On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation

In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2\pi \Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b \in S^m (\mathbb{T}^n \times \mathbb{R}^n)$. Subsequently, we consider $\mathrm{Op}^w_\hbar(H)$ when $H=\frac{1}{2} |\eta|^2 + V(x)$ where $V \in C^\infty (\Bbb T^n;\Bbb R)$ and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schr\"odinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schr\"odinger equation to the solution of the Liouville equation on $\Bbb T^n \times \Bbb R^n$ written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space $H^{1} (\mathbb{T}^n; \Bbb C)$ with phase functions in the class of Lipschitz continuous weak KAM solutions (of positive and negative type) of the Hamilton-Jacobi equation $\frac{1}{2} |P+ \nabla_x v_\pm (P,x)|^2 + V(x) = \bar{H}(P)$ for $P \in \ell \Bbb Z^n$ with $\ell >0$, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of $P+ \nabla_x v_\pm$

Measurement of the ϕ\phi meson parameters with CMD-2 detector at VEPP-2M Collider

About 300 000 $e^+e^-\to \phi\to K^0_L K^0_S$ events in the center of mass energy range from 984 to 1040 MeV were used for the measurement of the $\phi$ meson parameters. The following results have been obtained: $\sigma_0 = (1367 \pm 15 \pm 21) nb, m_{\phi}=(1019.504 \pm 0.011 \pm 0.033) MeV/c^2, \Gamma_\phi=(4.477 \pm 0.036 \pm 0.022) MeV, \Gamma_{e^+e^-}\cdot B(\phi\to K^0_L K^0_S) = (4.364 \pm 0.048 \pm 0.065)\cdot 10^{-4}$ MeV.Comment: 13 pages, 5 figures, 5 table