On Integrability of Dynamical Systems
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Integrability of Dynamical Systems I. V. Volovich a Received January 19, 2020; revised January 19, 2020; accepted May 8, 2020
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 Schr¨odinger equation in the multidimensional space under various fairly wide conditions on the potential. DOI: 10.1134/S0081543820050053
1. INTRODUCTION The concept of integrability of a dynamical system goes back to Newton, who integrated the twobody problem. An important contribution to the study of integrable systems was made by Euler, Bernoulli, Liouville, and many others (see [6]). According to Liouville’s theorem, if a Hamiltonian system with n degrees of freedom has n independent integrals of motion, then it can be integrated in quadratures. Integration in quadratures involves the calculation of integrals of “known” functions. However, the idea of which functions are considered known changes over time. When one tries to specify the concept of integrability in a precise way, it turns out that many different definitions are possible, each of which is of particular interest [6]. The same applies to the concept of non-integrability, the study of which was initiated by Poincar´e. The answer to the question of whether integrals of motion exist depends on the category of objects under consideration. For example, by the Kozlov theorem [5], a natural dynamical system on a compact Riemann surface does not have an analytic first integral independent of the energy integral if the genus of the surface is different from 0 and 1. At the same time, such integrals may exist in the category of smooth functions on a surface of the same topology. Thus, the condition of analyticity is an obstacle to the existence of integrals. Various aspects of the theory of integrable systems and the role of conservation laws are considered, in particular, in [1–4, 9, 10, 13, 16]. Deift [3] discusses an informal definition of an integrable dynamical system as a system for which we have enough information about its behavior at large times. Polynomial conservation laws in quantum mechanics are investigated in the recent papers [8, 11] (see also the references therein). The complete integrability of the Schr¨odinger equation with a non-degenerate spectrum in Cn was proved in [7]. A quantum dynamical system is a pair (H, Ut ), where H is a separable Hilbert space and Ut a continuous one-parameter group of unitary operators in H. It was recently shown [14, 15] that any quantum dynamical system is unitarily equivalent to a set of non-interacting classical harmonic oscillators and is therefore completely in
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