# Decay estimates for unitary representations with applications to continuous-
and discrete-time models

S. Richard1111Supported by the grant _Topological invariants through
scattering theory and noncommutative geometry_ from Nagoya University, and by
JSPS Grant-in-Aid for scientific research C no 18K03328 & 21K03292, and on
leave of absence from Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR
5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622
Villeurbanne cedex, France. and R. Tiedra de Aldecoa2222Partially supported by
the Chilean Fondecyt Grant 1210003.

###### Abstract

We present a new technique to obtain polynomial decay estimates for the matrix
coefficients of unitary operators. Our approach, based on commutator methods,
applies to nets of unitary operators, unitary representations of topological
groups, and unitary operators given by the evolution group of a self-adjoint
operator or by powers of a unitary operator. Our results are illustrated with
a wide range of examples in quantum mechanics and dynamical systems, as for
instance Schrödinger operators, Dirac operators, quantum waveguides, horocycle
flows, adjacency matrices, Jacobi matrices, quantum walks or skew products.

>   * 1
>
> Graduate school of mathematics, Nagoya University, Chikusa-ku, Nagoya
> 464-8602, Japan
>
>   * 2
>
> Facultad de Matemáticas, Pontificia Universidad Católica de Chile,
> Av. Vicuña Mackenna 4860, Santiago, Chile
>
>   * E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>>
>

2010 Mathematics Subject Classification: 22D10, 35Q40, 58J51, 81Q10.

Keywords: Decay estimates, unitary representations, self-adjoint operators,
unitary operators.

###### Contents

  1. 1 Introduction and main results
  2. 2 Decay estimates for unitary representations
    1. 2.1 General unitary representations
    2. 2.2 Unitary representations with self-adjoint generator
    3. 2.3 Unitary representations with unitary generator
  3. 3 Applications
    1. 3.1 Left regular representation
    2. 3.2 Schrödinger operator in $\mathbb{R}^{n}$
    3. 3.3 Dirac operator in $\mathbb{R}^{3}$
    4. 3.4 Quantum waveguides in $\mathbb{R}^{n}$
    5. 3.5 Stark Hamiltonian in $\mathbb{R}^{n}$
    6. 3.6 Fractional Laplacian in $\mathbb{R}^{n}$
    7. 3.7 Horocycle flow
    8. 3.8 Adjacency matrices
    9. 3.9 Jacobi matrices
    10. 3.10 Schrödinger operators on Fock spaces
    11. 3.11 Multiplication by $\lambda$ in $\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}_{+},\mathrm{d}\mu)$
    12. 3.12 $H=-\partial_{xx}+\partial_{yy}$ in $\mathbb{R}^{2}$
    13. 3.13 $H=-X^{2-s}\Delta-\Delta X^{2-s}$ in $\mathbb{R}_{+}$
    14. 3.14 Quantum walks on $\mathbb{Z}$
    15. 3.15 Quantum walks on trees
    16. 3.16 Skew products
  4. A Commutators and regularity classes
  5. B RAGE-type theorem for unitary operators

##  1 Introduction and main results

In recent papers [5, 26, 29, 33, 34, 35, 36, 37, 38], it has been shown that
one can combine certain tools from dynamical systems (averaging along the
dynamics, ergodic theorems) and quantum mechanics (commutator methods) to
determine spectral properties of various classes of continuous- and discrete-
time models. In particular, a new criterion for strong mixing has been put
into evidence in [34, 26]. Formally, it reads as follows: Let $(U_{j})$ be a
family of unitary operators in a Hilbert space $\mathcal{H}$, let $\ell_{j}$
be positive numbers such that $\ell_{j}\to\infty$, let $A$ be a self-adjoint
operator in $\mathcal{H}$, set $D_{j}:=\tfrac{1}{\ell_{j}}[A,U_{j}]U_{j}^{-1}$
and assume that the strong limit $D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}D_{j}$ exists. Then

$\lim_{j}|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|=0\quad\hbox{for all
$\varphi\in\ker(D)^{\perp}$ and $\psi\in\mathcal{H}$.}$ (1.1)

The operator $D$ can be interpreted as a topological degree of the map
$j\mapsto U_{j}$. Indeed, if one considers $[A,\;\\!\cdot\;\\!]$ as a
derivation along the set $(U_{j})$, then $D$ corresponds to a renormalised,
operator-valued, winding number for the map $j\mapsto U_{j}$ (the logarithmic
derivative $\frac{\mathrm{d}z}{z}$ in the usual definition of winding number
is replaced by the “logaritmic derivative" $[A,U_{j}]U_{j}^{-1}$ associated to
$[A,\;\\!\cdot\;\\!]$). See [11, 12, 13, 16, 17, 18, 26, 35, 36, 38] for more
details and examples.

In concrete situations, one usually seeks to get an explicit rate of decay in
estimates like (1.1) in order to quantify the time propagation of the wave
functions. This problem is a very broad and active field of research, with
numerous results in a variety of setups. Decay of correlations, local decay
estimates, pointwise decay estimates, $\mathrm{L}^{p}$ decay estimates,
Strichartz estimates, microlocal estimates, propagation estimates, Morawetz
estimates,… all are families of results related to this problem. In this
paper, we pursue the study initiated in [34, 26] and determine conditions that
guarantee a polynomial rate of decay in (1.1). Our results are general, in the
sense that they are stated first for general nets of unitary operators, then
for unitary representations of topological groups, and finally for unitary
operators given by the evolution group
$(\mathop{\mathrm{e}}\nolimits^{-itH})_{t\in\mathbb{R}}$ of a self-adjoint
operator $H$ or by the powers $(U^{n})_{n\in\mathbb{Z}}$ of a unitary operator
$U$. Moreover, they apply to a wide range of models both in quantum mechanics
and dynamical systems. And finally, they generalise to some extent the results
of [15] (see also [19]) where the authors use commutator methods to establish
abstract pointwise decay estimates for certain classes of self-adjoint
operators. We refer to [2, 6, 20] for related results about pointwise decay
estimates for self-adjoint and unitary operators.

Let us give a more detailed description of our results. In Section 2.1, we
introduce our framework and determine sufficient conditions that guarantee a
polynomial decay estimate

$|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|\leq\tfrac{1}{\ell_{j}^{n}}\;\\!c_{\varphi,\psi},\quad\ell_{j}>0,~{}n\in\mathbb{N}^{*},$

with $c_{\varphi,\psi}$ a constant depending on $\varphi,\psi$ (and $n$) but
not on $j$. A first set of conditions on $D_{j}$ and $\varphi,\psi$ leads to
this estimate in the case $n=1$, while more restrictive sets of conditions
lead to this estimate for any fixed $n\geq 1$ (Theorem 2.1). In addition, when
the unitary operators $U_{j}$ are given by a unitary representation
$\mathscr{U}$ of a topological group $X$ and the scalars $\ell_{j}$ are given
by a proper length function on $X$, then we provide conditions ensuring that
$D$ commutes with $\mathscr{U}$ and that $\mathscr{U}$ has no nontrivial
finite-dimensional unitary subrepresentation in $\ker(D)^{\perp}$ (Proposition
2.3).

These general results are then applied in Sections 2.2 & 2.3 to the case of a
representation of $\mathbb{R}$ given by an evolution group
$(\mathop{\mathrm{e}}\nolimits^{-itH})_{t\in\mathbb{R}}$ with self-adjoint
generator $H$ and to the case of a representation of $\mathbb{Z}$ given by the
powers $(U^{n})_{n\in\mathbb{Z}}$ of a unitary operator $U$. In the former
case, the operators $D_{j}$ can be written as Cesaro means

$D_{t}:=\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}[iH,A](H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H},\quad t>0,$

and our new results are the fact that $D$ is decomposable in the spectral
representation of $H$ and criteria for the continuity or absolute continuity
of the spectrum of $H$ in $\ker(D)^{\perp}$ (Lemma 2.4). In the latter case,
the operators $D_{j}$ can be written as Cesaro means

$D_{n}=\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}([A,U]U^{-1})U^{-m},\quad
n\in\mathbb{N}^{*},$

and our new results are the fact that $D$ is decomposable in the spectral
representation of $U$ and criteria for the continuity or absolute continuity
of the spectrum of $U$ in $\ker(D)^{\perp}$ (Lemma 2.7). Furthermore, in
Propositions 2.5 & 2.8 we pay a special attention to the particular cases
$[iH,A]=f(H)$ and $[A,U]=\gamma(U)$ (with $f$ and $\gamma$ functions) which
are important for applications.

In Section 3, we illustrate these abstract results with numerous examples. For
some of them, the decay estimates we obtain are known, while for others they
are new. A similar dichotomy holds for the spectral results we obtain when we
deal with representations admitting a self-adjoint or a unitary generator.
However, the most striking feature of our approach does not really rely on any
new result for a given example, but on its broad applicability. The whole
variety of examples introduced in Section 3 is conveniently covered with the
same philosophy and toolkit.

Since Section 3 contains a detailed presentation of each example, we just
highlight here a few noticeable facts. First, we note that several important
models of quantum mechanics and dynamical systems are discussed in Section 3.
This is for example the case of Schrödinger operators, Dirac operators,
quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices,
quantum walks and skew products. Next, as mentioned at the beginning of the
introduction, the operator $D$ can sometimes be interpreted as a topological
degree. This occurs for instance in the case of skew products, see Section
3.16. In other instances, the operator $D$ can be expressed in terms of the
square of an asymptotic velocity operator (a kinetic energy) for the unitary
group under study. This occurs for instance in the case of quantum walks on
$\mathbb{Z}$, see Section 3.14. Finally, in Section 3.1 we discuss the case of
the left regular representation of a $\sigma$-compact locally compact
Hausdorff group $X$ with left Haar measure $\mu$ and proper length function
$\ell$. In that case, we obtain for any net $(x_{j})$ in $X$ with
$x_{j}\to\infty$ and suitable
$\varphi,\psi\in\mathop{\mathrm{L}^{2}}\nolimits(X,\mu)$ the decay estimate

$|\langle\varphi,\mathscr{U}(x_{j})\psi\rangle_{\mathcal{H}}|\leq\tfrac{1}{\ell(x_{j})}\;\\!c_{\varphi,\psi}.$

This estimate is similar to others in this paper, but with the interesting
difference that in general the representation $\mathscr{U}$ doesn’t have
either a self-adjoint generator or a unitary generator. It thus illustrates
once again the fact that our approach is general, and not only applicable to
families of unitary operators admiting a self-adjoint generator or unitary
generator.

Finally, in Appendices A & B, we collect some technical results on commutators
and regularity classes and on RAGE-type theorems for unitary operators.

Notations: $\mathbb{N}:=\\{0,1,2,\ldots\\}$ is the set of natural numbers,
$\mathbb{N}^{*}:=\mathbb{N}\setminus\\{0\\}$, $\mathbb{R}_{+}:=(0,\infty)$,
$\mathbb{S}^{1}$ the complex unit circle, $\mathrm{U}(n)$ the group of
$n\times n$ unitary matrices, and $\langle\cdot\rangle:=\sqrt{1+|\cdot|^{2}}$.
Given a Hilbert space $\mathcal{H}$, we write $\|\cdot\|_{\mathcal{H}}$ for
its norm, $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ for its scalar product
(linear in the first argument), and $\mathrm{U}(\mathcal{H})$ for the set of
unitary operators on $\mathcal{H}$. Given two Hilbert spaces
$\mathcal{H}_{1},\mathcal{H}_{2}$, we write
$\mathscr{B}(\mathcal{H}_{1},\mathcal{H}_{2})$ (resp.
$\mathscr{K}(\mathcal{H}_{1},\mathcal{H}_{2})$) for the set of bounded (resp.
compact) operators from $\mathcal{H}_{1}$ to $\mathcal{H}_{2}$. We also write
$\|\cdot\|_{\mathscr{B}(\mathcal{H}_{1},\mathcal{H}_{2})}$ for the norm of
$\mathscr{B}(\mathcal{H}_{1},\mathcal{H}_{2})$, and use the shorthand
notations
$\mathscr{B}(\mathcal{H}_{1}):=\mathscr{B}(\mathcal{H}_{1},\mathcal{H}_{1})$
and
$\mathscr{K}(\mathcal{H}_{1}):=\mathscr{K}(\mathcal{H}_{1},\mathcal{H}_{1})$.

##  2 Decay estimates for unitary representations

###  2.1 General unitary representations

We start with a general theorem on decay estimates for the matrix coefficients
of unitary operators $U_{j}$ in a Hilbert space $\mathcal{H}$. In the proof,
we use standard results about commutators of operators recalled in Appendix A.

######  Theorem 2.1 (Decay estimates).

Let $(U_{j})_{j\in J}$ be a net in $\mathrm{U}(\mathcal{H})$, let
$(\ell_{j})_{j\in J}\subset[0,\infty)$ satisfy $\ell_{j}\to\infty$, assume
there exists a self-adjoint operator $A$ in $\mathcal{H}$ such that $U_{j}\in
C^{1}(A)$ for each $j\in J$, and suppose that the strong limit

$D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}D_{j}\quad\hbox{with}\quad
D_{j}:=\tfrac{1}{\ell_{j}}[A,U_{j}]U_{j}^{-1}$

exists. Then

  1. (a)

For each $\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and
$\psi\in\mathcal{D}(A)$ there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|\leq\|(D-D_{j})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{\ell_{j}}\;\\!c_{\varphi,\psi},\quad\ell_{j}>0.$

In particular, $\lim_{j}\langle\xi,U_{j}\zeta\rangle_{\mathcal{H}}=0$ for all
$\xi\in\ker(D)^{\perp}$ and $\zeta\in\mathcal{H}$.

  2. (b)

Assume that $D=D_{j}$ for all $j\in J$. Then for each $\varphi\in
D\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$ there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|\leq\tfrac{1}{\ell_{j}}\;\\!c_{\varphi,\psi},\quad\ell_{j}>0.$

  3. (c)

Assume that $D=D_{j}$ for all $j\in J$, that $D\in C^{1}(A)$, and that
$[A,D]=DB$ with $B\in C^{(n-1)}(A)$ ($n\in\mathbb{N}^{*}$) and $[D,B]=0$. Then
for each $\varphi\in D^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|\leq\tfrac{1}{\ell_{j}^{n}}\;\\!c_{\varphi,\psi},\quad\ell_{j}>0.$

###### Proof.

(a) Take $\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$,
$\psi\in\mathcal{D}(A)$, and $j\in J$ such that $\ell_{j}>0$. Then we have

$\displaystyle|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|$
$\displaystyle=\big{|}\langle(D-D_{j})\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}+\langle
D_{j}\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}\big{|}$
$\displaystyle\leq\|(D-D_{j})\widetilde{\varphi}\|_{\mathcal{H}}\\!\;\|\psi\|_{\mathcal{H}}+\tfrac{1}{\ell_{j}}\big{|}\langle[A,U_{j}]U_{j}^{-1}\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}\big{|}$
$\displaystyle\leq\|(D-D_{j})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{\ell_{j}}\big{|}\langle
A\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}\big{|}+\tfrac{1}{\ell_{j}}\big{|}\langle\widetilde{\varphi},U_{j}A\psi\rangle_{\mathcal{H}}\big{|}$
$\displaystyle\leq\|(D-D_{j})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{\ell_{j}}\;\\!c_{\varphi,\psi}$

with
$c_{\varphi,\psi}:=\|A\widetilde{\varphi}\|_{\mathcal{H}}\|\psi\|_{\mathcal{H}}+\|\widetilde{\varphi}\|_{\mathcal{H}}\|A\psi\|_{\mathcal{H}}$.
This proves the first part of the claim. Since $D=\operatorname*{s\hskip
0.1pt-\hskip 0.1ptlim}_{j}D_{j}$ and $\ell_{j}\to\infty$, we infer that
$\lim_{j}\langle\varphi,U_{j}\psi\rangle=0$, and thus the second part of the
claim follows by the density of $D\mathcal{D}(A)$ in
$\overline{D\mathcal{H}}=\ker(D)^{\perp}$ and the density of $\mathcal{D}(A)$
in $\mathcal{H}$.

(b) The claim is a direct consequence of point (a) in the case $D=D_{j}$ for
all $j\in J$.

(c) We prove the claim by induction on $n\in\mathbb{N}^{*}$. For $n=1$, the
claim is true due to point (b). For $n-1\geq 1$, we make the induction
hypothesis that the claim is true. For $n$, we take $\varphi\in
D^{n}\mathcal{D}(A^{n})$, $\psi\in\mathcal{D}(A^{n})$ and $j\in J$ such that
$\ell_{j}>0$. Then, since $D=D_{j}$ and $\varphi=D\widetilde{\varphi}$ with
$\widetilde{\varphi}\in D^{n-1}\mathcal{D}(A^{n})$, we get

$\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}=\langle
D_{j}\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}=\tfrac{1}{\ell_{j}}\langle
A\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}-\tfrac{1}{\ell_{j}}\langle\widetilde{\varphi},U_{j}A\psi\rangle_{\mathcal{H}}.$
(2.1)

The induction hypothesis applies to the second term in (2.1) since
$\widetilde{\varphi}\in D^{n-1}\mathcal{D}(A^{n})\subset
D^{n-1}\mathcal{D}(A^{n-1})$ and $A\psi\in\mathcal{D}(A^{n-1})$. So there
exists $c_{\widetilde{\varphi},A\psi}\geq 0$ such that

$\big{|}\langle\widetilde{\varphi},U_{j}A\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{\ell_{j}^{n-1}}\;\\!c_{\widetilde{\varphi},A\psi}.$
(2.2)

For the first term in (2.1), we have
$\widetilde{\varphi}=D^{n-1}\tilde{\tilde{\varphi}}$ with
$\tilde{\tilde{\varphi}}\in\mathcal{D}(A^{n})$. So, using the relations
$[A,D]=DB$ and $[D,B]=0$, we get that

$\displaystyle A\widetilde{\varphi}$
$\displaystyle=\big{(}D^{n-1}A+[A,D^{n-1}]\big{)}\tilde{\tilde{\varphi}}$
$\displaystyle=\big{(}D^{n-1}A+\mathop{\textstyle\sum}\nolimits_{m=0}^{n-2}D^{n-2-m}[A,D]D^{m}\big{)}\tilde{\tilde{\varphi}}$
$\displaystyle=D^{n-1}(A+(n-1)B)\tilde{\tilde{\varphi}}$

with $(A+(n-1)B)\tilde{\tilde{\varphi}}\in\mathcal{D}(A^{n-1})$ due to the
inclusions $\tilde{\tilde{\varphi}}\in\mathcal{D}(A^{n})$ and $B\in
C^{(n-1)}(A)$. Therefore $A\widetilde{\varphi}\in D^{n-1}\mathcal{D}(A^{n-1})$
and $\psi\in\mathcal{D}(A^{n})\subset\mathcal{D}(A^{n-1})$, and we infer from
the induction hypothesis that there exists $c_{A\widetilde{\varphi},\psi}\geq
0$ such that

$\big{|}\langle
A\widetilde{\varphi},U_{j}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{\ell_{j}^{n-1}}\;\\!c_{A\widetilde{\varphi},\psi}.$
(2.3)

Finally, combining (2.1), (2.2) and (2.3), we obtain that

$|\langle\varphi,U_{j}\psi\rangle_{\mathcal{H}}|\leq\tfrac{1}{\ell_{j}}\tfrac{1}{\ell_{j}^{n-1}}\;\\!c_{A\widetilde{\varphi},\psi}+\tfrac{1}{\ell_{j}}\tfrac{1}{\ell_{j}^{n-1}}\;\\!c_{\widetilde{\varphi},A\psi}=\tfrac{1}{\ell_{j}^{n}}\;\\!c_{\varphi,\psi}$

with
$c_{\varphi,\psi}:=c_{A\widetilde{\varphi},\psi}+c_{\widetilde{\varphi},A\psi}$.
∎

######  Remark 2.2.

(a) If the operators $U_{j}$ are given by a unitary representation, then the
property of Theorem 2.1(a)
$\lim_{j}\langle\xi,U_{j}\zeta\rangle_{\mathcal{H}}=0$ for
$\xi\in\ker(D)^{\perp}$ and $\zeta\in\mathcal{H}$ amounts to a strong mixing
property of the unitary representation in $\ker(D)^{\perp}$. See [26] for more
information on this point.

(b) The set $D^{n}\mathcal{D}(A^{n})$ in Theorem 2.1(c) is always dense in
$\ker(D)^{\perp}$, independently of the value of $n\in\mathbb{N}^{*}$. Indeed,
since $\mathcal{D}(A^{n})$ is dense in $\mathcal{H}$, we have that
$D^{n}\mathcal{D}(A^{n})$ is dense in
$\overline{D^{n}\mathcal{H}}=\ker(D^{n})^{\perp}$. But $D$ is self-adjoint. So
$\ker(D^{n})=\ker(D)$, and thus $D^{n}\mathcal{D}(A^{n})$ is dense in
$\ker(D)^{\perp}$.

(c) Sometimes the unitary operators $U_{j}$ are given by the evolution group
of a self-adjoint operator $H$, namely, $(U_{j})_{j\in
J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$. In this situation, a
convenient operator $A$ in Theorem 2.1 is often of the form
$\widetilde{A}=(H+i)^{-1}A(H-i)^{-1}$ with $A$ some self-adjoint operator such
that $(H-i)^{-1}\in C^{1}(A)$. In that case, the estimates of Theorem 2.1 hold
for vectors $\varphi\in D^{n}\mathcal{D}((\widetilde{A})^{n})$ and
$\psi\in\mathcal{D}((\widetilde{A})^{n})$. However, since $A$ is simpler than
$\widetilde{A}$ and since
$\mathcal{D}(A^{n})\subset\mathcal{D}((\widetilde{A})^{n})$, in concrete
examples we will only present the estimates for vectors $\varphi\in
D^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$ for the sake of
simplicity (see Sections 3.2, 3.4, 3.7, 3.8 and 3.9).

In the sequel, we assume that the unitary operators $U_{j}$ are given by a
unitary representation $\mathscr{U}:X\to\mathrm{U}(\mathcal{H})$ of a
topological group $X$. We also assume that the scalars $\ell_{j}$ are given by
a proper length function on $X$, that is, a function $\ell:X\to[0,\infty)$
satisfying the following properties (with $e$ the identity of $X$):

  1. (L1)

$\ell(e)=0$,

  2. (L2)

$\ell(x^{-1})=\ell(x)$ for all $x\in X$,

  3. (L3)

$\ell(xy)\leq\ell(x)+\ell(y)$ for all $x,y\in X$,

  4. (L4)

if $K\subset[0,\infty)$ is compact, then $\ell^{-1}(K)\subset X$ is relatively
compact.

Finally, we recall that a net $(x_{j})_{j\in J}$ in a topological space $X$
diverges to infinity, with notation $x_{j}\to\infty$, if $(x_{j})_{j\in J}$
has no limit point in $X$. This implies that for each compact set $K\subset
X$, there exists $j_{K}\in J$ such that $x_{j}\notin K$ for $j\geq j_{K}$. In
particular, $X$ is not compact.

In this situation, the existence of the strong limit $D$ leads to additional
properties of the unitary operators given by $\mathscr{U}$. Namely,
$\mathscr{U}$ has no nontrivial finite-dimensional unitary subrepresentation
in $\ker(D)^{\perp}$, and the operator $D$ commutes with $\mathscr{U}:$

######  Proposition 2.3.

Let $X$ be a topological group equipped with a proper length function $\ell$,
let $\mathscr{U}:X\to\mathrm{U}(\mathcal{H})$ be a unitary representation of
$X$, let $(x_{j})_{j\in J}$ be a net in $X$ with $x_{j}\to\infty$, assume
there exists a self-adjoint operator $A$ in $\mathcal{H}$ such that
$\mathscr{U}(x_{j})\in C^{1}(A)$ for each $j\in J$, and suppose that the
strong limit

$D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}D_{j}\quad\hbox{with}\quad
D_{j}:=\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x_{j})]\mathscr{U}(x_{j})^{-1}$

exists. Then

  1. (a)

$\mathscr{U}$ has no nontrivial finite-dimensional unitary subrepresentation
in $\ker(D)^{\perp}$.

  2. (b)

Let $x\in X$, assume that $\mathscr{U}(x)\in C^{1}(A)$, and suppose that the
strong limit

$\widetilde{D}:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\widetilde{D}_{j}\quad\hbox{with}\quad\widetilde{D}_{j}:=\tfrac{1}{\ell(x^{-1}x_{j})}[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}$

exists and satisfies $D=\widetilde{D}$. Then $[D,\mathscr{U}(x)]=0$.

###### Proof.

(a) The claim follows from Theorem 2.1(a) and the fact that matrix
coefficients of finite-dimensional unitary representations of a group do not
vanish at infinity (see for instance [3, Rem. 2.15(iii)]).

(b) First, note that the commutator $[A,\mathscr{U}(x^{-1}x_{j})]$ in the
expression for $\widetilde{D}$ is well-defined for each $j\in J$ because
$\mathscr{U}(x^{-1}x_{j})=\mathscr{U}(x)^{-1}\mathscr{U}(x_{j})$ with
$\mathscr{U}(x)\in C^{1}(A)$ and $\mathscr{U}(x_{j})\in C^{1}(A)$. Next, we
have

$\displaystyle D\mathscr{U}(x)$ $\displaystyle=\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x)\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}$
$\displaystyle=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x)]\mathscr{U}(x^{-1}x_{j})\mathscr{U}(x^{-1}x_{j})^{-1}$
$\displaystyle\quad+\mathscr{U}(x)\cdot\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}$
$\displaystyle=\lim_{j}\tfrac{1}{\ell(x_{j})}\cdot[A,\mathscr{U}(x)]+\mathscr{U}(x)\cdot\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}$

with the first term vanishing because $\lim_{j}\tfrac{1}{\ell(x_{j})}=0$ and
with the second term satisfying

$\displaystyle\mathscr{U}(x)\cdot\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}$
$\displaystyle=\mathscr{U}(x)\widetilde{D}+\mathscr{U}(x)\cdot\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{j}\left(\tfrac{1}{\ell(x_{j})}-\tfrac{1}{\ell(x^{-1}x_{j})}\right)[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}.$

Thus, to conclude the proof, it is sufficient to show that

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\left(\tfrac{1}{\ell(x_{j})}-\tfrac{1}{\ell(x^{-1}x_{j})}\right)[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}=0.$
(2.4)

Let $\varphi\in\mathcal{H}$. Then we have

$\displaystyle\lim_{j}\left\|\left(\tfrac{1}{\ell(x_{j})}-\tfrac{1}{\ell(x^{-1}x_{j})}\right)[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}\varphi\right\|_{\mathcal{H}}$
$\displaystyle=\lim_{j}\left|\tfrac{\ell(x^{-1}x_{j})-\ell(x_{j})}{\ell(x_{j})}\right|\cdot\left\|\tfrac{1}{\ell(x^{-1}x_{j})}[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}\varphi\right\|_{\mathcal{H}}$
$\displaystyle\leq\big{(}\|\widetilde{D}\varphi\|_{\mathcal{H}}+1\big{)}\lim_{j}\left|\tfrac{\ell(x^{-1}x_{j})-\ell(x_{j})}{\ell(x_{j})}\right|.$

Since $|\ell(x^{-1}x_{j})-\ell(x_{j})|\leq\ell(x)$ due to the triangle
inequality for $\ell$, we infer that

$\lim_{j}\left\|\left(\tfrac{1}{\ell(x_{j})}-\tfrac{1}{\ell(x^{-1}x_{j})}\right)[A,\mathscr{U}(x^{-1}x_{j})]\mathscr{U}(x^{-1}x_{j})^{-1}\varphi\right\|_{\mathcal{H}}\leq\big{(}\|\widetilde{D}\varphi\|_{\mathcal{H}}+1\big{)}\ell(x)\lim_{j}\tfrac{1}{\ell(x_{j})}=0,$

which proves (2.4). ∎

###  2.2 Unitary representations with self-adjoint generator

In this section, we consider the important case where the representation is a
strongly continuous unitary representation
$\mathscr{U}:\mathbb{R}\to\mathrm{U}(\mathcal{H})$ of the additive group
$\mathbb{R}$. In such a case, Stone’s theorem implies the existence of a self-
adjoint operator $H$ in $\mathcal{H}$ such that
$\mathscr{U}(t)=\mathop{\mathrm{e}}\nolimits^{-itH}$ for each
$t\in\mathbb{R}$. One could also consider the higher-dimensional case of a
strongly continuous unitary representation of the additive group
$\mathbb{R}^{d}$ for $d\geq 1$. But we refrained from doing it for the sake of
simplicity.

We use the notation $P_{\rm p}(H)$ (resp. $P_{\rm c}(H)$, $P_{\rm ac}(H)$) for
the projection onto the pure point (resp. continuous, absolutely continuous)
subspace $\mathcal{H}_{\rm p}(H)$ (resp. $\mathcal{H}_{\rm c}(H)$,
$\mathcal{H}_{\rm ac}(H)$) of $H$, $E^{H}(\cdot)$ for the spectral projections
of $H$, and $\chi_{\mathcal{B}}$ for the characteristic function of a Borel
set $\mathcal{B}\subset\mathbb{R}$.

######  Lemma 2.4 (Properties of $D$).

Let $H$ and $A$ be self-adjoint operators in a Hilbert space $\mathcal{H}$
with $(H-i)^{-1}\in C^{1}(A)$, and let

$D_{t}:=\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}[iH,A](H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H},\quad t>0.$

Then

  1. (a)

$\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{t\to\infty}D_{t}P_{\rm
p}(H)=0$.

  2. (b)

If there exists $B\in\mathscr{B}(\mathcal{H})$ such that
$(H+i)^{-1}([iH,A]-B)(H-i)^{-1}\in\mathscr{K}(\mathcal{H})$ and

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}B(H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm
c}(H)~{}~{}\hbox{exists,}$

then

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{t}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}B(H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H).$

Furthermore, if $D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}$ exists, then

  1. (c)

$[D,\mathop{\mathrm{e}}\nolimits^{isH}]=0$ for all $s\in\mathbb{R}$. In
particular, $D$ is decomposable in the spectral representation of $H$.

  2. (d)

$D=DP_{\rm c}(H)$. In particular, $H|_{\ker(D)^{\perp}}$ has purely continuous
spectrum.

  3. (e)

If $D\mathcal{D}(A)\subset\mathcal{D}(A)$ and
$\int_{1}^{\infty}\mathrm{d}t\;\\!\|(D-D_{t})\varphi\|^{2}_{\mathcal{H}}<\infty$
for all $\varphi\in\mathcal{D}(A)$, then $H|_{\ker(D)^{\perp}}$ has purely
a.c. spectrum.

Point (c) implies that $\ker(D)^{\perp}$ is a reducing subspace for $H$.
Therefore the operator $H|_{\ker(D)^{\perp}}$ in points (d)-(e) is a well-
defined self-adjoint operator (see [40, Thm. 7.28]).

###### Proof.

(a) Let $\varphi\in\mathcal{H}$. Then $P_{\rm p}(H)\varphi=\sum_{j\geq
1}\alpha_{j}\varphi_{j}$ with $(\varphi_{j})_{j\geq 1}$ an orthonormal basis
of $\mathcal{H}_{\rm p}(H)$, $\alpha_{j}\in\mathbb{C}$, and
$H\varphi_{j}=\lambda_{j}\varphi_{j}$ for some $\lambda_{j}\in\mathbb{R}$.
Thus we obtain

$\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{t\to\infty}D_{t}P_{\rm
p}(H)\varphi=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\sum_{j\geq
1}\alpha_{j}\left(\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau(H-\lambda_{j})}\right)(H+i)^{-1}[iH,A](H-i)^{-1}\varphi_{j}.$
(2.5)

Now
$\|\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau(H-\lambda_{j})}\|_{\mathscr{B}(\mathcal{H})}\leq
1$ for all $t>0$, and

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau(H-\lambda_{j})}=E^{H}(\\{\lambda_{j}\\})$

due to von Neumann’s mean ergodic theorem. Therefore we can exchange the limit
and the sum in (2.5) to get

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}P_{\rm p}(H)\varphi$ $\displaystyle=\sum_{j\geq
1}\alpha_{j}E^{H}(\\{\lambda_{j}\\})(H+i)^{-1}[iH,A](H-i)^{-1}\varphi_{j}$
$\displaystyle=\sum_{j\geq
1}\alpha_{j}\langle\lambda_{j}\rangle^{-2}E^{H}(\\{\lambda_{j}\\})[iH,A]E^{H}(\\{\lambda_{j}\\})\varphi_{j}.$

But $E^{H}(\\{\lambda_{j}\\})[iH,A]E^{H}(\\{\lambda_{j}\\})=0$ for each
$\lambda_{j}$ due to the virial theorem for self-adjoint operators [1, Prop.
7.2.10]. Thus we obtain that $\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}P_{\rm p}(H)\varphi=0$, which proves the claim.

(b) Let $K:=(H+i)^{-1}([iH,A]-B)(H-i)^{-1}\in\mathscr{K}(\mathcal{H})$. Then
it follows from point (a) that

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{t\to\infty}D_{t}$
$\displaystyle=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}B(H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H)$
$\displaystyle\quad+\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}K\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H).$

with $\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}K\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H)=0$ due to [30, Thm.
5.9].

(c) The claim follows from Proposition 2.3(b) in the case of the additive
group $X=\mathbb{R}$ and the auxiliary operator

$\widetilde{A}\varphi:=(H+i)^{-1}A(H-i)^{-1}\varphi,\quad\varphi\in\mathcal{D}(A).$

Indeed, we know from [26, Cor. 2.7 & Rem. 2.8] that $\widetilde{A}$ is
essentially self-adjoint (with closure denoted by the same symbol) and that
$\mathop{\mathrm{e}}\nolimits^{-itH}\in C^{1}(\widetilde{A})$ with
$[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{-itH}]=tD_{t}\mathop{\mathrm{e}}\nolimits^{-itH}$
for any $t>0$. Therefore, if we take the proper length function
$\ell:\mathbb{R}\to[0,\infty)$ given by $\ell(t):=|t|$, the unitary
representation $\mathscr{U}:\mathbb{R}\to\mathrm{U}(\mathcal{H})$ given by
$\mathscr{U}(t):=\mathop{\mathrm{e}}\nolimits^{-itH}$, and the net
$(x_{j})_{j\in J}=(t)_{t>0}$, then we get
$\mathscr{U}(s)=\mathop{\mathrm{e}}\nolimits^{-isH}\in C^{1}(\widetilde{A})$
for all $s\in\mathbb{R}$ and

$\displaystyle\widetilde{D}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t-s}[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{-i(t-s)H}]\mathop{\mathrm{e}}\nolimits^{i(t-s)H}=\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{-itH}]\mathop{\mathrm{e}}\nolimits^{itH}=\operatorname*{s\hskip
0.1pt-\hskip 0.1ptlim}_{t\to\infty}D_{t}=D.$

So all the assumptions of Proposition 2.3(b) are verified, and thus
$[D,\mathop{\mathrm{e}}\nolimits^{isH}]=0$ for all $s\in\mathbb{R}$.

Finally, since $[D,\mathop{\mathrm{e}}\nolimits^{isH}]=0$ for all
$s\in\mathbb{R}$, we have $D\chi_{\mathcal{B}}(H)=\chi_{\mathcal{B}}(H)D$ for
each Borel set $\mathcal{B}\subset\mathbb{R}$, and thus $D$ is decomposable in
the spectral representation of $H$ [4, Thm. 7.2.3(b)].

(d) The equality $D=DP_{\rm c}(H)$ follows from point (a). As a consequence,
we get that $\ker(D)^{\perp}\subset\mathcal{H}_{\rm c}(H)$, and thus the
operator $H|_{\ker(D)^{\perp}}$ has purely continuous spectrum.

(e) Take $\varphi\in\mathcal{D}(A)$. Then $\psi=D\varphi\in
D\mathcal{D}(\widetilde{A})\cap\mathcal{D}(\widetilde{A})$, and it follows
from Theorem 2.1(a) that there exists $c_{\psi}\geq 0$ such that

$\big{|}\langle\psi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\big{\|}(D-D_{t})\varphi\big{\|}_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{t}\;\\!c_{\psi},\quad
t>0.$

So, we infer from the assumption and Cauchy-Schwarz inequality that
$\int_{1}^{\infty}\mathrm{d}t\;\\!|\langle\psi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}|^{2}<\infty$,
and thus that
$t\mapsto\langle\psi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}$
belongs to $\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})$. Therefore,
Plancherel’s theorem for the group $X=\mathbb{R}$ [9, Thm. 4.26] implies that
$\psi\in D\mathcal{D}(A)$ belongs to the a.c. subspace $\mathcal{H}_{\rm
ac}(H)$ of $H$. Thus $H|_{\ker(D)^{\perp}}$ has purely a.c. spectrum, since
$D\mathcal{D}(A)$ is dense in $\overline{D\mathcal{H}}=\ker(D)^{\perp}$ and
$\mathcal{H}_{\rm ac}(H)$ is closed in $\mathcal{H}$. ∎

In the next proposition, we consider the particular case where $[iH,A]=f(H)$
for some Borel function $f:\mathbb{R}\to\mathbb{R}$. Our results in this case
generalise the results of [36, Cor. 4.3-4.4].

######  Proposition 2.5 (The case $[iH,A]=f(H)$).

Let $H$ and $A$ be self-adjoint operators in a Hilbert space $\mathcal{H}$,
assume that $(H-i)^{-1}\in C^{1}(A)$ with $[iH,A]=f(H)$ for some Borel
function $f:\mathbb{R}\to\mathbb{R}$, and set $g:=f\langle\cdot\rangle^{-2}$.
Then

  1. (a)

For each $\varphi\in g(H)\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$ there
exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$

  2. (b)

If $g(H)\mathcal{D}(A)\subset\mathcal{D}(A)$, then $H|_{\ker(f(H))^{\perp}}$
has purely a.c. spectrum.

  3. (c)

Suppose that $f\in C^{n}(\mathbb{R})$ ($n\in\mathbb{N}^{*}$) with
$g^{(k)}\in\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})\cap\mathop{\mathrm{L}^{\infty}}\nolimits(\mathbb{R})$
for all $k=0,\dots,n$ and $g^{(n)}$ uniformly continuous. Then for each
$\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$ there
exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

###### Proof.

(a) We know from the proof of Lemma 2.4(c) that the auxiliary operator

$\widetilde{A}\varphi=(H+i)^{-1}A(H-i)^{-1}\varphi,\quad\varphi\in\mathcal{D}(A),$

is essentially self-adjoint (with closure denoted by the same symbol) and that
$\mathop{\mathrm{e}}\nolimits^{-itH}\in C^{1}(\widetilde{A})$ with
$[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{-itH}]=tD_{t}\mathop{\mathrm{e}}\nolimits^{-itH}$
for any $t>0$. So the strong limit $D$ exists and satisfies the equalities

$D_{t}=\tfrac{1}{t}[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{-itH}]\mathop{\mathrm{e}}\nolimits^{itH}=\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}[iH,A](H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}=g(H)=D.$
(2.6)

Therefore the assumptions of Theorem 2.1(b) are satisfied for the net
$(U_{j})_{j\in J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set
$(\ell_{j})_{j\in J}=(t)_{t>0}$, the operator $\widetilde{A}$, and $D=g(H)$.
Thus for each $\varphi\in g(H)\mathcal{D}(A)\subset
g(H)\mathcal{D}(\widetilde{A})$ and
$\psi\in\mathcal{D}(A)\subset\mathcal{D}(\widetilde{A})$ there exists a
constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$

(b) The claim follows from Lemma 2.4(e) because $D=g(H)=D_{t}$ for all $t>0$
and

$\ker(g(H))=\chi_{g^{-1}(\\{0\\})}(H)\mathcal{H}=\chi_{f^{-1}(\\{0\\})}(H)\mathcal{H}=\ker(f(H)).$

(c) Since $g$ is uniformly continuous and belongs to
$\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})\cap\mathop{\mathrm{L}^{\infty}}\nolimits(\mathbb{R})$,
we have

$\lim_{\varepsilon\searrow
0}\big{\|}g(H)-g^{\varepsilon}(H)\big{\|}_{\mathscr{B}(\mathcal{H})}=0$ (2.7)

with

$g^{\varepsilon}(H):=\int_{\mathbb{R}}\mathrm{d}t\;\\!(\mathcal{F}g)(t)\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
t)^{2}}\mathop{\mathrm{e}}\nolimits^{2\pi itH}\quad\hbox{(strong or Bochner
integral)}$

and
$\mathcal{F}:\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})\to\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})$
the Fourier transform (see [8, Thm. 8.35(b)]). Using successively the fact
that $D=g(H)$, equation (2.7), the inclusion
$\mathop{\mathrm{e}}\nolimits^{2\pi
itH}\mathcal{D}(A)\subset\mathcal{D}(\widetilde{A})$, equation (2.6), the
relation $2\pi it(\mathcal{F}g)(t)=(\mathcal{F}g^{\prime})(t)$, and (2.7) with
$g$ replaced by $g^{\prime}$, we get for $\varphi\in\mathcal{D}(A)$ the
equalities

$\displaystyle\langle\widetilde{A}\varphi,D\varphi\rangle_{\mathcal{H}}-\langle\varphi,D\widetilde{A}\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}}\mathrm{d}t\,\overline{(\mathcal{F}g)(t)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
t)^{2}}\big{(}\langle\widetilde{A}\varphi,\mathop{\mathrm{e}}\nolimits^{2\pi
itH}\varphi\rangle_{\mathcal{H}}-\langle\varphi,\mathop{\mathrm{e}}\nolimits^{2\pi
itH}\widetilde{A}\varphi\rangle_{\mathcal{H}}\big{)}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}}\mathrm{d}t\,\overline{(\mathcal{F}g)(t)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
t)^{2}}\langle\varphi,[\widetilde{A},\mathop{\mathrm{e}}\nolimits^{2\pi
itH}]\varphi\rangle_{\mathcal{H}}$ $\displaystyle=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}}\mathrm{d}t\,\overline{(\mathcal{F}g)(t)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
t)^{2}}\langle\varphi,(-2\pi t)D\mathop{\mathrm{e}}\nolimits^{2\pi
itH}\varphi\rangle_{\mathcal{H}}$ $\displaystyle=\lim_{\varepsilon\searrow
0}\int_{\mathbb{R}}\mathrm{d}\tau\,\overline{i(\mathcal{F}g^{\prime})(t)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
t)^{2}}\langle\varphi,D\mathop{\mathrm{e}}\nolimits^{2\pi
itH}\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\langle\varphi,iDg^{\prime}(H)\varphi\rangle_{\mathcal{H}}.$

Since $g^{\prime}(H)\in\mathscr{B}(\mathcal{H})$ due to the inclusion
$g^{\prime}\in\mathop{\mathrm{L}^{\infty}}\nolimits(\mathbb{R})$, we infer
that $D\in C^{1}(\widetilde{A})$ with $[\widetilde{A},D]=iDg^{\prime}(H)$.
Thus, in the present case, the operator $B\in\mathscr{B}(\mathcal{H})$
appearing in the statement of Theorem 2.1(c) is $B=ig^{\prime}(H)$. So we
trivially get that $[D,B]=0$, and by reproducing $(n-1)$ times the previous
argument we obtain that $B\in C^{n-1}(\widetilde{A})$.

Summing up, the assumptions of Theorem 2.1(c) are satisfied for the net
$(U_{j})_{j\in J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set
$(\ell_{j})_{j\in J}=(t)_{t>0}$, the operator $\widetilde{A}$, and $D=g(H)$.
Thus for each $\varphi\in g(H)^{n}\mathcal{D}(A^{n})\subset
g(H)^{n}\mathcal{D}((\widetilde{A})^{n})$ and
$\psi\in\mathcal{D}(A^{n})\subset\mathcal{D}((\widetilde{A})^{n})$ there
exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

∎

######  Remark 2.6.

If the operators $H$ and $A$ satisfy the commutation relation $[iH,A]=f(H)$
for some Borel function $f:\mathbb{R}\to\mathbb{R}$, then they satisfy the
relation $[iH,A]=\widetilde{f}(H)$ for any Borel function
$\widetilde{f}:\mathbb{R}\to\mathbb{R}$ such that
$\widetilde{f}|_{\sigma(H)}=f|_{\sigma(H)}$ due to functional calculus. This
basic observation will be useful in some applications in which the initial
function $f$ fails to satisfy the assumptions of Proposition 2.5(b) or 2.5(c)
(see for instance Section 3.3).

###  2.3 Unitary representations with unitary generator

In this section, we consider the important case where the representation is a
unitary representation $\mathscr{U}:\mathbb{Z}\to\mathrm{U}(\mathcal{H})$ of
the additive group $\mathbb{Z}$. In such a case, the fact that $\mathbb{Z}$
has generator $1$ implies the existence of a unitary operator $U$ in
$\mathcal{H}$ such that $\mathscr{U}(m)=U^{m}$ for each $m\in\mathbb{Z}$. One
could also consider the higher-dimensional case of a unitary representation of
the additive group $\mathbb{Z}^{d}$ for $d\geq 1$. But we refrained from doing
it for the sake of simplicity.

We use the notation $P_{\rm p}(U)$ (resp. $P_{\rm c}(U)$, $P_{\rm ac}(U)$) for
the projection onto the pure point (resp. continuous, absolutely continuous)
subspace $\mathcal{H}_{\rm p}(U)$ (resp. $\mathcal{H}_{\rm c}(U)$,
$\mathcal{H}_{\rm ac}(U)$) of $U$, $E^{U}(\cdot)$ for the spectral projections
of $U$, and $\chi_{\Theta}$ for the characteristic function of a Borel set
$\Theta\subset\mathbb{S}^{1}$.

######  Lemma 2.7 (Properties of $D$).

Let $U$ and $A$ be a unitary and a self-adjoint operator in a Hilbert space
$\mathcal{H}$ with $U\in C^{1}(A)$, and let

$D_{n}:=\tfrac{1}{n}[A,U^{n}]U^{-n},\quad n\in\mathbb{N}^{*}.$

Then

  1. (a)

$\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{n\to\infty}D_{n}P_{\rm
p}(U)=0$.

  2. (b)

If there exists $B\in\mathscr{B}(\mathcal{H})$ such that
$[A,U]U^{-1}-B\in\mathscr{K}(\mathcal{H})$ and $\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}BU^{-m}P_{\rm c}(U)$
exists, then

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}BU^{-m}P_{\rm c}(U).$

Furthermore, if $D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}$ exists, then

  1. (c)

$[D,U^{m}]=0$ for all $m\in\mathbb{Z}$. In particular, $D$ is decomposable in
the spectral representation of $U$.

  2. (d)

$D=DP_{\rm c}(U)$. In particular, $U|_{\ker(D)^{\perp}}$ has purely continuous
spectrum.

  3. (e)

If $D\mathcal{D}(A)\subset\mathcal{D}(A)$ and $\sum_{n\geq
1}\|(D-D_{n})\varphi\|^{2}_{\mathcal{H}}<\infty$ for all
$\varphi\in\mathcal{D}(A)$, then $U|_{\ker(D)^{\perp}}$ has purely a.c.
spectrum.

Point (c) implies that $\ker(D)^{\perp}$ is a reducing subspace for $U$.
Therefore the operator $U|_{\ker(D)^{\perp}}$ in points (d)-(e) is a well-
defined unitary operator (see [40, Example 5.39(b)]).

###### Proof.

(a) Let $\varphi\in\mathcal{H}$. Then $P_{\rm p}(U)\varphi=\sum_{j\geq
1}\alpha_{j}\varphi_{j}$ with $(\varphi_{j})_{j\geq 1}$ an orthonormal basis
of $\mathcal{H}_{\rm p}(U)$, $\alpha_{j}\in\mathbb{C}$, and
$U\varphi_{j}=\theta_{j}\varphi_{j}$ for some $\theta_{j}\in\mathbb{S}^{1}$.
Furthermore, we have for any $n\in\mathbb{N}^{*}$ the equalities

$D_{n}=\tfrac{1}{n}[A,U^{n}]U^{-n}=\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}([A,U]U^{-1})U^{-m}.$

Therefore, we obtain

$\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{n\to\infty}D_{n}P_{\rm
p}(U)\varphi=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\sum_{j\geq
1}\alpha_{j}\left(\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}\right)[A,U]U^{-1}\varphi_{j}.$
(2.8)

Now
$\|\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}\|_{\mathscr{B}(\mathcal{H})}\leq
1$ for all $n\in\mathbb{N}^{*}$, and

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}=E^{U}(\\{\theta_{j}\\})$

due to von Neumann’s mean ergodic theorem. Therefore we can exchange the limit
and the sum in (2.8) to get

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}P_{\rm p}(U)\varphi=\sum_{j\geq
1}\alpha_{j}E^{U}(\\{\theta_{j}\\})[A,U]U^{-1}\varphi_{j}=\sum_{j\geq
1}\alpha_{j}E^{U}(\\{\theta_{j}\\})[A,U]U^{-1}E^{U}(\\{\theta_{j}\\})\varphi_{j}.$

But $E^{U}(\\{\theta_{j}\\})[A,U]U^{-1}E^{U}(\\{\theta_{j}\\})=0$ for each
$\theta_{j}$ due to the virial theorem for unitary operators [7, Prop. 2.3].
Thus we obtain that $\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}P_{\rm p}(U)\varphi=0$, which proves the claim.

(b) Let $K:=[A,U]U^{-1}-B\in\mathscr{K}(\mathcal{H})$. Then it follows from
point (a) that

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{n\to\infty}D_{n}$
$\displaystyle=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}([A,U]U^{-1})U^{-m}P_{\rm
c}(U)$ $\displaystyle=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}BU^{-m}P_{\rm
c}(U)+\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}P_{\rm c}(U).$

with $\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}P_{\rm c}(U)=0$
due to Theorem B.2.

(c) The claim follows from Proposition 2.3(b) in the case of the additive
group $X=\mathbb{Z}$ and the operator $A$. Indeed, if we take the proper
length function $\ell:\mathbb{Z}\to[0,\infty)$ given by $\ell(n):=|n|$, the
unitary representation $\mathscr{U}:\mathbb{Z}\to\mathrm{U}(\mathcal{H})$
given by $\mathscr{U}(n):=U^{n}$, and the net $(x_{j})_{j\in
J}=(n)_{n\in\mathbb{N}^{*}}$, then we get $\mathscr{U}(m)=U^{m}\in C^{1}(A)$
for all $m\in\mathbb{Z}$ and

$\widetilde{D}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n-m}[A,U^{n-m}]U^{-(n-m)}=\operatorname*{s\hskip
0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}[A,U^{n}]U^{-n}=\operatorname*{s\hskip
0.1pt-\hskip 0.1ptlim}_{n\to\infty}D_{n}=D.$

So all the assumptions of Proposition 2.3(b) are verified, and thus
$[D,U^{m}]=0$ for all $m\in\mathbb{Z}$.

Finally, since $[D,U^{m}]=0$ for all $m\in\mathbb{Z}$, we have
$D\chi_{\Theta}(U)=\chi_{\Theta}(U)D$ for each Borel set
$\Theta\subset\mathbb{S}^{1}$, and thus $D$ is decomposable in the spectral
representation of $U$ [4, Thm. 7.2.3(b)].

(d) The equality $D=DP_{\rm c}(U)$ follows from point (a). As a consequence,
we get that $\ker(D)^{\perp}\subset\mathcal{H}_{\rm c}(U)$, and thus the
operator $U|_{\ker(D)^{\perp}}$ has purely continuous spectrum.

(e) Take $\varphi\in\mathcal{D}(A)$. Then $\psi=D\varphi\in
D\mathcal{D}(A)\cap\mathcal{D}(A)$, and it follows from Theorem 2.1(a) that
there exists $c_{\psi}\geq 0$ such that

$\big{|}\langle\psi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{n})\varphi\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{n}\;\\!c_{\psi},\quad
n\in\mathbb{N}^{*}.$

So, we infer from the assumption and Cauchy-Schwarz inequality that
$\sum_{n\geq 1}|\langle\psi,U^{n}\psi\rangle_{\mathcal{H}}|^{2}<\infty$, and
thus that $n\mapsto\langle\psi,U^{n}\psi\rangle_{\mathcal{H}}$ belongs to
$\ell^{2}(\mathbb{Z})$. Therefore, Plancherel’s theorem for the group
$X=\mathbb{Z}$ [9, Thm. 4.26] implies that $\psi\in D\mathcal{D}(A)$ belongs
to the a.c. subspace $\mathcal{H}_{\rm ac}(U)$ of $U$. Thus
$U|_{\ker(D)^{\perp}}$ has purely a.c. spectrum, since $D\mathcal{D}(A)$ is
dense in $\overline{D\mathcal{H}}=\ker(D)^{\perp}$ and $\mathcal{H}_{\rm
ac}(U)$ is closed in $\mathcal{H}$. ∎

In the next proposition, we consider the particular case where
$[A,U]=\gamma(U)$ for some Borel function
$\gamma:\mathbb{S}^{1}\to\mathbb{C}$. Our results in this case generalise the
results of [36, Cor. 3.3-3.4].

######  Proposition 2.8 (The case $[A,U]=\gamma(U)$).

Let $U$ and $A$ be a unitary and a self-adjoint operator in a Hilbert space
$\mathcal{H}$, assume that $U\in C^{1}(A)$ with $[A,U]=\gamma(U)$ for some
Borel function $\gamma:\mathbb{S}^{1}\to\mathbb{C}$, and set
$\eta(U):=\gamma(U)U^{-1}$. Then

  1. (a)

For each $\varphi\in\eta(U)\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$ there
exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{n}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$

  2. (b)

If $\eta(U)\mathcal{D}(A)\subset\mathcal{D}(A)$, then
$U|_{\ker(\gamma(U))^{\perp}}$ has purely a.c. spectrum.

  3. (c)

Suppose that $\gamma\in C^{k}(\mathbb{S}^{1})$ ($k\in\mathbb{N}^{*}$). Then
for each $\varphi\in\eta(U)^{k}\mathcal{D}(A^{k})$ and
$\psi\in\mathcal{D}(A^{k})$ there exists a constant $c_{\varphi,\psi}\geq 0$
such that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{n^{k}}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$

###### Proof.

(a) The strong limit $D$ exists and satisfies for any $n\in\mathbb{N}^{*}$ the
equalities

$D_{n}=\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}([A,U]U^{-1})U^{-m}=\eta(U)=D.$ (2.9)

Therefore the assumptions of Theorem 2.1(b) are satisfied for the net
$(U_{j})_{j\in J}=(U^{n})_{n\in\mathbb{N}^{*}}$, the set $(\ell_{j})_{j\in
J}=(n)_{n\in\mathbb{N}^{*}}$, the operator $A$, and $D=\eta(U)$. Thus for each
$\varphi\in\eta(U)\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$ there exists a
constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{n}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$

(b) The claim follows from Lemma 2.7(e) because $D=\eta(U)=D_{n}$ for all
$n\in\mathbb{N}^{*}$ and

$\ker(\eta(U))=\chi_{\eta^{-1}(\\{0\\})}(U)\mathcal{H}=\chi_{\gamma^{-1}(\\{0\\})}(U)\mathcal{H}=\ker(\gamma(U)).$

(c) Since $\eta\in C(\mathbb{S}^{1})$, we have that

$\lim_{\varepsilon\searrow
0}\big{\|}\eta(U)-\eta^{\varepsilon}(U)\big{\|}_{\mathscr{B}(\mathcal{H})}=0$
(2.10)

with

$\eta^{\varepsilon}(U):=\sum_{m\in\mathbb{Z}}(\mathcal{F}\eta)(m)\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
m)^{2}}U^{m}\quad\hbox{(strong or operator norm sum)}$

and
$\mathcal{F}:\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{S}^{1})\to\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{Z})$
the Fourier transform (see [8, Thm. 8.36(a)]). Using successively the fact
that $D=\eta(U)$, equation (2.10), the inclusion
$U^{m}\mathcal{D}(A)\subset\mathcal{D}(A)$, equation (2.9), the relation
$m(\mathcal{F}\eta)(m)=(\mathcal{F}(\mathrm{id}_{\mathbb{S}^{1}}\cdot\eta^{\prime}))(m)$
with $\mathrm{id}_{\mathbb{S}^{1}}$ the identity function on $\mathbb{S}^{1}$,
and (2.10) with $\eta$ replaced by
$\mathrm{id}_{\mathbb{S}^{1}}\cdot\eta^{\prime}$, we get for
$\varphi\in\mathcal{D}(A)$ the equalities

$\displaystyle\langle
A\varphi,D\varphi\rangle_{\mathcal{H}}-\langle\varphi,A\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\sum_{m\in\mathbb{Z}}\overline{(\mathcal{F}\eta)(m)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
m)^{2}}\big{(}\langle
A\varphi,U^{m}\varphi\rangle_{\mathcal{H}}-\langle\varphi,U^{m}A\varphi\rangle_{\mathcal{H}}\big{)}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\sum_{m\in\mathbb{Z}}\overline{(\mathcal{F}\eta)(m)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
m)^{2}}\langle\varphi,[A,U^{m}]\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\sum_{m\in\mathbb{Z}}\overline{(\mathcal{F}\eta)(m)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
m)^{2}}\langle\varphi,mDU^{m}\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\lim_{\varepsilon\searrow
0}\sum_{m\in\mathbb{Z}}\overline{\big{(}\mathcal{F}(\mathrm{id}_{\mathbb{S}^{1}}\cdot\eta^{\prime})\big{)}(m)}\mathop{\mathrm{e}}\nolimits^{-(\varepsilon
m)^{2}}\langle\varphi,DU^{m}\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\langle\varphi,D(\mathrm{id}_{\mathbb{S}^{1}}\cdot\eta^{\prime})(U)\varphi\rangle_{\mathcal{H}}$
$\displaystyle=\langle\varphi,DU\eta^{\prime}(U)\varphi\rangle_{\mathcal{H}}.$

Since $\eta^{\prime}(U)\in\mathscr{B}(\mathcal{H})$ due to the inclusion
$\eta^{\prime}\in C^{k-1}(\mathbb{S}^{1})$, we infer that $D\in C^{1}(A)$ with
$[A,D]=DU\eta^{\prime}(U)$. Thus, in the present case, the operator
$B\in\mathscr{B}(\mathcal{H})$ appearing in the statement of Theorem 2.1(c) is
$B=U\eta^{\prime}(U)$. So we trivially get that $[D,B]=0$, and by reproducing
$(k-1)$ times the previous argument we obtain that $B\in C^{k-1}(A)$.

Summing up, the assumptions of Theorem 2.1(c) are satisfied for the net
$(U_{j})_{j\in J}=(U^{n})_{n\in\mathbb{N}^{*}}$, the set $(\ell_{j})_{j\in
J}=(n)_{n\in\mathbb{N}^{*}}$, the operator $A$, and $D=\eta(U)$. Thus for each
$\varphi\in\eta(U)^{k}\mathcal{D}(A^{k})$ and $\psi\in\mathcal{D}(A^{k})$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{n^{k}}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$

∎

##  3 Applications

In this section, we apply our results to various models in quantum mechanics
and dynamical systems. First, we consider an example of unitary representation
with no self-adjoint or unitary generator. Then we consider examples of
unitary representations having a self-adjoint generator. And finally we
consider examples of unitary representations having a unitary generator.

###  3.1 Left regular representation

This example is motivated by [26, Ex. 2.9]. Let $X$ be a $\sigma$-compact
locally compact Hausdorff group with left Haar measure $\mu$ and proper length
function $\ell$ (see properties (L1)-(L4) in Section 2.1). Let
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(X,\mu)$, let
$\mathscr{D}\subset\mathcal{H}$ be the set of functions $X\to\mathbb{C}$ with
compact support, let $\mathscr{U}:X\to\mathrm{U}(\mathcal{H})$ be the left
regular representation of $X$ on $\mathcal{H}$

$\mathscr{U}(x)\varphi:=\varphi(x^{-1}\;\\!\cdot),\quad x\in
X,~{}\varphi\in\mathcal{H},$

and let $A$ be the maximal multiplication operator by $\ell$ in $\mathcal{H}$

$A\varphi:=\ell\varphi,\quad\varphi\in\mathcal{D}(A):=\\{\varphi\in\mathcal{H}\mid\|\ell\varphi\|_{\mathcal{H}}<\infty\\}.$

For any $x\in X$, one has on $\mathscr{D}$ the equality

$A\mathscr{U}(x)-\mathscr{U}(x)A=\big{(}\ell(\cdot)-\ell(x^{-1}\;\\!\cdot)\big{)}\mathscr{U}(x),$

with $|(\ell(\cdot)-\ell(x^{-1}\;\\!\cdot))|\leq\ell(x)$ due to properties
(L2)-(L3). Since $\mathscr{D}$ is dense in $\mathcal{D}(A)$, it follows that
$\mathscr{U}(x)\in C^{1}(A)$ with

$[A,\mathscr{U}(x)]\mathscr{U}(x)^{-1}=\ell(\cdot)-\ell(x^{-1}\;\\!\cdot).$
(3.1)

In addition, we know from [26, Ex. 2.9] that for any net $(x_{j})_{j\in J}$ in
$X$ with $x_{j}\to\infty$ we have

$D:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}D_{j}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{j}\tfrac{1}{\ell(x_{j})}[A,\mathscr{U}(x_{j})]\mathscr{U}(x_{j})^{-1}=-1.$

Therefore, Theorem 2.1(a) applies for the net $(U_{j})_{j\in
J}=(\mathscr{U}(x_{j}))_{j\in J}$, the set $(\ell_{j})_{j\in
J}=(\ell(x_{j}))_{j\in J}$, the operator $A$, and $D=-1$. Thus, for each
$\varphi,\psi\in\mathcal{D}(A)$ there exists a constant $c_{\varphi,\psi}\geq
0$ such that

$\big{|}\langle\varphi,\mathscr{U}(x_{j})\psi\rangle_{\mathcal{H}}\big{|}\leq\big{\|}(D-D_{j})\varphi\big{\|}_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{\ell(x_{j})}\;\\!c_{\varphi,\psi},\quad\ell(x_{j})>0.$

But (3.1) and the bound
$|\ell(\cdot)-\ell(x_{j}^{-1}\;\\!\cdot)+\ell(x_{j})|\leq 2\ell(\cdot)$ (which
follows from properties (L2)-(L3)) imply that

$\big{\|}(D-D_{j})\varphi\big{\|}_{\mathcal{H}}=\Big{\|}\tfrac{\ell(\cdot)-\ell(x_{j}^{-1}\;\\!\cdot)+\ell(x_{j})}{\ell(x_{j})}\varphi\Big{\|}_{\mathcal{H}}\leq\tfrac{2}{\ell(x_{j})}\|\ell\varphi\|_{\mathcal{H}}.$

Therefore, by setting
$\widetilde{c}_{\varphi,\psi}:=2\|\ell\varphi\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+c_{\varphi,\psi}$,
we obtain that

$\big{|}\langle\varphi,\mathscr{U}(x_{j})\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{\ell(x_{j})}\;\\!\widetilde{c}_{\varphi,\psi},\quad\ell(x_{j})>0.$

This estimate is similar to the estimates of Propositions 2.5(a) and 2.8(a),
but with the interesting difference that in general the representation
$\mathscr{U}$ doesn’t have neither a self-adjoint generator nor a unitary
generator. In addition, we note that one can even obtain higher order decay
estimates by carrying on the above calculations. But we refrained from
presenting them here for the sake of simplicity.

###  3.2 Schrödinger operator in $\mathbb{R}^{n}$

In the Hilbert space
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}^{n})$, consider the
Schrödinger operator and the generator of dilations given by

$H\varphi:=(-\Delta+V)\varphi\quad\hbox{and}\quad
A\varphi:=\tfrac{1}{2}(Q\cdot P+P\cdot Q),\quad\varphi\in\SS(\mathbb{R}^{n}),$

with $V\in\mathop{\mathrm{L}^{\infty}}\nolimits(\mathbb{R}^{n},\mathbb{R})$,
$Q:=(Q_{1},\dots,Q_{n})$ the position operator, $P:=-i\nabla$ the momentum
operator and $\SS(\mathbb{R}^{n})$ the Schwartz space on $\mathbb{R}^{n}$.
Both operators are essentially self-adjoint with closures denoted by the same
symbols. If we assume that

$x\mapsto x\cdot(\nabla
V)(x)\in\mathop{\mathrm{L}^{\infty}}\nolimits(\mathbb{R}^{n},\mathbb{R})\quad\hbox{and}\quad\lim_{|x|\to\infty}\big{|}2V(x)+x\cdot(\nabla
V)(x)\big{|}=0,$

then we have $(H-i)^{-1}\in C^{1}(A)$ with $[iH,A]=2H-2V-Q\cdot\nabla V$, we
have

$(H+i)^{-1}(2V+Q\cdot\nabla V)(H-i)^{-1}\in\mathscr{K}(\mathcal{H})$

due to the standard theorem [1, Prop. 4.1.3], and

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}2H(H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm
c}(H)=2H\langle H\rangle^{-2}P_{\rm c}(H).$

Therefore, it follows from Lemma 2.4 that

$D=\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{t\to\infty}D_{t}=2H\langle
H\rangle^{-2}P_{\rm c}(H)$

and that $H|_{\ker(D)^{\perp}}$ has purely continuous spectrum. Moreover,
Theorem 2.1(a) applies for the net $(U_{j})_{j\in
J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set $(\ell_{j})_{j\in
J}=(t)_{t>0}$, the operator $\widetilde{A}=(H+i)^{-1}A(H-i)^{-1}$, and $D$ as
above. Therefore, for each $\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$
and $\psi\in\mathcal{D}(A)$ there exists a constant $c_{\varphi,\psi}\geq 0$
such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{t})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$ (3.2)

In this example, one cannot easily improve the decay estimate (3.2) when
$V\neq 0$. Indeed, in order to establish the convergence
$D_{t}\stackrel{{\scriptstyle\rm s}}{{\to}}D$, we used Lemma 2.4(b). But the
proof of Lemma 2.4(b) relies on the RAGE theorem for self-adjoint operators,
whose proof relies in turn on Wiener’s theorem. And as sad as it is, in
general one cannot infer an explicit rate of convergence from Wiener’s
theorem.

On the other hand, if $V=0$, then we have $[iH,A]=2H$, and thus the stronger
decay estimates of Proposition 2.5(c) are satisfied.

###  3.3 Dirac operator in $\mathbb{R}^{3}$

This example is motivated by [15, Ex. 7.7] and [25, Sec. 7.3]. Consider in the
Hilbert space
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}^{3},\mathbb{C}^{4})$
the Dirac operator for a spin-$\tfrac{1}{2}$ particle of mass $m>0$

$H\varphi:=(\alpha\cdot P+\beta
m)\varphi,\quad\varphi\in\SS(\mathbb{R}^{3},\mathbb{C}^{4}),$

with $\alpha:=(\alpha_{1},\alpha_{2},\alpha_{3})$ and $\beta$ the usual
$4\times 4$ Dirac matrices. Then $H$ is essentially self-adjoint (with closure
denoted by the same symbol), and we have the following result:333In [15, Ex.
7.7], the authors say that it is shown in [25] that $(H-i)^{-1}\in
C^{1}(X_{\rm NW})$ with  $[iH,X_{\rm NW}]=\sqrt{H^{2}-m^{2}}\;\\!H^{-1}.$
(3.3) This is not what is written in [25, Sec. 7.3], and it cannot be since
$X_{\rm NW}$ on the l.h.s. of (3.3) is a vector operator with three components
whereas $\sqrt{H^{2}-m^{2}}H^{-1}$ on the r.h.s. of (3.3) is a scalar operator
with one component.

######  Lemma 3.1.

Let $X_{\rm NW}:=(X_{\rm NW,1},X_{\rm NW,2},X_{\rm NW,3})$ be the Newton-
Wigner position operator. Then the operator

$A\varphi:=\tfrac{1}{2}(X_{\rm NW}\cdot PH^{-1}+PH^{-1}\cdot X_{\rm
NW})\varphi,\quad\varphi\in\SS(\mathbb{R}^{3},\mathbb{C}^{4}),$

is essentially self-adjoint (with closure denoted by the same symbol) and
$(H-i)^{-1}\in C^{1}(A)$ with $[iH,A]=(H^{2}-m^{2})H^{-2}$.

###### Proof.

First, we recall that $X_{\rm NW}=\mathcal{F}_{\rm FW}^{-1}Q\mathcal{F}_{\rm
FW}$, with $\mathcal{F}_{\rm FW}$ the (unitary) Foldy-Wouthuysen transform,
and that $\mathcal{F}_{\rm FW}$ leaves $\SS(\mathbb{R}^{3},\mathbb{C}^{4})$
invariant [31, Sec. 1.4.3]. Therefore, one gets on
$\SS(\mathbb{R}^{3},\mathbb{C}^{4})$ the equalities

$\displaystyle A$ $\displaystyle=\tfrac{1}{2}(X_{\rm NW}\cdot
PH^{-1}+PH^{-1}\cdot X_{\rm NW})$ $\displaystyle=\tfrac{1}{2}\mathcal{F}_{\rm
FW}^{-1}(Q\cdot\mathcal{F}_{\rm FW}PH^{-1}\mathcal{F}_{\rm
FW}^{-1}+\mathcal{F}_{\rm FW}PH^{-1}\mathcal{F}_{\rm FW}^{-1}\cdot
Q)\mathcal{F}_{\rm FW}$ $\displaystyle=\tfrac{1}{2}\mathcal{F}_{\rm
FW}^{-1}(Q\cdot P\beta|H|^{-1}+P\beta|H|^{-1}\cdot Q)\mathcal{F}_{\rm FW}$

with the operator within the parenthesis equal to a direct sum of operators of
the form

$\pm\big{(}Q\cdot P(P^{2}+m^{2})^{-1/2}+P(P^{2}+m^{2})^{-1/2}\cdot Q\big{)}.$

Since each of these operators is essentially self-adjoint on
$\SS(\mathbb{R}^{3})$ due to Nelson’s lemma, one infers that $A$ is
essentially self-adjoint on $\SS(\mathbb{R}^{3},\mathbb{C}^{4})$.

For the second claim, we recall from [25, Sec. 7.3] that $(H-i)^{-1}\in
C^{1}(X_{\rm NW,j})$ with $[iH,X_{\rm NW,j}]=P_{j}H^{-1}$. Therefore, a
calculation in the form sense on $\SS(\mathbb{R}^{3},\mathbb{C}^{4})$ gives

$\displaystyle[(H-i)^{-1},A]$
$\displaystyle=\tfrac{1}{2}\sum_{j=1}^{3}\big{(}[(H-i)^{-1},X_{\rm
NW,j}]P_{j}H^{-1}+P_{j}H^{-1}[(H-i)^{-1},X_{\rm NW,j}]\big{)}$
$\displaystyle=\tfrac{i}{2}(H-i)^{-1}\sum_{j=1}^{3}\big{(}[iH,X_{\rm
NW,j}]P_{j}H^{-1}+P_{j}H^{-1}[iH,X_{\rm NW,j}]\big{)}(H-i)^{-1}$
$\displaystyle=i(H-i)^{-1}(H^{2}-m^{2})H^{-2}(H-i)^{-1}.$

Since $\SS(\mathbb{R}^{3},\mathbb{C}^{4})$ is a core for $A$, this implies
that $(H-i)^{-1}\in C^{1}(A)$ with $[iH,A]=(H^{2}-m^{2})H^{-2}$. ∎

Now, take any function $f\in C^{\infty}(\mathbb{R})$ such that
$f(x)=(x^{2}-m^{2})x^{-2}$ if $|x|\geq m$. Then Remark 2.6 and Lemma 3.1 imply
that $(H-i)^{-1}\in C^{1}(A)$ with $[iH,A]=f(H)$, and $g(H)=f(H)\langle
H\rangle^{-2}\in C^{1}(A)$. Thus $g(H)\mathcal{D}(A)\subset\mathcal{D}(A)$ and
Proposition 2.5 applies. Since an application of the Foldy-Wouthuysen
transform shows that $H$ has spectrum equal to $(-\infty,-m]\cup[m,\infty)$
and $\ker(f(H))=\\{0\\}$, it follows that $H$ has purely a.c. spectrum equal
to $(-\infty,-m]\cup[m,\infty)$ (which is standard knowledge) and for each
$\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$
($n\in\mathbb{N}^{*}$) there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

###  3.4 Quantum waveguides in $\mathbb{R}^{n}$

This example is motivated by [32]. Let $\Sigma$ be a bounded open connected
set in $\mathbb{R}^{n-1}$ ($n\geq 2$), let $\Omega:=\Sigma\times\mathbb{R}$ be
the corresponding waveguide with coordinates $x\equiv(\omega,x_{n})$, let
$-\Delta^{\Sigma}_{\rm D}$ be the Dirichlet Laplacian in
$\mathop{\mathrm{L}^{2}}\nolimits(\Sigma)$, let

$H_{0}:=1\otimes P_{n}^{2}+(-\Delta^{\Sigma}_{\rm D})\otimes
1\quad\hbox{in}\quad\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\Omega)\simeq\mathop{\mathrm{L}^{2}}\nolimits(\Sigma)\otimes\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}),$

and let $H:=H_{0}+V$ with
$V\in\mathop{\mathrm{L}^{\infty}}\nolimits(\Omega,\mathbb{R})$ satisfying

$(1\otimes\langle
Q_{n}\rangle^{1+\varepsilon})V\in\mathscr{B}(\mathcal{H})\quad\hbox{and}\quad(1\otimes\langle
Q_{n}\rangle^{1+\varepsilon})(\partial_{n}V)\in\mathscr{B}(\mathcal{H})\quad\hbox{for
some $\varepsilon>0$.}$

Then $V\in\mathscr{K}(\mathcal{D}(H_{0}),\mathcal{H})$. Indeed, if we use the
strongly commuting self-adjoint operators $X_{1}:=1\otimes\langle
P_{n}\rangle^{2}$ and $X_{2}:=(-\Delta^{\Sigma}_{\rm D}+1)\otimes 1$, we get

$V(H_{0}+2)^{-1}=(1\otimes\langle Q_{n}\rangle)V\cdot(-\Delta^{\Sigma}_{\rm
D}+1)^{-1/2}\otimes\langle Q_{n}\rangle^{-1}\langle P_{n}\rangle^{-1}\cdot
X_{1}^{1/2}X_{2}^{1/2}(X_{1}+X_{2})^{-1}.$

But $(1\otimes\langle Q_{n}\rangle)V\in\mathscr{B}(\mathcal{H})$ by
assumption,
$X_{1}^{1/2}X_{2}^{1/2}(X_{1}+X_{2})^{-1}\in\mathscr{B}(\mathcal{H})$ due to
the bound

$\big{\|}X_{1}^{1/2}X_{2}^{1/2}(X_{1}+X_{2})^{-1}\big{\|}_{\mathscr{B}(\mathcal{H})}\leq\sup_{x_{1},x_{2}\geq
1}(x_{1}x_{2})^{1/2}(x_{1}+x_{2})^{-1}<\infty,$

and $(-\Delta^{\Sigma}_{\rm D}+1)^{-1/2}\otimes\langle
Q_{n}\rangle^{-1}\langle P_{n}\rangle^{-1}\in\mathscr{K}(\mathcal{H})$ because
$-\Delta^{\Sigma}_{\rm D}$ has compact resolvent and $\langle
Q_{n}\rangle^{-1}\langle P_{n}\rangle^{-1}$ is compact. Hence
$V(H_{0}+2)^{-1}\in\mathscr{K}(\mathcal{H})$, and thus
$V\in\mathscr{K}(\mathcal{D}(H_{0}),\mathcal{H})$. Therefore, the assumptions
of [32] are satisfied and the following holds true: The operator $H$ has no
singular continuous spectrum, the eigenvalues of $H$ (if any) are of finite
multiplicity and can accumulate only at the set of eigenvalues of
$-\Delta^{\Sigma}_{\rm D}$, and the wave operators
$W_{\pm}:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\pm\infty}\mathop{\mathrm{e}}\nolimits^{itH}\mathop{\mathrm{e}}\nolimits^{-itH_{0}}$
exist and are complete.

Now, set $A:=1\otimes A_{n}$ with $A_{n}$ the generator of dilations in
$\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R})$. Then $(H-i)^{-1}\in C^{1}(A)$
with

$[iH,A]=2(1\otimes P_{n}^{2})-(1\otimes Q_{n})(\partial_{n}V),$

and an argument as above shows that the following three operators belong to
$\mathscr{K}(\mathcal{H})$:

$\displaystyle(H+i)^{-1}(1\otimes Q_{n})(\partial_{n}V)(H-i)^{-1},$
$\displaystyle\big{(}(H+i)^{-1}-(H_{0}+i)^{-1}\big{)}(1\otimes
P_{n}^{2})(H-i)^{-1},$ $\displaystyle(H_{0}+i)^{-1}(1\otimes
P_{n}^{2})\big{(}(H-i)^{-1}-(H_{0}-i)^{-1}\big{)}.$

Furthermore, since $P_{\rm c}(H)=P_{\rm ac}(H)$, one has

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{\tau\to\infty}\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H_{0}+i)^{-1}2(1\otimes
P_{n}^{2})(H_{0}-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H)$
$\displaystyle=2\;\\!\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{\tau\to\infty}\mathop{\mathrm{e}}\nolimits^{-i\tau
H}\mathop{\mathrm{e}}\nolimits^{i\tau H_{0}}(1\otimes P_{n}^{2})\langle
H_{0}\rangle^{-2}\mathop{\mathrm{e}}\nolimits^{-i\tau
H_{0}}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm ac}(H)$
$\displaystyle=2W_{-}(1\otimes P_{n}^{2})\langle H_{0}\rangle^{-2}W_{-}^{*}.$

And since strong convergence implies strong Cesaro convergence, this implies
that

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H_{0}+i)^{-1}2(1\otimes
P_{n}^{2})(H_{0}-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau H}P_{\rm c}(H)$
$\displaystyle=2W_{-}(1\otimes P_{n}^{2})\langle H_{0}\rangle^{-2}W_{-}^{*}.$

Thus it follows from Lemma 2.4(b) that

$D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}=2W_{-}(1\otimes P_{n}^{2})\langle
H_{0}\rangle^{-2}W_{-}^{*}.$

So, Theorem 2.1(a) applies for the net $(U_{j})_{j\in
J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set $(\ell_{j})_{j\in
J}=(t)_{t>0}$, the operator $\widetilde{A}=(H+i)^{-1}A(H-i)^{-1}$, and $D$ as
above. Therefore, for each $\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$
and $\psi\in\mathcal{D}(A)$ there exists a constant $c_{\varphi,\psi}\geq 0$
such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{t})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$

Of course, one obtains stronger decay estimates when $V=0$. Indeed, one gets
in such a case $[iH,A]=2(1\otimes P_{n}^{2})$ and $D=D_{t}=2(1\otimes
P_{n}^{2})\langle H\rangle^{-2}$ for all $t>0$, from which one can infer the
estimates of Theorem 2.1(c).

###  3.5 Stark Hamiltonian in $\mathbb{R}^{n}$

This example is motivated by [15, Sec. 7.3] and [25, Sec. 7.1]. Let
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}^{n})$ and take a
unit vector $v\in\mathbb{R}^{n}$. Then the Stark Hamiltonian $H$ with electric
field along $v$ and the operator $A$ given by

$(H\varphi)(x):=-(\Delta\varphi)(x)+(v\cdot x)\varphi(x)\quad\hbox{and}\quad
A\varphi:=i(v\cdot\nabla)\varphi,\quad\varphi\in\SS(\mathbb{R}^{n}),~{}x\in\mathbb{R}^{n}.$

are essentielly self-adjoint (with closures denoted by the same symbols), and
$(H-i)^{-1}\in C^{\infty}(A)$ with $[iH,A]=1$. Thus Proposition 2.5 applies
with $f(H)=1$, $g(H)=\langle H\rangle^{-2}\in C^{\infty}(A)$ and
$\ker(f(H))=\\{0\\}$. It follows that $H$ has purely a.c. spectrum (which is
standard knowledge) and for each $\varphi\in\langle
H\rangle^{-2n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$
($n\in\mathbb{N}^{*}$) there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

This estimate is similar to the result of [15, Sec. 7.3] (see [15, Thm. 6.1]).

###  3.6 Fractional Laplacian in $\mathbb{R}^{n}$

This example is motivated by [15, Sec. 7.5]. For $s\in(0,2)$, let
$H:=(-\Delta)^{s/2}$ be the fractional Laplacian in the Hilbert space
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}^{n})$ and let $A$ be
the generator of dilations in $\mathcal{H}$. Then $(H-i)^{-1}\in
C^{\infty}(A)$ with $[iH,A]=sH$. Thus Proposition 2.5 applies with $f(H)=sH$,
$g(H)=sH\langle H\rangle^{-2}\in C^{\infty}(A)$ and $\ker(f(H))=\ker(H)$.
Since an application of the Fourier transform shows that $H$ has spectrum
equal to $[0,\infty)$ and $\ker(H)=\\{0\\}$, it follows that $H$ has purely
a.c. spectrum equal to $[0,\infty)$ (which is standard knowledge) and for each
$\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$
($n\in\mathbb{N}^{*}$) there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

This estimate improves the result of [15, Sec. 7.5] (see [15, Thm. 6.3]).

###  3.7 Horocycle flow

This example is motivated by [33, 36], but see also [34, 37]. Let $\Sigma$ be
a finite volume Riemann surface of genus $\geq 2$ and let $M:=T^{1}\Sigma$ be
the unit tangent bundle of $\Sigma$. The $3$-manifold $M$ carries a
probability measure $\mu_{\Omega}$, induced by a canonical volume form
$\Omega$, which is preserved by two distinguished one-parameter groups of
diffeomorphisms : the horocycle flow $F_{1}:=(F_{1,t})_{t\in\mathbb{R}}$ and
the geodesic flow $F_{2}:=(F_{2,t})_{t\in\mathbb{R}}$. Each flow admits a
self-adjoint generator $H_{j}$ in
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(M,\mu_{\Omega})$ essentially
self-adjoint on $C^{\infty}_{\rm c}(M)$ and given by

$H_{j}\varphi:=-i\mathscr{L}_{X_{j}}\varphi,\quad\varphi\in C^{\infty}_{\rm
c}(M),$

with $X_{j}$ the divergence-free vector field associated to $F_{j}$ and
$\mathscr{L}_{X_{j}}$ the corresponding Lie derivative. Moreover, one has
$(H_{1}-i)^{-1}\in C^{\infty}(H_{2})$ with $[iH_{1},H_{2}]=H_{1}$, see [33,
Sec. 3]. Thus Proposition 2.5 applies with $f(H_{1})=H_{1}$,
$g(H_{1})=H_{1}\langle H_{1}\rangle^{-2}\in C^{\infty}(H_{2})$ and
$\ker(f(H_{1}))=\ker(H_{1})$. It follows that $H_{1}|_{\ker(H_{1})^{\perp}}$
has purely a.c. spectrum (which is standard knowledge) and for each
$\varphi\in g(H_{1})^{n}\mathcal{D}(H_{2}^{n})$ and
$\psi\in\mathcal{D}(H_{2}^{n})$ ($n\in\mathbb{N}^{*}$) there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\psi\circ
F_{1,t}\rangle_{\mathcal{H}}\big{|}=\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH_{1}}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

Roughly, this estimate shows that if the vectors $\varphi,\psi$ are $n$ times
differentiable along the geodesic flow, then their correlation along the
horocycle flow decays as $\tfrac{1}{t^{n}}$. It thus provides a new version of
the polynomial decay of correlations for the horocycle flow on the unit
tangent bundle of a finite volume surface of constant negative curvature. And
its (rather short) proof did not use the identification of $M$ with a
homogeneous space $\Gamma\setminus{\rm PSL}(2,\mathbb{R})$ and the
representation theory associated to it as is customary (see for instance
[22]).

Now, assume that $\Sigma$ is compact and consider a $C^{1}$ vector field
proportional to $X_{1}$, that is, $fX_{1}$ with $f\in C^{1}(M,(0,\infty))$.
The vector field $fX_{1}$ admits a complete flow
$\widetilde{F}_{1}:=(\widetilde{F}_{1,t})_{t\in\mathbb{R}}$ uniquely ergodic
with respect to the measure
$\widetilde{\mu}_{\Omega}:=\frac{f^{-1}\mu_{\Omega}}{\int_{M}f^{-1}\mathrm{d}\mu_{\Omega}}$
and with generator $H:=fH_{1}$ essentially self-adjoint on
$C^{1}(M)\subset\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(M,f^{-1}\mu_{\Omega})$.
Furthermore, the following holds true [36, Ex. 4.8]: The operator
$A:=f^{1/2}H_{2}f^{-1/2}$ is essentially self-adjoint on
$C^{1}(M)\subset\mathcal{H}$, one has $(H-i)^{-1}\in C^{1}(A)$ with

$[iH,A]=H\xi+\xi
H,\quad\xi:=\tfrac{1}{2}-\tfrac{1}{2}f^{-1}\mathscr{L}_{X_{2}}(f),$

and $D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}=H\langle H\rangle^{-2}$. Thus Theorem 2.1(a)
applies for the net $(U_{j})_{j\in
J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set $(\ell_{j})_{j\in
J}=(t)_{t>0}$, the operator $\widetilde{A}=(H+i)^{-1}A(H-i)^{-1}$, and
$D=H\langle H\rangle^{-2}$. Therefore, for each
$\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\psi\circ\widetilde{F}_{1,t}\rangle_{\mathcal{H}}\big{|}=\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{t})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$ (3.4)

In this case, one cannot easily improve the decay estimate (3.4) with the
tools of this paper. Indeed, in order to establish the convergence
$D_{t}\stackrel{{\scriptstyle\rm s}}{{\to}}D$, one uses the ergodic theorem
for uniquely ergodic flows (see [36]) which does not come with an explicit
rate of convergence. We refer to [10, Thm. 19] for a more quantitative decay
estimate obtained using the representation theory of ${\rm SL}(2,\mathbb{R})$.

###  3.8 Adjacency matrices

This example is motivated by [21] and [36, Ex. 4.7]. Let $(X,\sim)$ be a graph
of finite degree, with symmetric relation $\sim$, and with no multiple edges
or loops. Then the adjacency matrix of $(X,\sim)$ is the bounded self-adjoint
operator in the Hilbert space $\mathcal{H}:=\ell^{2}(X)$ given by

$(H\varphi)(x):=\sum_{y\sim x}\varphi(y),\quad\varphi\in\mathcal{H},~{}x\in
X.$

A directed graph $(X,<)$ subjacent to $(X,\sim)$ is the graph $(X,\sim)$
together with a relation $<$ on $X$ such that, for each $x,y\in X$, $x\sim y$
is equivalent to $x<y$ or $y<x$ (and one cannot have both $x<y$ and $y<x$).
When using drawings, we draw an arrow $x\leftarrow y$ if $x<y$. For each $x\in
X$, we define the sets $N^{-}(x):=\\{y\in X\mid x<y\\}$ and $N^{+}(x):=\\{y\in
X\mid y<x\\}$. Then a directed graph $(X,<)$ is called admissible if [21, Def.
5.1]:

  1. (i)

each closed path in $X$ has the same number of positively and negatively
oriented edges,

  2. (ii)

for each $x,y\in X$, one has $\\#\\{N^{-}(x)\cap
N^{-}(y)\\}=\\#\\{N^{+}(x)\cap N^{+}(y)\\}$.

Figure 1: Example of admissible graph

In the case of admissible graphs, it is shown in [21, Secs. 3-5] that there
exist a self-adjoint operator $A$ and a bounded self-adjoint operator $K$ in
$\mathcal{H}$ such that $H\in C^{1}(A)$ with $[iH,A]=K^{2}$, $K^{2}\in
C^{1}(A)$ with $[iK^{2},A]=-2HK^{2}$, and $[K,H]=0$. Thus Lemma 2.4 applies
with $D=D_{t}=K^{2}\langle H\rangle^{-2}\in C^{1}(A)$ for all $t>0$ and
$\ker(D)=\ker(K)$. So $H|_{\ker(K)^{\perp}}$ has purely a.c. spectrum, as was
first proved in [21, Thm. 1.1] with a more complicated method.

Now, since $H,K^{2}\in C^{1}(A)$, we have $(H\pm
i)^{-1}\mathcal{D}(A)\subset\mathcal{D}(A)$ and
$K^{2}\mathcal{D}(A)\subset\mathcal{D}(A)$. Thus a calculation using the above
commutation relations and the operator $\widetilde{A}:=(H+i)^{-1}A(H-i)^{-1}$
gives on $\mathcal{D}(A)$:

$\displaystyle[\widetilde{A},D]$
$\displaystyle=[(H+i)^{-1}A(H-i)^{-1},K^{2}\langle H\rangle^{-2}]$
$\displaystyle=K^{2}(H+i)^{-1}[A,\langle
H\rangle^{-2}](H-i)^{-1}+(H+i)^{-1}[A,K^{2}](H-i)^{-1}\langle H\rangle^{-2}$
$\displaystyle=K^{2}(H+i)^{-1}(-2iK^{2}H\langle
H\rangle^{-4})(H-i)^{-1}+(H+i)^{-1}(-2iHK^{2})(H-i)^{-1}\langle H\rangle^{-2}$
$\displaystyle=-2iK^{2}H\langle H\rangle^{-4}(K^{2}\langle H\rangle^{-2}+1)$
$\displaystyle=-2iDH\langle H\rangle^{-2}(D+1).$

Since $DH\langle H\rangle^{-2}(D+1)\in\mathscr{B}(\mathcal{H})$ and
$\mathcal{D}(A)$ is a core for $\widetilde{A}$, we infer that $D\in
C^{1}(\widetilde{A})$ with $[\widetilde{A},D]=-2iDH\langle
H\rangle^{-2}(D+1)$. So, in the present case, the operator
$B\in\mathscr{B}(\mathcal{H})$ appearing in the statement of Theorem 2.1(c) is
$B=-2iH\langle H\rangle^{-2}(D+1)$ and it satisfies $[D,B]=0$. Furthermore, a
repeated use of the information gathered so far shows that each factor
appearing in the expression for $B$ belongs to $C^{(n-1)}(\widetilde{A})$ for
any $n\in\mathbb{N}^{*}$. Thus $B\in C^{(n-1)}(\widetilde{A})$ for any
$n\in\mathbb{N}^{*}$. It follows that the assumptions of Theorem 2.1(c) are
satisfied for the net $(U_{j})_{j\in
J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set $(\ell_{j})_{j\in
J}=(t)_{t>0}$, the operator $\widetilde{A}$, and $D=K^{2}\langle
H\rangle^{-2}$. Thus for each $\varphi\in D^{n}\mathcal{D}(A^{n})$ and
$\psi\in\mathcal{D}(A^{n})$ ($n\in\mathbb{N}^{*}$) there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

###  3.9 Jacobi matrices

This example is motivated by [27]. Let
$\mathcal{H}:=\ell^{2}(\mathbb{N}^{*})$, let $\ell^{2}_{0}\subset\mathcal{H}$
be the set of functions $\mathbb{N}^{*}\to\mathbb{C}$ with compact support,
and let $H$ be the Jacobi matrix

$(H\varphi)(n):=a_{n-1}\varphi(n-1)+b_{n}\varphi(n)+a_{n}\varphi(n+1),\quad\varphi\in\ell^{2}_{0},~{}n\in\mathbb{N}^{*},~{}\varphi(0):=0,$

with coefficients

$a_{n}:=n^{\alpha}+\eta_{n},\quad
b_{n}:=\lambda(n^{\alpha}+(n-1)^{\alpha})+\beta_{n},\quad\alpha>0,~{}\lambda,\eta_{n},\beta_{n}\in\mathbb{R}.$

For any sequence $(r_{n})_{n\in\mathbb{N}^{*}}$ and $k\in\mathbb{N}^{*}$,
define by induction $(\partial^{k+1}r)_{n}:=(\partial\partial^{k}r)_{n}$ with
$(\partial r)_{n}:=r_{n+1}-r_{n}$, set

$\beta_{n}^{\prime}:=\beta_{n}-\lambda(\eta_{n}+\eta_{n-1}),\quad
n\in\mathbb{N}^{*},~{}\eta_{0}:=0,$

and assume the following:

######  Assumption 3.2.

Suppose that $|\lambda|<1$, $\alpha\in(0,1)$, and assume that the sequences
with elements

$\tfrac{\eta_{n}}{n^{\alpha}},\quad(\partial\eta)_{n},\quad
n^{1-2\alpha}(\partial\eta)_{n},\quad
n^{1-\alpha}(\partial^{2}\eta)_{n},\quad\beta_{n}^{\prime},\quad
n^{1-\alpha}(\partial\beta^{\prime})_{n},\quad(n\in\mathbb{N}^{*})$

vanish as $n\to\infty$.

Then it is shown in [27, Thms 1.1-1.4] that $H$ is essentially self-adjoint
(with closure denoted by the same symbol), that $\sigma_{\rm
ess}(H)=\mathbb{R}$, and that the eigenvalues of $H$ (if any) are of finite
multiplicity and can accumulate only at $\pm\infty$. Furthermore, it is shown
in the proof of [27, Cor. 8.1] that there exists a self-adjoint operator $A$
in $\mathcal{H}$ such that $(H-i)^{-1}\in C^{1}(A)$ with

$[iH,A]=4(1-\lambda^{2})(1-\alpha)+K,\quad
K\in\mathscr{K}(\mathcal{D}(H),\mathcal{H}).$

Since $(H+i)^{-1}K(H-i)^{-1}\in\mathscr{K}(\mathcal{H})$ and

$\displaystyle\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}\tfrac{1}{t}\int_{0}^{t}\mathrm{d}\tau\,\mathop{\mathrm{e}}\nolimits^{-i\tau
H}(H+i)^{-1}4(1-\lambda^{2})(1-\alpha)(H-i)^{-1}\mathop{\mathrm{e}}\nolimits^{i\tau
H}P_{\rm c}(H)$ $\displaystyle=4(1-\lambda^{2})(1-\alpha)\langle
H\rangle^{-2}P_{\rm c}(H),$

it follows from Lemma 2.4 that

$D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{t\to\infty}D_{t}=4(1-\lambda^{2})(1-\alpha)\langle
H\rangle^{-2}P_{\rm c}(H)$

and that $H|_{\ker(D)^{\perp}}=H|_{P_{\rm c}(H)\mathcal{H}}$ has (trivially)
purely continuous spectrum. Moreover, Theorem 2.1(a) applies for the net
$(U_{j})_{j\in J}=(\mathop{\mathrm{e}}\nolimits^{-itH})_{t>0}$, the set
$(\ell_{j})_{j\in J}=(t)_{t>0}$, the operator
$\widetilde{A}=(H+i)^{-1}A(H-i)^{-1}$, and $D$ as above. Therefore, for each
$\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{t})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{t}\;\\!c_{\varphi,\psi},\quad
t>0.$

One can obtain stronger decay estimates in particular cases. For instance, it
is shown in [27, Appx. A] that for certain choices of $a_{n}$ and $b_{n}$ one
gets the relation $[iH,A]=aH+b$ with
$(a,b)\in\mathbb{R}^{2}\setminus\\{(0,0)\\}$, from which one can infer the
estimates of Theorem 2.1(c).

###  3.10 Schrödinger operators on Fock spaces

This example is motivated by [14]. Let $U$ be an isometry in a Hilbert space
$\mathcal{H}$ and let $N$ be a number operator for $U$, that is, a self-
adjoint operator in $\mathcal{H}$ such that
$U^{*}\mathcal{D}(N)\subset\mathcal{D}(N)$ and $UNU^{*}=N-1$ on
$\mathcal{D}(N)$. Number operators do not always exist, but if $U$ is
completely non-unitary, namely if $\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}(U^{*})^{n}=0$, then $N$ exists and is unique. Next,
consider the bounded self-adjoint operators
$H:=\mathop{\mathrm{Re}}\nolimits(U)$ and
$S:=\mathop{\mathrm{Im}}\nolimits(U)$, and set

$A\varphi:=\tfrac{1}{2}(SN+NS)\varphi,\quad\varphi\in\mathcal{D}(N).$

Then it is shown in [14, Secs. 2-3] that $A$ is essentially self-adjoint (with
closure denoted by the same symbol) and that $H\in C^{\infty}(A)$ with
$[iH,A]=1-H^{2}$. Thus Proposition 2.5 applies with $f(H)=1-H^{2}$,
$g(H)=(1-H^{2})\langle H\rangle^{-2}\in C^{\infty}(A)$ and
$\ker(f(H))=E^{H}(\\{-1,1\\})\mathcal{H}$. It follows that $H$ has purely a.c.
spectrum in $(-1,1)$ and for each $\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and
$\psi\in\mathcal{D}(A^{n})$ ($n\in\mathbb{N}^{*}$) there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

Operators like $H$ (and additive perturbations of it) appear naturally in the
framework of Fock spaces and their applications to Schrödinger operators on
trees. Indeed, let $\mathfrak{h}$ be a complex Hilbert space and
$\mathcal{H}:=\bigoplus_{n=0}^{\infty}\mathfrak{h}^{\otimes n}$ the complete
Fock space associated to it (with $\mathfrak{h}^{\otimes 0}:=\mathbb{C}$ and
$\mathfrak{h}^{\otimes n}:=\\{0\\}$ if $n<0$). For any $u\in\mathfrak{h}$ with
$\|u\|_{\mathfrak{h}}=1$, let $U\in\mathscr{B}(\mathcal{H})$ act on the sector
$\mathfrak{h}^{\otimes n}$ as

$U(h_{1}\otimes\dots\otimes h_{n}):=h_{1}\otimes\dots\otimes h_{n}\otimes
u,\quad h_{1}\otimes\dots\otimes h_{n}\in\mathfrak{h}^{\otimes n}.$

Then $U$ is completely non-unitary, the operator
$H=\mathop{\mathrm{Re}}\nolimits(U)$ acts on the sector $\mathfrak{h}^{\otimes
n}$ as

$\begin{cases}H(h_{1}\otimes\dots\otimes
h_{n})=\tfrac{1}{2}\big{(}h_{1}\otimes\dots\otimes h_{n}\otimes
u+h_{1}\otimes\dots\otimes h_{n-1}\langle
h_{n},u\rangle_{\mathfrak{h}}\big{)}&\hbox{if $n\geq 1$,}\\\
Hh=\tfrac{1}{2}hu&\hbox{if $h\in\mathfrak{h}^{\otimes
0}=\mathbb{C}$,}\end{cases}$

and the number operator $N$ for $U$ can be explicitly described (see [14, Sec.
4] for more details).

###  3.11 Multiplication by $\lambda$ in
$\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}_{+},\mathrm{d}\mu)$

This example is motivated by [15, Sec. 7.6]. Let $H$ be the maximal
multiplication operator by the variable $\lambda\in\mathbb{R}_{+}$ in the
Hilbert space
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}_{+},\mathrm{d}\mu)$,
with $\mathrm{d}\mu:=h\;\\!\mathrm{d}\lambda$ and $h\in
C^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$, and assume that the function

$q:\mathbb{R}_{+}\to\mathbb{R},~{}~{}\lambda\mapsto\tfrac{\lambda
h^{\prime}(\lambda)}{h(\lambda)}+1,$

is bounded. Then the operator

$A\varphi:=-\tfrac{i}{4}(2H\varphi^{\prime}+q\varphi),\quad\varphi\in
C^{\infty}_{\rm c}(\mathbb{R}_{+}),$

is essentially self-adjoint (with closure denoted by the same symbol) and
$(H-i)^{-1}\in C^{\infty}(A)$ with $[iH,A]=-\tfrac{1}{2}H$.444In [15, Sec.
7.6], there is a small mistake in the calculation of the commutator $[iH,A]$.
This is why our operator $A$ and commutator $[iH,A]$ slightly differ from the
ones appearing in [15, Sec. 7.6]. Thus Proposition 2.5 applies with
$f(H)=-\tfrac{1}{2}H$, $g(H)=-\tfrac{1}{2}H\langle H\rangle^{-2}\in
C^{\infty}(A)$ and $\ker(f(H))=\ker(H)$. Furthermore, since

$\mu\big{(}\\{\lambda\in[0,\infty)\mid\lambda=0\\}\big{)}=0,$

we have that $\ker(H)=\\{0\\}$. It follows that $H$ has purely a.c. spectrum
equal to $[0,\infty)$ and for each $\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and
$\psi\in\mathcal{D}(A^{n})$ ($n\in\mathbb{N}^{*}$) there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

This estimate improves the result of [15, Sec. 7.6] (see [15, Thm. 6.3]).

###  3.12 $H=-\partial_{xx}+\partial_{yy}$ in $\mathbb{R}^{2}$

This example is motivated by [15, Sec. 7.2]. Consider in the Hilbert space
$\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}^{2})$ the partial
differential operator

$H\varphi:=(-\partial_{xx}+\partial_{yy})\varphi,\quad\varphi\in\SS(\mathbb{R}^{2}),$

which is essentially self-adjoint (with closure denoted by the same symbol),
and $A$ the generator of dilations in $\mathcal{H}$. Then we have
$(H-i)^{-1}\in C^{\infty}(A)$ with $[iH,A]=2H$. Thus Proposition 2.5 applies
with $f(H)=2H$, $g(H)=2H\langle H\rangle^{-2}\in C^{\infty}(A)$ and
$\ker(f(H))=\ker(H)$. Since an application of the Fourier transform shows that
$H$ has spectrum equal to $\mathbb{R}$ and $\ker(H)=\\{0\\}$, it follows that
$H$ has purely a.c. spectrum equal to $\mathbb{R}$, and for each $\varphi\in
g(H)^{n}\mathcal{D}(A^{n})$ and $\psi\in\mathcal{D}(A^{n})$
($n\in\mathbb{N}^{*}$) there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

This estimate improves the result of [15, Sec. 7.2] (see [15, Thm. 6.3]).

###  3.13 $H=-X^{2-s}\Delta-\Delta X^{2-s}$ in $\mathbb{R}_{+}$

This example is motivated by [15, Sec. 7.4]. Let $X$ be the maximal
multiplication operator by the variable $x\in\mathbb{R}_{+}$ in the Hilbert
space $\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(\mathbb{R}_{+})$ and let
$A$ be the generator of dilations in $\mathcal{H}$. Then the operator

$H\varphi:=(-X^{2-s}\Delta-\Delta X^{2-s})\varphi,\quad\varphi\in
C^{\infty}_{\rm c}(\mathbb{R}_{+}),~{}s\in(0,2),$

is essentially self-adjoint (with closure denoted by the same symbol) and
$(H-i)^{-1}\in C^{\infty}(A)$ with $[iH,A]=sH$. In particular, one obtains
that
$\mathop{\mathrm{e}}\nolimits^{-itA}H=\mathop{\mathrm{e}}\nolimits^{st}H\mathop{\mathrm{e}}\nolimits^{-itA}$
for all $t\in\mathbb{R}$ which implies that $\ker(H)=\\{0\\}$, since otherwise
$\ker(H)$ would be a nontrivial invariant subspace of the irreducible
representation $\mathbb{R}\ni
t\mapsto\mathop{\mathrm{e}}\nolimits^{-itA}\in\mathrm{U}(\mathcal{H})$. Thus
Proposition 2.5 applies with $f(H)=sH$, $g(H)=sH\langle H\rangle^{-2}\in
C^{\infty}(A)$ and $\ker(f(H))=\ker(H)=\\{0\\}$. It follows that $H$ has
purely a.c. spectrum, and for each $\varphi\in g(H)^{n}\mathcal{D}(A^{n})$ and
$\psi\in\mathcal{D}(A^{n})$ ($n\in\mathbb{N}^{*}$) there exists a constant
$c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,\mathop{\mathrm{e}}\nolimits^{-itH}\psi\rangle_{\mathcal{H}}\big{|}\leq\tfrac{1}{t^{n}}\;\\!c_{\varphi,\psi},\quad
t>0.$

This estimate improves the result of [15, Sec. 7.4] (see [15, Thm. 6.3]).

###  3.14 Quantum walks on $\mathbb{Z}$

This example is motivated by [23, 24]. Consider a quantum walk on $\mathbb{Z}$
with evolution operator $U:=SC$ in the Hilbert space
$\mathcal{H}:=\ell^{2}(\mathbb{Z},\mathbb{C}^{2})$, where $S$ is the shift
operator defined as

$(S\varphi)(x):=\left(\begin{smallmatrix}\varphi^{(0)}(x+1)\\\
\varphi^{(1)}(x-1)\end{smallmatrix}\right),\quad\varphi=\left(\begin{smallmatrix}\varphi^{(0)}\\\
\varphi^{(1)}\end{smallmatrix}\right)\in\mathcal{H},~{}x\in\mathbb{Z},$

and $C$ the coin operator defined as

$(C\varphi)(x):=C(x)\varphi(x),\quad\varphi\in\mathcal{H},~{}x\in\mathbb{Z},~{}C(x)\in\mathrm{U}(2).$

Assume that $C$ is anisotropic, namely, converging with short-range rate to an
asymptotic coin on the left and to an asymptotic coin on the right:

######  Assumption 3.3.

There exist $C_{\ell},C_{\sf r}\in\mathrm{U}(2)$, $\kappa_{\ell},\kappa_{\sf
r}>0$, and $\varepsilon_{\ell},\varepsilon_{\sf r}>0$ such that

$\displaystyle\|C(x)-C_{\ell}\|_{\mathscr{B}(\mathbb{C}^{2})}\leq\kappa_{\ell}\;\\!|x|^{-1-\varepsilon_{\ell}},\quad
x<0,$ $\displaystyle\|C(x)-C_{\sf
r}\|_{\mathscr{B}(\mathbb{C}^{2})}\leq\kappa_{\sf
r}\;\\!|x|^{-1-\varepsilon_{\sf r}},\quad x>0,$

where the indexes $\ell$ and ${\sf r}$ stand for “left" and “right".

Under this assumption, it is shown in [23, Sec. 4] that $U$ has no singular
continuous spectrum and that the eigenvalues of $U$ (if any) are of finite
multiplicity and can accumulate only at a finite set of threshold values.
Furthermore, there exist a self-adjoint operator $A$ in $\mathcal{H}$, a
unitary operator $U_{0}$ in an auxiliary Hilbert space $\mathcal{H}_{0}$, a
self-adjoint operator $A_{0}$ in $\mathcal{H}_{0}$ and
$J\in\mathscr{B}(\mathcal{H}_{0},\mathcal{H})$ such that $U\in C^{1}(A)$,
$U_{0}\in C^{1}(A_{0})$ and

$[A_{0},U_{0}]U_{0}^{-1}=V_{0}^{2}\quad\hbox{and}\quad[A,U]U^{-1}-J[A_{0},U_{0}]U_{0}^{-1}J^{*}\in\mathscr{K}(\mathcal{H})$

with $V_{0}\in\mathscr{B}(\mathcal{H}_{0})$ an asymptotic velocity operator
for $U_{0}$ satisfying $[V_{0},U_{0}]=0$. Since $P_{\rm c}(U)=P_{\rm ac}(U)$,
one then infers from [24, Sec. 3] that

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}U^{n}JV_{0}^{2}J^{*}U^{-n}P_{\rm
c}(U)=\left(\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}U^{n}JU_{0}^{-n}P_{\rm
ac}(U_{0})\right)V_{0}^{2}\left(\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}U_{0}^{n}J^{*}U^{-n}P_{\rm
ac}(U)\right)=W_{-}V_{0}^{2}W_{-}^{*}$

with

$W_{\pm}:=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\pm\infty}U^{-n}JU_{0}^{n}P_{\rm ac}(U_{0})$ (3.5)

the wave operators for the triple $(U,U_{0},J)$. Since strong convergence
implies strong Cesaro convergence, we obtain that

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}JV_{0}^{2}J^{*}U^{-m}P_{\rm
c}(U)=W_{-}V_{0}^{2}W_{-}^{*},$

and thus it follows from Lemma 2.7(b) that

$D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}=W_{-}V_{0}^{2}W_{-}^{*}.$

So, Theorem 2.1(a) applies for the net $(U_{j})_{j\in
J}=(U^{n})_{n\in\mathbb{N}^{*}}$, the set $(\ell_{j})_{j\in
J}=(n)_{n\in\mathbb{N}^{*}}$, the operator $A$, and
$D=W_{-}V_{0}^{2}W_{-}^{*}$. Therefore, for each
$\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{n})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{n}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$ (3.6)

In this example, one cannot easily improve the decay estimate (3.6). Indeed,
in order to establish the convergence $D_{n}\stackrel{{\scriptstyle\rm
s}}{{\to}}D$, we used Lemma 2.7(b). But the proof of Lemma 2.7(b) relies on
the RAGE theorem B.1, whose proof relies in turn on a discrete version of
Wiener’s theorem. And as in the continuous case, in general one cannot infer
an explicit rate of convergence from Wiener’s theorem. In addition, we used
the strong limits (3.5) which do not come with an explicit rate of convergence
either.

###  3.15 Quantum walks on trees

This example is motivated by [39]. Let $\mathcal{T}$ be a homogeneous tree of
odd degree $d\geq 3$, that is, a finitely generated group $\mathcal{T}$ with
generators $a_{1},\dots,a_{d}$, identity $e$, and presentation
$\mathcal{T}:=\langle a_{1},\dots,a_{d}\mid
a_{1}^{2}=\dots=a_{d}^{2}=e\rangle$.

Homogeneous tree $\mathcal{T}$ of degree $d=3$

Using the word length $|\cdot|$ on $\mathcal{T}$, we define sets of even/odd
elements of $\mathcal{T}$

$\mathcal{T}_{\sf e}:=\\{x\in\mathcal{T}\mid|x|\in
2\mathbb{N}\\}\quad\hbox{and}\quad\mathcal{T}_{\sf
o}:=\\{x\in\mathcal{T}\mid|x|\in 2\mathbb{N}+1\\}$

with corresponding characteristic functions $\chi_{\sf e}$ and $\chi_{\sf o}$.
Then we consider a quantum walk on $\mathcal{T}$ with evolution operator
$U:=SC$ in the Hilbert space
$\mathcal{H}:=\ell^{2}(\mathcal{T},\mathbb{C}^{d})$, where $S$ is the shift
operator defined as

$\displaystyle S:=\left(\begin{smallmatrix}S_{1+1,1+2}&&&\\\
&S_{2+1,2+2}&&\mbox{\large$0$}\\\ \mbox{\large$0$}&&\ddots&\\\
&&&S_{d+1,d+2}\\\ \end{smallmatrix}\right),\quad
S_{d,d+1}:=S_{d,1},~{}S_{d+1,d+2}:=S_{1,2},$ $\displaystyle
S_{i,j}f:=\chi_{\sf e}f(\;\\!\cdot\;\\!a_{i})+\chi_{\sf
o}f(\;\\!\cdot\;\\!a_{j}),\quad
i,j\in\\{1,\dots,d\\},~{}f\in\ell^{2}(\mathcal{T}),$

and $C$ the coin operator defined as

$(C\varphi)(x):=C(x)\varphi(x),\quad\varphi\in\mathcal{H},~{}x\in\mathcal{T},~{}C(x)\in\mathrm{U}(d).$

Assume that $C$ is anisotropic, namely, converging with short-range rate to a
diagonal asymptotic coin on each main branch of $\mathcal{T}$:

######  Assumption 3.4.

For $i=1,\dots,d$, there exist a diagonal matrix $C_{i}\in\mathrm{U}(d)$ and
$\varepsilon_{i}>0$ such that

$\|C(x)-C_{i}\|_{\mathscr{B}(\mathbb{C}^{d})}\leq{\rm Const.}\;\\!\langle
x\rangle^{-(1+\varepsilon_{i})},\quad x\in\mathcal{T}_{i},$

where $\mathcal{T}_{i}:=\\{x\in\mathcal{T}\mid|a_{i}x|=|x|-1\\}$.

Under this assumption, it is shown in [39, Sec. 5] that the spectrum of $U$
covers the whole unit circle and is purely absolutely continuous, outside
possibly a finite set where $U$ may have eigenvalues of finite multiplicity.
Furthermore, there exist a self-adjoint operator $A$ in $\mathcal{H}$, a
unitary operator $U_{0}$ in an auxiliary Hilbert space $\mathcal{H}_{0}$, a
self-adjoint operator $A_{0}$ in $\mathcal{H}_{0}$ and
$J\in\mathscr{B}(\mathcal{H}_{0},\mathcal{H})$ such that $U\in C^{1}(A)$,
$U_{0}\in C^{\infty}(A_{0})$ and

$JJ^{*}=1_{\mathcal{H}},\quad[A_{0},U_{0}]U_{0}^{-1}=2,\quad[A,U]U^{-1}-J[A_{0},U_{0}]U_{0}^{-1}J^{*}\in\mathscr{K}(\mathcal{H}).$

It then follows from Lemma 2.7(b) that

$D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}(J[A_{0},U_{0}]U_{0}^{-1}J^{*})U^{-m}P_{\rm
c}(U)=2P_{\rm c}(U).$

Thus, Theorem 2.1(a) applies for the net $(U_{j})_{j\in
J}=(U^{n})_{n\in\mathbb{N}^{*}}$, the set $(\ell_{j})_{j\in
J}=(n)_{n\in\mathbb{N}^{*}}$, the operator $A$, and $D=2P_{\rm c}(U)$. So for
each $\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and
$\psi\in\mathcal{D}(A)$ there exists a constant $c_{\varphi,\psi}\geq 0$ such
that

$\big{|}\langle\varphi,U^{n}\psi\rangle_{\mathcal{H}}\big{|}\leq\|(D-D_{n})\widetilde{\varphi}\|_{\mathcal{H}}\;\\!\|\psi\|_{\mathcal{H}}+\tfrac{1}{n}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$ (3.7)

As in the example of the previous section, one cannot easily improve the decay
estimate (3.7).

###  3.16 Skew products

This example is motivated by [38], but see also [5, 34, 35, 36]. Let $X$ be a
smooth compact second countable Hausdorff manifold with Borel probability
measure $\mu_{X}$, and let $(F_{t})_{t\in\mathbb{R}}$ be a $C^{1}$ measure-
preserving flow on $X$. The operators
$V_{t}:\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X})\to\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X})$
given by $V_{t}\varphi:=\varphi\circ F_{t}$ define a strongly continuous one-
parameter unitary group with self-adjoint generator $H$ in
$\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X})$ essentially self-adjoint on
$C^{1}(X)$ and given by

$H\varphi:=i\mathscr{L}_{Y}\varphi,\quad\varphi\in C^{1}(X),$

with $Y$ the $C^{0}$ vector field associated to $F$ and $\mathscr{L}_{Y}$ the
corresponding Lie derivative.

Let $G$ be a compact Lie group with identity $e_{G}$, normalised Haar measure
$\mu_{G}$, and Lie algebra $\mathfrak{g}$. Then any measurable function
$\phi:X\to G$ induces a measurable cocycle
$X\times\mathbb{Z}\ni(x,n)\mapsto\phi^{(n)}(x)\in G$ over $F_{1}$ given by

$\phi^{(n)}(x):=\begin{cases}\phi(x)(\phi\circ F_{1})(x)\cdots(\phi\circ
F_{n-1})(x)&\hbox{if $n\geq 1$}\\\ \hfil e_{G}&\hbox{if $n=0$}\\\
\hfil(\phi^{(-n)}\circ F_{n})(x)^{-1}&\hbox{if $n\leq-1$.}\end{cases}$

The skew product $T_{\phi}$ defined by

$T_{\phi}:X\times G\to X\times
G,~{}~{}(x,g)\mapsto\big{(}F_{1}(x),g\;\\!\phi(x)\big{)},$

is an automorphism of $(X\times G,\mu_{X}\otimes\mu_{G})$, and the
corresponding Koopman operator

$U_{\phi}\psi:=\psi\circ
T_{\phi},\quad\psi\in\mathcal{H}:=\mathop{\mathrm{L}^{2}}\nolimits(X\times
G,\mu_{X}\otimes\mu_{G}),$

is a unitary operator in $\mathcal{H}$.

Let $\widehat{G}$ be the set of (equivalence classes of) finite-dimensional
irreducible unitary representations of $G$. Then each $\pi\in\widehat{G}$ is a
$C^{\infty}$ group homomorphism from $G$ to the unitary group
$\mathrm{U}(d_{\pi})$ of degree $d_{\pi}:=\dim(\pi)$, and Peter-Weyl’s theorem
implies that the set of all matrix elements $\\{\pi_{jk}\\}_{j,k=1}^{d_{\pi}}$
of all $\pi\in\widehat{G}$ forms an orthogonal basis of
$\mathop{\mathrm{L}^{2}}\nolimits(G,\mu_{G})$. Accordingly, one has the
orthogonal decomposition

$\mathcal{H}=\bigoplus_{\pi\in\widehat{G}}\,\bigoplus_{j=1}^{d_{\pi}}\mathcal{H}^{(\pi)}_{j},\quad\mathcal{H}^{(\pi)}_{j}:=\bigoplus_{k=1}^{d_{\pi}}\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X})\otimes\\{\pi_{jk}\\},$
(3.8)

and $U_{\phi}$ is reduced by the decomposition (3.8), with restriction
$U_{\phi,\pi,j}:=U_{\phi}\big{|}_{\mathcal{H}^{(\pi)}_{j}}$ given by

$U_{\phi,\pi,j}\sum_{k=1}^{d_{\pi}}\varphi_{k}\otimes\pi_{jk}=\sum_{k,\ell=1}^{d_{\pi}}(\varphi_{k}\circ
F_{1})(\pi_{\ell
k}\circ\phi)\otimes\pi_{j\ell},\quad\varphi_{k}\in\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X}).$

Furthermore, the following holds true [38, Sec. 3]: The operator $A$ given by

$A\sum_{k=1}^{d_{\pi}}\varphi_{k}\otimes\pi_{jk}:=\sum_{k=1}^{d_{\pi}}H\varphi_{k}\otimes\pi_{jk},\quad\varphi_{k}\in
C^{1}(X),$

is essentially self-adjoint in $\mathcal{H}^{(\pi)}_{j}$ (with closure denoted
by the same symbol). If
$\mathscr{L}_{Y}(\pi\circ\phi)\in\mathop{\mathrm{L}^{\infty}}\nolimits(X,\mathscr{B}(\mathbb{C}^{d_{\pi}}))$,
then $U_{\phi,\pi,j}\in C^{1}(A)$ with

$[A,U_{\phi,\pi,j}]=iM_{\pi\circ\phi}\;\\!U_{\phi,\pi,j}$

and $M_{\pi\circ\phi}$ the bounded matrix-valued multiplication operator in
$\mathcal{H}^{(\pi)}_{j}$ given by

$M_{\pi\circ\phi}\sum_{k=1}^{d_{\pi}}\varphi_{k}\otimes\pi_{jk}:=\sum_{k,\ell=1}^{d_{\pi}}\big{(}\mathscr{L}_{Y}(\pi\circ\phi)\cdot(\pi\circ\phi)^{-1}\big{)}_{k\ell}\;\\!\varphi_{\ell}\otimes\pi_{jk},\quad\varphi_{k}\in\mathop{\mathrm{L}^{2}}\nolimits(X,\mu_{X}).$

Finally, if $\mathscr{L}_{Y}\phi$ exists $\mu_{X}$-almost everywhere and
$M_{\phi}\in\mathop{\mathrm{L}^{2}}\nolimits(X,\mathfrak{g})$, then

$D=\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}D_{n}=i(\mathrm{d}\pi)_{e_{G}}\big{(}(P_{\phi}M_{\phi})(\cdot)\big{)}$

where $(\mathrm{d}\pi)_{e_{G}}\big{(}(P_{\phi}M_{\phi})(\cdot)\big{)}$ is a
bounded operator in $\mathcal{H}^{(\pi)}_{j}$ that can be interpreted as a
matrix-valued degree of the cocycle $\pi\circ\phi:X\to\pi(G)$ (see [38, Rem.
3.12] for more details). Thus, Theorem 2.1(a) applies for the net
$(U_{j})_{j\in J}=(U_{\phi,\pi,j}^{n})_{n\in\mathbb{N}^{*}}$, the set
$(\ell_{j})_{j\in J}=(n)_{n\in\mathbb{N}^{*}}$, the operator $A$, and
$D=i(\mathrm{d}\pi)_{e_{G}}((P_{\phi}M_{\phi})(\cdot))$. So for each
$\varphi=D\widetilde{\varphi}\in D\mathcal{D}(A)$ and $\psi\in\mathcal{D}(A)$
there exists a constant $c_{\varphi,\psi}\geq 0$ such that

$\big{|}\big{\langle}\varphi,U_{\phi,\pi,j}^{n}\psi\big{\rangle}_{\mathcal{H}^{(\pi)}_{j}}\big{|}\leq\|(D-D_{n})\widetilde{\varphi}\|_{\mathcal{H}^{(\pi)}_{j}}\|\psi\|_{\mathcal{H}^{(\pi)}_{j}}+\tfrac{1}{n}\;\\!c_{\varphi,\psi},\quad
n\in\mathbb{N}^{*}.$ (3.9)

In this example, one cannot easily improve the decay estimate (3.9). Indeed,
the convergence $D_{n}\stackrel{{\scriptstyle\rm s}}{{\to}}D$ follows from
[38, Lemma 3.3], whose proof relies on Birkhoff’s pointwise ergodic theorem
for Banach-valued functions. And without additional information, one cannot
infer an explicit rate of convergence from that theorem. That being said, one
can exhibit various examples where (3.9) is satisfied, and one can even prove
that $U_{\phi}$ has purely a.c. spectrum in appropriate subspaces of
$\mathcal{H}$. We refer to [38, Sec. 4] for more details.

##  Appendix A Commutators and regularity classes

In this appendix, we recall the definitions of commutators of operators and
regularity classes associated with them that we use in this work. We refer to
Chapters 5-6 of the monograph [1] for more details.

Let $A$ be a self-adjoint operator in a Hilbert space $\mathcal{H}$ with
domain $\mathcal{D}(A)$, and take a bounded operator
$S\in\mathscr{B}(\mathcal{H})$. For any $k\in\mathbb{N}$, we say that $S$
belongs to $C^{k}(A)$, with notation $S\in C^{k}(A)$, if the map

$\mathbb{R}\ni
t\mapsto\mathop{\mathrm{e}}\nolimits^{-itA}S\mathop{\mathrm{e}}\nolimits^{itA}\in\mathscr{B}(\mathcal{H})$
(A.1)

is strongly of class $C^{k}$. The sets
$C^{k}(A)\subset\mathscr{B}(\mathcal{H})$ satisfy the inclusions

$C^{\infty}(A):=\cap_{k\in\mathbb{N}}C^{k}(A)\subset\cdots\subset
C^{2}(A)\subset C^{1}(A)\subset C^{0}(A)=\mathscr{B}(\mathcal{H}).$

In the case $k=1$, one has $S\in C^{1}(A)$ if and only if the quadratic form

$\mathcal{D}(A)\ni\varphi\mapsto\langle\varphi,iSA\varphi\rangle_{\mathcal{H}}-\langle
A\varphi,iS\varphi\rangle_{\mathcal{H}}\in\mathbb{C}$

is continuous for the topology induced by $\mathcal{H}$ on $\mathcal{D}(A)$.
We denote by $[iS,A]$ the bounded operator associated with the continuous
extension of this form, or equivalently the strong derivative of the map (A.1)
at $t=0$.

If $H$ is a self-adjoint operator in $\mathcal{H}$ with domain
$\mathcal{D}(H)$ and spectrum $\sigma(H)$, we say that $H$ is of class
$C^{k}(A)$ if $(H-z)^{-1}\in C^{k}(A)$ for some
$z\in\mathbb{C}\setminus\sigma(H)$. In particular, $H$ is of class $C^{1}(A)$
if and only if the quadratic form

$\mathcal{D}(A)\ni\varphi\mapsto\langle\varphi,(H-z)^{-1}A\varphi\rangle_{\mathcal{H}}-\langle
A\varphi,(H-z)^{-1}\varphi\rangle_{\mathcal{H}}\in\mathbb{C}$

extends continuously to a bounded form with corresponding bounded operator
denoted by $[(H-z)^{-1},A]\in\mathscr{B}(\mathcal{H})$. In such a case, the
set $\mathcal{D}(H)\cap\mathcal{D}(A)$ is a core for $H$ and the quadratic
form

$\mathcal{D}(H)\cap\mathcal{D}(A)\ni\varphi\mapsto\langle
H\varphi,A\varphi\rangle-\langle A\varphi,H\varphi\rangle\in\mathbb{C}$

is continuous in the graph norm topology of $\mathcal{D}(H)$ [1, Thm.
6.2.10(a)]. This form then extends uniquely to a continuous quadratic form on
$\mathcal{D}(H)$ which can be identified with a continuous operator $[H,A]$
from $\mathcal{D}(H)$ to the adjoint space $\mathcal{D}(H)^{*}$. In addition,
the following relation holds in $\mathscr{B}(\mathcal{H})$ [1, Thm.
6.2.10(b)]:

$[(H-z)^{-1},A]=-(H-z)^{-1}[H,A](H-z)^{-1}.$

##  Appendix B RAGE-type theorem for unitary operators

In this appendix, we give the proof of a RAGE-type theorem for unitary
operators that we use in Section 2.3. The theorem is surely well-known, but
since we did not find it in this form in the literature we present its proof
for completeness.

We start by recalling the usual RAGE theorem for a unitary operator $U$. As in
the previous sections, we use the notation $P_{\rm p}(U)$ for the projection
onto the pure point subspace $\mathcal{H}_{\rm p}(U)$ of $U$ and $P_{\rm
c}(U)$ for the projection onto the continuous subspace $\mathcal{H}_{\rm
c}(U)$ of $U$.

######  Theorem B.1 (RAGE theorem, page 320 of [28]).

Let $U$ be a unitary operator in a Hilbert space $\mathcal{H}$ and
$K\in\mathscr{K}(\mathcal{H})$. Then

$\lim_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}\|KU^{-m}\varphi\|_{\mathcal{H}}^{2}=\|KP_{\rm
p}(U)\varphi\|_{\mathcal{H}}^{2}\quad\hbox{for all $\varphi\in\mathcal{H}$.}$

######  Theorem B.2.

Let $U$ be a unitary operator in a Hilbert space $\mathcal{H}$ and
$K\in\mathscr{K}(\mathcal{H})$. Then

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}=\sum_{\theta\in\scriptsize\\{\hbox{eigenvalues
of $U$}\\}}E^{U}(\\{\theta\\})KE^{U}(\\{\theta\\}).$

###### Proof.

We mimic the proof of the analogous theorem in the self-adjoint case [30, Thm.
5.9]. Any $\varphi\in\mathcal{H}$ admits an orthogonal decomposition
$\varphi=\varphi_{\rm p}+\varphi_{\rm c}$ with $\varphi_{\rm
p}\in\mathcal{H}_{\rm p}(U)$ and $\varphi_{\rm c}\in\mathcal{H}_{\rm c}(U)$.
For the component $\varphi_{\rm c}$, we get from the Cauchy-Schwarz inequality
and Theorem B.1 that

$\lim_{n\to\infty}\left\|\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}\varphi_{\rm
c}\right\|_{\mathcal{H}}\leq\lim_{n\to\infty}\left(\sum_{m=0}^{n-1}\tfrac{1}{n}\right)^{1/2}\left(\tfrac{1}{n}\sum_{m=0}^{n-1}\big{\|}KU^{-m}\varphi_{\rm
c}\big{\|}_{\mathcal{H}}^{2}\right)^{1/2}=0.$

For the component $\varphi_{\rm p}$, we write $\varphi_{\rm p}=\sum_{j\geq
1}\alpha_{j}\varphi_{j}$ with $(\varphi_{j})_{j\geq 1}$ an orthonormal basis
of $\mathcal{H}_{\rm p}(U)$, $\alpha_{j}\in\mathbb{C}$, and
$U\varphi_{j}=\theta_{j}\varphi_{j}$ for some $\theta_{j}\in\mathbb{S}^{1}$.
Then we get

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}\varphi_{\rm
p}=\operatorname*{s\hskip 0.1pt-\hskip 0.1ptlim}_{n\to\infty}\sum_{j\geq
1}\alpha_{j}\left(\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}\right)K\varphi_{j}.$
(B.1)

Now we have
$\|\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}\|_{\mathscr{B}(\mathcal{H})}\leq
1$ for all $n\in\mathbb{N}^{*}$, and

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}(U\theta_{j}^{-1})^{m}=E^{U}(\\{\theta_{j}\\})$

due to von Neumann’s mean ergodic theorem. Therefore we can exchange the limit
and the sum in (B.1) to obtain

$\operatorname*{s\hskip 0.1pt-\hskip
0.1ptlim}_{n\to\infty}\tfrac{1}{n}\sum_{m=0}^{n-1}U^{m}KU^{-m}\varphi_{\rm
p}=\sum_{j\geq
1}\alpha_{j}E^{U}(\\{\theta_{j}\\})K\varphi_{j}=\sum_{\theta\in\scriptsize\\{\hbox{eigenvalues
of $U$}\\}}E^{U}(\\{\theta\\})KE^{U}(\\{\theta\\})\varphi,$

as desired. ∎

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