In the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long-range interaction. Fractional generalizations of the sine-Gordon, nonlinear Schrödinger, and Hilbert–Schrödinger equations have been found.
There exists a critical value of the power s of the long-range potential. Below the critical value (s<3, s≠1, 2) we obtain equations with fractional derivatives while for s3 we have the well-known nonlinear dynamical equations with space derivatives of integer order.
Long-range interaction impact on the quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the well-known Gaussian-like behavior to a power-law decay due to the long-range interaction. A link between 1D quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed.