# Explicit form of S-matrix on the line

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.

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.