Abstract
We propose a sketch for a proof of an interesting theorem on the evolution of coherent states, whose statement has been first presented in Paul (Séminaire: Équations aux Dérivées Partielles. 2007–2008, pages Exp. No. IV, 21. École Polytech., Palaiseau, 2009), and give some further insights on the asymptotic behavior, involving weak KAM theory.
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Notes
We just recall that \(\varLambda\subset T^{*}\mathbb{T}^{n}\) is Lagrangian if (i) \(\dim \varLambda =\dim \mathbb{T}^{n}=n\) and (ii) the symplectic 2-form \(\omega=\sum_{i=1}^{n} dp_{i}\wedge dq^{i}\) restricted on Λ is vanishing: ω| Λ =0.
By Maslov-Hörmander theory.
In the present case we are concerning with a Hamiltonian vector field: \(X_{h}: T^{*}\mathbb{T}^{n}\to TT^{*}\mathbb{T}^{n}\).
At the first order of some hierarchical sequence or of the approximation of the Madelung equation.
For the definition of Tonelli Hamiltonian see [6].
To be precise, in [1] the authors considered the mathematical setting of the forward viscosity solutions: in the mechanical case under consideration here, the standard backward solution setting is easily restored by exchanging S by −S: this is the motivation for which their Stable Manifold becomes here Unstable Manifold.
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Cardin, F., Vazzoler, S. Coherent States and Quantum Asymptotic Features by Weak KAM Theory. Acta Appl Math 132, 189–197 (2014). https://doi.org/10.1007/s10440-014-9895-y
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DOI: https://doi.org/10.1007/s10440-014-9895-y