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Volume 31, Issue 5
The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation

Renato Spigler

Commun. Comput. Phys., 31 (2022), pp. 1341-1361.

Published online: 2022-05

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  • Abstract

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 $(ω_c\tau)^{−1},$ 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 $\tau$ 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 occurrence more generally, for some classes of linear singularly perturbed partial differential equations.

  • AMS Subject Headings

81Q05, 35Q40, 35J10, 35B25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1341, author = {Spigler , Renato}, title = {The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {1341--1361}, abstract = {

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 $(ω_c\tau)^{−1},$ 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 $\tau$ 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 occurrence more generally, for some classes of linear singularly perturbed partial differential equations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0001}, url = {http://global-sci.org/intro/article_detail/cicp/20507.html} }
TY - JOUR T1 - The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation AU - Spigler , Renato JO - Communications in Computational Physics VL - 5 SP - 1341 EP - 1361 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2022-0001 UR - https://global-sci.org/intro/article_detail/cicp/20507.html KW - Schrödinger equation, hyperbolic Schrödinger equation, Dirac equations, quantum Lattice Boltzmann, Klein-Gordon equation, singular perturbations. AB -

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 $(ω_c\tau)^{−1},$ 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 $\tau$ 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 occurrence more generally, for some classes of linear singularly perturbed partial differential equations.

Renato Spigler. (2022). The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation. Communications in Computational Physics. 31 (5). 1341-1361. doi:10.4208/cicp.OA-2022-0001
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