The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic field
type: Item
instance of: scholarly article
title: The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic field
zbMATH Open document ID: 0618.46063
DOI: 10.1016/0022-1236(87)90067-X
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: The nonrelativistic limit, i.e. the limit as the light velocity c tends to \(\infty\), is discussed of the solution \(\psi^ c(t,x)\) for the imaginary-time Schrödinger equation \[ \partial_ t\psi^ c(t,x)=- [H^ c-mc^ 2]\psi^ c(t,x),\quad t>0,\quad x\in {\mathbb{R}}^ d, \] on the basis of its path integral or probabilistic representation obtained by \textit{T. Ichinose} and \textit{H. Tamura} [Proc. Japan Acad. Ser. A 62, 91-93 (1986; Zbl 0597.58038), Commun. Math. Phys. 105, 239-257 (1986; Zbl 0606.60060)]. Here \(H^ c\) is the Weyl quantized Hamiltonian corresponding to the relativistic classical Hamiltonian \([c^ 2(p- eA(x))^ 2+m^ 2c^ 4]^{1/2}+e\Phi(x)\) of a spinless particle with mass \(m>0\) and charge e in an electromagnetic field. The vector and scalar potentials \(A(x)\) and \(\Phi(x)\) are assumed to be respectively \({\mathbb{R}}^ d\)- and \({\mathbb{R}}\)-valued \(C^{\infty}\) functions which together with all their derivatives are bounded in \({\mathbb{R}}^ d\). It is shown that \(\psi^ c(t,\cdot)\) converges, in both \(L^ 2({\mathbb{R}}^ d)\) and \(C_{\infty}({\mathbb{R}}^ d)\) uniformly on every finite interval in \(t\geq 0\), to the solution \(\psi\) (t,\(\cdot)\) for the heat equation \[ \partial_ t\psi (t,x)=-[(1/2m)(-i\partial-eA(x))^ 2+e\Phi(x)]\psi (t,x),\quad t>0,\quad x\in {\mathbb{R}}^ d, \] i.e. to its Feynman-Kac-Itô representation.
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 81V10
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35S10
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 81S40
zbMATH DE Number: 4002843
zbMATH Keywords: Weyl pseudo-differential operator
zbMATH Keywords: nonrelativistic limit
zbMATH Keywords: imaginary-time Schrödinger equation
zbMATH Keywords: path integral or probabilistic representation
zbMATH Keywords: Weyl quantized Hamiltonian
zbMATH Keywords: spinless particle
zbMATH Keywords: electromagnetic field
zbMATH Keywords: heat equation
zbMATH Keywords: Feynman-Kac-Itô representation
author: Takashi Ichinose
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4002843