You can't solve the problem without specifying the functional form of V[x]. The Schrödinger equation is

1/2 \[Psi]''[x] + V[x] \[Psi][x] = k^2/2 \[Psi][x]

with a scattering solution of the form

\[Psi][x] = E^(I k x) + r[k, x] E^(-I k x) + t[k, x] E^(I k x)

Given V[x], you can separate the real and imaginary parts to the equation using ComplexExpand[ ], you get two rather complicated equations which determine r[k,x] and t[k,x].

Thanks to both for your answers and ideas.

I have checked Ulrich's Wolfram demostrations and look impressive. Yet I should have been more explicit in my needs, which are very special. It is the problem of a "surface state" in a metal surface.

The surface state problem can be reduced to quantum confinement within an asymmetric Quantum Well (QW), in one dimension. On one side (say, the left side) confinement is produced by reflection at the gap of a bulk crystal potential. So, there is a finite energy range within which the "surface electron" cannot penetrate the bulk crystal (if it does penetrate, it is not a surface state anymore!!). At the other side (right side of the QW) we have a vacuum barrier (image potential barrier). Traditionally, this problem is solved in the Born-Oppenheimer approach (also called phase accumulation model), because one can find a very handy description of the reflection phase in the gap of a crystal (phiC). Hence we also need the reflection phase at the other side of the quantum well, i.e., at the vacuum side, phiB. With both,one simply makes phiC+phiB=2 n Pi and finds the bound states. Note that both phiC and phiB will depend on the energy. There is abundant literature on this approach (see for example "Phase analysis of image states and surface states associated with nearly-free-electron band gaps", N. V. Smith, Phys. Rev. B 32, 3549 (1985))

One can avoid the BO approach, but this requires to solve the whole complex Hamiltonian inside the crystal...my theory colleagues like that, but I am an experimentalist looking for a model to play with and test my results...My investigated systems are "modified surfaces", such as adsorbed monolayers, of any kind of materials (metals, organic molecules, etc.), and hence I have to model the "vacuum side" of my QW, but still retrieve phiB' to apply the BO sum rule phiC+phi_B'=2 n Pi.

So, you may understand now why I need the "reflection phase" for an electron wave...