Solving the time-dependent Klein-Gordon and square-root Klein-Gordon equations with Krylov-subspace methods

Abstract

For relativistic spinless particles, the standard equation with Lorentz covariance is the Klein-Gordon equation, while the square-root Klein-Gordon equation, which is usually referred to be semirelativistic since it lacks Lorentz covariance, is another frequently used equation. We numerically solve these two equations coupled with time-dependent electromagnetic fields using the non-Hermitian Arnoldi and the Hermitian Lanczos propagator, respectively, in the three-dimensional real space. The non-Hermitian Arnoldi method is shown to be ideally suited for the Klein-Gordon equation. A different method to treat the nonlocal operator in the square-root Klein-Gordon equation is developed, allowing the calculation of the propagator to be more efficient. Furthermore, some strong-field problems such as the photoionization, high harmonic generation, and the pair production are studied using our methods. The results of both equations are compared with those of the nonrelativistic Schr\"odinger equation, which demonstrates the relativistic effects in various regimes. We show that observables of the Klein-Gordon equation and the square-root Klein-Gordon equation are identical up to the first order of relativistic corrections and confirm it numerically.