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

The one-dimensional Schrödinger equation with the point potential in the form of the derivative of Dirac's delta function, λδ′(x)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)ψ(x) discontinuous at the origin under the two-sided (at x=±0x=±0) boundary conditions given through the transfer matrix (A00A−1) where A=2+λ2−λ. However, the recent studies, where a resonant non-zero transmission across this potential has been established to occur on discrete sets {λn}n=1∞ 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.