Scattering by a Delta Function Well in 1D 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


*

*What is scattering in 1D quantum mechanics and what is the scattered wave here? 

*How to calculate the scattering phase shift $\delta_0$ for this scattering process?

*Is the scattering phase shift $\delta_0$ is given by the phase of $\frac{c_3}{c_1}$?

*The reflected wave should also have a phase shift which is different from the scattering phase shift $\delta_0$ . Is it correct? 

 A: $\let\d=\delta \def\de{\d_{\mathrm e}} \def\do{\d_{\mathrm o}} 
\def\ps#1{\psi_{\mathrm{#1}}} \def\sgn{{\mathrm{sgn}}}$
About the meaning of scattering phaseshift. I too never saw this
quantity used in 1D scattering. In 3D phaseshifts for central
potential are related to angular momentum: there is one phaseshift
for each $l$ quantum number. This is because angular momentum is
conserved and a scattering process can be separately analized for each
$l$.
Moreover phase factor $e^{i\d_l}$ is defined between ingoing and
outgoing wave. Actually $e^{2i\d_l}$ is an eigenvalue of scattering
matrix, which is diagonal in this representation. Ingoing wave is
defined as the one solution having asymptotic behaviour
$e^{-ikr}\!/r$, outgoing $e^{ikr}\!/r$.
In 1D we can't use angular momentum, but if potential is an even
function of $x$ then parity is a good quantum number and makes sense
to study even and odd states separately. Using Hanting Zhang's
notations an even state has $A=G$, $B=F$ whereas an odd state has
$A=-G$, $B=-F$. Furthermore an ingoing state has $B=F=0$, an outgoing
one has $A=G=0$.
That is:
$$\eqalign{
    \ps{+i} &= A\,e^{-ik|x|} \cr
    \ps{+o} &= B\,e^{ik|x|}\cr
    \ps{-i} &= A\,\sgn(x)\,e^{-ik|x|}\cr
    \ps{-o} &= B\,\sgn(x)\,e^{ik|x|}.\cr}$$
From Hanting Zhang I take
$$F - G - A + B = 2 i z\,(A + B).$$
For even states
$$-A + B = i z\,(A + B)$$
$$B = A\,{1 + iz \over 1 - iz}$$
and defining $\de$ by 
$$e^{2i\de} = {B \over A}$$
we find
$$e^{2i\de} = {1 + iz \over 1 - iz}$$
$$\de = \arctan z.$$
For odd states
$B = -A$
$$\do = \pi/2.$$
Caution 1: I don't know what is meant for $\d_0$ in the questions. Almost certainly the meaning differs from the one I assumed.
Caution 2: I didn't thoroughly check my equations. Possible wrong signs or other errors.
A: 
What is scattering in 1D quantum mechanics and what is the scattered wave here?

Scattering in quantum mechanics usually refers to how a wavefunction behaves when its energy is greater than the potential $V(x)$ as $x \rightarrow \pm\infty$, normally called a scattering state. Normally we choose $V(x)$ to go to zero at infinity so that a state is scattering simply if it has positive energy. 
For the delta function well you've given, the scattered wave is just the incoming wave for the left, which scatters into a reflected wave and a transmitted wave.

How to calculate the scattering phase shift $\delta_0$ for this scattering process?
Is the scattering phase shift $\delta_0$ is given by the phase of $\frac{c_3}{c_1}$?

Explicitly, let 
$$\psi(x) = Ae^{ikx} + Be^{-ikx}, \kern{3mm} x<0$$
$$\psi(x) = Fe^{ikx} + Ge^{-ikx}, \kern{3mm} x>0$$
Continuity of $\psi(x)$ requires that, 
$$A + B = F + G.$$
For $\psi(x)$'s derivative, the delta well requires a discontinuity proportional to its strength:
$$ik(F-G-A+B) = \left.\Delta\left(\frac{d\psi}{dx}\right)\right|_0 = -\frac{2m\alpha}{\hbar^2}(A+B).$$
Now we kill off incoming waves from the right, and set $G = 0$, and rearrange to get:
$$B = \frac{iz}{1-iz}A, \kern{3mm}F = \frac{1}{1-iz}A,$$
where $z = \frac{m\alpha}{\hbar^2k}$. The the phase shift is just the phases of the coefficients in front of $A$.

The reflected wave should also have a phase shift which is different from the scattering phase shift $\delta_0$. Is it correct?

It would seem that yes, the reflected wave does have a different phase.
