Some generic fractal properties of bounded self-adjoint operators
aa r X i v : . [ m a t h . SP ] J u l Some generic fractal properties of boundedself-adjoint operators
M. Aloisio ∗ Departamento de Matem´atica, UFAM, Manaus, AM, 69067-005 Brazil
S. L. Carvalho
Departamento de Matem´atica, UFMG, Belo Horizonte, MG, 30161-970 Brazil
C. R. de Oliveira
Departamento de Matem´atica, UFSCar, S˜ao Carlos, SP, 13560-970 Brazil
July 2020
Abstract
We study generic fractal properties of bounded self-adjoint operators through lower andupper generalized fractal dimensions of spectral measures. Two groups of results are presented.Firstly, it is shown that the set of vectors whose associated spectral measures have lower(upper) generalized fractal dimension equal to zero (one) for every q > < q <
1) is eitherempty or generic. The second one gives sufficient conditions, for separable regular spaces ofoperators, for the presence of generic extreme dimensional values; in this context, we have anew proof of the celebrated Wonderland Theorem.
Keywords : Spectral theory, generalized dimensions, spectrum and quantum dynamics.
AMS classification codes : 81Q10 (primary), 47A10 (secondary).
Spectral and dynamical properties of self-adjoint operators have a fundamental role in quantummechanics, and there are many subtleties among them; for instance: 1) any self-adjoint operatormay be approximated by a pure point operator (this is the Weyl-von Neumann Theorem [27, 28]);2) in some topological spaces of self-adjoint operators, the set of elements with purely singularcontinuous spectra is generic (the conclusion of the so-called Wonderland Theorem [25]); 3) densepoint spectrum imply a form of dynamical instability [1]; etc. Here, we present two new subtleproperties related to generic dimensional properties of spectral measures, which are summarizedin Theorems 1.1 and 1.2. For technical simplicity, we restrict ourselves to bounded self-adjoint ∗ Corresponding author. Email: [email protected] T acting on the complex and separable Hilbert space H ; we denote by µ Tψ the spectralmeasure of T associated with the state ψ ∈ H ; for each Borel set Λ ⊂ R , P T (Λ) represents thespectral resolution of T over Λ; by µ we always mean a finite nonnegative Borel measure on R .Here, for every complete metric space X , we say that R ⊂ X is residual if it contains a generic(i.e., a dense G δ ) set in X .The main results are described in Subsections 1.1 and 1.2, along with some examples anddynamical consequences. In Section 2 we recall some concepts and results regarding dimensionalproperties of nonnegative Borel measures. The proofs of Theorems 1.1 and 1.2 are left to Section 3. Let T be a bounded self-adjoint operator on H , pick two vectors ψ, ϕ ∈ H and, for each k ∈ N ,set ψ k = ψ + 1 k ϕ ;although ψ k → ψ as k → ∞ , it is not clear which properties of ψ and/or ψ k are inherited from ϕ .E.g., if ϕ belongs to the point subspace of T , this property is clearly not preserved if ψ belongsto the continuous subspace; moreover, ψ k is a “mixed vector.” The first result in this work saysthat for each k ∈ N , (some) values of the generalized fractal dimensions of µ Tψ k satisfy the samebounds as the values of µ Tϕ , being therefore, held by a large set of spectral measures associatedwith T . Roughly, the idea is to show that µ Tψ k inherits such dimensional properties from µ Tϕ , so theset of the associated vectors is dense in H (since ψ is arbitrary in H ), and then combine this withsuitable G δ properties, proven in [1], to show that they hold for generic sets.Before we present a precise formulation of this result (see Subsection 3.1 for its proof), weneed a small preparation. For q >
0, let D − µ ( q ) and D + µ ( q ) (see Definition 2.3 ahead) denote the lower and the upper generalized fractal dimensions of µ , respectively; recall also that the functions q D ∓ µ ( q ) are nonincreasing and if µ has bounded support, then 0 ≤ D − µ ( q ) ≤ D + µ ( q ) ≤
1, forall q >
Theorem 1.1.
Let T be a bounded self-adjoint operator on H and α, β ≥ .1. Let s > . If there exists = ϕ ∈ H such that D − µ Tϕ ( s ) ≤ β , then Λ − ( T, s, β ) := { ψ ∈ H | D − µ Tψ ( s ) ≤ β } is a dense G δ set (i.e., a generic set) in H .2. Let < q < . If there exists = ϕ ∈ H such that D + µ Tϕ ( q ) ≥ α , then Λ + ( T, q, α ) := { ψ ∈ H | D + µ Tψ ( q ) ≥ α } is a dense G δ set in H . Example 1.1 (Rank-one perturbation of the almost-Mathieu operator) . Write δ = ( δ ,n ) n ∈ Z andlet H be a rank-one perturbation of a quasi-periodic operator, acting on ℓ ( Z ), given by the law( Hu ) n = ( H λ,α,θ,κ u ) n := u n +1 + u n − + κ cos( παn + θ ) + λ h· , δ i δ , λ ∈ [0 , α ∈ [0 , π ), θ ∈ [0 , π ) and κ >
2. It was shown in [19] that there exists a dense G δ set of irrational numbers Ω ⊂ [0 , π ) such that, for every α ∈ Ω, every θ, λ, κ > q ∈ (0 , D + µ Hδ ( q ) = 1. It follows from Theorem 1.1 that, for each α ∈ Ω and each 0 < q < (cid:8) ψ ∈ ℓ ( Z ) | D + µ Hψ ( q ) = 1 (cid:9) is a generic set in ℓ ( Z ). This example is particularly interesting because, for such parameter values,the spectrum of H λ,α,θ,κ is always purely singular (see [19]), and generically with maximum valueof the upper dimensions (0 < q < Example 1.2 (Continuous one-dimensional free Hamiltonian) . Let H : H ( R ) ⊂ L ( R ) → L ( R )be given by the law ( H ψ )( x ) = − ψ ′′ ( x ), and set H := H P H ([0 , θ n = − n +2 , let ψ n ∈ L ( R ) be such that its Fourier transform satisfies, foreach t > c ψ n ( t ) = χ [0 , ( t ) t − θ n . It turns out that ([15], Section 8.4.1)d µ Hψ n ( x ) = 12 χ [0 , ( x ) x − ( θ n +1 / d x. It is straightforward to check that, for each n ∈ N and each s > D ∓ µ Hψn ( s ) = 1 − θ n = 2 n + 2 . Therefore, by Theorem 1.1, for every s > (cid:8) ψ ∈ L ( R ) | D − µ Hψ ( s ) = 0 (cid:9) = \ n ≥ Λ − ( H, s, θ n )is a dense G δ set in L ( R ). This example is interesting because H has purely absolutely continuousspectrum and, generically, with minimum values of such lower dimensions, a result intuitivelyassociated with singular spectrum. Remark 1.1.
To the best knowledge of the present authors, the result presented in Example 1.2leads to a phenomenon which has never been discussed: there exist an operator whose spectrum ispurely absolutely continuous and a generic set of vectors whose time-average return probabilitiesdecay with arbitrarily slow polynomial rates (for sequences of time t j → ∞ ). Namely, in this case,generically in ψ ∈ L ( R ) (by (1) just ahead), for every k ≥ t →∞ t − /k t t Z |h ψ, e − iHt ψ i| d s = 0 . We note that this is, in some sense, the counterpart of the following situation: an operator withpure point spectrum and a generic set of states whose spectral measures have maximal uppergeneralized dimension (such is the case of the operator discussed in Example 1.1); such states are,therefore, delocalized (see Subsection 1.2.1 for details and [1, 19]).Next we turn to the second group of generic results in this work.3 .2 Generic fractal properties of spectral measures
Recall that a metric space (
X, d ) of self-adjoint operators acting in H is regular [25] if itis complete and convergence in the metric d implies strong resolvent convergence. Denote by C p = C p ( X ) the set of operators T ∈ X with pure point spectrum and by C ac = C ac ( X ) the setof operators T ∈ X with purely absolutely continuous spectrum.Under some assumptions, a version of the Wonderland Theorem related to extreme correlationdimensional values (i.e., D − µ Tψ (2) = 0 and D + µ Tψ (2) = 1) of spectral measures, was proven in [7], anddynamical consequences were explored. In the following, we extend this result to all dimensions D ∓ µ Tψ ( q ), q >
0, for separable regular sets of bounded self-adjoint operators. These fine dimensionalproperties will also imply generic singular continuous spectrum; such results are gathered in thenext statements. As a spinoff, for such spaces we have a new proof of the Wonderland Theorem(see Corollary 1.1).
Theorem 1.2.
Let X be a separable regular space of bounded self-adjoint operators. If both sets C p and C ac are dense in X , then there exists a generic set M in H such that, for each ψ ∈ M ,the set X ( ψ ) := { T ∈ X | D − µ Tψ ( q ) = 0 and D + µ Tψ ( q ) = 1 , for all q > } is residual in X . Remark 1.2.
We note that even for T with pure point spectrum, it may occur that D − µ Tψ ( q ) > < q < T has purely absolutely continuous spectrum,it may happen that D + µ Tψ ( q ) < q > Corollary 1.1 (Wonderland Theorem) . Let X be as in Theorem 1.2. If both sets C p and C ac aredense in X , then the set C sc = C sc ( X ) := { T ∈ X | T has purely singular continuous spectrum } isresidual in X . Remark 1.3. (a) The proof of Corollary 1.1 presented below is entirely based on the conclusionsof Theorem 1.2, that is, it is based on the existence of the residual sets M and X ( ψ ), for ψ ∈ M .It is, therefore, a different proof from the one presented in [25]. (b) Naturally, one may combineTheorem 1.2 and Corollary 1.1 to conclude that, for each ψ ∈ M , the set X sc01 ( ψ ) := { T ∈ X | T has purely singular continuous spectrum, D − µ Tψ ( q ) = 0 and D + µ Tψ ( q ) = 1, for all q > } is residualin X . Indeed, it is enough to note that X sc01 ( ψ ) = X ( ψ ) ∩ C sc . Remark 1.4.
Since for bounded self-adjoint operators on H strong convergence implies strongresolvent convergence [15], each space of bounded self-adjoint operators endowed with the strongoperator topology is a regular space, and by [10] it is also separable.Let dim +H ( µ ) denote the upper Hausdorff dimension of µ (such notion is recalled in Section 2).The next result, presented in [4], relates this quantity to the lower generalized fractal dimensions. Proposition 1.1.
Let µ be a finite nonnegative Borel measure on R and < q < < s . Then, D − µ ( q ) ≥ dim +H ( µ ) ≥ D − µ ( s ) . Lemma 1.1.
Let T be a bounded self-adjoint operator on H and = ψ ∈ H . If there exist < q ′ < < s ′ such that D − µ Tψ ( q ′ ) < and D + µ Tψ ( s ′ ) > , then µ Tψ is a purely singular continuousmeasure. roof. If µ Tψ has an atom, that is, if there exists λ ∈ R such that µ Tψ ( { λ } ) >
0, then it is easy toshow that for each s > D + µ Tψ ( s ) = 0 (see (3) ahead). On the other hand, if µ Tψ has an absolutelycontinuous component, then dim +H ( µ Tψ ) = 1 and, therefore, it follows from Proposition 1.1 thatfor each 0 < q < D − µ Tψ ( q ) = 1. Hence, if there exist 0 < q ′ < < s ′ with D + µ Tψ ( s ′ ) > D − µ Tψ ( q ′ ) <
1, then µ Tψ is singular continuous. Proof. (Corollary 1.1) Let M be as in the statement of Theorem 1.2 and let { ψ j } j ∈ Z ⊂ M be adense sequence in H . It follows from Lemma 1.1 that for each φ ∈ M and each S ∈ X ( φ ), thespectral measure µ Sφ is purely singular continuous. Then, since the singular continuous subspaceassociated with each self-adjoint operator is a closed subspace of H [15], one has C sc ⊃ ∩ j ∈ Z X ( ψ j ) . The result is now a consequence of Theorem 1.2.The following result is a direct consequence of Remark 1.3 (b) and Proposition 1.1.
Corollary 1.2.
Let X be as in Theorem 1.2. If both sets C p and C ac are dense in X , then thereexists a generic set M in H such that, for each ψ ∈ M , the set { T ∈ X | T has purely singularcontinuous spectrum, dim +H ( µ Tψ ) = 0 } is residual in X . Corollary 1.2 is also a consequence of the results recently presented in [9]. However, the resultsand methods of this paper are different from those of [9]. Namely, the main technical ingredientsin the present paper involve some decompositions of spectral measures with respect to the fractalgeneralized dimensions, whereas in [9] the main idea is to directly show that for each 0 = ψ ∈ H , { T ∈ X | dim +H ( µ Tψ ) = 0 } is a G δ set in X .There are in the literature (see, for instance, [7, 8, 12, 15, 25]) numerous important examplesfor which our general results apply. As an illustration, we present the following application. Example 1.3.
Consider the class of Schr¨odinger operators with analytic quasiperiodic potentials,acting on ℓ ( Z ), generated by a nonconstant real analytic function v ∈ C ω ( T , R ), that is,( H vλ,α,θ u ) n := u n +1 + u n − + λv ( θ + αn ) u n , where 0 = λ ∈ R , α ∈ T is the frequency and θ ∈ T is the phase (an important example is givenby the almost Mathieu operator, for which v ( x ) = 2 cos(2 πx ); see Example 1.1).For each nonconstant v ∈ C ω ( T , R ), 0 = λ ∈ R and θ ∈ T , consider the space of self-adjointoperators X vλ,θ := { H vλ,α,θ | α ∈ T } endowed with the following metric (whose induced topology is equivalent to the strong operatortopology) d ( H vλ,α,θ , H vλ,α ′ ,θ ) := (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) α − α ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . Since for λ ∈ R , θ ∈ T and α ∈ Q / Z , the operator H vλ,α,θ is purely absolutely continuous, andthere exists λ ( v ) > λ = 1) so that, for every λ > λ ( v ) >
0, every θ ∈ T and for all α outside a set of zero Lebesgue measure, H vλ,α,θ is pure5oint, it follows from Theorem 1.2 (see also Remark 1.3) that there exists a generic set M ( v ) in ℓ ( Z ) such that, for each ψ ∈ M ( v ), the set { H = H vλ,α,θ ∈ X vλ,θ has purely singular continuousspectrum with D − µ Hψ ( q ) = 0 and D + µ Hψ ( q ) = 1, for all q > } is residual in X vλ,θ . Remark 1.5.
We note that, under the above assumptions, Bourgain has shown in [6] that forevery α outside a set of zero Lebesgue measure, H vλ,α,θ has dynamical localization; therefore, theconclusions regarding the lower dimensions in Example 1.3 follow from Theorem 4.3 in [8]. Here, we explore some dynamical consequences of the general above results. We recall that if T is a bounded self-adjoint operator acting on H , then R ∋ t e − itT is a one-parameter stronglycontinuous unitary evolution group and, for each ψ ∈ H , ( e − itT ψ ) t ∈ R is the unique wave packetsolution to the Schr¨odinger equation ∂ t ψ = − iT ψ, t ∈ R ,ψ (0) = ψ ∈ H . (SE)Next, we list two quantities usually considered to probe the large time behaviour of the dynamics e − itT ψ . The (time-average) quantum return probability h γ Tψ i ( t ), which gives the (time-average)probability of finding the particle at time t > ψ , h γ Tψ i ( t ) := 1 t t Z |h ψ, e − isT ψ i| d s ;its lower and upper decaying exponents, respectively, are given by [3, 23]lim inf t →∞ ln h γ Tψ i ( t )ln t = − D + µ Tψ (2) , lim sup t →∞ ln h γ Tψ i ( t )ln t = − D − µ Tψ (2) . (1)In order to probe dynamical (de)localization associated with an initial state ψ with respect toa general orthonormal basis B = { η j } of H , one may quantify the “travel to large dimensions j ”by considering the time evolution of the (time-average) p -moments of ψ , p >
0, that is, r ψ,Tp, B ( t ) := (cid:18) t t Z X j | n | p |h η j , e − isT ψ i| d s (cid:19) p . If one thinks of a polynomial growth r ψ,Tp, B ( t ) ∼ t β ( p ) , then the lower and upper p -moment growthexponents are then naturally introduced, respectively, by β − ψ,T ( p, B ) := lim inf t →∞ ln r ψ,Tp, B ( t )ln t , β + ψ,T ( p, B ) := lim sup t →∞ ln r ψ,Tp, B ( t )ln t . The following inequality, due to Barbaroux, Germinet and Tcheremchantsev [5], independentlyobtained by Guarneri and Schultz-Baldes [22], β ∓ ψ ( p, B ) ≥ D ∓ µ Tψ (cid:18)
11 + p (cid:19) , (2)6olds for all orthonormal bases and all p >
0. Such notions are particularly interesting when T isa Schr¨odinger operator acting in ℓ ( Z ν ), ν ∈ N , { η j } is a basis of ℓ ( Z ν ) and ψ = f ( T ) η , with f ∈ C ∞ ( R ), so displaying some locality condition [14, 20].The corresponding dynamical consequences of Theorem 1.2 come from (1) and (2); namely, thetypical dynamical situation is characterized by the fact that the decay rates of the quantum returnprobability assume their extreme values, and by occurrence of weak dynamical delocalization : for atypical T ∈ X , for every ψ ∈ M and for all p > β + ψ ( p, B ) ≥ r ψ,Tp, B ( t j ) ∼ t β + ψ ( p, B ) j for a sequence of instants of time t j → ∞ . The term weak is due to the possibility of β − ψ ( p, B ) = 0,for all p > Remark 1.6.
We note that a natural strategy to prove Theorem 1.2 consists in showing that thereexist a dense subset of T ∈ X such that µ Tψ is 1-H¨older continuous (since in this case, D − µ Tψ ( q ) = 1for all q > X with dynamical localization, that is, satisfyingfor each ψ ∈ H and each p >
0, lim sup t →∞ r ψ,Tp, B ( t ) < + ∞ (since in this case, by (2), D + µ Tψ ( q ) = 0 for all q > ψ ∈ ℓ ( Z ν ) satisfying some locality condition (for instance, ψ ∈ ℓ ( Z ν )) such that µ Tψ is at most 1 / ψ [14, 20]).Thus, although natural, this strategy does not seem to be suitable for the rather general setting ofthis work. Remark 1.7.
It is worth underlying that by combining some results of [7, 8], one gets Theorem 1.2for the particular space of one-dimensional Jacobi matrices (with a necessarily nontrivial restrictionof the spectrum to obtain 1-H¨older continuity) endowed with the topology of pointwise convergence.However, the strategy followed in [7, 8] does not seem to be adequate to prove this theorem in suchgenerality.
Definition 2.1.
Let µ be a finite nonnegative Borel measure on R . The pointwise lower scalingexponent of µ at x ∈ R is defined as d − µ ( x ) := lim inf ǫ ↓ ln µ ( B ( x, ǫ ))ln ǫ if, for all ǫ > µ ( B ( x ; ǫ )) >
0; otherwise, one sets d − µ ( x ) := ∞ . Definition 2.2.
The upper Hausdorff dimension of µ is defined asdim +H ( µ ) := µ - ess . sup d − µ . Proposition 2.1.
Let µ be a nonnegative Borel measure on R which is absolutely continuous withrespect to the Lebesgue measure. Then, µ - ess . inf d − µ = 1 . The study of fractal dimensions of spectral measures in the context of quantum mechanicsappeared as an attempt to answer the following question: “What determines the spreading of awave packet?” For a broader discussion, we highlight the works [5, 11, 19, 21, 22, 24].
Definition 2.3.
Let µ be a finite positive Borel measure on R . The lower and upper q -generalizedfractal dimensions, q > q = 1, of µ are defined, respectively, as D − µ ( q ) := lim inf ǫ ↓ ln (cid:20) R µ ( B ( x, ǫ )) q − d µ ( x ) (cid:21) ( q −
1) ln ǫ and D + µ ( q ) := lim sup ǫ ↓ ln (cid:20) R µ ( B ( x, ǫ )) q − d µ ( x ) (cid:21) ( q −
1) ln ǫ , with the integrals taken over the support of µ . The lower and upper 1-generalized fractal dimensionsof µ are defined, respectively, as D − µ (1) := lim inf ǫ ↓ ln (cid:20) R ln µ ( B ( x, ǫ ))d µ ( x ) (cid:21) ( q −
1) ln ǫ and D + µ (1) := lim sup ǫ ↓ ln (cid:20) R ln µ ( B ( x, ǫ ))d µ ( x ) (cid:21) ( q −
1) ln ǫ ;again, the integrals are taken over the support of µ .Other important quantities related to the q -generalized fractal dimensions, q > q = 1, arethe so-called mean q -dimensions. Definition 2.4.
Let µ be a finite positive Borel measure on R and q >
0. The lower and uppermean q -dimensions of µ are defined, respectively, as m − µ ( q ) := lim inf ǫ ↓ ln[ ǫ − R µ ( B ( x, ǫ )) q d x ]( q −
1) ln ǫ and m + µ ( q ) := lim sup ǫ ↓ ln[ ǫ − R µ ( B ( x, ǫ )) q d x ]( q −
1) ln ǫ .
Proposition 2.2 (Theorem 2.1. and Propositions 3.1 and 3.3 in [4]) . Let µ be as before. Then,1. For every q > , q = 1 , D ∓ µ ( q ) = m ∓ µ ( q ) .2. D − µ ( q ) and D + µ ( q ) are nonincreasing functions of q > .3. If µ has bounded support, then for all q > , ≤ D − µ ( q ) ≤ D + µ ( q ) ≤ . The next results play a fundamental role in the proof of Theorem 1.2.
Proposition 2.3 (Proposition 3.1 in [1]) . Let T be a bounded self-adjoint operator on H and q > , q = 1 . Then, for every Γ ≥ ,1. { ψ ∈ H | D − µ Tψ ( q ) ≤ Γ } is a G δ set in H ,2. { ψ ∈ H | D + µ Tψ ( q ) ≥ Γ } is a G δ set in H . roposition 2.4. Let ( X, d ) be a regular space of bounded self-adjoint operators and let q > , q = 1 . Then, for every = ψ ∈ H and every Γ ≥ ,1. { T | D − µ Tψ ( q ) ≤ Γ } is a G δ set in X ,2. { T | D + µ Tψ ( q ) ≥ Γ } is a G δ set in X . Since the proof of Proposition 2.4 is based on the same arguments of the proof of Proposition 2.3(presented in details in [1]), it will be omitted. Case 1. T has an eigenvalue. Then, there exist λ ∈ R and 0 = η ∈ H such that T η = λη . Set,for each ψ ∈ H and each k ∈ N , ψ k := ψ + k η and note that lim k →∞ k ψ k − ψ k = 0; for each k ≥ P T p ψ k = 0, where P T p is the orthogonal projection onto the pure point subspace of T . Therefore,since { ψ ∈ H | k P T p ψ k > } is open and dense, the result follows from the set inclusions { ψ ∈ H | D − µ Tψ ( s ) ≤ β } ⊃ { ψ ∈ H | D − µ Tψ ( s ) = 0 } ⊃ { ψ ∈ H | k P p ψ k > } . Namely, if ξ := P T p ψ = 0, then µ Tξ has an atom, i.e, there exists ζ ∈ R such that µ Tξ ( { ζ } ) > s > D + µ Tξ ( s ) = lim sup ǫ → ln hR µ Tξ ( B ( x, ǫ )) s − d µ Tξ ( x ) i ( s −
1) ln ǫ ≤ lim sup ǫ → ln h µ Tξ ( { ζ } ) s i ( s −
1) ln ǫ = 0 . (3) Case 2.
The spectrum of T is purely continuous. Firstly, let us build a sequence of decreasingcompact sets ( A k ) so that, for each k ≥ D − µ k ( s ) ≤ β , where, for every Borel set Λ ⊂ R , µ k (Λ) := µ Tϕ (Λ ∩ A k ).Let r > µ Tϕ ) ⊂ [ − r, r ]; set I := [ − r,
0] and I := [0 , r ]. Since D − µ Tϕ ( s ) ≤ β ,one has, for every σ > ∞ = lim sup ǫ → ǫ ( σ + β )(1 − s ) r Z − r µ Tϕ ( B ( x, ǫ )) s − d µ Tϕ ( x ) ≤ X j =1 lim sup ǫ → ǫ ( σ + β )(1 − s ) Z I j µ Tϕ ( B ( x, ǫ )) s − d µ Tϕ ( x ) , and so, there exists j ∈ { , } such thatlim sup ǫ → ǫ ( σ + β )(1 − s ) Z I j µ Tϕ ( B ( x, ǫ )) s − d µ Tϕ ( x ) = ∞ . Now, write I j = [ a , b ] and define A := L ∪ I j ∪ L ′ , where L = [ −| I j | / a , a ] and9 ′ = [ b , b + | I j | / µ ( · ) := µ Tϕ ( · ∩ A ). Then, for every 0 < ǫ < | I j | / Z A µ ( B ( x, ǫ )) s − d µ ( x ) = Z A µ Tϕ ( A ∩ B ( x, ǫ )) s − d µ Tϕ ( x ) ≥ Z I j µ Tϕ ( A ∩ B ( x, ǫ )) s − d µ Tϕ ( x )= Z I j µ Tϕ ( B ( x, ǫ )) s − d µ Tϕ ( x ) . Thus, for every σ >
0, lim sup ǫ → ǫ ( σ + β )(1 − s ) Z A µ ( B ( x, ǫ )) s − d µ ( x ) = ∞ , and so, D − µ ( s ) ≤ β . Using the same reasoning as before, there is a closed interval [ a , b ] =: I j ⊂ A such that | I j | = | A | andlim sup ǫ → ǫ ( σ + β )(1 − s ) Z I j µ Tϕ ( B ( x, ǫ )) s − d µ Tϕ ( x ) = ∞ ( I j is “one half” of A ). Then, define A := L ∪ I j ∪ L ′ , where L = [ −| I j | / a , a ] and L ′ = [ b , b + | I j | / D − µ ( s ) ≤ β , where µ ( · ) := µ Tϕ ( · ∩ A ); note that | A | = 12 | I j | + | I j | = 32 | I j | = 34 | A | = 34 3 r . Proceeding in this way, one builds a decreasing sequence of closed intervals A k +1 ⊂ A k suchthat | A k | → k → ∞ (namely, | A k | = (3 / k − (3 r/ k ≥ D − µ k ( s ) ≤ β , with µ k ( · ) := µ Tϕ ( · ∩ A k ). Since each set A k is a compact interval, there exists Γ ∈ [ − r, r ] such that A k ↓ { Γ } .Finally, for every ψ ∈ H and every k ≥
1, set ψ k := P T ( R \ A k ) ψ + k ϕ . Since T has purelycontinuous spectrum, lim k →∞ k ψ k − ψ k = 0. Now, one has, for every k ≥
1, every 0 < ǫ < x ∈ R , µ Tψ k ( B ( x, ǫ )) ≥ µ Tψ k ( B ( x, ǫ ) ∩ A k ) ≥ k Re h P T ( B ( x, ǫ ) ∩ A k ) P T ( R \ A k ) ψ, ϕ i + 1 k µ Tϕ ( B ( x, ǫ ) ∩ A k )= 1 k µ Tϕ ( B ( x, ǫ ) ∩ A k ) = 1 k µ k ( B ( x, ǫ )) , from which follows thatln h ǫ R µ Tψ k ( B ( x, ǫ )) s d x i ( s −
1) ln ǫ ≤ ln (cid:2) k s ǫ R µ k ( B ( x, ǫ )) s d x (cid:3) ( s −
1) ln ǫ ;thus, for every k ≥
1, by Proposition 2.2 , D − ψ k ( s ) ≤ D − µ k ( s ) ≤ β . Since ψ is arbitrary, { ξ ∈ H | D − µ Tξ ( s ) ≤ β } is dense in H and so, by Proposition 2.3, it is a dense G δ set in H .10 . Since the case α = 0 is trivial we let α >
0. Again, let r > µ Tϕ ) ⊂ [ − r, r ];set I := [ − r,
0] and I := [0 , r ]. Since D + µ Tϕ ( q ) ≥ α , it follows that, for every 0 < σ < α , ∞ = lim sup ǫ → ǫ ( α − σ )(1 − q ) r Z − r µ Tϕ ( B ( x, ǫ )) q − d µ Tϕ ( x ) ≤ X j =1 lim sup ǫ → ǫ ( α − σ )(1 − q ) Z I j µ Tϕ ( B ( x, ǫ )) q − d µ Tϕ ( x ) . Thus, for some j ∈ { , } ,lim sup ǫ → ǫ ( α − σ )(1 − q ) Z I j µ Tϕ ( B ( x, ǫ )) q − d µ Tϕ ( x ) = ∞ . Let B := I j and set λ ( · ) := µ Tϕ ( · ∩ B ). Then, for every ǫ > Z B λ ( B ( x, ǫ )) q − d λ ( x ) = Z B λ ( B ( x, ǫ )) q − d µ Tϕ ( x )= Z B µ Tϕ ( B ∩ B ( x, ǫ )) q − d µ Tϕ ( x ) ≥ Z B µ Tϕ ( B ( x, ǫ )) q − d µ Tϕ ( x ) , where we have used in the last inequality the fact that 0 < q <
1. Thus, for every 0 < σ < α ,lim sup ǫ → ǫ ( α − σ )(1 − q ) Z B λ ( B ( x, ǫ )) q − d λ ( x ) = ∞ and so, D + λ ( q ) ≥ α . Proceeding in this way, we build a decreasing sequence of closed intervals B k +1 ⊂ B k so that | B k | → k → ∞ and, for each k ≥ D + λ k ( q ) ≥ α , where λ k ( · ) := µ Tϕ ( · ∩ B k ).Since each set B k is a compact interval, there exists a Γ ∈ [ − r, r ] such that B k ↓ { Γ } .Now set, for every k ≥ A k := B k \ { Γ } , and note that A k ↓ ∅ . Moreover, using the samereasoning as before, it follows that for every k ≥ D + µ k ( q ) ≥ α , where µ k ( · ) := µ Tϕ ( ·∩ A k ). Namely,given that 0 < q <
1, one has for every k ≥ ǫ > ǫ ( α − σ )(1 − q ) Z A k µ k ( B ( x, ǫ )) q − d µ k ( x ) = ǫ ( α − σ )(1 − q ) Z A k µ k ( B ( x, ǫ )) q − d λ k ( x ) ≥ ǫ ( α − σ )(1 − q ) Z A k λ k ( B ( x, ǫ )) q − d λ k ( x )= ǫ ( α − σ )(1 − q ) Z B k λ k ( B ( x, ǫ )) q − d λ k ( x ) − ǫ ( α − σ )(1 − q ) λ k ( B (Γ , ǫ )) q − λ k ( { Γ } ) . We also have thatlim ǫ → ǫ ( α − σ )(1 − q ) λ k ( B (Γ , ǫ )) q − λ k ( { Γ } ) ≤ lim ǫ → ǫ ( α − σ )(1 − q ) λ k ( { Γ } ) q = 011f Γ is an atom, and that ǫ ( α − σ )(1 − q ) λ k ( B (Γ , ǫ )) q − λ k ( { Γ } ) = 0 , otherwise. Hence, lim sup ǫ → ǫ ( α − σ )(1 − q ) Z A k µ k ( B ( x, ǫ )) q − d µ k ( x ) = ∞ . Finally, for every ψ ∈ H and every k ≥
1, define ψ k := P T ( R \ A k ) ψ + k ϕ , and note thatlim k →∞ k ψ k − ψ k = 0. Now, one has, for every k ≥
1, every 0 < ǫ < x ∈ R , µ Tψ k ( B ( x, ǫ )) ≥ µ Tψ k ( B ( x, ǫ ) ∩ A k ) ≥ k µ Tϕ ( B ( x, ǫ ) ∩ A k ) = 1 k µ k ( B ( x, ǫ )) , from which follows thatln h ǫ R µ Tψ k ( B ( x, ǫ )) q d x i ( q −
1) ln ǫ ≥ ln (cid:2) k q ǫ R µ k ( B ( x, ǫ )) q d x (cid:3) ( q −
1) ln ǫ ;thus, for every k ≥
1, by Proposition 2.2 , D + ψ k ( q ) ≥ D + µ k ( q ) ≥ α . Since ψ is arbitrary, { ξ ∈ H | D + µ Tξ ( q ) ≥ α } is dense in H and so, by Proposition 2.3, it is a dense G δ set in H . We need the following claims.
Claim I:
Let T be a bounded self-adjoint operator in H . If T has purely absolutely continuousspectrum, then N + ( T ) := { ψ | D + µ Tψ ( q ) = 1 , for all q > , q = 1 } is a dense G δ set in H . Claim II:
Let T be a bounded self-adjoint operator in H . If T has pure point spectrum, then N − ( T ) := { ψ | D − µ Tψ ( q ) = 0 , for all q > , q = 1 } is a dense G δ set in H .Since ( X, d ) is a separable space, C ac ⊂ X (with the induced topology) is also separable.Let { T j } be a dense sequence in C ac (being, therefore, dense in X , since C ac is dense in X byhypothesis). By Claim I , N + := \ j N + ( T j )is a dense G δ set in H . Then, by Proposition 2.4, for every ψ ∈ N + , X + ( ψ ) := { T | D + µ Tψ ( q ) = 1 , for all q > , q = 1 } ⊃ ∪ j { T j } is a dense G δ set in X . 12imilarly, since C p ⊂ X is a separable space, let { S j } be a dense sequence in C p . By Claim II , N − := \ j N − ( S j )is a dense G δ set in H . Then, it follows again by Proposition 2.4 that, for every ψ ∈ N − , { T | D − µ Tψ ( q ) = 0 , for all q > , q = 1 } ⊃ ∪ j { S j } is a dense G δ set in X .Combining the previous results, it follows that for every ψ ∈ M := N + ∩ N − , { T | D − µ Tψ ( q ) = 0 and D + µ Tψ ( q ) = 1 for all q > , q = 1 } is a dense G δ set in X . Hence, as the functions q D ∓ µ ( q ) are nonincreasing, follows that, for each ψ ∈ M , the set X ( ψ ) = { T ∈ X | D − µ Tψ ( q ) = 0 and D + µ Tψ ( q ) = 1, for all q > } is residual in X .Now, it remains to prove the claims. Proof of Claim I:
Let 0 = ψ ∈ H . It follows from Proposition 2.1 that for µ Tψ -a.e. x , d − µ Tψ ( x ) = 1.So, there exists Ω ⊂ R such that µ Tψ (Ω) = k ψ k and such that, for every x ∈ Ω, d − µ Tψ ( x ) = 1. Nowlet, for every ǫ > f ǫ : Ω −→ R be the measurable function given by the law f ǫ ( x ) := inf ǫ>r ln µ Tψ ( B ( x, r ))ln r . The sequence ( f ǫ ( x )) converges pointwise to d − µ Tψ ( x ); so, by Egoroff’s Theorem, there exist Borelsets S k ↑ Ω such that, for every k ≥ µ Tψ ( S ck ) < /k and such that lim ǫ ↓ f ǫ ( x ) = d − µ Tψ ( x ) uniformlyon S k . But then, given 0 < σ <
1, there exists 0 < ǫ σ,k < < ǫ < ǫ σ,k andfor every x ∈ S k , µ Tψ ( B ( x, ǫ )) ≤ ǫ − σ . Now, set ψ k := P T ( S k ) ψ and note that:lim k →∞ k ψ k − ψ k = lim k →∞ k P T ( S ck ) ψ k ≤ lim k →∞ /k = 0;for every k ≥
1, every q > < ǫ < R µ Tψ k ( B ( x, ǫ )) q − d µ Tψ k ( x )( q −
1) ln ǫ ≥ k ψ k k ( q −
1) ln ǫ + (1 − σ ) , from which follows that D ∓ µ Tψk ( q ) = 1, since 0 < σ ≤ N + ( T ) = { ψ | D + µ Tψ ( q ) = 1 for all q > , q ∈ N } , is a dense G δ set in H (note that the above equality holds because r D + µ ( r ) is a nonincreasingfunction). Proof of Claim II:
We note that this claim has been proven in Theorem 3.1. in [1]. For theconvenience of the reader, we present its proof in details. Let ( η j ) be an orthonormal family ofeigenvectors of T , that is, T η j = λ j η j for every j ≥
1. Let, for every 0 < q <
1, ( b j ) ⊂ C be a13equence such that | b j | >
0, for all j ≥
1, and P ∞ j =1 | b j | q < ∞ . Given ψ ∈ H , write ψ = P ∞ j =1 a j η j ,and then consider, for each k ≥ ψ k := k X j =1 a j η j + ∞ X j = k +1 b j η j . It is clear that ψ k → ψ . Moreover, for k ≥ ǫ > Z µ Tψ k ( B ( x, ǫ )) q − d µ Tψ k ( x ) = ∞ X j =1 µ Tψ k ( B ( λ j , ǫ )) q − µ Tψ k ( { λ j } ) ≤ ∞ X j =1 µ Tψ k ( { λ j } ) q = k X j =1 | a j | q + ∞ X j = k +1 | b j | q , from which follows that D ∓ µ Tψk ( q ) = 0. Hence, for every 0 < q < { ψ | D − µ Tψ ( q ) = 0 } is a dense setin H ; so, by Proposition 2.3, N − ( T ) = { ψ | D − µ Tψ ( q ) = 0 for all 0 < q < q ∈ Q } is a dense G δ set in H (again, the above equality holds because r D − µ ( r ) is a nonincreasingfunction). Acknowledgments
SLC thanks the partial support by FAPEMIG (Minas Gerais state agency; Universal Projectunder contract 001/17/CEX-APQ-00352-17) and CRdO thanks the partial support by CNPq (aBrazilian government agency, under contract 303503/2018-1).
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