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

The one-dimensional Schrodinger 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 = +/- 0) boundary conditions given through the transfer matrix ((A)(0) (0)(A-1)) where A = 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)(infinity) 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. (c) 2010 Elsevier B.V. All rights reserved.