新古典电荷理论中的若干数学问题
Some mathematical problems in a neoclassical theory of electric charges
摘要：
我们在这里研究了一些数学问题，我们最近推出的新古典理论的电磁现象，其中收费表示的复值波函数在薛定谔波动力学。在非相对论性的情况下，电荷的动力学是由一个系统的非线性薛定谔方程耦合的电磁场，我们证明了它的波函数的中心收敛在宏观制度的轨迹由牛顿方程与洛伦兹力提供的波函数保持本地化。对一类同场方程的时间多谐解的研究表明，当自由电荷尺寸远大于玻尔半径时，它们满足Planck-Einstein关系，并且氢原子的非线性本征值问题的能级收敛到已知的线性薛定谔算符的能级.
英文摘要：
We study here a number of mathematical problems related to our recently introduced neoclassical theory for the electromagnetic phenomena in which charges are represented by complex valued wave functions as in the Schrodinger wave mechanics. Dynamics of charges in the non-relativistic case is governed by a system of nonlinear Schrodinger equations coupled with the electromagnetic fields, and we prove for it that the centers of wave functions converge in macroscopic regimes to trajectories of points governed by the Newton's equations with the Lorentz forces provided the wave functions remain localized. Exact solutions in the form of localized accelerating solitons are found. Our studies of a class of time multiharmonic solutions of the same field equations show that they satisfy Planck-Einstein relation and that the energy levels of the nonlinear eigenvalue problem for the hydrogen atom converge to the well-known energy levels of the linear Schrodinger operator when the free charge size is much larger than the Bohr radius.
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