Controlling a resonant transmission across the $δ'$-potential: the inverse problem

Recently, the non-zero transmission of a quantum particle through the one-dimensional singular potential given in the form of the derivative of Dirac's delta function, $λδ'(x)$, with $λ\in \R$, being a potential strength constant, has been discussed by several authors. The transmission occurs at certain discrete values of $λ$ forming a resonance set ${λ_n}_{n=1}^\infty$. For $λ\notin {λ_n}_{n=1}^\infty$ this potential has been shown to be a perfectly reflecting wall. However, this resonant transmission takes place only in the case when the regularization of the distribution $δ'(x)$ is constructed in a specific way. Otherwise, the $δ'$-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence $Δ'_\varepsilon(x)$ that tends to $δ'(x)$ in the sense of distributions as $\varepsilon \to 0$. Therefore, from a practical point of view, it would be interesting to have an inverse solution, i.e. for a given $\barλ \in \R$ to construct such a regularizing sequence $Δ'_\varepsilon(x)$ that the $δ'$-potential at this value is transparent. If such a procedure is possible, then this value $\barλ$ has to belong to a corresponding resonance set. The present paper is devoted to solving this problem and, as a result, the family of regularizing sequences is constructed by tuning adjustable parameters in the equations that provide a resonance transmission across the $δ'$-potential.

No reviews yet.

No comments yet.