Let us consider a scattering process by a delta function well in 1D:
$$
V(x) = -\alpha \, \delta(x), \quad \alpha > 0.
$$

I solve the Schrödinger equation for the scattering states and get the followings:

$$
\psi\_1(x) = c\_1 e^{ikx} + c\_2 e^{-ikx}, \quad x < 0, \quad k^2 = \frac{2mE}{\hbar^2}, \quad E > 0, 
$$ 

and,

$$
\psi\_2(x) = c\_3 e^{ikx}, \quad x > 0, \quad k^2 = \frac{2mE}{\hbar^2}, \quad E > 0. 
$$ 

Here $c\_1 e^{ikx}$ is the incoming wave from the left, $c\_2 e^{-ikx}$ is the reflected wave and, $c\_3 e^{ikx}$ is the transmitted wave. 

**Questions**

1. What is scattering in 1D quantum mechanics and what is the scattered wave here?
2. How to calculate the scattering phase shift $\delta\_0$ for this scattering process?
3. Is the scattering phase shift $\delta\_0$ is given by the phase of $\frac{c\_3}{c\_1}$?
4. The reflected wave should also have a phase shift which is different from the scattering phase shift $\delta\_0$ . Is it correct?