Consider the Hamiltonian $H$ on functions on the line with \begin{eqnarray} H=H_0+V,\\ H_0=-\frac{1}{2m}\frac{d^2}{dx^2} \end{eqnarray} where $V$ is a potential vanishing outside of a bounded interval and $m>0$. To avoid discrete spectrum of $H$ one may assume $V\geq 0$. One even may assume that $V(x)=V_0>0$ for $x\in [0,a]$ and $V(x)=0$ otherwise.

How to write down explicitly the $S$-matrix for this Hamiltonian? Namely how does $S$ act on $e^{ipx}$?

**Remark.** Since $S$ commutes with $H_0$ then necessarily
$$S(e^{ipx})=A(p)e^{ipx}+B(p)e^{-ipx}.$$

My question is how to write down $A(p),B(p)$ explicitly.

**ADDED.** Let me ask a more precise question. The equation $H\psi=\frac{p^2}{2m}\psi$ has a solution $\psi_p$ such that
\begin{eqnarray}
\psi_p(x)=\left\{\begin{array}{cc}
e^{ipx}+\tilde B(p)e^{-ipx},&x<\inf supp(V)\\
\tilde A(p)e^{ipx},&x>\sup supp(V)
\end{array}\right.
\end{eqnarray}
where $\tilde A(p),\tilde B(p)$ are appropriate constants.

Is it true that for $p>0$ one has $\tilde A(p)=A(p)$ and $\tilde B(p)=B(p)$?

(If my understanding is correct, $\psi_p$ is the IN state corrresponding to plane wave $e^{ipx}$ provided $p>0$.)

I believe this should be a basic example in the scattering theory, so a reference will be most helpful.