Spectral characterization of mixing evolutions in classical statistical physics (Q2708573)

Let \(V_t= e^{itL}\) be a unitary group acting on a Hilbert space \({\mathcal H}\), generated by the selfadjoint operator \(L\) with spectral representation \(L= \int^\infty_{-\infty}\lambda dE_\lambda\) and let \(\mu_h\), \(h\in{\mathcal H}\), denote the spectral measure determined by the nondecreasing function \(F_h(\lambda)=\langle h,E_\lambda h\rangle\), \(\lambda\in\mathbb{R}\).NEWLINENEWLINENEWLINEThe direct sum decomposition \({\mathcal H}_{sc}={\mathcal H}^D_{sc}\oplus {\mathcal H}^{ND}_{sc}\) of a singular continuous subspace of \({\mathcal H}\) is introduced, where for each \(h\in{\mathcal H}^D_{sc}\), \(\mu_h\in M_0\) (the Fourier transform of \(\mu_h\) converges to \(0\) at infinity) and for each \(h\in{\mathcal H}^{ND}_{sc}\), \(\nu\not\in M_0\) for each (not identically \(0\)) measure \(\nu\gg |\mu_h|\). It is proved that these subspaces are closed and \(L\)-invariant. Then, denote the corresponding spectra of reduced operators by \(\sigma^D_{sc}\) and \(\sigma^{ND}_{sc}\). A new decomposition of the spectrum \(\sigma\) of any selfadjoint operator into point, absolutely continuous, decaying singular, and nondecaying singular components is obtained: \(\sigma= \sigma_p\cup \sigma_{ac}\cup \sigma^D_{sc}\cup \sigma^{ND}_{sc}\).NEWLINENEWLINENEWLINEIn these terms the spectral characterization of classical dynamical systems (groups of measure-preserving transformations) \(\{S_t\}\) which are mixing is proposed. The necessary and sufficient condition proved is that the spectra of the corresponding unitary group \(V_tf(\omega)= f(S_t\omega)\) on \({\mathcal H}= L^2\ominus\{1\}\) were consistent of purely decaying part, i.e. \(\sigma(L)= \sigma_{ac}\cup \sigma^D_{sc}\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].