The non-relativistic Schrödinger equation can be derived as a limit of the Klein-Gordon equation
The claim that the non-relativistic Schrödinger equation can be derived as a limit of the Klein-Gordon equation is fully supported by multiple mathematical and physical analyses.
The retrieved literature consistently confirms that under appropriate asymptotic limits (such as low frequency, low velocity, or taking the speed of light to infinity), both linear and nonlinear Klein-Gordon equations reduce to or can be approximated by the Schrödinger equation.
Yong Lu, Zhifei Zhang. Higher order asymptotic analysis of the Klein–Gordon equation in the non-relativistic limit regime. 2017. https://doi.org/10.3233/asy-171414
The study demonstrates that the nonlinear Klein-Gordon equation can be approximated by nonlinear Schrödinger equations in the non-relativistic limit regime.
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P. -A. Gourdain. The Klein-Gordon equation with relativistic mass: a relativistic Schrödinger equation. 2025. http://arxiv.org/abs/2501.07798v2
The work notes that the non-relativistic, quasi-static limit of the Klein-Gordon equation simply yields the Schrödinger equation under general conditions.
Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy. Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Asymptotics. 2025. http://arxiv.org/abs/2511.08724v1
The research analyzes the low-frequency phase space regime where the Klein-Gordon equation is approximable by the usual Schrödinger equation.
Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy. Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates. 2025. http://arxiv.org/abs/2509.09518v1
The study establishes robust global estimates showing the Klein-Gordon equation approaches the Schrödinger equation in the non-relativistic limit.
Dario Bambusi, Andrea Belloni, Filippo Giuliani. Non relativistic limit of the nonlinear Klein-Gordon equation: Uniform in time approximation of KAM solutions. 2025. http://arxiv.org/abs/2501.17691v2
The investigation proves that solutions to the cubic nonlinear Klein-Gordon equation converge to solutions of the cubic non-linear Schrödinger equation in the non-relativistic limit.
Zhen Lei, Yifei Wu. Non-relativistic limit for the cubic nonlinear Klein-Gordon equations. 2023. http://arxiv.org/abs/2309.10235v1
The paper confirms the non-relativistic limit of defocusing cubic nonlinear Klein-Gordon equations where modulated profiles are solutions to a Schrödinger equation.
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