Quantum time delay for unitary operators: general theory

arXiv:1812.10718v1 [math-ph] 27 Dec 2018

Quantum time delay for unitary operators: general theory

D. Sambou^∗ and R. Tiedra de Aldecoa^†

Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, Santiago, Chile

E-mails:disambou@mat.puc.cl, rtiedra@mat.puc.cl Abstract

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey’s imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Our approach is general and model-independent.

2010 Mathematics Subject Classification:46N50, 47A40, 81Q10, 81Q12.

Keywords:Quantum mechanics, scattering theory, quantum time delay, unitary operators.

Contents

1 Introduction and main results 1

2 Free propagator and position operators 5

3 LocallyU0-smooth operators 12

4 Summation formula 15

4.1 Averaged localisation functions . . . . 16 4.2 Proof of the summation formula . . . . 17 4.3 Interpretation of the summation formula . . . . 24

5 Quantum time delay 27

5.1 Symmetrised time delay . . . . 27 5.2 Non-symmetrised time delay . . . . 29

1 Introduction and main results

One can find a large literature on the notion of quantum time delay in the setup of scattering theory for self-adjoint operators (see for instance [2, 3, 5, 6, 7, 13, 14, 16, 17, 18, 25, 27, 28, 34, 35]). But as far as we know, there is no mathematical work on quantum time delay in the setup of scattering theory for unitary operators. The purpose of this paper is to fill in this gap by developing a general

∗Supported by the Chilean Fondecyt Grant 3170411.

†Supported by the Chilean Fondecyt Grant 1170008.

theory of quantum time delay for unitary operators in a two-Hilbert spaces setting. Namely, we present a suitable framework for the definition of time delay in terms of sojourn times in dilated regions for quantum scattering systems consisting in two unitary operators acting in two (a priori different) Hilbert spaces.

Then, we prove under appropriate conditions the existence of this time delay and its equality with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator appearing in the self-adjoint theory. This establishes in a general unitary setup, as was established before in a general self- adjoint setup [25], the identity between the time-dependent definition of time delay (with sojourn times) and the time-independent definition of time delay (with expectation value).

Our framework is the following. Assume that we have a scattering system (U0, U, J) consisting in a unitary operatorU in a Hilbert spaceH(a full propagator), a unitary operatorU0 in a Hilbert spaceH0

(a free propagator), and a bounded operatorJ:H0→ H(an identification operator) such that the wave operators

W±:= s-lim

n→±∞U⁻ⁿJ U₀ⁿPac(U0)

exist and are partial isometries with initial subspaces H^±₀ ⊂ H0 and final subspaces Hac(U). Then, the scattering operator

S:=W₊^∗W−:H⁻₀ → H⁺₀

is a well-defined unitary operator which commutes with U0 and decomposes into a family of unitary operators{S(z)}z∈σ(U0)in the spectral representation ofU0. Assume also that we have a family of mutually commuting self-adjoint operators Q := (Q1, . . . , Qd) in H0 (position operators) satisfying appropriate commutations relations with respect to U0. Finally, assume that we have a non-negative even Schwarz functionf ∈S(R^d) equal to 1 on a neighbourhood of 0∈ R^d (a localisation function). Then, for any fixedr >0 and suitableϕ∈ Hac(U0), we define the sojourn time of the freely evolving stateU₀ⁿϕin the region defined by the localisation operatorf(Q/r) as

T_r⁰(ϕ) :=X

n∈Z

U₀ⁿϕ, f(Q/r)U₀ⁿϕ

H0.

Now, the evolution group {Uⁿ}n∈Z acts inHwhereas f(Q/r) acts in H0. Therefore, in order to define a sojourn time for the corresponding fully evolving state UⁿW−ϕ, one needs to introduce a family of operatorsLn : H → H0 to inject the operator f(Q/r) in the Hilbert space H (the operators Ln must satisfy some natural conditions which in simple cases are verified ifLn=J^∗for alln∈Z). We then define the sojourn time of the fully evolving state UⁿW−ϕ in the region defined by the localisation operator f(Q/r) as

Tr,1(ϕ) :=X

n∈Z

LnUⁿW−ϕ, f(Q/r)LnUⁿW−ϕ

H0.

An additional sojourn time appears naturally in this two-Hilbert spaces setting: the time spent by the fully evolving stateUⁿW−ϕinside the time-dependent subset (1−L^∗_nLn)HofH

T2(ϕ) =X

n∈Z

UⁿW−ϕ, 1−L^∗_nLn

UⁿW−ϕ

H0.

The symmetrised time delayτ_r^sym(ϕ) for the scattering system (U0, U, J) is then defined as the difference between the sojourn times for the fully evolving stateUⁿW−ϕand the sojourn times for the freely evolving state U₀ⁿϕbefore and after scattering

τ_r^sym(ϕ) := Tr,1(ϕ) +T2(ϕ)

−¹₂ T_r⁰(ϕ) +T_r⁰(Sϕ) .

Our main result, properly stated in Theorem5.3and Remark5.4, is the proof of the existence of the limit limr→∞τ_r^sym(ϕ) and its identity with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator, namely,

r→∞lim τ_r^sym(ϕ) =

ϕ,−S^∗U0

dS dU0

ϕ

H0

. (1.1)

To our knowledge, this formula is completely new. It establishes in the unitary setup the identity between the time-dependent definition of time delay and the time-independent definition of time delay. Under the additional assumption that the scattering system is elastic in some appropriate sense (namely, that the operatorS commutes with some function of a velocity operator V defined in terms of U0 and Q), we show in Theorem5.9that the simpler, non-symmetrised time delay

τ_r^nsym(ϕ) := Tr,1(ϕ) +T2(ϕ)

−T_r⁰(ϕ) also exists in the limitr → ∞and satisfies

r→∞lim τ_r^nsym(ϕ) = lim

r→∞τ_r^sym(ϕ) =

ϕ,−S^∗U0

dS dU0

ϕ

H0

. (1.2)

We give now a more detailed description of the paper. In Section2, we introduce the free propagator U0 and the family of position operatorsQ = (Q1, . . . , Qd), and we present the general conditions of regularity and commutation that we impose onU0 andQ(Assumptions2.2and2.4). These conditions, formulated in terms of the family of unitary operators

U0(x) := e^{−i x·Q}U0e^{i x·Q}, x∈R^d,

imply the existence of a family of mutually commuting self-adjoint operatorsV = (V1, . . . , Vd) with Vj

given by

Vj := s- d dxj

x=0

i U0(x)U₀⁻¹

on some appropriate core. The operatorsVj can be interpreted as the components of the velocity vector associated to U0 and Qj. Accordingly, the set κ(U0) of values in the spectrum σ(U0) where V = 0 (precisely defined in Definition2.7) plays an important role and is called the set of critical values ofU0. It is a unitary analogue of the set of critial values introduced in [24, Def. 2.5] in the self-adjoint setup.

In Section3, we use commutator methods for unitary operators [9] to construct a conjugate operator for U0 and to prove a Mourre estimate forU0 on the set σ(U0)\κ(U0) (Lemmas 3.1 and 3.3). As a consequence, we obtain in Theorem3.4 a class locallyU0-smooth operators onS¹\κ(U0) and we show that the operator U0 has purely absolutely continuous in σ(U0)\κ(U0). To illustrate these results, we present in Examples3.5and3.6the cases where the velocity vectorV is constant andU0 is the time-one propagator for the Laplacian inR^d.

In Section4, we prove a formula which relates the evolution of the localisation operatorf(Q) under U0 to a time operatorTf. First, we recall in Section4.1the properties of averaged localisation functions Rf :R^d\ {0} →Cwhich appears naturally when dealing with quantum time delay. Then, in Section4.2, we prove for appropriate vectorsϕ∈ Hac(U0) the summation formula

r→∞lim

1 2

X

n≥0

ϕ, U₀⁻ⁿf(Q/r)U₀ⁿ−U₀ⁿf(Q/r)U₀⁻ⁿ ϕ

H0 =

ϕ, Tfϕ

H0. (1.3)

The proof, given in Theorem4.6, is not trivial: It involves commutator methods for families of self-adjoint and unitary operators, the class of locallyU0-smooth operators obtained in Section3, operator identities following from the Trotter-Kato formula (Lemma 4.5) and a repeated use of Lebesgue’s dominated convergence theorem. The operatorTf is similar to an operator appearing in the self-adjoint setup [24, Prop. 5.2]. Its precise definition is given in terms of a quadratic form (see Proposition4.3), but formally

Tf =¹₂ Q·(∇Rf)(V) + (∇Rf)(V)·Q .

In Section4.3, Lemma4.7, we show that the operatorsTf andU0 satisfy under appropriate conditions the relation

U₀⁻¹ Tf, U0

=−1,

which is the unitary analogue of the canonical time-energy commutation relation of the self-adjoint setup.

Therefore, the operatorTf can be interpreted as a time operator forU0, andTf is equal in some suitable sense to the operator−U0 d

dU0. Indeed, by applying Mackey’s imprimitivity theorem [19], we are able to show that Tf acts as the differential operator−z _dz^d (z ∈S¹) in a Hilbert space isomorphic toH0 (see Remark 4.8). In consequence, the formula (1.3) can be seen as an equality between on the l.h.s. the difference of times spent by the evolving state U₀ⁿϕ in the future (first term) and in the past (second term) within dilated regions defined by the localisation operatorsf(Q/r) and on the r.h.s. the expectation value inϕof the time operatorTf. At the end of Section4.3, we illustrate these results once again with the cases where the velocity vectorV is constant andU0is the time-one propagator for the Laplacian in R^d (see Examples4.9and4.10).

In Section5.1, we prove (1.1), that is, the existence of the symmetrised time delay τ_r^sym(ϕ) in the limitr → ∞and its identity with the expectation value of the unitary analogue of the Eisenbud-Wigner time delay operator (see Theorem5.3and Remark5.4). The main ingredient of the proof is the summation formula (1.3). In Section5.2, under the additional assumption that the scattering operatorS commutes with some appropriate function of the velocity operatorV, we prove that the simpler, non-symmetrised time delay τ_r^nsym(ϕ) also exists in the limit r → ∞ and satisfies the same identity (that is, (1.2), see Theorem5.9).

Before concluding, we would like to emphasize that our results here in the unitary setup are not a mere consequence of the corresponding results [24,25] in the self-adjoint setup. The standard tools allowing one to go from unitary operators to self-adjoint operators (such as the Cayley transform, functional calculus or operator logarithms) are not suited for the problem of quantum time delay. Even more, various proofs in the present paper turn out to be more subtle than the corresponding proofs in the self-adjoint setup.

Some of the reasons explaining this fact are the following:

• In the unitary setup, the sojourn times are defined as sums over a discrete time n∈Z, whereas in the self-adjoint setup the sojourn times are defined as integrals over a continuous timet ∈R. So, in order to obtain results in the unitary setup, one has to evaluate infinite sums, which in general is more challenging than to evaluate improper integrals. In particular, the proof of the summation formula (1.3) is more technical than the proof of the corresponding integral formula of the self- adjoint setup [24, Thm. 5.5], and the proof of the existence of the non-symmetrised time delay (1.2) relies on a preliminary result based on the Poisson summmation formula (Lemma 5.5) not needed in the self-adjoint setup.

• The unitary operatorsU0andUthat we consider are completely general. In particular, there are not supposed to be time-one propagators of some self-adjoint operatorsH0andH. In consequence, one cannot apply all the technics coming from the self-adjoint theory. Moreover, one does not have at disposal a predefined dense setD⊂ H0(such as the domain ofH0) where to perform the necessary the calculations for the free theory. Instead, one has to come up with an assumption onU0specific enough to put into evidence a dense set of D ⊂ H0 appropriate for the calculations, but general enough not to oversimplify the theory. Assumption2.2fullfils these requirements.

• In the self-adjoint setup, the unitary groups generated by the free Hamiltonian and the time operator satisfy in favorable situations the Weyl relation. Thus, one can apply Stone-von Neumann theorem to conclude that the time operator acts as the energy derivative in the spectral representation of the free Hamiltonian (see [24, Sec. 6]). In the unitary setup, the unitary groups generated by the free propagator and the time operator satisfy at best only an imprimitivity relation. Thus, one has to apply Mackey’s imprimitivity theorem, which is more complex than Stone-von Neumann theorem, to conclude that the time operator acts as the energy derivative in the spectral representation of the free propagator (see Section4.3).

To conclude, we point out that the theory presented here is general, adapted to cover a variety of unitary scattering systems, both in the one and two-Hilbert spaces setting. Therefore, we plan in the

future to apply it to various unitary scattering systems, as for instance anisotropic quantum walks as presented in [22,23].

Acknowledgements. The authors express their gratitude to the referees who dedicated time to review this manuscript and suggested various improvements to the text. In particular, we are really grateful to one referee who pointed to us possible alternative proofs for some of our results.

2 Free propagator and position operators

In this section, we recall needed facts on commutators methods, we introduce our assumptions on the free propagator and the position operators, and we describe a set of critical values of the free propagator which appears naturally under our assumptions.

We start by recalling some facts on commutators methods borrowed from [1, 11]. LetH be a self- adjoint operator with domain D(H) and spectrum σ(H) in a Hilbert space H0, and let A be a second self-adjoint operator with domainD(A) inH0. We say thatH is of class C^k(A) withk ∈Nif for some ω∈C\σ(H) the map

R∋t7→e^{−i tA}(H−ω)⁻¹e^{i tA}∈B(H0) (2.1) is strongly of classC^k. In the casek = 1, the quadratic form

D(A)∋ϕ7→

ϕ,(H−ω)⁻¹A ϕ

H0−

A ϕ,(H−ω)⁻¹ϕ

H0 ∈C extends continuously to a bounded operator denoted by

(H−ω)⁻¹, A

. Furthermore, the setD(H)∩D(A) is a core forH and the quadratic form

D(H)∩ D(A)∋ϕ7→

Hϕ, A ϕ

H0−

A ϕ, Hϕ

H0∈C

is continuous in the topology ofD(H). Thus, it extends uniquely to a continuous quadratic form [H, A]

onD(H) which can be identified with a continuous operator fromD(H) to the adjoint spaceD(H)^∗, and the following equality holds:

(H−ω)⁻¹, A

=−(H−ω)⁻¹[H, A](H−ω)⁻¹. (2.2)

In [11, Lemma 2], it has been shown that if [H, A]D(H) ⊂ H, then e^{i tA}D(H) ⊂ D(H) for eacht ∈R. Accordingly, we say in the sequel thati[H, A] is essentially self-adjoint onD(H) if [H, A]D(H)⊂ Hand ifi[H, A] is essentially self-adjoint onD(H) in the usual sense.

Now, let Q:= (Q1, . . . , Qd) be a family of mutually strongly commuting self-adjoint operators in H0(the position operators). Then, any functionf ∈L^∞(R^d) defines byd-variables functional calculus a bounded operatorf(Q) inH0. In particular, the operator e^{i x·Q}, withx·Q:=Pd

j=1xjQj, is unitary inH0

for eachx ∈R^d. In this context, we say thatH is of class C^k(Q) withk ∈Nif for someω∈C\σ(H) the map

R^d ∋x7→e^{−i x·Q}(H−ω)⁻¹e^{i x·Q}∈B(H0) (2.3) is strongly of classC^k. Clearly, ifHis of classC^k(Q), thenH is of classC^k(Qj) for eachj.

Remark 2.1. The definitions are similar if we consider a bounded operator B ∈ B(H0) instead of a self-adjoint operatorH. In such a case, we use the notationB∈C^k(A)if the map(2.1), with(H−ω)⁻¹ replaced by B, is strongly of class C^k, and we use the notation B ∈ C^k(Q) if the map (2.3), with (H−ω)⁻¹ replaced byB, is strongly of class C^k.

In the sequel, we assume the existence of a unitary operatorU0∈B(H0) (the free propagator) with associated family of unitary operators

U0(x) := e^{−i x·Q}U0e^{i x·Q}, x∈R^d, regular with respect toQin the following sense:

Assumption 2.2(Regularity). The map

R^d ∋x7→U0(x)U₀⁻¹∈B(H0)

is strongly differentiable on a coreD⊂ H0of the operatorQ², and for eachj∈ {1, . . . , d} the operator

Vjϕ:= s- d dxj

x=0

i U0(x)U₀⁻¹ϕ, ϕ∈D,

is essentially self-adjoint, with self-adjoint extension denoted by the same symbol. The operatorVj is of classC¹(Q), and for eachk ∈ {1, . . . , d}, i

Vj, Qk

is essentially self-adjoint onD(Vj), with self-adjoint extension denoted byV_{j k}^′. The operatorV_{j k}^′ is of classC¹(Q), and for each ℓ∈ {1, . . . , d}, i

V_{j k}^′, Qℓ is essentially self-adjoint onD V_{j k}^′

, with self-adjoint extension denoted byV_{j kℓ}^′′ .

Assumption 2.2implies that the set D is a core for all the operators Vj, V_{j k}^′, andV_{j kℓ}^′′ . Assumption 2.2 also implies the invariance of the domainsD(Vj) under the action of the unitary group{e^{i x·Q}}_x∈R^d. Indeed, the condition

Vj, Qk

D(Vj) ⊂ H and [11, Lemma 2] imply that e^{i tQk}D(Vj) ⊂ D(Vj) for each t∈R. Thus e^{i tQk}D(Vj) =D(Vj) for eacht∈R, and since this holds for eachk ∈ {1, . . . , d}one obtains that e^{i x·Q}D(Vj) =D(Vj) for eachx ∈R^d. As a consequence, the operators

Vj(x) := e^{−i x·Q}Vje^{i x·Q}, x∈R^d, are self-adjoint operators with domainsD Vj(x)

=D(Vj). Similarly, the domainsD V_{j k}^′

are left invariant by the unitary group{e^{i x·Q}}_x∈R^d, and the operators

V_{j k}^′(x) := e^{−i x·Q}V_{j k}^′ e^{i x·Q}, x∈R^d, are self-adjoint operators with domainsD V_{j k}^′(x)

=D V_{j k}^′ .

Remark 2.3. The operators Vj andV_{j k}^′ can be interpreted as the components of the velocity vector and the acceleration matrix associated to the propagatorU0and the position operatorsQj.

Our second main assumption on the operatorsU0(x) is a commutation assumption:

Assumption 2.4(Commutation).

U0(x), U0(y)

= 0for allx, y ∈R^d.

Assumption2.4 implies that the operatorsU0(x) mutually commute in the strong sense, namely, if E^U0(x)denotes the spectral measure ofU0(x) on the complex unit circleS¹:={e^{i θ}|θ∈[0,2π)}, then

E^U0(x)(Θ), E^U0(y)(Θ^′)

= 0 for allx, y ∈R^d and all Borel sets Θ,Θ^′⊂S¹ (see [30, Prop. 5.27]).

Additional commutation relations are obtained in the following lemma:

Lemma 2.5. Let Assumptions2.2and2.4be satisfied. Then, the operatorsU0(x), Vj(y)+i−1

, V_kℓ^′(z)+

i−1

mutually commute for allx, y , z ∈R^d and allj, k, ℓ∈ {1, . . . , d}.

Proof. Letj, k, ℓ, m ∈ {1, . . . , d},x, y ∈R^d, and set R^Vj(x):= Vj(x) +i−1

and R^Vkℓ′(x):= V_kℓ^′(x) +i−1

. For anyϕ, ψ∈ H0, there exist (ϕn)n∈N,(ψn)n∈N⊂D such that

n→∞lim

ϕn− R^Vj∗

ϕ

D(Vj)= 0 and lim

n→∞

ψn−R^Vjψ

D(Vj)= 0.

This, together with the facts that

U0(x), i U0(y)U₀⁻¹

= 0 and s-_dy^d_j

y=0 i U0(y)U₀⁻¹∗

ϕm=Vjϕm, implies that

ϕ, R^VjU0(x)VjR^Vjψ

H0 = lim

m→∞ lim

n→∞

ϕm, U0(x)Vjψn

H0

= lim

m→∞ lim

n→∞

d dyj

y=0

i U0(y)U₀⁻¹∗

ϕm, U0(x)ψn

H0

= lim

m→∞ lim

n→∞

Vjϕm, U0(x)ψn

H0

=

ϕ, R^VjVjU0(x)R^Vjψ

H0. (2.4)

Sinceϕ, ψare arbitrary, this implies that U0(x), R^Vj

=−R^Vj

U0(x), Vj

R^Vj = 0.

Therefore, we obtain that

U0(x), R^Vj(y)

= e^{−i y·Q}

U0(x−y), R^Vj

e^{i y·Q}= 0 and thus the operatorsU0(x) andR^Vj(y)commute.

A calculation as in (2.4) using the commutation ofU0(x)U₀⁻¹andR^Vj(y)implies that D

ϕ, R^VjVjR^Vk(y)R^VjψE

H0

=D

ϕ, R^VjR^Vk(y)VjR^VjψE

H0

. Sinceϕ, ψare arbitrary, this implies that

R^Vj, R^Vk(y)

=−R^Vj

Vj, R^Vk(y)

R^Vj = 0.

Therefore, we obtain that

R^Vj(x), R^Vk(y)

= e^{−i x·Q}

R^Vj, R^Vk(y)

e^{i x·Q}= 0, and thus the operatorsR^Vj(x) andR^Vk(y)commute.

Leteℓ be theℓ-th standard orthonormal vector in R^d. Then, the commutation ofR^Vj(x) andR^Vk(y) implies that

R^Vj(x)R^Vk(y+εeℓ)−R^Vk(y)

ε =R^Vk(y+εeℓ)−R^Vk(y)

ε R^Vj(x), ε∈R\ {0}.

Taking the limitε→0 and using (2.2) and the strong commutation ofVj(x) andVk(y), one obtains R^Vk(y)

R^Vj(x), V_kℓ^′(y)

R^Vk(y)= 0.

Since the resolventR^Vk(y)on the left is injective, this implies that

R^Vj(x), V_kℓ^′(y)

= 0 on D Vk(x) , and sinceD Vk(x)

is a core forV_kℓ^′(y) the last equality extends toD V_kℓ^′(y)

. Therefore, we obtain that R^Vj(x), R^Vkℓ′(y)

=−R^Vkℓ′(y)

R^Vj(x), V_kℓ^′(y)

R^Vkℓ′(y)= 0, and thus the operatorsR^Vj(x) andR^Vkℓ′(y) commute.

The commutation ofU0(x) andR^Vk(y)implies that U0(x)R^Vk(y+εeℓ)−R^Vk(y)

ε =R^Vk(y+εeℓ)−R^Vk(y)

ε U0(x), ε∈R\ {0}.

Taking the limitε→0, and using (2.2), the commutation ofU0(x) andR^Vk(y), and the commutation of R^Vk(y)yR^Vkℓ′(y), one obtains that

R^Vk(y)

U0(x), V_kℓ^′(y)

R^Vk(y)= 0

⇐⇒ R^Vk(y)R^Vkℓ′(y)

U0(x), V_kℓ^′(y)

R^Vkℓ′(y)R^Vk(y)= 0

⇐⇒ R^Vkℓ′(y)

U0(x), V_kℓ^′(y)

R^Vkℓ′(y)= 0.

Therefore, we obtain that

U0(x), R^Vkℓ′(y)

=−R^Vkℓ′(y)

U0(x), V_kℓ^′(y)

R^Vkℓ′(y)= 0, and thus the operatorsU0(x) andR^Vkℓ′(y)commute.

Finally, the commutation ofR^Vjk′(x)andR^{Vℓm′ (y)}is proved in a similar way. The details are left to the reader.

Lemma2.5and [30, Prop. 5.27] imply that the operatorsU0(x),Vj(y) andV_kℓ^′(z) mutually commute in the strong sense for all x, y , z ∈ R^d and all j, k, ℓ ∈ {1, . . . , d}, that is, the spectral projections of U0(x), Vj(y) andV_kℓ^′(z) mutually commute for all x, y , z ∈ R^d and allj, k, ℓ∈ {1, . . . , d}. Using these commutation relations, we can establish other useful facts:

Lemma 2.6. Let Assumptions 2.2and2.4be satisfied.

(a) For allj ∈ {1, . . . , d},x ∈R^d andϕ∈D, one has s- d

dxj

i U0(x)U₀⁻¹ϕ=Vj(x)U0(x)U₀⁻¹ϕ.

(b) For alln∈Zandj∈ {1, . . . , d}, one has

U₀ⁿ D(Qj)∩ D(Vj)

=D(Qj)∩ D(Vj).

(c) For alln∈Z,j∈ {1, . . . , d} andϕ∈ D(Qj)∩ D(Vj), one has U₀ⁿQjU₀⁻ⁿϕ= (Qj−n Vj)ϕ,

and the operator(Qj−n Vj)is essentially self-adjoint onD(Qj)∩ D(Vj).

(d) For allx ∈R^dandn∈Z, the operator x·Q−n(x·V)

is essentially self-adjoint onD(x·Q)∩D(x·V), with self-adjoint extension

x·Q−n(x·V)

↾D(x·Q)∩ D(x·V) =U₀ⁿ(x·Q)U₀⁻ⁿ. (e) For allj, k ∈ {1, . . . , d}, one hasV_{j k}^′ =V_kj^′.

The result of point (c) has the following interpretation: After time n, the positionU₀ⁿQjU₀⁻ⁿ of the quantum system with propagatorU0 is equal to the valueQof its initial position minusntimes the value V of its initial velocity.

Proof. (a) Letϕ∈D andψ∈e^{−i x·Q}D. Then, one has ψ,s-_dx^d_ji U0(x)U₀⁻¹ϕ

H0 = lim

ε→0

ψ,_εⁱ e^{−i x·Q}U0(εej) e^{i x·Q}U₀⁻¹−e^{−i x·Q}U0e^{i x·Q}U₀⁻¹ ϕ

H0

= lim

ε→0

−_εⁱ U0(εej)U₀⁻¹−1∗

e^{i x·Q}ψ, U0e^{i x·Q}U₀⁻¹ϕ

H0. Since e^{i x·Q}ψ∈D, Assumption2.2implies that

−s-lim

ε→0 i

ε U0(εej)U₀⁻¹−1∗

e^{i x·Q}ψ=Vje^{i x·Q}ψ.

Thus,

ψ,s-_dx^d_ji U0(x)U₀⁻¹ϕ

H0 =

Vje^{i x·Q}ψ, U0e^{i x·Q}U₀⁻¹ϕ

H0 =

ψ, Vj(x)U0(x)U₀⁻¹ϕ

H0