Two exact lattice propagators

Exact Schrödinger and heat propagators are given for a particle hopping on a one-dimensional rectangular lattice, assuming a uniform field ${\mathit{V}}_{\mathit{n}\mathit{n}}$=λn and a δ-function potential ${\mathit{V}}_{\mathit{n}\mathit{n}}$=λ${\mathrm{\delta }}_{\mathit{n}0}$. In its quantum form, the uniform-field propagator is the general solution of the Wannier-Stark problem for a discrete lattice, describing a particle moving in the superposition of a homogeneous field and a discrete periodic potential created by the lattice. A disentangled form for the uniform-field propagator is obtained by using the transformation properties of the Hamiltonian under the Lie algebra iso(1,1). Using this result, it is shown that the expected position and spatial extension of a lattice wave packet oscillate in phase with equal amplitudes. The discrete δ-function heat propagator is related by a Lyapunov transformation to the solution of the lattice Smoluchowski equation for the cusp potential ${\mathit{V}}_{\mathit{n}\mathit{n}}$∝‖n‖. It is shown that the implied discrete-time Smoluchowski evolution operator generates a Markov process in which a pair of nonsymmetric random walks on the right and left half-axes are coupled at cell 0 by a partly reflecting, partly transmitting, sticky barrier. The interaction term in the lattice δ-function heat propagator is a Poisson weighted superposition of nonsymmetric random walks.