Geodesic
Liouville's equation as a~Schr\"odinger equation
Izvestiya. Mathematics , Tome 78 (2014) no. 4, pp. 744-757.

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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: Hermitian operator, invariant measure.
Mots-clés : Weyl quantization, non-selfadjoint extension, invariant manifold 
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V. V. Kozlov. Liouville's equation as a~Schr\"odinger equation. Izvestiya. Mathematics , Tome 78 (2014) no. 4, pp. 744-757. http://geodesic.mathdoc.fr/item/IM2_2014_78_4_a3/

[1] A. A. Kirillov, “Geometricheskoe kvantovanie”, Dinamicheskie sistemy – 4, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 4, VINITI, M., 1985, 141–176 | MR | Zbl

[2] M. A. Naĭmark, Linear differential operators, v. I, II, Frederick Ungar Publishing Co., New York, 1967, 1968, xiii+144 pp., xv+352 pp. | MR | MR | MR | Zbl | Zbl

[3] I. Prigogine, Non-equilibrium statistical mechanics, Monographs in Statistical Physics and Thermodynamics, I, Interscience Publishers John Wiley Sons, Inc., New York–London, 1962, viii+319 pp. | MR | Zbl | Zbl

[4] I. M. Gel'fand, G. E. Shilov, Generalized functions, v. 3, Theory of differential equations, Academic Press, New York–London, 1967, x+222 pp. | MR | MR | Zbl | Zbl

[5] M. V. Berry, J. P. Keating, “$H=xp$ and the Riemann zeros”, Supersymmetry and trace formulae: chaos and disorder, Cambridge, UK, September 8–20, 1997, NATO ASI Series. Series B. Physics, 370, eds. I. V. Lerner et al., Kluwer Academic Press, New York, NY, 1999, 355–367 | Zbl

[6] A. N. Kolmogorov, “O dinamicheskikh sistemakh s integralnym invariantom na tore”, Dokl. AN SSSR, 93:5 (1953), 763–766 | MR | Zbl

[7] V. V. Kozlov, “Dynamical systems with multivalued integrals on a torus”, Proc. Steklov Inst. Math., 256 (2007), 188–205 | DOI | MR | Zbl

[8] J. Moser, “On the volume elements on a manifold”, Trans. Amer. Math. Soc., 120:2 (1965), 286–294 | DOI | MR | Zbl

[9] V. V. Kozlov, “On Bohl's argument theorem”, Math. Notes, 93:1 (2013), 83–89 | DOI | DOI | Zbl

[10] V. V. Kozlov, “Thermal equilibrium in the sense of Gibbs and Poincaré”, Dokl. Math., 65:1 (2002), 125–128 | MR | Zbl

[11] D. V. Treshchev, “An estimate of irremovable nonconstant terms in the reducibility problem”, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl. Ser. 2, 168, ed. V. V. Kozlov, Amer. Math. Soc., Providence, RI, 1995, 91–128 | MR | Zbl