Hartree型方程的Escherichfest系统和静止点谱
The Ehrenfest system and the rest point spectrum for a Hartree-type Equation
摘要：
按照Escherfest的方法，量子-经典对应的问题可以在近似为$\h\to 0$量子力学态的相干函数类中处理。这个想法导致了一族常微分方程组，称为Escherichfest M-系统（M= 0，1，2，.），形式上等价于线性薛定谔方程的半经典近似。在本文中，一个类似的方法进行了非线性Hartree型方程的光滑积分核。它演示了如何量子特性可以直接从相应的Escherichfest系统中检索，而无需求解量子方程：光谱的半经典渐近从静止点解获得。其中一个关键步骤是推导出适用于相干量子态的修正的非线性叠加原理。
英文摘要：
Following Ehrenfest's approach, the problem of quantum-classical correspondence can be treated in the class of trajectory-coherent functions that approximate as $\h\to 0$ a quantum-mechanical state. This idea leads to a family of systems of ordinary differential equations, called Ehrenfest M-systems (M=0,1,2,...), formally equivalent to the semiclassical approximation for the linear Schroedinger equation. In this paper a similar approach is undertaken for a nonlinear Hartree-type equation with a smooth integral kernel. It is demonstrated how quantum characteristics can be retrieved directly from the corresponding Ehrenfest systems, without solving the quantum equation: the semiclassical asymptotics for the spectrum are obtained from the rest point solution. One of the key steps is derivation of a modified nonlinear superposition principle valid in the class of trajectory-coherent quantum states.
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