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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. To avoid bound states 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 $e^{ipx}$?

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

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The $\mathcal{S}$-matrix can be obtained by the Dyson series:

$$\mathcal{S}=\operatorname{T\exp}-\frac{i}{\hbar}\int_{-\infty}^{\infty} V_I(t')dt'\tag{1}$$

Here $V_I(t)$ is the interaction potential in the interaction picture:

$$V_I(t) = e^{iH_0t}V e^{-iH_0t}.\tag{2}$$

So you should know (2) and then use it in (1) and expand to the desired order in perturbation theory by expanding the time-ordered exponential.

In that case taking $\mathcal{S}$-matrix elements between in and out states amounts to evaluating $$\langle p'|\mathcal{S}|p\rangle=\sum_{N=0}^\infty \frac{(-i)^n}{\hbar^n} \int_{-\infty}^\infty dt_1\cdots dt_N\langle p'|T\{V_I(t_1)\cdots V_I(t_N)\}|p\rangle,\tag{3}$$

to the desired order in perturbation theory.

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