  We describe two approximation methods designed to capture the leading
behavior of the low-energy tail of the momentum-dependent spectral density
$A(\mathbf{k}, E)$ and the tunneling density of states $D(E)$ for an injected
particle, such as an electron or an exciton, interacting with a bath of phonons
at a non-zero initial temperature $T$, including quantum corrections due to the
non-zero frequencies of the relevant phonons. In our imaginary-time-dependent
Hartree (ITDH) approximation, we consider a situation where the particle is
injected into a specified coherent state of the phonon system, and we show how
one can use the ITDH approximation to obtain the correlation function $C(\tau)$
for that initial state. The thermal average $C(\tau)$ is obtained, in
principle, by integrating the result over all possible initial phonon coherent
states, weighted by a thermal distribution. However, in the low-energy tail,
one can obtain a good first approximation by considering only initial states
near the one that maximizes the integrand. Our second approximation, the
fixed-wave-function (FWF) approximation, assumes that the wave function of the
injected particle evolves instantaneously to a wave function which then is
independent of time, while the phonon system continues to evolve due to
interaction with the particle. We discuss how to invert the Laplace transform
and how to obtain $A(\mathbf{k},E)$ as well as $D(E)$ from the imaginary-time
analysis. The FWF approximation is used to calculate $D(E)$ for a
one-dimensional continuum model of a particle interacting with acoustic
phonons, and effects due to the quantum motion of phonons are observed. In the
classical phonon limit, the dominant behaviors of both the ITDH and FWF
approximations in the low-energy tail reduce to that found in the past for a
particle in a random potential with a Gaussian statistical distribution.
