Boundary conditions for the states with resonant tunnelling across the ${\delta }^{\prime }$-potential

Abstract

The one-dimensional Schrödinger equation with the point potential in the form of the derivative of Dirac's delta function, $\lambda {\delta }^{\prime }(x)$ 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 $cc\mathcal{A}00{\mathcal{A}}^{-1})$ where $\mathcal{A}=\frac{2+\lambda }{2-\lambda }$. 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.

Submitted Phys. Lett. A. Essentially revised and extended version, 1 figure added. 12 pages

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