aa r X i v : . [ m a t h . SP ] J un A note on spectrum and quantum dynamics
Moacir AloisioMay 2019
Abstract
We show, in the same vein of Simon’s Wonderland Theorem, that, typically in Baire’ssense, the rates with whom the solutions of the Schr¨odinger equation escape, in time average,from every finite-dimensional subspace, depend on subsequences of time going to infinite.
Keywords : spectral theory, Schr¨odinger equation, decaying rates.
AMS classification codes : 46L60 (primary), 81Q10 (secondary).
There is a vast literature concerning the large time asymptotic behaviour of wave packet solutionsof the Schr¨odinger equation ∂ t ξ = − iT ξ, t ∈ R ,ξ (0) = ξ, ξ ∈ H , (1)where T is a self-adjoint operator in a separable complex Hilbert space H . Namely, the relationsbetween the quantum dynamics of solutions of (1) and the spectral properties of T are a classicalsubject of quantum mechanics. In this context, we refer to [2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 16],among others. We recall that, for every ξ ∈ H , the one-parameter unitary group R ∋ t e − itT is so that thecurve e − itT ξ , in some sense, solves (1). Next we list two quantities usually considered to probe thelarge time behaviour of the dynamics e − itT ξ .1. Let A be a positive operator such that, for each t ∈ R , e − itT D ( A ) ⊂ D ( A ). For each ξ ∈ D ( A ), the expectation value of A in the state ξ at time t is defined as A Tξ := h e − itT ξ, Ae − itT ξ i . (A)1. The time average for A is defined as h A Tξ i t := 1 t t Z h e − isT ξ, Ae − isT ξ i ds. (B)In this paper, we are interested in studying the relations between the large time behaviour ofthe time average of expectation value of A in the state ξ and the spectral properties of the spectralmeasure µ Tξ of T associated with ξ . In this context, firstly we refer to notorious RAGE Theorem,named after Ruelle, Amrein, Georgescu, and Enss [8]. Theorem 1.1 (RAGE Theorem) . Let A be a compact operator on H . For every ξ ∈ H , lim t →∞ h| A Tξ |i t = 0 , if, and only if, µ Tξ is purely continuous. We note that since any projector onto a finite-dimensional subspace of H satisfies the hypothesesof RAGE Theorem, initial states whose spectral measures are purely continuous can be interpretedas those whose trajectories escape, in time average, from every finite-dimensional subspace. A complete metric space (
X, d ) of self-adjoint operators, acting in the separable complex Hilbertspace H , is said to be regular if convergence with respect to d implies strong resolvent convergenceof operators (see Definition 2.2 ahead). One of the results stated in [15], the so-called WonderlandTheorem, says that, for some regular spaces X , { T ∈ X | T has purely singular continuousspectrum } is a dense G δ set in X . Hence, for these spaces, by RAGE Theorem, for each compactoperator A and each ξ ∈ H , { T ∈ X | lim t →∞ h| A Tξ |i t = 0 } contains a dense G δ set in X . In this workwe present refinements of this result for three different classes of self adjoint operators.1. General operators.
Let a > X a := { T | T self-adjoint, k T k ≤ a } with themetric d ( T, T ′ ) := ∞ X j =0 min(2 − j , k ( T − T ′ ) e j k ) , where ( e j ) is an orthonormal basis of H . Then, X a is a complete metric space such thatmetric convergence implies strong resolvent convergence.2. Jacobi matrices.
For every fixed b >
0, let the family of Jacobi matrices, M , given on ℓ ( Z )by the action ( M u ) j := u j − + u j +1 + v j u j , where ( v j ) is a sequence in ℓ ∞ ( Z ), such that, for each j ∈ Z , | v j | ≤ b . Denote by X b theset of these matrices endowed with the topology of pointwise convergence on ( v j ). Then, onehas that X b is (by Tychonoff’s Theorem) a compact metric space so that metric convergenceimplies strong resolvent convergence. 2. Schr¨odinger operators.
Fix
C > H V ,defined in the Sobolev space H ( R ) by the action( H V u )( x ) := −△ u ( x ) + V ( x ) u ( x ) , with V ∈ B ∞ ( R ) (the space of bounded Borel functions) so that, for each x ∈ R , | V ( x ) | ≤ C .Denote by X C the set of these operators endowed with the metric d ( H V , H U ) := ∞ X j =0 min(2 − j , k V − U k j ) , where k V − U k j := sup x ∈ B (0 ,j ) | V ( x ) − U ( x ) | . Then, one has that X C is (again by Tychonoff’sTheorem) a compact metric space such that convergence metric implies strong resolventconvergence (Definition 2.2). Namely, if H V k → H V in X C , then, for every x ∈ R , one hasthat lim k →∞ V k ( x ) = V ( x ). Therefore, for each u ∈ L ( R ), by the second resolvent identity anddominated convergence, k ( R i ( H V k ) − R i ( H V )) u k L ( R ) = k R i ( H V k )( V k − V ) R i ( H V ) u k L ( R ) ≤ k ( V k − V ) R i ( H V ) u k L ( R ) −→ k → ∞ .Next, we introduce a dynamic quantity, the time average of quantum return probability, andits lower and upper decaying exponents, respectively, given by [3]lim inf t →∞ log (cid:18) t R t |h ξ, e − isT ξ i| ds (cid:19) log t = − D +2 ( µ Tξ ) , (2)lim sup t →∞ log (cid:18) t R t |h ξ, e − isT ξ i| ds (cid:19) log t = − D − ( µ Tξ ) , (3)where D − ( µ Tξ ) and D +2 ( µ Tξ ) denote, respectively, the lower and upper correlation dimensions of µ Tξ (see Theorem 2.2 in [3]).For every { e n } n ∈ Z orthonormal basis of H and every q > q -moment of the position operator at time t >
0, with initial condition ξ , hh| X | q ii t,ξ := 1 t t Z X n ∈ Z | n | q |h e − isT ξ, e n i| d s. Let the lower and upper transport exponents, respectively, given by β − ( ξ, q ) := lim inf t →∞ log hh| X | q ii t,ξ q log t , (4) β + ( ξ, q ) := lim sup t →∞ log hh| X | q ii t,ξ q log t . (5)3n [6, 7] Carvalho and de Oliveira has been proven refinements of Wonderland Theorem forsome classes of discrete self-adjoint operators; in particular, they have shown, for the spaces X a and X b , that the set of operators whose lower and upper correlation dimensions are simultaneouslyzero and one (this corresponds to the distinct polynomially decaying rates for the quantum returnprobability (2)-(3)), respectively, is generic [6] and that is true a dual result for the transportexponents (which implies in distinct polynomially growth rates for the q -moments (4)-(5)) [7].Thus, for X a and X b (from the topological viewpoint) the phenomenon of quantum intermittency isexceptional. 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]; morespecifically, thanks to the RAGE Theorem, we evaluate arbitrary decaying rates of (B) (Theorem1.2). Moreover, in this work, we also developed an argument involving separability to extendpartially such results to the class of (continuous) Schr¨odinger operators X C (Theorem 1.3).We shall prove following Theorem 1.2.
Let α : R −→ R such that lim sup t →∞ α ( t ) = ∞ . For every compact operator A and every ξ ∈ H non null, { T ∈ X Γ | T has purely singular continuous spectrum, lim sup t →∞ α ( t ) h| A Tξ |i t = ∞ and lim inf t →∞ t h| A Tξ |i t = 0 } is a dense G δ set in X Γ for Γ ∈ { a, b } . Remark 1.1.
1. Theorem 1.2 one says that, for X a and X b , typically Baire’s sense, expectation values ofcompact operators, in time average, have distinct decaying rates; in particular, in this case,typically, the rates with whom the trajectories escape, in time average, from every finite-dimensional subspace, depend on subsequences of time going to infinite. We emphasize that α , in statement of Theorem 1.2, can be chosen arbitrarily and that is well known that, aboutrather general conditions on A , T and ξ , for every ǫ > t →∞ t ǫ h| A Tξ |i t = ∞ .
2. The proof from that { T ∈ X Γ | lim inf t →∞ t h| A Tξ |i t = 0 } is a dense G δ set in X Γ is a direct application of (2) and (3) (Theorem 2.2 in [3]), Lemma 3.2and Theorem 3.2 (Theorem 2.1) in [13] and Theorems 1.2 and 1.4 in [6]. For the convenienceof the reader, we presented in details a simple proof of this fact here.3. The proof from that { T ∈ X Γ | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞}
4s a dense G δ set in X Γ is more delicate since that involves an arbitrary growth of α . To provesuch result we use the RAGE Theorem combined with the fact that, for some γ (Γ) > D γ := { T ∈ X Γ | σ ( T ) = [ − γ, γ ] is pure point } is a dense set in X Γ . For X Γ = X a , is a direct consequence from the Weyl-von NeumannTheorem [18, 19] that D a is a dense set in X a (see proof of Theorem in 3.2 [15] for details),whereas for X Γ = X b is known that this can be obtained by using Anderson’s localization.Namely, for every fixed b >
0, consider Ω = [ − b, b ] Z be endowed with the product topologyand with the respective Borel σ -algebra. Assume that ( ω j ) j ∈ Z = ω ∈ Ω is a set of independent,identically distributed real-valued random variables with a common probability measure ρ not concentrated on a single point and so that R | ω j | θ d ρ ( w j ) < ∞ for some θ >
0. Denoteby ν := ρ Z the probability measure on Ω. The Anderson model is a random Hamiltonian on ℓ ( Z ), defined for each ω ∈ Ω by( h ω u ) j := u j − + u j +1 + ω j u j . It turns out that [1, 9, 16] σ ( h ω ) = [ − ,
2] + supp( ρ ) , and ν -a.s. ω , h ω has pure point spectrum [5, 17] (see also Theorem 4.5 in [9]). Thus, if µ denotes the product of infinite copies of the normalized Lebesgue measure on [ − b, b ], that is,(2 b ) − ℓ , then D b +2 = { M ∈ X b | σ ( M ) = [ − b − , b + 2] , σ ( M ) is pure point } is so that µ ( X b \ D b +2 ) = 0.4. We note that for X C the ideas presented above do not apply. Namely, this comes from thefact that this is a space of unbounded operators and, to the best of our knowledge, still nohas been detailed in the literature arguments that show that { H ∈ X C | σ ( H ) is pure point } is a dense set in X C . In this context, in this work, we use the separability of X C to prove thefollowing result. Theorem 1.3.
Take α be as before. Then, for every compact operator A , there exists a dense G δ set G α ( A ) in L ( R ) such that, for every ξ ∈ G α ( A ) , { H ∈ X C | H has purely singular continuous spectrum on (0 , ∞ ) and lim sup t →∞ α ( t ) h| A Hξ |i t = ∞} is a dense G δ set in X C . The paper is organized as follows. In Section 2, we present the proof of Theorem 1.2. In section3, we prove Theorem 1.3. 5ome words about notation: H denotes a separable complex Hilbert space. If A is a linearoperator in H , we denote its domain by D ( A ), its spectrum by σ ( A ), its point spectrum by σ p ( A ).If ̺ ( A ) denotes its resolvent set, then the resolvent operator of A at λ ∈ ̺ ( A ) is denoted by R λ ( A ).For every set Ω ⊂ R , χ Ω denotes the characteristic function of the set Ω. Finally, for every x ∈ R and r > B ( x, r ) denotes the open interval ( x − r, x + r ). We need some preliminaries to present the proof of Theorem 1.2.
Definition 2.1.
A sequence of bounded linear operators ( T n ) strongly converges to T in H if, forevery ξ ∈ H , T n ξ −→ T ξ in H .Next, we present definitions of a sequence of (unbounded) self-adjoint operators ( T n ) approach-ing another one T . Definition 2.2.
Let ( T n ) be a sequence of self-adjoint operators and let T be another self-adjointoperator. One says that:1. T n converges to T in the strong resolvent sense (SR) if R i ( T n ) strongly converges to R i ( T ).2. T n converges to T in the strong dynamical sense (SD) if, for each t ∈ R , e itT n stronglyconverges to e itT .The next result shows that both notions of convergence are equivalent. Proposition 2.1 (Proposition 10.1.9 in [8]) . The SR and SD convergences of self-adjoint operatorsare equivalent.
Definition 2.3.
Let µ be a σ -finite positive Borel measure on R . One says that µ is Lipschitzcontinuous if there exists a constant C > I with ℓ ( l ) < µ ( I )
Proposition 2.2.
Let α as in the statement of Theorem 1.2 and let ξ ∈ H non null. Let X be aregular space of self-adjoint operators. Suppose that:1. For some γ > , D γ := { T ∈ X | σ ( T ) = [ − γ, γ ] is pure point } is a dense set in X . . L := { T ∈ X | µ Tξ is Lipschitz continuous } is a dense set in X .Then, for any compact operator A , { T ∈ X | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞ and lim inf t →∞ t h| A Tξ |i t = 0 } is a dense G δ set in X .Proof. Since, for each t ∈ R , by Proposition 2.1 and dominated convergence, the mapping X ∋ T α ( t ) h| A Tξ |i t is continuous, it follows that, for each k ≥ n ≥
1, the set [ t ≥ k { T ∈ X | α ( t ) h| A Tξ |i t > n } is open, from which follows that { T ∈ X | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞} = \ n ≥ \ k ≥ [ t ≥ k { T ∈ X | α ( t ) h| A Tξ |i t > n } is a G δ set in X . By RAGE Theorem, D γ ⊂ { T ∈ X | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞} . Hence, { T ∈ X | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞} is a dense G δ set in X .We note that, for each j ≥ L j := { T ∈ X | lim inf t →∞ t − j h| A Tξ |i t = 0 } is also a G δ set in X . Since, by Theorem 2.1, for each j ≥ L ⊂ L j , it follows that, by Baire’sTheorem, { T ∈ X | lim inf t →∞ t h| A Tξ |i t = 0 } = \ j ≥ L j is a dense G δ set in X , concluding the proof of proposition. Proof (Theorem 1.2) . As each T ∈ X Γ can be approximated by an operator whose spectral mea-sures are Lipschitz continuous (see proof of Theorems 1.2 and 1.4 in [6] for details). The theorem isnow a direct consequence of Remark 1.1, Proposition 2.2, Theorems 3.1 and 4.1 in [15] and Baire’sTheorem. In order to prove Theorem 1.3 we need of result to follow and a fine separability argument (checkbelow).
Lemma 3.1.
Let T be a self-adjoint operator so that σ p ( T ) = ∅ and let α as in the statement ofTheorem 1.2 . Then, for any compact operator AG α ( A, T ) := { ξ ∈ H | lim sup t →∞ α ( t ) h| A Tξ |i t = ∞} is a dense G δ set in H . roof. Since, for each t ∈ R , by dominated convergence, the mapping H ∋ ξ α ( t ) h| A Tξ |i t is continuous, it follows that G α ( T, A ) = \ n ≥ \ k ≥ [ t ≥ k { ξ ∈ H | α ( t ) h| A Tξ |i t > n } is a G δ set in H .Given ξ ∈ H , write ξ = ξ + ξ , with ξ ∈ Span { ξ } ⊥ and ξ ∈ Span { ξ } , where ξ , with k ξ k H = 1, is an eigenvector of T associated with an eigenvalue λ . If ξ = 0, then µ Tξ ( { λ } ) = k P T ( { λ } ) ξ k H ≥ Re h P T ( { λ } ) ξ , P T ( { λ } ) ξ i + k P T ( { λ } ) ξ k H = k ξ k H > , where P T ( { λ } ) represents the spectral resolution of T over the set { λ } . Now, if ξ = 0, define, foreach k ≥ ξ k := ξ + ξ k . It is clear that ξ k → ξ . Moreover, by the previous arguments, for each k ≥
1, one has µ Tξ k ( { λ } ) > . Thus, G := { ξ ∈ H | µ Tξ has point component } is a dense set in H . Nevertheless, by RAGETheorem, G ⊂ G α ( T, A ). Hence, G α ( T, A ) is a dense G δ set in H . Proof (Theorem 1.3) . By the arguments presented in the proof of Proposition 2.2, for every ξ ∈ L ( R ), { H V ∈ X C | lim sup t →∞ α ( t ) h| A H V ξ |i t = ∞} is a G δ set in X C .Now given H V ∈ X C , we define, for every k ≥ V k ( x ) := kk + 1 χ B (0 ,k ) V ( x ) − C ( k + 1)( | x | + 1) . We note that, for each k ≥ x ≥ k , V k ( x ) ≤
0. Moreover, for each k ≥ ∞ Z k V k ( x ) = −∞ . Therefore, for every k ≥ H V k has at least a negative eigenvalue [14]; in particular, σ p ( H V k ) = ∅ .Since H V k → H V in X C , it follows that Y := { H V ∈ X C | σ p ( H V ) = ∅} is a dense set in X C . 8ow, let ( H V k ) be an enumerable dense subset in Y (which is separable, since X C is separable);then, by Lemma 3.1 and Baire’s Theorem, T k ≥ G α ( H V k , A ) is a dense G δ set in L ( R ). Moreover,for every ξ ∈ T k ≥ G α ( H V k , A ), { H V ∈ X C | lim sup t →∞ α ( t ) h| A H V ξ |i t = ∞} ⊃ [ k ≥ { H V k } is a dense G δ set in X C . The theorem is now a consequence of Theorem 4.5 in [15] and Baire’sTheorem. Remark 3.1.
Note as this separability argument used in the proof of Theorem 1.2, in a sense,has allowed to use a typical behaviour in L ( R ) (Lemma 3.1) to find a typical behaviour in X C . Acknowledgments
M.A. was supported by CAPES (a Brazilian government agency; Contract 88882.184169/2018-01). The author is grateful to C´esar R. de Oliveira and Silas L. Carvalho for fruitful discussionsand helpful remarks.
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