  We investigate the low-energy properties of the Holstein polaron through
calculation of the q-dependent phonon spectral function using an improved
exact-diagonalization technique, defined over a variational Hilbert space. We
perform a comprehensive study of the low-energy excitations of the polaron.
Beside the energy range, where the additional phonon excitation is unbound, we
observe separate coherent peaks which correspond to bound and antibound states
of a polaron and additional phonon quanta. These novel states can be observed
for intermediate and strong electron-phonon coupling strengths, as well as
below and above the unbound one-phonon excitation spectrum. A detailed
investigation of their properties is presented. We find good agreement between
the phonon spectral function obtained from the first-order strong-coupling
perturbation theory and numerical results.
