The Hyperbolic Schr ¨odinger 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) Schr ¨odinger equation. We point out that the qlB method actually approximates the hyperbolic version of the non-relativistic Schr ¨odinger equation, whose solution is thus obtained at the price of an additional small error. Such an error is of order of (wCt)-1, where wC := mc2¯h is the Compton frequency, ¯h being the reduced Planck constant, m the rest mass of the electrons, c the speed of light, and t 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 boundary layer and, consequently, the approximation holds uniformly, which fact is relevant in view of numerical approximations. We also discuss this occurrencemore generally, for some classes of linear singularly perturbed partial differential equations