Salpeter equation with complex Hamiltonian.

Abstract

We use integral representations and analytical solutions for the propagator of the (1+1)-dimensional Salpeter Hamiltonian to describe the wave propagation of a non-spin-relativistic quantum particle. An explicit expression of the propagator, or Green function, is derived in terms of special functions. The exact Green function is convoluted with the initial wave function by using suitable boundary conditions to find the wave propagation for both massive and massless particles. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent stochastic problem, namely the Bäumer equation. This equation describes relativistic stochastic processes with time-changing anomalous diffusion. Given the known crossover between Cauchy and Gaussian diffusion, we analytically calculate the propagator to interpolate between these regimes. This Bäumer propagator corresponds to the Green function of a relativistic diffusion process that interpolates between Cauchy distributions for small times and Gaussian diffusion for large times, providing a framework for stochastic processes where anomalous diffusion is time-dependent.