Quantum quasiballistic dynamics and thick point spectrum

Abstract

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schrödinger operators. MSC (2010): primary 81Q10. Secondary: 28A80, 35J10.

Hydrogen atom bound states whose spectral measures have positive upper fractal dimensions

“…Recently, the present authors have shown in [2] that for systems with thick point spectrum, Baire generally, each initial state has 0-lower and 1-upper generalized fractal dimensions, 0 < q < 1. Here, we are in a different setting.…”

“…Finally, in Section 3 the proof of Theorem 1.1 is presented. The ingredients of the proof are: some robust arguments developed in [2], combined with a fine analysis of generalized fractal dimensions of pure point measures of the H-atom (Proposition 3.1).…”

“…Proposition 3.2 (Proposition 3.1 in [2]). For every q ∈ (0, 1) and every 0 ≤ γ ≤ 1, each of the sets…”

Slow dynamics for self-adjoint semigroups and unitary evolution groups

“…for purely absolutely continuous spectrum can be seen as the counterpart of the following situation detailed in [2]: for operators with pure point spectrum dense in an interval, there exists a generic set of states ψ whose spectral measures have maximal upper generalized dimension D + µ T ψ (q) = 1 (0 < q < 1); such states are, therefore, (weakly) delocalized (see [2,4,15,17]).…”

Some generic fractal properties of bounded self-adjoint operators

“…2) in some topological spaces of self-adjoint operators, the set of elements with purely singular continuous spectra is generic (the conclusion of the so-called Wonderland Theorem [25]); 3) dense point spectrum imply a form of dynamical instability [1]; etc. Here, we present two new subtle properties related to generic dimensional properties of spectral measures, which are summarized in Theorems 1.1 and 1.2.…”