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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     author = {V. V. Kozlov},
     title = {Liouville's equation as {a~Schr\"odinger} equation},
     journal = {Izvestiya. Mathematics },
     pages = {744--757},
     publisher = {mathdoc},
     volume = {78},
     number = {4},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2014_78_4_a3/}
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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/