Abstract
In one dimension, many-body matrix elements can be calculated exactly by the transfer integral (TI) method if the wave functions are assumed to be products of single-particle functions and nearest-neighbor correlation functions. In the frame of this TI approximation the phonons can be calculated without any further approximation. The theory is first developed for the case of a harmonic potential. Here the TI phonons are an excellent approximation of the well-known exact results, except for very long wavelengths. However, a Bogoliubov transformation finally yields the exact phonons. The whole procedure can be generalized for arbitrary two-body interactions, including those containing a hard core. In the generalized theory the hard-core problem has vanished completely.
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