Correspondence of anharmonic localized vibrations to N-phonon bound states

A discussion of the interrelation of anharmonic localized modes and their quantum-mechanical analogs, the so-called N-phonon bound states, is presented. For small systems and moderate quantum numbers, the ''exact'' eigenvalue spectrum is obtained by direct numerical diagonalization. A variational ansatz is presented which allows one to estimate analytically the energy levels of strongly anharmonic systems, exemplified here by a single quartic oscillator and a dimer model. It is shown that expectation values and level spacings for N-phonon bound states are in close agreement with estimates derived from the classical anharmonic localized vibrations. Finally, an estimate is given for the action of anharmonic localized modes in infinite higher-dimensional systems. This estimate suggests a threshold for anharmonic localized mode existence in three-dimensional lattices.