The Hyperbolic Schrodinger Equation and the Quantum Lattice Boltzmann Approximation

The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic) Schrodinger equation. We point out that the qlB method actually approximates the hyperbolic version of the non-relativistic Schrodinger equation, whose solution is thus obtained at the price of an additional small error. Such an error is of order of (omega(c)tau)(-1), where omega(c) := mc(2)/h is the Compton frequency, h being the reduced Planck constant, m the rest mass of the electrons, c the speed of light, and tau a chosen reference time (i.e., 1 s), and hence it vanishes in the non-relativistic limit c -> +infinity. This asymptotic result comes from a singular perturbation process which does not require any boundary layer and, consequently, the approximation holds uniformly, which fact is relevant in view of numerical approximations. We also discuss this occurrence more generally, for some classes of linear singularly perturbed partial differential equations.

Automated summary

The quantum lattice Boltzmann (qlB) algorithm is used to approximate the 1D Dirac equations and the non-relativistic Schrodinger equation. This method provides accurate solutions in the non-relativistic limit, but with a small error.