A quantum field theory of localized and resonant modes in 3D nonlinear lattices at finite temperatures II

We solve the eigenvalue equation derived in paper I under the assumption that a localized or resonant mode is well-localized at a lattice site. Localized modes appear above the top of the phonon band for an appropriate positive quartic potential at any temperatures. Resonant modes appear in the middle of the phonon band for an appropriate negative quartic potential at low temperatures due to the frequency dependent coupling of the mode. This fact is essentially different from that in the force constant defect, where resonant modes appear in the lower frequency regions due to the frequency independent couplings. Both modes are investigated numerically using the tables of the Greenʹs functions for monoatomic simple cubic lattices in terms of the Bessel functions. Resonant modes are also investigated in the Debye approximation