Controlling a resonant transmission across the ${\delta }^{\prime }$-potential: the inverse problem

Abstract

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, $\lambda {\delta }^{\prime }(x)$, with $\lambda \in \mathbb{R}$, being a potential strength constant, has been discussed by several authors. The transmission occurs at certain discrete values of $\lambda$ forming a resonance set ${{\lambda }_n}_{n=1}^{\infty }$. For $\lambda \notin {{\lambda }_n}_{n=1}^{\infty }$ 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 ${\delta }^{\prime }(x)$ is constructed in a specific way. Otherwise, the ${\delta }^{\prime }$-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence ${\Delta }_{\epsilon }^{\prime }(x)$ that tends to ${\delta }^{\prime }(x)$ in the sense of distributions as $\epsilon \to 0$. Therefore, from a practical point of view, it would be interesting to have an inverse solution, i.e. for a given $\bar{\lambda }\in \mathbb{R}$ to construct such a regularizing sequence ${\Delta }_{\epsilon }^{\prime }(x)$ that the ${\delta }^{\prime }$-potential at this value is transparent. If such a procedure is possible, then this value $\bar{\lambda }$ 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 ${\delta }^{\prime }$-potential.

21 pages, 4 figures. Corrections to the published version added; http://iopscience.iop.org/1751-8121/44/37/375305

References in corpus (1)

• Coupling in the singular limit of thin quantum waveguides

Cited by in corpus (3)

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• Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian
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