The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation (Communications in Computational Physics, 2022)
The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic)Schrödinger equation. We point out that the qlB method actually approximates thehyperbolic version of the non-relativistic Schrödinger equation, whose solution is thusobtained at the price of an additional small error. Such an error is of order of $(ω_cτ)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $τ$ a chosen reference time (i.e.,1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundarylayer and, consequently, the approximation holds uniformly, which fact is relevant inview of numerical approximations. We also discuss this occurrence more generally, forsome classes of linear singularly perturbed partial differential equations.
Keywords: Schrödinger equation; hyperbolic Schrödinger equation; Dirac equations; quantum Lattice Boltzmann; Klein-Gordon equation; singular perturbations.