Universal lower bounds for quantum diffusion
Let \(H\) be a self-adjoint operator in \({\mathcal H}=L^2 (\mathbb{R}^d)\) or in \({\mathcal H}= \ell^2(\mathbb{Z}^d)\). The authors study the asymptotic behaviour of the Cesaro mean \(\langle\langle|X|^p \rangle_{\psi(t)} \rangle(T)\) at time \(T\) of the moments of order \(p\) for \(\psi \in{\mathcal H}\). With the aid of the so-called double-space method an upper bound for \[ B_\psi (T,N):= T^{-1} \int^T_0 \bigl\|\psi_N(t) \bigr\|^2_{\mathcal H} dt \] is proved, where \(\psi_N(t) =F_{<N}e^{-itH} \psi\) and \(F_{<N}\) is the characteristic function of the closed ball of radius \(N\). This key estimate holds without assuming any continuity properties of the spectral measure \(\mu_\psi\) and yields lower bounds for \(\langle\langle|X|^p \rangle_{ \psi(t)} \rangle (T)\) in terms of the Fourier transform of \(\mu_\psi\). The obtained estimates are applied to derive necessary conditions for dynamic localization, provided \(H\) has pure point spectrum. In an appendix the converse of Strichartz' theorem on Fourier asymptotics of uniformly \(\alpha\)-Hölder continuous measures is proved.
scientific article; zbMATH DE number 1389440
scientific article