Correlation dimension Wonderland theorems

Abstract

Existence of generic sets of self-adjoint operators, related to correlation dimensions of spectral measures, is investigated in separable Hilbert spaces. Typical results say that, given an orthonormal basis, the set of operators whose corresponding spectral measures are both 0-lower and 1-upper correlation dimensional is generic. The proofs rely on details of the relations among Fourier transform of spectral measures and Hausdorff and packing measures on the real line. Then such results are naturally combined with the Wonderland theorem. Applications are to classes of discrete one-dimensional Schrödinger operators and general (bounded) self-adjoint operators as well. Physical consequences include a proof of exotic dynamical behavior of singular continuous spectrum in some settings.

A characterization of singular packing subspaces with an application to limit-periodic operators

“…However, the hypotheses in Theorem 4.2 below are weaker than in Theorem 4.1 in [5], and therefore easier to meet. In particular, we were not able to apply the method discussed in [5] to the class (1.1) of limit periodic operators, since the estimation of upper correlation dimension seems to be far from trivial in this case.…”

A note on spectrum and quantum dynamics

“…In this paper we go beyond. Namely, we presented results that show this phenomenon, for X a and X b , through of a more robust dynamic quantity (B) than these discussed in [6,7]; more specifically, thanks to the RAGE Theorem, we evaluate arbitrary decaying rates of (B) (Theorem 1.2). Moreover, in this work, we also developed an argument involving separability to extend partially such results to the class of (continuous) Schrödinger operators X C (Theorem 1.3).…”

“…is a dense G δ set in X Γ is a direct application of (2) and (3) [6]. For the convenience of the reader, we presented in details a simple proof of this fact here.…”

Refined scales of weak-mixing dynamical systems: typical behaviour

“…Take any dense countable collection in , say {S −1 j R S j } (such collection exists since G is separable), and define B := j A S −1 j RS j ; since, by the Claim, each A S −1 j RS j is a dense G δ subset of [1] ⊥ , it follows that B satisfies the properties required in the statement of the proposition. 4. Generic sets PROPOSITION 4.1.…”

“…Since the proofs of both items are similar, we will only discuss item (a). By combining adapted versions of [4,Lemmas 3.3 and 4.6], one has…”