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

The one-dimensional Schrödinger equation with the point potential in the form of the derivative of Dirac's delta function, $λδ'(x)$ with $λ$ being a coupling constant, is investigated. This equation is known to require an extension to the space of wave functions $ψ(x)$ 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 ${\cal A} = {2+λ\over 2-λ}$. However, the recent studies, where a resonant non-zero transmission across this potential has been established to occur on discrete sets $\{λ_n \}_{n=1}^\infty$ in the $λ$-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.