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].
