Complete Integrability of Quantum and Classical Dynamical Systems

@article{Volovich2019CompleteIO,
  title={Complete Integrability of Quantum and Classical Dynamical Systems},
  author={Igor V. Volovich},
  journal={p-Adic Numbers, Ultrametric Analysis and Applications},
  year={2019},
  volume={11},
  pages={328-334}
}
  • I. Volovich
  • Published 1 October 2019
  • Physics, Mathematics
  • p-Adic Numbers, Ultrametric Analysis and Applications
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… 
View on Springer
arxiv.org
3 Citations
Background Citations
1

3 Citations

On Integrability of Dynamical Systems

  • I. Volovich
  • Mathematics
    Proceedings of the Steklov Institute of Mathematics
  • 2020
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…

Effective Heisenberg equations for quadratic Hamiltonians

  • A. Teretenkov
  • Physics, Mathematics
    International Journal of Modern Physics A
  • 2022
We discuss effective quantum dynamics obtained by averaging projector with respect to free dynamics. For unitary dynamics generated by quadratic fermionic Hamiltonians we obtain effective Heisenberg…

Quadratic conservation laws for equations of mathematical physics

  • V. Kozlov
  • Mathematics
    Russian Mathematical Surveys
  • 2020
Linear systems of differential equations in a Hilbert space are considered that admit a positive-definite quadratic form as a first integral. The following three closely related questions are the…

References

SHOWING 1-10 OF 47 REFERENCES

Complete Integrability of Quantum and Classical Dynamical Systems

  • I. Volovich
  • Physics, Mathematics
    p-Adic Numbers, Ultrametric Analysis and Applications
  • 2019
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…

Polynomial Conservation Laws in Quantum Systems

We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized…

Classical and quantum superintegrability with applications

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum…

Hamiltonian methods in the theory of solitons

The Nonlinear Schrodinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples…

Linear systems with quadratic integral and complete integrability of the Schrödinger equation

  • V. Kozlov
  • Mathematics
    Russian Mathematical Surveys
  • 2019
where m ∈ Z+. If the operators A and B are non-singular, then n is even and (1) is a Hamiltonian system, with symplectic structure given by the antisymmetric operator BA−1 and with Hamiltonian f [1].…

Quantum stochastic equation for the low density limit

A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for…

On the Hamiltonian property of linear dynamical systems in Hilbert space

  • D. V. TreshchevA. A. Shkalikov
  • Mathematics
  • 2017
Conditions for the operator differential equation $$\dot x = Ax$$x˙=Ax possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional case, it suffices…

Hamiltonian aspects of quantum theory

This paper considers Hamiltonian structures related to quantum mechanics. We do not discuss structures used in the quantization of classical Hamil� tonian or Lagrangian systems (which was first done…

Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their…

Quantum Physics: A Functional Integral Point of View

This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modem physics: statistical physics, quantum mechanics, and quantum fields. The unity of…