Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time

Article type: Research Article

Authors: Bambusi, Dario | Graffi, Sandro^; | Paul, Thierry

Affiliations: Dipartimento di Matematica, Università di Milano, Italy E‐mail: [email protected] | Dipartimento di Matematica, Università di Bologna, Italy E‐mail: [email protected] | CEREMADE, Université de Paris‐Dauphine, France E‐mail: [email protected]

Note: [] Investigation is supported by the University of Bologna, Funds for selected research topics.

Abstract: Let \mathcal{H} be a holomorphic Hamiltonian of quadratic growth on \mathbb{R}^{2n}, b a holomorphic exponentially localized observable, H,B the corresponding operators on L^2(\mathbb{R}^n) generated by Weyl quantization, and U(t)=\exp{\mathrm{i}Ht/\hbar}. It is proved that the L^2 norm of the difference between the Heisenberg observable B_t=U(t)BU(-t) and its semiclassical approximation of order N-1 is majorized by K^N N^{(6n+1)N}\hbar^{-4/9}(-\hbar\log\hbar)^N for t\in [0,T_n(\hbar)], where T_n(\hbar):=-2\log\hbar/[\alpha(6n+3)(N-1)] and \alpha:=\Vert\mathrm{Hess}_{(x,\xi)}\,\mathcal{H}\Vert. Choosing a suitable N(\hbar) the error is majorized by C\hbar^{\log\vert\log\hbar\vert}, 0\leq t\leq \vert\log\hbar\vert/\log\vert\log\hbar\vert (here K and C are explicit constants independent of N,\hbar).

Journal: Asymptotic Analysis, vol. 21, no. 2, pp. 149-160, 1999