Integrability, chaos and Galois groups in quantum dynamical systems

The talk will discuss a theorem stating that any quantum dynamical system is integrable. It will be explained how this theorem is consistent with ergodicity, mixing and other properties of quantum chaos.
In more detail, a quantum dynamical system is given by a unitary representation of the additive group of real numbers, called the evolution operator. By means of the spectral theorem we prove that this unitary representation is unitary equivalent to a system of harmonic oscillators which is integrable. As a criterion for the efficiency of constructing integrals of motion, we consider the solvability of the corresponding Galois group, by analogy with the efficiency of computing the roots of a polynomial in the main theorem of algebra.
A brief discussion of integrability of classical dynamical systems, wave operators in scattering theory, open quantum systems, the hypothesis of thermalization of eigenstates, and a categorical formulation will also be given.

References

1. I. V. Volovich, “Complete integrability of quantum and classical dynamical systems”, $p$-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334        
2. I. V. Volovich, “Ob integriruemosti dinamicheskikh sistem”, Izbrannye voprosy matematiki i mekhaniki, Sbornik statei. K 70-letiyu so dnya rozhdeniya akademika Valeriya Vasilevicha Kozlova, Trudy MIAN, 310, MIAN, M., 2020, 78–85      ; I. V. Volovich, “On the integrability of dynamical systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77