Fermi's golden rule, continuuous spectrum

Usually when the Fermi’s golden rule is derived using time-dependent perturbation theory, the notation suggests that the system under consideration has discrete spectrum (quantum harmonic oscillator, infinite square well etc.). In my course, when a system involving free particle was considered, it was put inside a large box, with periodic boundary conditions, before the perturbation theory was applied. Only at the end, the size of the box was assumed to approach infinity and the sums over discrete states were converted into integrals. During this process, the density of states, which for a particle inside a box with periodic b. c. is a grid of delat functions, was approximated by a continuous funciton.

My question is, can we derive the Fermi’s golden rule for a system with a continuous spectrum (e.g. a free particle subject to a weak pulse that lasts for a finite time), without resorting to the box argument? How do we define the density of states in this case? Is it even possible to apply time-dependent perturbation theory in the case of continuous spectrum?

Example Hamiltonian:
$$hat{H} = frac{hat{p}^2}{2m} + Ae^{-ahat{x}^2}e^{-bt^2}$$