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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 4, Pages 109–122 (Mi izv8143)  

Liouville's equation as a Schrödinger equation

V. V. Kozlov

Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We show that every non-negative solution of Liouville's equation for an arbitrary (possibly non-Hamiltonian) dynamical system admits a factorization $\psi\psi^*$, where $\psi$ satisfies a Schrödinger equation of special form. The corresponding quantum system is obtained by Weyl quantization of a Hamiltonian system whose Hamiltonian is linear in the momenta. We discuss the structure of the spectrum of the special Schrödinger equation on a multidimensional torus and show that the eigenfunctions may have finite smoothness in the analytic case. Our generalized solutions of the Schrödinger equation are natural examples of non-selfadjoint extensions of Hermitian differential operators. We give conditions for the existence of a smooth invariant measure of a dynamical system. They are expressed in terms of stability conditions for the conjugate equations of variations.

Keywords: Weyl quantization, Hermitian operator, non-selfadjoint extension, invariant manifold, invariant measure.

DOI: https://doi.org/10.4213/im8143

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English version:
Izvestiya: Mathematics, 2014, 78:4, 744–757

Bibliographic databases:

Document Type: Article
UDC: 517.43
MSC: 70G60, 70H14, 70K42, 81Q10
Received: 04.07.2013

Citation: V. V. Kozlov, “Liouville's equation as a Schrödinger equation”, Izv. RAN. Ser. Mat., 78:4 (2014), 109–122; Izv. Math., 78:4 (2014), 744–757

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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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