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

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

2012-02-06

arxiv.org/abs/1202.1117v1

J. Phys. A: Math. Theor. vol. 44 (2011) 375305

Physics

Mathematical Physics

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

Scientific paper

10.1088/1751-8113/44/37/375305

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'(x) $, with $\lambda \in \R$, 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'(x) $ is constructed in a specific way. Otherwise, the $\delta'$-potential is fully non-transparent. Moreover, when the transmission is non-zero, the structure of a resonant set depends on a regularizing sequence $\Delta'_\varepsilon(x)$ that tends to $\delta'(x)$ 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 $\bar{\lambda} \in \R$ to construct such a regularizing sequence $\Delta'_\varepsilon(x)$ that the $\delta'$-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 $\delta'$-potential.

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