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

1 Introduction and results

A study of packing continuity and singularity of bounded self-adjoint operators is performed. Our main goals here are to present a dynamical characterization of singular α-packing subspaces (Theorem 3.9), and use this result (along with the well-known equivalence between strong convergence and strong dynamical convergence of operators) to prove, for some metric space of self-adjoint operators, that the set of operators whose spectral measures have upper packing dimension equal to 1 is a Gδ set (Theorem 4.2). The dynamical characterization will be obtained in Section 3 through the notion of uniformly α-Hölder singular measures and Lemma 3.4 (which is a “singular version” of Strichartz’s Theorem [15]).

To put our work into perspective, we mention that, although important, it is not always easy to present dynamical characterizations of spectral subspaces of a self-adjoint operator T. For a vector ξ, denote the so-called return probability at time t by pξ⁢(t):=|〈ξ,e-i⁢t⁢T⁢ξ〉|2, clearly a dynamical quantity. It is well known (see, for instance, [13, Theorem 13.5.5]) that the absolutely continuous subspace of T is the closure of the vectors ξ so that pξ∈L1⁢(ℝ). By using Strichartz’s Theorem, recalled in Section 3, Last [11, Theorem 5.3] has presented a dynamical characterization of the α-Hausdorff continuous subspace ℋα⁢HcT of T (the definition is similar to ℋα⁢PcT in Proposition 2.5) in terms of averages

that is, given 0<α<1, then for all ε>0,

Here we have a parallel of this result (Theorem 3.9) for the α-packing singular subspace; for this, we have introduced an appropriate quantity in equation (3.1) and the concept of UαHS measures (see Definition 3.1).

We apply our general packing results to a class of limit-periodic operators. These are discrete one-dimensional ergodic Schrödinger operators, denoted by Hg,τκ, acting in l2⁢(ℤ), whose action is given by

with

here, κ belongs to a Cantor group Ω, τ:Ω→Ω is a minimal translation on Ω and g:Ω→ℝ a continuous sampling function, i.e., g∈C⁢(Ω,ℝ) with the norm of uniform convergence. For more details, see [1].

For each κ∈Ω, let Xκ be the set of limit-periodic operators Hg,τκ given by (1.1) and (1.2), with metric

We shall prove the following

We stress that in a recent work [5], it was proven, again for some metric spaces of self-adjoint operators, that the set of operators whose spectral measures have upper correlation dimension equal to 1 is a Gδ. Since every Borel and finite measure on ℝ whose upper correlation dimension is one has upper packing dimension equal to 1, one can conclude that if the hypotheses in [5, Theorem 4.1] are fulfilled, then Theorem 4.2 follows. However, the hypotheses in Theorem 4.2 below are weaker than in [5, Theorem 4.1], 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.

The notorious Wonderland Theorem [14] gives sufficient conditions for a set of self-adjoint operators, whose spectrum is purely singular continuous, to be generic. Theorem 4.2 is another step towards a better comprehension of the typical (in a topological sense) behavior of the spectral measures of a self-adjoint operator, since it generalizes Wonderland theorem in the sense that every application of Wonderland Theorem discussed in [14] for bounded operators can be extended to results about generic sets of operators whose spectral measures have upper packing dimension equal to 1 (on top of singular continuous spectrum).

The proof of Theorem 1.1 is presented in Section 5, since a previous preparation is required. Let us briefly discuss some dynamical consequences of Theorem 1.1 within the context of the unitary evolution group e-i⁢t⁢T, that is, the solution to the corresponding Schrödinger equation [13]. Regarding the dynamics generated by a self-adjoint operator T acting on l2⁢(ℤ), the growth of the width of “quantum wave packets” is usually probed by the algebraic growth of the (time-averaged) q-moments, q>0,

of the position operator at time t>0; such wave packets are represented here by the initial state δ0. To describe this algebraic growth 〈MTq〉⁢(t)∼tq⁢β⁢(q) for large t, one usually considers the lower and the upper transport exponents, given respectively by

The following result, extracted from [8], gives basic properties of moments within the setting of bounded self-adjoint operators T acting on l2⁢(ℤ), in particular, for bounded discrete Schrödinger operators (that is, operators whose action is given by (1.1) with bounded potentials (Vn)).

In case βT+⁢(q)=1 (resp. βT-⁢(q)=1), for all q>0, the corresponding dynamics is called quasi-ballistic (resp. ballistic). We shall make use of the general inequality (see Definition 2.2 for the description of the upper packing dimension dimP+⁡(⋅))

proven in [10]. It is worth mentioning that βT-⁢(q) is related to the upper Hausdorff dimension [9, 2], but we will not use them in this work. Following the discussion above and Theorem 1.1, one has:

In Section 2, we recall suitable results on decompositions of Borel measures with respect to packing measures and dimensions, and how these components are related to pointwise scaling exponents and generalized dimensions. In Section 3, we present a singular version-like of Strichartz’s Theorem [15, 11] for finite measures, which is used to give a dynamical characterization of packing singular subspaces of bounded self-adjoint operators in general separable Hilbert spaces (see Theorem 3.9). The very same arguments lead to a version for spectral measures restricted to any given subinterval of the real line.

In Section 4, we state and prove Theorem 4.2. In the last section, as previously stated, we present the proof of Theorem 1.1. It is important to emphasize that the results of Section 4 can be used to prove, for every general class of bounded operators (including classes of not necessarily Schrödinger-like operators) discussed in [14], existence of generic sets of operators whose spectral measures have upper packing dimensions equal to 1.

Some words about notation: ℋ will always denote a complex separable Hilbert space. The spectrum of a self-adjoint operator T is denoted by σ⁢(T). Unless explicitly stated, μ will always indicate a finite nonnegative Borel measure on ℝ, and its restriction to a Borel set E will be denoted by μ;E; it is singular if μ and the Lebesgue measure are mutually singular; it is supported on a Borel set S if μ⁢(ℝ∖S)=0. We denote by supp⁡(μ) the support of μ, that is, the complement of the largest open set B with μ⁢(B)=0, and PT⁢(E) represents the spectral projection of T associated with the Borel set E⊂ℝ.

2 Basic tools

In this section, we recall important decompositions of Borel measures on ℝ with respect to packing measures and dimensions, along with the corresponding spectral decompositions of self-adjoint operators. We also recollect how these decompositions are related to the upper and generalized dimensions. This discussion parallels the rather well-known corresponding Hausdorff properties.

3 Dynamical characterization of ℋα⁢PsT

In Definition 3.1, we introduce a class of special measures for which a singular version-like of Strichartz’s Theorem (Theorem 3.3) will be deduced (see Lemma 3.4). The arguments there will be used to prove the main result of this section, that is, Theorem 3.9. We assume, in what follows, that 0≠T represents a bounded self-adjoint operator on ℋ.

Besides the application to limit-periodic operators ahead, the next results have interest on their own (as well as the results in Section 4). Next a description of α-packing singular measures in terms of UαHS ones.

Now we introduce another quantity that has proven useful. For a finite and positive Borel measure μ on ℝ and every t∈ℝ, write

If the measure μ is a spectral measure μψT, we denote Ξμ⁢(t) by ΞψT⁢(t).

Recall that μ is uniformly α-Hölder continuous [11] if there are positive and finite constants C and r0, so that for each 0<r<r0 and for μ a.e. x, μ⁢(B⁢(x;r))≤C⁢rα. The following result is known as Strichartz’s Theorem (it is, in fact, an adapted version of the Theorem presented in [11] for f≡1∈L2⁢(ℝ;d⁢μ); see also [15] for the original result).

The proof of Theorem 3.3 in [11], after some preparation, essentially consists of showing that there exist constants D~ and t0>0 so that

In such proofs, in case μ=μψT, the parameter “t” comes from the time evolution e-i⁢t⁢T⁢ψ and the left hand side of (3.2) coincides with the average return probability 〈pψ〉⁢(t), so one may look at Ξμ⁢(t) as a dynamical quantity. Equation (3.3), related to Hausdorff continuity, was our main motivation to introduce Ξμ⁢(t), and we have got the following singular version of this result; although simple, it will be very useful ahead.

By combining Propositions 3.5 and 3.8, we obtain the following characterization of the α-packing singular subspace.

4 Generic quasiballistic and upper packing sets

Let (X,d) be a complete metric space of bounded self-adjoint operators, acting on the infinite-dimensional separable Hilbert space ℋ, such that the metric d convergence implies strong convergence of operators. We denote its elements by T. In order to obtain the main results of this section (i.e., Propositions 4.3 and 4.5), we will prove, for each fixed vector ψ∈ℋ, that the set

is a Gδ set in X. Recall that here, μψ;(a,b)T denotes the restriction of μψT to the open interval (a,b), where -∞≤a<b≤+∞. If (a,b)=ℝ, we simply denote C1⁢u⁢P⁢dψ;(a,b) by C1⁢u⁢P⁢dψ.

We remark that Lemma 4.1 holds true for restrictions to intervals (a,b). The choice (a,b)=ℝ was just for simplicity. However, we will keep the interval in Theorem 4.2 to deal with some subtleties there.