Abstract

A classical dynamical system may have smooth integrals of motion and not have analytic ones; i.e., the integrability property depends on the category of smoothness. Recently it has been shown that any quantum dynamical system is completely integrable in the category of Hilbert spaces and, moreover, is unitarily equivalent to a set of classical harmonic oscillators. The same statement holds for classical dynamical systems in the Koopman formulation. Here we construct higher conservation laws in an explicit form for the Schrödinger equation in the multidimensional space under various fairly wide conditions on the potential.

中文翻译：

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动力系统的可积性

摘要

经典的动力学系统可能具有平滑的运动积分，而没有解析的积分。即，可积性取决于光滑度的类别。最近，已经证明，任何量子动力学系统在希尔伯特空间范畴内都是完全可积分的，而且，其整体等效于一组经典的谐振子。对于考夫曼公式中的经典动力系统，也存在相同的说法。在这里，我们以各种形式在相当广泛的电势条件下，以多维形式的Schrödinger方程以显式形式构造了更高的守恒律。