Revisiting double Dirac delta potential

TL;DR

This paper analyzes a double Dirac delta potential, demonstrating its versatility in modeling quantum phenomena like resonances and perfect transmission, and clarifies the conditions for zero reflectivity at zero energy.

Contribution

It introduces a comprehensive analysis of the double Dirac delta potential, highlighting its ability to model complex quantum behaviors and clarifying the distinction between resonant and perfect transmission energies.

Findings

Perfect transmission energies correspond to eigenvalues of a symmetric delta potential.

Zero reflectivity at zero energy is a critical phenomenon with specific parameter conditions.

Resonant energies differ from perfect transmission energies and have different origins.

Abstract

We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ($T=1$). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, $R(E)$] at energy $E=0$ is non-intuitive. However, earlier this has been found and called "threshold anomaly". Here we show that it is a critical phenomena and we can have $ 0 \le R(0)<1$ when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate ($\psi(x)=Ax+B$) of delta well between two symmetric rigid walls for $R(0)=0$. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.