A. V. Zolotaryuk, Y. Zolotaryuk
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.
View original:
http://arxiv.org/abs/1202.1117
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