Neat and nice sample of a real-world application of spectral theory.
The \v{S}eba's dry statements (cf. [27]) which characterize the
derivative Dirac's delta-function interaction as an extremely boring
(viz., opaque) barrier in one dimension inspired the authors who
reanalyzed the problem. Their paper reviews the more recent relevant
literature, offers a few new theorems and explains to the reader why
the \v{S}eba's scepticism does not apply in the so called resonant
cases where the transmission does not vanish  (the persuasive
numerical illustration given in the (shortest) last section may be
read first).

MR2608272 Golovaty, Yu. D.; Hryniv, R. O. On norm resolvent
convergence of Schrödinger operators with $\delta'$-like potentials.
J. Phys. A 43 (2010), no. 15, 155204, 14 pp. 81Qxx (34L40 47E05)