= Solution
Subtract the common sector <energy> and call the remaining <eigenvalue> $\hbar\lambda$. Its characteristic equation is
$$
\lambda^2-\frac{\Delta^2}{4}-g^2(n+1)=0.
$$
The dressed-state energies are consequently
$$
\boxed{E_{\pm n}=\hbar\omega\left(n+\frac12\right)\pm\frac\hbar2\Omega_n,
\qquad\Omega_n=\sqrt{\Delta^2+4g^2(n+1)}.}
$$
On resonance their splitting is $2\hbar|g|\sqrt{n+1}$; far off resonance the <eigenstates> approach the uncoupled <quantum states> with small dispersive <energy> shifts. The <eigenvalue> splitting is an angular-frequency difference $\Omega_n$, so an ordinary <frequency> would be $\Omega_n/(2\pi)$.
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