Boundary conditions for the states with resonant tunnelling across the $\delta'$-potential

Abstract

The one-dimensional Schr\"odinger equation with the point potential in the form of the derivative of Dirac's delta function, $\lambda \delta'(x)$ 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 ${cc} {\cal A} 0 0 {\cal A}^{-1})$ where ${\cal A} = {2+\lambda \over 2-\lambda}$. 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.