equation with a $ abla$ interaction, revealing transparency resonances, multiple bound states, and a negative electron density drop, challenging previous beliefs about the $ abla$ potential.

Contribution

It introduces new exact solutions and uncovers phenomena like resonance levels and multiple bound states for the $ abla$ interaction, contrary to prior assumptions.

Findings

Existence of infinite transparency resonance levels.

Multiple bound states are possible, not just one.

Negative stepwise drop in electron density observed.

Abstract

Exact positive and negative energy solutions for the eigenvalue problem of the Schr\"{o}dinger equation in one dimension with a $\delta^\prime$ interaction are found and analyzed. An infinite series of transparency resonance levels in the strength of this interaction is shown to exist. This result is against the actual belief that the $\delta^\prime$ potential acts as a totally reflecting wall. A finite number of bound states is obtained, contrary to the previous result on the existence of only one bound state. A new effect of a {\it negative stepwise} drop in the electron density across the $\delta^\prime$ junction is observed. The solutions are also applied to the propagation of the electromagnetic field in dielectric media.