Nonrelativistic limit of Klein-Gordon-Maxwell to Schrödinger-Poisson

<abstract abstract-type="TeX"><p>We prove that in the nonrelativistic limit <i>c</i> → ∞, where <i>c</i> is the speed of light, solutions of the Klein-Gordon-Maxwell system on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] converge in the energy space <i>C</i>([0, <i>T</i>];<i>H</i><sup>1</sup>) to solutions of a Schrödinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons. The proof relies on bilinear spacetime estimates related to the Klainerman-Machedon estimates, but taking into account the variation of the parameter <i>c</i>. A crucial fact is that the system has a null form structure in Coulomb gauge, as proved by Klainerman-Machedon.

[1]  Paul R. Chernoff,et al.  Dirac and Klein-Gordon equations: Convergence of solutions in the nonrelativistic limit , 1981 .

[2]  H. C. Corben,et al.  Classical Mechanics (2nd ed.) , 1961 .

[3]  T. Tao,et al.  Endpoint Strichartz estimates , 1998 .

[4]  Sergiu Klainerman,et al.  Space-time estimates for null forms and the local existence theorem , 1993 .

[5]  Christopher D. Sogge,et al.  Fourier Integrals in Classical Analysis , 1993 .

[6]  K Veselić.  On the nonrelativistic limit of the bound states of the Klein-Gordon equation , 1983 .

[7]  N. Mauser,et al.  (Semi)-Nonrelativistic Limits of the Dirac Equation with External Time-Dependent Electromagnetic Field , 1998 .

[8]  町原 秀二,et al.  The Nonrelativistic Limit of the Nonlinear Klein-Gordon Equation (調和解析学と非線形偏微分方程式) , 2000 .

[9]  T. Sideris,et al.  Local regularity of nonlinear wave equations in three space dimensions , 1993 .

[10]  S. Klainerman,et al.  On the Maxwell-Klein-Gordon equation with finite energy , 1994 .

[11]  F. Castella,et al.  L2 Solutions to the Schrödinger–Poisson System: Existence, Uniqueness, Time Behaviour, and Smoothing Effects , 1997 .

[12]  Kenji Nakanishi,et al.  From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrödinger equations , 2002 .

[13]  K. Nakanishi,et al.  Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger , 2003 .

[14]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[15]  N. Mauser,et al.  The Selfconsistent Pauli Equation , 2001 .

[16]  K. Nakanishi.  Nonrelativistic limit of scattering theory for nonlinear Klein-Gordon equations (Spectral and Scattering Theory and Related Topics) , 2001 .

[17]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[18]  Kenji Nakanishi,et al.  Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations , 2002 .

[19]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[20]  Branko Najman.  The nonrelativistic limit of the nonlinear Klein-Gordon equation , 1990 .