A note on spectrum and quantum dynamics

TL;DR

This paper investigates the quantum dynamics of the Schrödinger equation, demonstrating that the rate at which solutions escape from finite-dimensional subspaces varies along subsequences of time, highlighting complex temporal behaviors.

Contribution

It extends the understanding of quantum dynamical systems by analyzing the subsequential escape rates of solutions, inspired by Simon's Wonderland Theorem.

Findings

Escape rates depend on subsequences of time going to infinity

Solutions typically escape from finite-dimensional subspaces in a Baire generic sense

The behavior parallels phenomena described by Simon's Wonderland Theorem

Abstract

We show, in the same vein of Simon's Wonderland Theorem, that, typically in Baire's sense, 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.

Equations

$$\{\begin{array}{l}{\partial }_t\xi =-iT\xi ,t\in \mathbb{R},\\ \xi (0)=\xi ,\xi \in \mathcal{H},\end{array}$$

$$A_{\xi }^T:=⟨e^{-itT}\xi ,A{e}^{-itT}\xi ⟩.$$

$$⟨A_{\xi }^T{⟩}_t:=\frac{1}{t}{\int }_0^t⟨e^{-isT}\xi ,A{e}^{-isT}\xi ⟩ds.$$

$$\underset{t\to \infty }{\lim }⟨|A_{\xi }^T|{⟩}_t=0,$$

$$d(T,T^{\prime }):=\sum _{j=0}^{\infty }\min (2^{-j},\Vert (T-T^{\prime })e_j\Vert ),$$

$$(Mu{)}_j:=u_{j-1}+u_{j+1}+v_j{u}_j,$$

$$(H_{V}u)(x):=-△u(x)+V(x)u(x),$$

$$d(H_V,H_U):=\sum _{j=0}^{\infty }\min (2^{-j},\Vert V-U{\Vert }_j),$$

$$\Vert (R_i(H_{V_k})-R_i(H_V))u{\Vert }_{{\mathrm{L}}^2(\mathbb{R})}$$

$$\liminf_{t\to\infty}\frac{\log\biggr{(}\frac{1}{t}\int_{0}^{t}|\langle\xi,e^{-isT}\xi\rangle|^{2}ds\biggr{)}}{\log t}=-D_{2}^{+}(\mu_{\xi}^{T}),$$

$$\limsup_{t\to\infty}\frac{\log\biggr{(}\frac{1}{t}\int_{0}^{t}|\langle\xi,e^{-isT}\xi\rangle|^{2}ds\biggr{)}}{\log t}=-D_{2}^{-}(\mu_{\xi}^{T}),$$

$$⟨⟨|X{|}^q⟩{⟩}_{t,\xi }:=\frac{1}{t}{\int }_0^t\sum _{n\in \mathbb{Z}}|n{|}^q|⟨e^{-isT}\xi ,e_n⟩{|}^2\mathrm{d}s.$$

$${\beta }^{-}(\xi ,q):=\underset{t\to \infty }{\mathrm{liminf}}\frac{\log ⟨⟨|X{|}^q⟩{⟩}_{t,\xi }}{q\log t},$$

$${\beta }^{+}(\xi ,q):=\underset{t\to \infty }{\mathrm{limsup}}\frac{\log ⟨⟨|X{|}^q⟩{⟩}_{t,\xi }}{q\log t}.$$

$$\underset{t\to \infty }{\mathrm{limsup}}\alpha (t)=\infty .$$

$$\{T\in X_{\Gamma }\mid Thaspurelysingularcontinuousspectrum,$$

$$\underset{t\to \infty }{\mathrm{limsup}}\alpha (t)⟨|A_{\xi }^T|{⟩}_t=\infty and\underset{t\to \infty }{\mathrm{liminf}}t⟨|A_{\xi }^T|{⟩}_t=0\}$$

$$\underset{t\to \infty }{\mathrm{liminf}}{t}^{1+ϵ}⟨|A_{\xi }^T|{⟩}_t=\infty .$$

$$\{T\in X_{\Gamma }\mid \underset{t\to \infty }{\mathrm{liminf}}t⟨|A_{\xi }^T|{⟩}_t=0\}$$

$$\{T\in X_{\Gamma }\mid \underset{t\to \infty }{\mathrm{limsup}}\alpha (t)⟨|A_{\xi }^T|{⟩}_t=\infty \}$$

$$D_{\gamma }:=\{T\in X_{\Gamma }\mid \sigma (T)=[-\gamma ,\gamma ]\mathrm{is}\mathrm{pure}\mathrm{point}\}$$

$$(h_{\omega }u{)}_j:=u_{j-1}+u_{j+1}+{\omega }_j{u}_j.$$

$$\sigma (h_{\omega })=[-2,2]+\text{supp}(\rho ),$$

$$D_{b+2}=\{M\in X_b\mid \sigma (M)=[-b-2,b+2],\sigma (M)\text{ is pure point}\}$$

$$\{H\in X_C\mid \sigma (H)\mathrm{is}\mathrm{pure}\mathrm{point}\}$$

$$\{H\in X_C\mid Hhaspurelysingularcontinuousspectrumon(0,\infty )$$

$$and\underset{t\to \infty }{\mathrm{limsup}}\alpha (t)⟨|A_{\xi }^H|{⟩}_t=\infty \}$$

$$⟨|A_{\xi }^T|{⟩}_t<C_{\xi }\Vert A{\Vert }_1{t}^{-1},$$

$$\{T\in X\mid \underset{t\to \infty }{\mathrm{limsup}}\alpha (t)⟨|A_{\xi }^T|{⟩}_t=\infty and\underset{t\to \infty }{\mathrm{liminf}}t⟨|A_{\xi }^T|{⟩}_t=0\}$$

$$X\ni T\mapsto \alpha (t)⟨|A_{\xi }^T|{⟩}_t$$

$$\bigcup _{t\ge k}\{T\in X\mid \alpha (t)⟨|A_{\xi }^T|{⟩}_t>n\}$$

Full text

Keywords: spectral theory, Schrödinger equation, decaying rates.

AMS classification codes: 46L60 (primary), 81Q10 (secondary).

1 Introduction and results

There is a vast literature concerning the large time asymptotic behaviour of wave packet solutions of the Schrödinger equation

[TABLE]

where $T$ is a self-adjoint operator in a separable complex Hilbert space $\cal{H}$. Namely, the relations between the quantum dynamics of solutions of (1) and the spectral properties of $T$ are a classical subject of quantum mechanics. In this context, we refer to [2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 16], among others.

1.1 Spectrum and dynamics

We recall that, for every $\xi\in\cal{H}$, the one-parameter unitary group ${\mathbb{R}}\ni t\mapsto e^{-itT}$ is so that the curve $e^{-itT}\xi$, in some sense, solves (1). Next we list two quantities usually considered to probe the large time behaviour of the dynamics $e^{-itT}\xi$.

Let $A$ be a positive operator such that, for each  $t\in\mathbb{R}$,  $e^{-itT}{\cal{D}}(A)\subset{\cal{D}}(A)$. For each $\xi\in{\cal{D}}(A)$, the expectation value of $A$ in the state $\xi$ at time $t$ is defined as

[TABLE] 2. 2.

The time average for $A$ is defined as

[TABLE]

In this paper, we are interested in studying the relations between the large time behaviour of the time average of expectation value of $A$ in the state $\xi$ and the spectral properties of the spectral measure $\mu_{\xi}^{T}$ of $T$ associated with $\xi$. 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 $\cal{H}$. For every $\xi\in\cal{H}$,

[TABLE]

if, and only if, $\mu_{\xi}^{T}$ is purely continuous.

We note that since any projector onto a finite-dimensional subspace of $\cal{H}$ satisfies the hypotheses of RAGE Theorem, initial states whose spectral measures are purely continuous can be interpreted as those whose trajectories escape, in time average, from every finite-dimensional subspace.

1.2 Fine scales of decay

A complete metric space $(X,d)$ of self-adjoint operators, acting in the separable complex Hilbert space $\mathcal{H}$, is said to be regular if convergence with respect to $d$ implies strong resolvent convergence of operators (see Definition 2.2 ahead). One of the results stated in [15], the so-called Wonderland Theorem, says that, for some regular spaces $X$, $\{T\in X\mid T$ has purely singular continuous spectrum$\}$ is a dense $G_{\delta}$ set in $X$. Hence, for these spaces, by RAGE Theorem, for each compact operator $A$ and each $\xi\in\cal{H}$, $\{T\in X\mid\displaystyle\lim_{t\to\infty}\langle|A_{\xi}^{T}|\rangle_{t}=0\}$ contains a dense $G_{\delta}$ set in $X$. In this work we present refinements of this result for three different classes of self adjoint operators.

General operators. Let $a>0$ and endow $X_{a}:=\{T\mid T$ self-adjoint, $\|T\|\leq a\}$ with the metric

[TABLE]

where $(e_{j})$ is an orthonormal basis of $\cal{H}$. Then, $X_{a}$ is a complete metric space such that metric convergence implies strong resolvent convergence. 2. 2.

Jacobi matrices. For every fixed $b>0$, let the family of Jacobi matrices, $M$, given on ${\ell}^{2}({{\mathbb{Z}}})$ by the action

[TABLE]

where $(v_{j})$ is a sequence in ${\ell}^{\infty}({\mathbb{Z}})$, such that, for each $j\in{\mathbb{Z}}$, $|v_{j}|\leq b$. Denote by $X_{b}$ the set of these matrices endowed with the topology of pointwise convergence on $(v_{j})$. Then, one has that $X_{b}$ is (by Tychonoff’s Theorem) a compact metric space so that metric convergence implies strong resolvent convergence. 3. 3.

Schrödinger operators. Fix $C>0$ and consider the family of Schrödinger operators, $H_{V}$, defined in the Sobolev space ${\mathcal{H}}^{2}({{\mathbb{R}}})$ by the action

[TABLE]

with $V\in{\cal{B}}^{\infty}({{\mathbb{R}}})$ (the space of bounded Borel functions) so that, for each $x\in{\mathbb{R}}$, $|V(x)|\leq C$. Denote by $X_{C}$ the set of these operators endowed with the metric

[TABLE]

where $\|V-U\|_{j}:=\displaystyle\sup_{x\in B(0,j)}|V(x)-U(x)|$. Then, one has that $X_{C}$ is (again by Tychonoff’s Theorem) a compact metric space such that convergence metric implies strong resolvent convergence (Definition 2.2). Namely, if $H_{V_{k}}\rightarrow H_{V}$ in $X_{C}$, then, for every $x\in{\mathbb{R}}$, one has that $\displaystyle\lim_{k\to\infty}V_{k}(x)=V(x)$. Therefore, for each $u\in{\mathrm{L}}^{2}({\mathbb{R}})$, by the second resolvent identity and dominated convergence,

[TABLE]

as $k\rightarrow\infty$.

Next, we introduce a dynamic quantity, the time average of quantum return probability, and its lower and upper decaying exponents, respectively, given by [3]

[TABLE]

[TABLE]

where $D_{2}^{-}(\mu_{\xi}^{T})$ and $D_{2}^{+}(\mu_{\xi}^{T})$ denote, respectively, the lower and upper correlation dimensions of $\mu_{\xi}^{T}$ (see Theorem 2.2 in [3]).

For every $\{e_{n}\}_{n\in{\mathbb{Z}}}$ orthonormal basis of $\mathcal{H}$ and every $q>0$ we also introduce the (time-average) $q$-moment of the position operator at time $t>0$, with initial condition $\xi$,

[TABLE]

Let the lower and upper transport exponents, respectively, given by

[TABLE]

[TABLE]

In [6, 7] Carvalho and de Oliveira has been proven refinements of Wonderland Theorem for some 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 simultaneously zero and one (this corresponds to the distinct polynomially decaying rates for the quantum return probability (2)-(3)), respectively, is generic [6] and that is true a dual result for the transport exponents (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 is exceptional. 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).

We shall prove following

Theorem 1.2**.**

Let $\alpha:{\mathbb{R}}\longrightarrow{\mathbb{R}}$ such that

[TABLE]

For every compact operator $A$ and every $\xi\in{\cal{H}}$ non null,

[TABLE]

is a dense $G_{\delta}$ set in $X_{\Gamma}$ for $\Gamma\in\{a,b\}$.

Remark 1.1**.**

Theorem 1.2 one says that, for $X_{a}$ and $X_{b}$, typically Baire’s sense, expectation values of compact 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 $\alpha$, in statement of Theorem 1.2, can be chosen arbitrarily and that is well known that, about rather general conditions on $A$, $T$ and $\xi$, for every $\epsilon>0$,

[TABLE] 2. 2.

The proof from that

[TABLE]

is a dense $G_{\delta}$ set in $X_{\Gamma}$ is a direct application of (2) and (3) (Theorem 2.2 in [3]), Lemma 3.2 and Theorem 3.2 (Theorem 2.1) in [13] and Theorems 1.2 and 1.4 in [6]. For the convenience of the reader, we presented in details a simple proof of this fact here. 3. 3.

The proof from that

[TABLE]

is a dense $G_{\delta}$ set in $X_{\Gamma}$ is more delicate since that involves an arbitrary growth of $\alpha$. To prove such result we use the RAGE Theorem combined with the fact that, for some $\gamma(\Gamma)>0$,

[TABLE]

is a dense set in $X_{\Gamma}$. For $X_{\Gamma}=X_{a}$, is a direct consequence from the Weyl-von Neumann Theorem [18, 19] that $D_{a}$ is a dense set in $X_{a}$ (see proof of Theorem in 3.2 [15] for details), whereas for $X_{\Gamma}=X_{b}$ is known that this can be obtained by using Anderson’s localization. Namely, for every fixed $b>0$, consider $\Omega=[-b,b]^{\mathbb{Z}}$ be endowed with the product topology and with the respective Borel $\sigma$-algebra. Assume that $(\omega_{j})_{j\in\mathbb{Z}}=\omega\in\Omega$ is a set of independent, identically distributed real-valued random variables with a common probability measure $\rho$ not concentrated on a single point and so that $\int|\omega_{j}|^{\theta}{\mathrm{d}}\rho(w_{j})<\infty$ for some $\theta>0$. Denote by $\nu:=\rho^{\mathbb{Z}}$ the probability measure on $\Omega$. The Anderson model is a random Hamiltonian on $\ell^{2}(\mathbb{Z})$, defined for each $\omega\in\Omega$ by

[TABLE]

It turns out that [1, 9, 16]

[TABLE]

and $\nu$-a.s. $\omega$, $h_{\omega}$ has pure point spectrum [5, 17] (see also Theorem 4.5 in [9]). Thus, if $\mu$ denotes the product of infinite copies of the normalized Lebesgue measure on $[-b,b]$, that is, $(2b)^{-1}\ell$, then

[TABLE]

is so that $\mu(X_{b}\backslash D_{b+2})=0$. 4. 4.

We note that for $X_{C}$ the ideas presented above do not apply. Namely, this comes from the fact that this is a space of unbounded operators and, to the best of our knowledge, still no has been detailed in the literature arguments that show that

[TABLE]

is a dense set in $X_{C}$. In this context, in this work, we use the separability of $X_{C}$ to prove the following result.

Theorem 1.3**.**

Take $\alpha$ be as before. Then, for every compact operator $A$, there exists a dense $G_{\delta}$ set $G_{\alpha}(A)$ in ${\mathrm{L}}^{2}({\mathbb{R}})$ such that, for every $\xi\in G_{\alpha}(A)$,

[TABLE]

is a dense $G_{\delta}$ set in $X_{C}$.

The paper is organized as follows. In Section 2, we present the proof of Theorem 1.2. In section 3, we prove Theorem 1.3.

Some words about notation: $\mathcal{H}$ denotes a separable complex Hilbert space. If $A$ is a linear operator in $\mathcal{H}$, we denote its domain by ${\mathcal{D}}(A)$, its spectrum by $\sigma(A)$, its point spectrum by $\sigma_{p}(A)$. If $\varrho(A)$ denotes its resolvent set, then the resolvent operator of $A$ at $\lambda\in\varrho(A)$ is denoted by $R_{\lambda}(A)$. For every set $\Omega\subset{\mathbb{R}}$, $\chi_{\Omega}$ denotes the characteristic function of the set $\Omega$. Finally, for every $x\in{\mathbb{R}}$ and $r>0$, $B(x,r)$ denotes the open interval $(x-r,x+r)$.

2 General operators and Jacobi Matrices

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 $\cal{H}$ if, for every $\xi\in\cal{H}$, $T_{n}\xi\longrightarrow T\xi$ in $\cal{H}$.**

Next, we present definitions of a sequence of (unbounded) self-adjoint operators $(T_{n})$ approaching another one $T$.

Definition 2.2**.**

Let $(T_{n})$ be a sequence of self-adjoint operators and let $T$ be another self-adjoint operator. One says that:

$T_{n}$  converges to  $T$ in the strong resolvent sense (SR) if $R_{i}(T_{n})$ strongly converges to $R_{i}(T)$. 2. 2.

$T_{n}$  converges to  $T$ in the strong dynamical sense (SD) if, for each $t\in\mathbb{R}$, $e^{itT_{n}}$ strongly converges 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 operators are equivalent.

Definition 2.3**.**

Let $\mu$ be a $\sigma$-finite positive Borel measure on $\mathbb{R}$. One says that $\mu$ is Lipschitz continuous if there exists a constant $C>0$ such that, for each interval $I$ with $\ell(l)<1$, $\mu(I)<C\,\ell(I)$, where $\ell(\cdot)$ denotes the Lebesgue measure on $\mathbb{R}$.**

We need the following

Theorem 2.1** (Theorem 3.2 in [13]).**

If $\mu_{\xi}^{T}$ is Lipschitz continuous, then there exists a constant $C_{\xi}$ such that for any compact operator $A$ and $t>0$

[TABLE]

where $\|A\|_{1}$ denotes the trace norm of $A$.

Now we confirm the phenomenon of quantum intermittency for regular spaces of self-adjoint operators about mild hypotheses.

Proposition 2.2**.**

Let $\alpha$ as in the statement of Theorem 1.2 and let $\xi\in\cal{H}$ non null. Let $X$ be a regular space of self-adjoint operators. Suppose that:

For some $\gamma>0$, $D_{\gamma}:=\{T\in X\mid\sigma(T)=[-\gamma,\gamma]~{}is~{}pure~{}point\}$ is a dense set in $X$. 2. 2.

$L:=\{T\in X\mid\mu_{\xi}^{T}~{}is~{}Lipschitz~{}continuous\}$* is a dense set in $X$.*

Then, for any compact operator $A$,

[TABLE]

is a dense $G_{\delta}$ set in $X$.

Proof.

Since, for each $t\in{\mathbb{R}}$, by Proposition 2.1 and dominated convergence, the mapping

[TABLE]

is continuous, it follows that, for each $k\geq 1$ and each $n\geq 1$, the set

[TABLE]

is open, from which follows that

[TABLE]

is a $G_{\delta}$ set in $X$. By RAGE Theorem, $D_{\gamma}\subset\{T\in X\mid\displaystyle\limsup_{t\to\infty}\alpha(t)\langle|A_{\xi}^{T}|\rangle_{t}=\infty\}$. Hence, $\{T\in X\mid\displaystyle\limsup_{t\to\infty}\alpha(t)\langle|A_{\xi}^{T}|\rangle_{t}=\infty\}$ is a dense $G_{\delta}$ set in $X$.

We note that, for each $j\geq 1$,

[TABLE]

is also a $G_{\delta}$ set in $X$. Since, by Theorem 2.1, for each $j\geq 1$, $L\subset L_{j}$, it follows that, by Baire’s Theorem,

[TABLE]

is a dense $G_{\delta}$ set in $X$, concluding the proof of proposition. ∎

Proof (Theorem 1.2).

As each $T\in X_{\Gamma}$ can be approximated by an operator whose spectral measures are Lipschitz continuous (see proof of Theorems 1.2 and 1.4 in [6] for details). The theorem is now a direct consequence of Remark 1.1, Proposition 2.2, Theorems 3.1 and 4.1 in [15] and Baire’s Theorem. ∎

3 Schrödinger operators

In order to prove Theorem 1.3 we need of result to follow and a fine separability argument (check below).

Lemma 3.1**.**

Let $T$ be a self-adjoint operator so that $\sigma_{p}(T)\not=\emptyset$ and let $\alpha$ as in the statement of Theorem 1.2 . Then, for any compact operator $A$

[TABLE]

is a dense $G_{\delta}$ set in $\cal{H}$.

Proof.

Since, for each $t\in{\mathbb{R}}$, by dominated convergence, the mapping

[TABLE]

is continuous, it follows that

[TABLE]

is a $G_{\delta}$ set in $\cal{H}$.

Given $\xi\in\cal{H}$, write $\xi=\xi_{1}+\xi_{2}$, with $\xi_{1}\in Span\{\xi_{0}\}^{\perp}$ and $\xi_{2}\in Span\{\xi_{0}\}$, where $\xi_{0}$, with $\|\xi_{0}\|_{\cal{H}}=1$, is an eigenvector of $T$ associated with an eigenvalue $\lambda$. If $\xi_{2}\not=0$, then

[TABLE]

where $P^{T}(\{\lambda\})$ represents the spectral resolution of $T$ over the set $\{\lambda\}$. Now, if $\xi_{2}=0$, define, for each $k\geq 1$,

[TABLE]

It is clear that $\xi_{k}\rightarrow\xi$. Moreover, by the previous arguments, for each $k\geq 1$, one has

[TABLE]

Thus, $G:=\{\xi\in{\cal{H}}\mid\mu_{\xi}^{T}$ has point component$\}$ is a dense set in $\cal{H}$. Nevertheless, by RAGE Theorem, $G\subset G_{\alpha}(T,A)$. Hence, $G_{\alpha}(T,A)$ is a dense $G_{\delta}$ set in $\cal{H}$. ∎

Proof (Theorem 1.3).

By the arguments presented in the proof of Proposition 2.2, for every $\xi\in{\mathrm{L}}^{2}({\mathbb{R}})$,

[TABLE]

is a $G_{\delta}$ set in $X_{C}$.

Now given $H_{V}\in X_{C}$, we define, for every $k\geq 1$,

[TABLE]

We note that, for each $k\geq 1$ and each $x\geq k$, $V_{k}(x)\leq 0$. Moreover, for each $k\geq 1$,

[TABLE]

Therefore, for every $k\geq 1$, $H_{V_{k}}$ has at least a negative eigenvalue [14]; in particular, $\sigma_{p}(H_{V_{k}})\not=\emptyset$. Since $H_{V_{k}}\rightarrow H_{V}$ in $X_{C}$, it follows that

[TABLE]

is a dense set in $X_{C}$.

Now, 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, $\bigcap_{k\geq 1}G_{\alpha}(H_{V_{k}},A)$ is a dense $G_{\delta}$ set in ${\mathrm{L}}^{2}({\mathbb{R}})$. Moreover, for every $\xi\in\bigcap_{k\geq 1}G_{\alpha}(H_{V_{k}},A)$,

[TABLE]

is a dense $G_{\delta}$ set in $X_{C}$. The theorem is now a consequence of Theorem 4.5 in [15] and Baire’s Theorem. ∎

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 ${\mathrm{L}}^{2}({\mathbb{R}})$ (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 César R. de Oliveira and Silas L. Carvalho for fruitful discussions and helpful remarks.