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We investigate the low-energy properties of the Holstein polaron throughcalculation of the q-dependent phonon spectral function using an improvedexact-diagonalization technique, defined over a variational Hilbert space. Weperform a comprehensive study of the low-energy excitations of the polaron.Beside the energy range, where the additional phonon excitation is unbound, weobserve separate coherent peaks which correspond to bound and antibound statesof a polaron and additional phonon quanta. These novel states can be observedfor intermediate and strong electron-phonon coupling strengths, as well asbelow and above the unbound one-phonon excitation spectrum. A detailedinvestigation of their properties is presented. We find good agreement betweenthe phonon spectral function obtained from the first-order strong-couplingperturbation theory and numerical results.

Emergence of states in the phonon spectral function of the Holstein polaron below and above the one-phonon continuum