Quantization of intrinsic localized modes: Effective linear lattice method

A working method is proposed to quantize intrinsic localized modes (ILMs) in lattices with hard quartic anharmonicity in the framework of the rotating-wave approximation. This is done by reducing a nonlinear eigenvalue problem to a linear one by averaging slowly varying reduced effective force constants over frequencies, enabling us to quantize the ILM in a straightforward manner. Such an effective linear lattice (ELL) method is first applied to an analytically tractable d-dimensional cubic lattice to show that the concept of the ELL holds exactly in the strong localization limit. Next, general lattice models are investigated to achieve quantization of the ILM in an approximate manner. The obtained analytical results are tested by solving numerically a model one-dimensional lattice to show that phase-space trajectory of an ILM-bearing atom is of elliptic type with finite but small width. The numerical result confirms the validity of the ELL leading to its semi-classical quantization. On the other hand, orbits of all the remaining atoms exhibits complex non-periodic trajectory to which a direct application of the semi-classical quantization rule appears impossible.