= Solution
Use the ordered <orthonormal basis> $(|1s\rangle,|2p_x\rangle)$. The projected <Hamiltonian> between pulses is $H_0=-\Delta\sigma_z$, so its free <time-evolution operator> through a period is
$$
U_\Delta=e^{i\alpha\sigma_z}=\begin{pmatrix}e^{i\alpha}&0\\0&e^{-i\alpha}\end{pmatrix},\qquad \alpha=\frac{\Delta\tau}{\hbar}.
$$
Across one instantaneous pulse, integrating the <Schrödinger equation> gives
$$
U_E=e^{-i\beta\sigma_x}=\begin{pmatrix}\cos\beta&-i\sin\beta\\-i\sin\beta&\cos\beta\end{pmatrix},\qquad \beta=\frac{Ex_0\tau}{\hbar},
$$
using the <exponential of a Pauli vector>. The finite free <Hamiltonian> contributes vanishing evolution during the zero-duration kick. A period measured immediately before successive pulses therefore consists of the kick followed by free evolution. Composition gives
$$
\boxed{U(N\tau-\epsilon,-\epsilon)=(U_\Delta U_E)^N,\qquad\epsilon\to0^+.}
$$
This is the <Floquet operator> at the specified phase of the pulse train. Actual hydrogenic <energy levels> are not opposite numbers: the displayed convention subtracts their midpoint, with $\Delta=(E_{2p}-E_{1s})/2$. Restoring that midpoint multiplies the result by an unobservable <global phase>. Likewise $E$ in the given <Hamiltonian> denotes the electric coupling per unit length; if it is instead a physical electric-field strength, its charge factor must be included in $Ex_0$.
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