Boundary conditions for the states with resonant tunnelling across the $\delta'$-potential

TL;DR

This paper investigates the boundary conditions for the Schrödinger equation with a $\, ext{delta'}$-potential, resolving discrepancies in previous models by proposing more general boundary conditions to account for resonant tunneling phenomena.

Contribution

It introduces a more general form of boundary conditions for the $\, ext{delta'}$-potential, addressing inconsistencies in earlier models regarding resonant tunneling.

Findings

Resonant non-zero transmission occurs at discrete $\, ext{lambda}_n$ values.

Traditional boundary conditions are insufficient to describe resonant tunneling.

Proposed boundary conditions reconcile previous discrepancies.

Abstract

The one-dimensional Schr\"odinger equation with the point potential in the form of the derivative of Dirac's delta function, $\lambda \delta'(x)$ with $\lambda$ being a coupling constant, is investigated. This equation is known to require an extension to the space of wave functions $\psi(x)$ discontinuous at the origin under the two-sided (at $x=\pm 0$) boundary conditions given through the transfer matrix ${cc} {\cal A} 0 0 {\cal A}^{-1})$ where ${\cal A} = {2+\lambda \over 2-\lambda}$. However, the recent studies, where a resonant non-zero transmission across this potential has been established to occur on discrete sets $\{\lambda_n \}_{n=1}^\infty$ in the $\lambda$-space, contradict to these boundary conditions used widely by many authors. The present communication aims at solving this discrepancy using a more general form of boundary conditions.