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Volovich, Igor V.

Complete integrability of quantum and classical dynamical systems. (English) Zbl 1436.81036

\(p\)-Adic Numbers Ultrametric Anal. Appl. 11, No. 4, 328-334 (2019).
Summary: It is proved that the Schrödinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Integrals of motion are presented. A similar statement is proved for classical dynamical systems in terms of Koopman’s approach to dynamical systems. Examples of explicit reduction of quantum and classical dynamics to the family of harmonic oscillators by using direct methods of scattering theory and wave operators are given.

Cited in 4 Documents

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
81Q80 Special quantum systems, such as solvable systems

Keywords:

complete integrability; quantum dynamical system; classical dynamical system
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References:

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