Schroedinger equation on a line is considered as a homework toy
terrain for interrelating the mathematical theory of self-adjoint
extensions (by returning, in particular, to the idea of Griffiths
[7] who contemplated a - for simplicity, central - potential in
the form, roughly speaking, of the n-th derivative of the delta
function) with something which I would call a non-Hermitian
extension of Quantum Mechanics [may be also called
pseudo-Hermitian or quasi-Hermitian or PT-symmetric - for a
compact introductory reading see, e.g., the fresh ``proceedings"
of the related internationl Workshop, to appear in the
September issue of Czechosl. J. Phys., vol. 55, pp. cca 1048 - 1196
(2005)] and which, in particular, admits the mathematically as
well as physically fully consistent work with an apparently
inconsistent concept of the energy-dependent quantum Hamiltonians
(sorry for self-citation: M. Znojil, Linear representation of
energy-dependent Hamiltonians, Phys. Lett. A 326 (2004) 70 - 76).
On this background the authors reveal that, unexpectedly, their
S-matrix remains unitary but that it still proves suitable for the
inverse-scattering reconstruction of the present specific singular
potential from any prescribed scattering data (due to the precise 
match of their number of the number of parameters in the interaction
in question). The authors argue that and why their recipes may be 
extended to the coupled-channel situation and/or, in some sense, to non-stationary states.

MR2148638
Coutinho, F. A. B. ; Nogami, Y. ; Tomio, Lauro ; Toyama, F. M. 
Energy-dependent point interactions in one dimension. 
J. Phys. A 38 (2005), no. 22, 4989--4998.