Closed-form solutions for the Salpeter equation

TL;DR

This paper derives exact integral and analytical solutions for the 1+1 dimensional Salpeter Hamiltonian's propagator, linking relativistic quantum mechanics with stochastic processes exhibiting time-dependent anomalous diffusion.

Contribution

It introduces closed-form solutions for the Salpeter equation and connects it to a stochastic B"aumer equation describing relativistic diffusion with time-varying anomalous behavior.

Findings

Green function for the Salpeter Hamiltonian derived

Exact solutions for specific initial conditions obtained

Relativistic diffusion interpolates between Cauchy and Gaussian distributions

Abstract

We propose integral representations and analytical solutions for the propagator of the $1+1$ dimensional Salpeter Hamiltonian, describing a relativistic quantum particle with no spin. We explore the exact Green function and an exact solution for a given initial condition, and also find the asymptotic solutions in some limiting cases. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent stochastic problem, namely the B\"aumer equation. This equation describes \textit{relativistic} stochastic processes with time-changing anomalous diffusion. This B\"aumer 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.