The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation  

作　　者：Renato Spigler 

机构地区：[1]Department of Mathematics and Physics,Roma Tre University,1 Largo S.Leonardo Murialdo,00146 Rome,Italy,and Complex Systems Institute ISC–CNR

出　　处：《Communications in Computational Physics》2022年第5期1341-1361,共21页计算物理通讯（英文）

摘　　要：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 the hyperbolic version of the non-relativistic Schrödinger equation,whose solution is thus obtained at the price of an additional small error.Such an error is of order of(ωCτ)-1,where ω_(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,andτa 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 occurrence more generally,for some classes of linear singularly perturbed partial differential equations.

关 键 词：Schrödinger equation hyperbolic Schrödinger equation Dirac equations quantum Lattice Boltzmann Klein-Gordon equation singular perturbations 

分 类 号：O17[理学—数学]