text: we know that Φ and Ψ correspond to the bottom and the top of the lattice of driver’s total (cid:96)i,τi (cid:96)i,τi payments among all CE outcomes, hence Lemma 3. Φ and Ψ are Optimal Potentials: N E Given the MCF problem (??) with graph G = ( , ) and an optimal integral solution f (which N E is guaranteed to exist), we ﬁrst construct the standard residual graph G˜ = ( , ˜) where the set of E E ∪ E ∪ E nodes are the same and the set of edges ˜ = ˜ ˜ ˜ is consisted of: • E { | ∈ R } ∪ { | ∈1 R 2 3 } ˜ = e j , f(e ) = 0 e˜ j , f(e ) = 1 . e = ((o , τ ), (d , τ + δ(o , d ))) 1 j j j j − j j j j j j j is the edge corresponding to rider j with γ(e ) = v , c(e ) = 0, and c¯(e ) = 1. e˜ = j j j j j ((d , τ + δ(o , d ), (o , τ ))) is the reversed edge corresponding to rider j s.t. f(e ) = 1, with j j j j j j j γ(e˜ ) = v , c(e˜ ) = 0 and c¯(e˜ ) = 1, j j j j • E E ∪ { | ∈ E } ∈ E ˜ = e˜ e , f(e) > 0 , where for all e = ((a, t), (b, t + δ(a, b))) , e˜ = ((b, t + 2 2 2 2 δ(a, b)), (a, t)) with γ(e˜) = 0, c(e˜) = 0 and c¯(e˜) = f(e), • E E ∪ { | ∈ E } ∈ E ˜ = e˜ e where for all e = ((a, T ), n ) , e˜ = (n , (a, T )) with γ(e˜) = 0, 3 3 3 0 3 0 c(e˜) = 0 and c¯(e˜) = f(e). From the standard argument on the residual graphs [?], we know that the cost of a feasible ﬂow in the residual graph is equal to the incremental cost of the same ﬂow in the original graph. For any node (a, t), the “shortest distance” from this node to the sink n refers to the smallest 0 total cost among all paths from (a, t) to n in the residual graph. Since the costs are negations of 0 rider values, the shortest distance corresponds to the negation of the maximum incremental welfare − created by an additional unit of driver ﬂow starting from (a, t), i.e. Φ . ∈ T (cid:44) − (cid:44) a,t{ − } For all (a, b, t) , deﬁne p Φ Φ , and let µ max v p , 0 for all ∈ R a,b,t a,t b,t+δ(a,b) j j oj,dj,τj j . We show that (Φ, µ) forms an optimal solution to (??). The argument is very similar to that of the reduced cost optimality, however, we include the proof here for completeness. We ﬁrst show the feasibility of (Φ, µ): 1. Constraint (??) holds by deﬁnition of µ. ∈ T ∈ E 2. For (??), observe that for all (a, b, t) , there exists an edge ((a, t), (b, t + δ(a, b))) ˜ with zero cost, thus the shortest distance from (a, t) to n is at most zero plus the shortest distance − ≤ − 0 ⇒ ≥ from (b, t + δ(a, b)) to n , implying Φ Φ Φ Φ , 0 a,t b,t+δ(a,b) a,t b,t+δ(a,b) ∈ L ∈ E 3. For (??), note that for all a , there exists ((a, T ), n ) ˜ s.t. γ(e) = 0. Therefore, the − 0 − ≤ ⇔ ≥ shortest distance Φ between (a, T ) and n is at most 0, i.e. Φ 0 Φ 0, a,T 0 a,T a,T 4. (??) holds by deﬁnition of µ. We now show the optimality by examining that the complementary slackness conditions hold between f and (Φ, µ): ∈ R ⇒ 1. For all j , µ > 0 f(e ) = 1. To show this, observe that when µ > 0, we know − − j j − − − j v (Φ Φ ) > 0, implying Φ > Φ v , i.e. in the residual j oj,τj dj,τj+δ(oj,dj) oj,τj dj,τj+δ(o−j,dj) j graph, the shortest distance from (o , τ ) to the sink is longer than v plus the shortest distance j j j from (d , τ +δ(o , d )). This means that the edge e = ((o , τ ), (d , τ +δ(o , d ))) with capacity j j− j j j j j j j j j 1 and cost v cannot be present in the residual graph, implying f(e ) = 1. j j ∈ R ⇒ − 2. For all j , f(e ) > 0 Φ Φ + µ = v : we know from the deﬁnition of µ j oj,τj dj,τj+δ(oj,dj) j j − − ≥ ⇔j that to show this, we only need to show that when f(e ) > 0, v (Φ Φ ) 0 j j oj,τj dj,τj+δ(oj,dj) 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: we

text: know

text: that

text: Φ

text: and

text: Ψ

text: correspond

text: to

text: the

text: bottom

text: and

text: the

text: top

text: of

text: the

text: lattice

text: of

text: driver’s

text: total

text: payments

text: among

text: all

text: CE

text: outcomes,

text: hence

text: Lemma

text: Φ

text: and

text: Ψ

text: are

text: Optimal

text: Potentials:

text: N

text: E

text: Given

text: the

text: MCF

text: problem

text: with

text: graph

text: G

text: and

text: an

text: optimal

text: integral

text: solution

text: f

text: (which

text: N

text: E

text: is

text: guaranteed

text: to

text: exist),

text: we

text: ﬁrst

text: construct

text: the

text: standard

text: residual

text: graph

text: G˜

text: where

text: the

text: set

text: of

text: E

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text: E

text: E

text: nodes

text: are

text: the

text: same

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text: the

text: set

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text: c(e

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text: c¯(e

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text: t

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text: b)),

text: (a,

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text: c¯(e˜)

text: f(e).

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text: the

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text: residual

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text: the

text: cost

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text: a

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text: in

text: the

text: residual

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text: equal

text: to

text: the

text: incremental

text: cost

text: of

text: the

text: same

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text: in

text: the

text: original

text: graph.

text: For

text: any

text: node

text: (a,

text: t),

text: the

text: “shortest

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text: the

text: sink

text: n

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text: the

text: smallest

text: total

text: cost

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text: (a,

text: t)

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text: in

text: the

text: residual

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text: Since

text: the

text: costs

text: are

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text: of

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text: the

text: shortest

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text: negation

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text: the

text: maximum

text: incremental

text: welfare

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text: by

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text: (a,

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text: that

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text: reduced

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text: however,

text: we

text: include

text: the

text: proof

text: here

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text: completeness.

text: We

text: ﬁrst

text: show

text: the

text: feasibility

text: of

text: (Φ,

text: µ):

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text: t),

text: (b,

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text: shortest

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text: γ(e)

text: Therefore,

text: the

text: shortest

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text: (a,

text: T

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text: n

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text: at

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text: Φ

text: Φ

text: a,T

text: a,T

text: a,T

text: holds

text: by

text: deﬁnition

text: of

text: µ.

text: We

text: now

text: show

text: the

text: optimality

text: by

text: examining

text: that

text: the

text: complementary

text: slackness

text: conditions

text: hold

text: between

text: f

text: and

text: (Φ,

text: µ):

text: R

text: For

text: all

text: j

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text: show

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text: observe

text: that

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text: µ

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text: j

text: j

text: j

text: v

text: (Φ

text: Φ

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text: Φ

text: Φ

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text: i.e.

text: in

text: the

text: residual

text: j

text: oj,τj

text: dj,τj+δ(oj,dj)

text: oj,τj

text: dj,τj+δ(o−j,dj)

text: j

text: graph,

text: the

text: shortest

text: distance

text: from

text: (o

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text: sink

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text: shortest

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text: f(e

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text: j

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text: the

text: deﬁnition

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text: µ

text: j

text: oj,τj

text: dj,τj+δ(oj,dj)

text: j

text: j

text: ⇔j

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text: this,

text: we

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text: need

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text: show

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text: when

text: f(e

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text: (Φ

text: Φ

text: j

text: j

text: oj,τj

text: dj,τj+δ(oj,dj)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Now, consider the scenario where the rest of the drivers all follow the plan at time t but driver i deviates and relocates to location b. Denote the state of the platform at time t + 1 as (cid:44) s˜ (s− , (b, t + δ(a, b))). Driver i is not paid at time t, and from time t + 1 onward, her t+1 i,t+1 ∗ (t+1) total payment under σ would be Φ (s˜ ), the welfare gain from replicating the driver at b,t+δ(a,b) t+1 (cid:48) ≥ (b, t + δ(a, b)) computed at time t + 1 given state s˜ . We prove Φ(t )(s (cid:48)) Φ(t+1) (s˜ ) by t+1 a,t t b,t+δ(a,b) t+1 showing: (cid:48) ≥ (i) Φ(t )(s (cid:48)) Φ(t)(s ), a,t t a,t t ≥ ∈ L ≤ (t) (t+1) (ii) Φ (s ) Φ (s ) for all b s.t. t + δ(a, b) T , and a,t t b,t+δ(a,b) t+1 ≥ (t+1) (t+1) (iii) Φ (s ) Φ (s˜ ). b,t+δ(a,b) t+1 b,t+δ(a,b) t+1 Part (i). This is straightforward, since the highest achievable welfare at state s (cid:48) with additional t driver at (a, t) is weakly higher than the welfare of the scenario where all drivers follow the plan (cid:48) (cid:48) (cid:48) (cid:48) (x(t )(s (cid:48)), z(t )(s (cid:48)), q(t )(s (cid:48)), π(t )(s (cid:48))) until time t and then jointly optimize at time t for all the t t t t existing drivers (whose states are now s ) and the additional driver at (a, t): t D (cid:48) ∪ ≥ D (cid:48) − D D ∪ W ( (t )(s (cid:48)) (t, T, a)) W ( (t )(s (cid:48))) W ( (t)(s )) + W ( (t)(s ) (t, T, a)). t t t t D (cid:48) − D Here, W ( (t )(s (cid:48))) W ( (t)(s )) is the welfare achieved by all agents from following the original t(cid:48) − t plan from time t to t 1. The welfare gain is therefore (cid:48) D (cid:48) ∪ − D (cid:48) Φ(t )(s (cid:48)) =W ( (t )(s (cid:48)) (t, T, a)) W ( (t )(s (cid:48))) a,t t t t ≥ D (cid:48) − D D ∪ − D (cid:48) W ( (t )(s (cid:48))) W ( (t)(s )) + W ( (t)(s ) (t, T, a)) W ( (t )(s (cid:48))) t t t t D ∪ − D =W ( (t)(s ) (t, T, a)) W ( (t)(s )) t t (t) =Φ (s ). a,t t Part (ii). This is similar to part (i), observing that the additional driver at (a, t) can simply relocate to b while the rest of the drivers follow the original plan at time t and then jointly optimize: D ∪ ≥ D − D D ∪ W ( (t)(s ) (t, T, a)) W ( (t)(s )) W ( (t+1)(s )) + W ( (t+1)(s ) (t + δ(a, b), T, b)). t t t+1 t+1 Therefore, D ∪ − D Φ(t)(s ) =W ( (t)(s ) (t, T, a)) W ( (t)(s )) a,t t t t ≥ D − D D ∪ − D W ( (t)(s )) W ( (t+1)(s )) + W ( (t+1)(s ) (t + δ(a, b), T, b)) W ( (t)(s )) t t+1 t+1 t D ∪ − D =W ( (t+1)(s ) (t + δ(a, b), T, b)) W ( (t+1)(s )) t+1 t+1 (t+1) =Φ (s ). b,t+δ(a,b) t+1 Part (iii). First, observe that the only possible diﬀerence between s and s˜ is the state of t+1 t+1 driver i. Fixing the state of the rest of the riders as s− , Φ(t+1) (s ) is the welfare gain from i,t+1 b,t+δ(a,b) t+1 (t+1) adding an additional driver at (b, t + δ(a, b)) where driver i is at s , whereas Φ (s˜ ) i,t+1 b,t+δ(a,b) t+1 is the welfare gain from adding an additional driver at (b, t + δ(a, b)) where driver i is also at (b, t + δ(a, b)), the state of driver i that had deviated while the replan happens at time t + 1. 51

text: Now,

label: paragraph

text: consider

text: the

text: scenario

text: where

text: the

text: rest

text: of

text: the

text: drivers

text: all

text: follow

text: the

text: plan

text: at

text: time

text: t

text: but

text: driver

text: i

text: deviates

text: and

text: relocates

text: to

text: location

text: b.

text: Denote

text: the

text: state

text: of

text: the

text: platform

text: at

text: time

text: t

text: as

text: s˜

text: (s−

text: (b,

text: t

text: δ(a,

text: b))).

text: Driver

text: i

text: is

text: not

text: paid

text: at

text: time

text: t,

text: and

text: from

text: time

text: t

text: onward,

text: her

text: total

text: payment

text: under

text: σ

text: would

text: be

text: Φ

text: (s˜

text: the

text: welfare

text: gain

text: from

text: replicating

text: the

text: driver

text: at

text: b,t+δ(a,b)

text: (b,

text: t

text: δ(a,

text: b))

text: computed

text: at

text: time

text: t

text: given

text: state

text: s˜

text: We

text: prove

text: Φ(t

text: )(s

text: (s˜

text: by

text: a,t

text: t

text: b,t+δ(a,b)

text: showing:

text: (i)

text: Φ(t

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text: Φ(t)(s

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text: a,t

text: t

text: L

text: (t)

text: (ii)

text: Φ

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text: t

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text: (iii)

text: Φ

text: (s

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text: b,t+δ(a,b)

text: b,t+δ(a,b)

text: Part

text: (i).

text: This

text: is

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text: since

text: the

text: highest

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text: welfare

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text: state

text: s

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text: t

text: driver

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text: (a,

text: t)

text: is

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text: the

text: welfare

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text: the

text: scenario

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text: plan

text: (x(t

text: )(s

text: z(t

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text: q(t

text: )(s

text: π(t

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text: t

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text: mechanism and that all drivers stay until at least the end of planning horizon, and do not have an intrinsic preference over location, including where they ﬁnish their last trip in the planning horizon. Each driver seeks to maximize the total payment received over the planning horizon. R Denote as the set of riders, each intending to take a single trip during the planning horizon. ∈ R The type of rider j is (o , d , τ , v ), where o and d are the trip origin and destination, τ the j≥ j j j j j j requested start time, and v 0 the value for the trip. We assume (S2) that riders are impatient, j only value trips starting at τ , are not willing to relocate or walk from a drop-oﬀ point to their j actual, intended destination. The value v models the willingness to pay of the rider, over-and- j above a base payment for a trip. This base payment is an amount that is collected by the platform, so that in aggregate drivers’ costs are covered. Accordingly, the prices we derive in this paper correspond to “surge” prices, over-and-above these base payments.16 Rider utility is quasi-linear, − with utility v p to rider j for a completed trip at (incremental to base) price p. j We assume the platform has complete information about supply and demand over the planning horizon (driver and rider types, including driver entry during the planning horizon). Thus, the challenge addressed in this paper is one of promoting desirable behavior in the absence of time- extended contracts, and not one of information asymmetry. We assume drivers have the same information, and that this is common knowledge amongst drivers.17 Unless otherwise noted, we assume properties (S1), (S2), and complete, symmetric information throughout the paper, and discuss the eﬀect of relaxing these assumptions in Section . (cid:48) At each time t, a driver is en route if she started her last trip from a to b at time t (with or (cid:48) ≥ without a rider), and t < t + δ(a, b). A driver is available if she has entered the platform (t τ ) i and is not en route. A driver who is available at time t and location a is able to complete a pick-up at this location and time. We allow a driver to drop-oﬀ a rider and pick-up another rider in the same location at the same time point. A path is a sequence of tuples (a, b, t), representing the trips taken by a driver over the planning horizon. A feasible path for driver i starts at ((cid:96) , τ ), with the starting time and location of each i i Z successive trip equal to the ending time and location of the previous trip. Let denote the set of ∈ Z i ∈ all feasible paths of driver i, with Z to denote the kth feasible path. Denote (a, b, t) Z i,k i i,k if path Z includes (or covers) trip (a, b, t). Driver i with path Z is able to pick up rider j if i,k∈ i,k (o , d , τ ) Z . j j j i,k L { } Example 1. The planning horizon is T = 2 and there are two locations = A, B , with distance δ(A, A) = δ(B, B) = 1 and δ(A, B) = δ(B, A) = 2. See Figure 1. There is one driver, who enters at time τ = 0 at location (cid:96) = a and leaves at time τ¯ = 2. There are three riders: 1 1 1 • Rider 1: o = A, d = A, τ = 0, v = 5, 1 1 1 1 • Rider 2: o = A, d = A, τ = 1, v = 6, 2 2 2 1 • Rider 3: o = A, d = B, τ = 0, v = 8. 3 3 3 3 There are two feasible paths for driver 1: Z = ((A, A, 0), (A, A, 1)) and Z = ((A, B, 0)). 1,1 1,2 A path speciﬁes only the movement of the driver in space and time. Each of “picking up riders 1 and 2” and “staying static at location A for two periods” are consistent with path Z . Path 1,1 ((A, A, 0), (A, B, 1)) is infeasible, since the last trip ends later than the driver’s leaving time. (cid:3) Similarly, paths ((A, B, 0), (B, B, 1)) and ((A, A, 0), (B, B, 1)) are infeasible. 16The base payment can be thought to cover, for example, the unit costs to drivers of driving and the opportunity cost for other outside employment options. We suppose the base payment is collected and distributed to drivers in some trip-based manner (e.g. through per mile and per minute rates, and may be non-zero even when the prices determined by the STP mechanism is zero). 17More generally, it would be suﬃcient that it be common knowledge amongst drivers that the platform has the correct information. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: mechanism

text: and

text: that

text: all

text: drivers

text: stay

text: until

text: at

text: least

text: the

text: end

text: of

text: planning

text: horizon,

text: and

text: do

text: not

text: have

text: an

text: intrinsic

text: preference

text: over

text: location,

text: including

text: where

text: they

text: ﬁnish

text: their

text: last

text: trip

text: in

text: the

text: planning

text: horizon.

text: Each

text: driver

text: seeks

text: to

text: maximize

text: the

text: total

text: payment

text: received

text: over

text: the

text: planning

text: horizon.

text: R

text: Denote

text: as

text: the

text: set

text: of

text: riders,

text: each

text: intending

text: to

text: take

text: a

text: single

text: trip

text: during

text: the

text: planning

text: horizon.

text: R

text: The

text: type

text: of

text: rider

text: j

text: is

text: (o

text: d

text: τ

text: v

text: where

text: o

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text: trip

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text: τ

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text: j≥

text: j

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text: j

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text: j

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text: requested

text: start

text: time,

text: and

text: v

text: the

text: value

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text: the

text: trip.

text: We

text: assume

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text: riders

text: are

text: impatient,

text: j

text: only

text: value

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text: at

text: τ

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text: relocate

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text: a

text: drop-oﬀ

text: point

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text: their

text: j

text: actual,

text: intended

text: destination.

text: The

text: value

text: v

text: models

text: the

text: willingness

text: to

text: pay

text: of

text: the

text: rider,

text: over-and-

text: j

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text: a

text: base

text: payment

text: for

text: a

text: trip.

text: This

text: base

text: payment

text: is

text: an

text: amount

text: that

text: is

text: collected

text: by

text: the

text: platform,

text: so

text: that

text: in

text: aggregate

text: drivers’

text: costs

text: are

text: covered.

text: Accordingly,

text: the

text: prices

text: we

text: derive

text: in

text: this

text: paper

text: correspond

text: to

text: “surge”

text: prices,

text: over-and-above

text: these

text: base

text: Rider

text: utility

text: is

text: quasi-linear,

text: with

text: utility

text: v

text: p

text: to

text: rider

text: j

text: for

text: a

text: completed

text: trip

text: at

text: (incremental

text: to

text: base)

text: price

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text: j

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text: assume

text: the

text: platform

text: has

text: complete

text: information

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text: supply

text: and

text: demand

text: over

text: the

text: planning

text: horizon

text: (driver

text: and

text: rider

text: types,

text: including

text: driver

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text: during

text: the

text: planning

text: horizon).

text: Thus,

text: the

text: challenge

text: addressed

text: in

text: this

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text: is

text: one

text: of

text: promoting

text: desirable

text: behavior

text: in

text: the

text: absence

text: of

text: time-

text: extended

text: contracts,

text: and

text: not

text: one

text: of

text: information

text: asymmetry.

text: We

text: assume

text: drivers

text: have

text: the

text: same

text: information,

text: and

text: that

text: this

text: is

text: common

text: knowledge

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text: Unless

text: otherwise

text: noted,

text: we

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text: properties

text: and

text: complete,

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text: the

text: paper,

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text: eﬀect

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text: a

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text: time.

text: We

text: allow

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text: driver

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text: drop-oﬀ

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text: same

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text: path

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text: Let

text: denote

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text: set

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text: Z

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text: (a,

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text: Driver

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text: driver,

text: who

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text: at

text: time

text: τ

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text: location

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text: time

text: τ¯

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text: three

text: riders:

text: Rider

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text: d

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text: τ

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text: driver

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text: time.

text: Each

text: of

text: “picking

text: up

text: riders

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text: 2”

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text: “staying

text: static

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text: location

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text: with

text: path

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text: ((A,

text: A,

text: (A,

text: B,

text: is

text: infeasible,

text: since

text: the

text: last

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text: than

text: the

text: driver’s

text: leaving

text: time.

text: Similarly,

text: paths

text: ((A,

text: B,

text: (B,

text: B,

text: and

text: ((A,

text: A,

text: (B,

text: B,

text: are

text: infeasible.

text: base

text: payment

text: can

text: be

text: thought

text: to

text: cover,

text: for

text: example,

text: the

text: unit

text: costs

text: to

text: drivers

text: of

text: driving

text: and

text: the

text: opportunity

text: cost

text: for

text: other

text: outside

text: employment

text: options.

text: We

text: suppose

text: the

text: base

text: payment

text: is

text: collected

text: and

text: distributed

text: to

text: drivers

text: in

text: some

text: trip-based

text: manner

text: (e.g.

text: through

text: per

text: mile

text: and

text: per

text: minute

text: rates,

text: and

text: may

text: be

text: non-zero

text: even

text: when

text: the

text: prices

text: determined

text: by

text: the

text: STP

text: mechanism

text: is

text: zero).

text: generally,

text: it

text: would

text: be

text: suﬃcient

text: that

text: it

text: be

text: common

text: knowledge

text: amongst

text: drivers

text: that

text: the

text: platform

text: has

text: the

text: correct

text: information.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The Three Layer Causal Hierarchy Level Typical Typical Questions Examples (Symbol) Activity 1. Association Seeing What is? What does a symptom tell me | P(y x) How would seeing X about a disease? change my belief inY ? What does a survey tell us about the election results? 2. Intervention Doing What if? What if I take aspirin, will my | P(y do(x), z) Intervening What if I do X? headache be cured? What if we ban cigarettes? 3. Counterfactuals Imagining, Why? Was it the aspirin that | (cid:48) (cid:48) P(y x , y ) Retrospection Was it X that caused Y ? stopped my headache? x What if I had acted Would Kennedy be alive had diﬀerently? Oswald not shot him? What if I had not been smok- ing the past 2 years? Figure 1: The Causal Hierarchy. Questions at level i can only be answered if information from level i or higher is available. An extremely useful insight unveiled by the logic of causal reasoning is the existence of a sharp classiﬁcation of causal information, in terms of the kind of questions that each class is capable of answering. The classiﬁcation forms a 3-level hierarchy in the sense that questions at level i (i = 1, 2, 3) can only be ≥ answered if information from level j (j i) is available. Figure 1 shows the 3-level hierarchy, together with the characteristic questions that can be answered at each level. The levels are titled 1. Association, 2. Intervention, and 3. Counterfactual. The names of these layers were chosen to emphasize their usage. We call the ﬁrst level Association, because it invokes purely statistical relationships, deﬁned by the naked data.1 For instance, observing a customer who buys toothpaste makes it more likely that he/she buys ﬂoss; such association can be inferred directly from the observed data using conditional expectation. Questions at this layer, because they require no causal information, are placed at the bottom level on the hierarchy. The second level, Intervention, ranks higher than Association because it involves not just seeing what is, but changing what we see. A typical question at this level would be: What happens if we double the price? Such questions cannot be answered from sales data alone, because they involve a change in customers behavior, in reaction to the new pricing. These choices may diﬀer substantially from those taken in previous price-raising situations. (Unless we replicate precisely the market conditions that existed when the price reached double its current value.) Finally, the top level is called Counterfactuals, a term that goes back to the philosophers David Hume and John Stewart Mill, and which has been given computer-friendly semantics in the past two decades. A typical question in the counterfactual category is “What if I had acted diﬀerently,” thus necessitating retrospective reasoning. Counterfactuals are placed at the top of the hierarchy because they subsume interventional and associational questions. If we have a model that can answer counterfactual queries, we can also answer questions about interventions and observations. For example, the interventional question, What will happen if we double the price? can be answered by asking the counterfactual question: What would happen had the price been twice its current value? Likewise, associational questions can be answered once we can answer interventional questions; we simply ignore the action part and let observations take over. The translation does not work in the opposite direction. Interventional questions cannot be answered from purely observational information (i.e., from statistical data alone). No counterfactual question involving retrospection can be answered from purely interventional information, such as that acquired from controlled experiments; we cannot re-run an experiment on subjects who were treated with a drug and see how they behave had they not given the drug. The hierarchy is therefore directional, with the top level being the most powerful one. 1Other names used for inferences at this layer are: “model-free,” “model-blind,” “black-box,” or “data-centric.” Darwiche (2017) used “function-ﬁtting,” for it amounts to ﬁtting data by a complex function deﬁned by the neural network architecture. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

label: section

text: Three

text: Layer

text: Causal

text: Hierarchy

text: Level

text: Typical

text: Typical

text: Questions

text: Examples

text: (Symbol)

text: Activity

text: Association

text: Seeing

text: What

text: is?

text: What

text: does

text: a

text: symptom

text: tell

text: me

text: P(y

text: x)

text: How

text: would

text: seeing

text: X

text: about

text: a

text: disease?

text: change

text: my

text: belief

text: inY

text: What

text: does

text: a

text: survey

text: tell

text: us

text: about

text: the

text: election

text: results?

text: Intervention

text: Doing

text: What

text: if?

text: What

text: if

text: I

text: take

text: aspirin,

text: will

text: my

text: P(y

text: do(x),

text: z)

text: Intervening

text: What

text: if

text: I

text: do

text: X?

text: headache

text: be

text: cured?

text: What

text: if

text: we

text: ban

text: cigarettes?

text: Counterfactuals

text: Imagining,

text: Why?

text: Was

text: it

text: the

text: aspirin

text: that

text: P(y

text: x

text: y

text: Retrospection

text: Was

text: it

text: X

text: that

text: caused

text: Y

text: stopped

text: my

text: headache?

text: x

text: What

text: if

text: I

text: had

text: acted

text: Would

text: Kennedy

text: be

text: alive

text: had

text: diﬀerently?

text: Oswald

text: not

text: shot

text: him?

text: What

text: if

text: I

text: had

text: not

text: been

text: smok-

text: ing

text: the

text: past

text: years?

text: Figure

text: The

text: Causal

text: Hierarchy.

text: Questions

text: at

text: level

text: i

text: can

text: only

text: be

text: answered

text: if

text: information

text: from

text: level

text: i

text: or

text: higher

text: is

text: available.

text: An

text: extremely

text: useful

text: insight

text: unveiled

text: by

text: the

text: logic

text: of

text: causal

text: reasoning

text: is

text: the

text: existence

text: of

text: a

text: sharp

text: classiﬁcation

text: of

text: causal

text: information,

text: in

text: terms

text: of

text: the

text: kind

text: of

text: questions

text: that

text: each

text: class

text: is

text: capable

text: of

text: answering.

text: The

text: classiﬁcation

text: forms

text: a

text: hierarchy

text: in

text: the

text: sense

text: that

text: questions

text: at

text: level

text: i

text: (i

text: can

text: only

text: be

text: answered

text: if

text: information

text: from

text: level

text: j

text: (j

text: i)

text: is

text: available.

text: Figure

text: shows

text: the

text: hierarchy,

text: together

text: with

text: the

text: characteristic

text: questions

text: that

text: can

text: be

text: answered

text: at

text: each

text: level.

text: The

text: levels

text: are

text: titled

text: Association,

text: Intervention,

text: and

text: Counterfactual.

text: The

text: names

text: of

text: these

text: layers

text: were

text: chosen

text: to

text: emphasize

text: their

text: usage.

text: We

text: call

text: the

text: ﬁrst

text: level

text: Association,

text: because

text: it

text: invokes

text: purely

text: statistical

text: relationships,

text: deﬁned

text: by

text: the

text: naked

text: For

text: instance,

text: observing

text: a

text: customer

text: who

text: buys

text: toothpaste

text: makes

text: it

text: more

text: likely

text: that

text: he/she

text: buys

text: ﬂoss;

text: such

text: association

text: can

text: be

text: inferred

text: directly

text: from

text: the

text: observed

text: data

text: using

text: conditional

text: expectation.

text: Questions

text: at

text: this

text: layer,

text: because

text: they

text: require

text: no

text: causal

text: information,

text: are

text: placed

text: at

text: the

text: bottom

text: level

text: on

text: the

text: hierarchy.

text: The

text: second

text: level,

text: Intervention,

text: ranks

text: higher

text: than

text: Association

text: because

text: it

text: involves

text: not

text: just

text: seeing

text: what

text: is,

text: but

text: changing

text: what

text: we

text: see.

text: A

text: typical

text: question

text: at

text: this

text: level

text: would

text: be:

text: What

text: happens

text: if

text: we

text: double

text: the

text: price?

text: Such

text: questions

text: cannot

text: be

text: answered

text: from

text: sales

text: data

text: alone,

text: because

text: they

text: involve

text: a

text: change

text: in

text: customers

text: behavior,

text: in

text: reaction

text: to

text: the

text: new

text: pricing.

text: These

text: choices

text: may

text: diﬀer

text: substantially

text: from

text: those

text: taken

text: in

text: previous

text: price-raising

text: situations.

text: (Unless

text: we

text: replicate

text: precisely

text: the

text: market

text: conditions

text: that

text: existed

text: when

text: the

text: price

text: reached

text: double

text: its

text: current

text: value.)

text: Finally,

text: the

text: top

text: level

text: is

text: called

text: Counterfactuals,

text: a

text: term

text: that

text: goes

text: back

text: to

text: the

text: philosophers

text: David

text: Hume

text: and

text: John

text: Stewart

text: Mill,

text: and

text: which

text: has

text: been

text: given

text: computer-friendly

text: semantics

text: in

text: the

text: past

text: two

text: decades.

text: A

text: typical

text: question

text: in

text: the

text: counterfactual

text: category

text: is

text: “What

text: if

text: I

text: had

text: acted

text: diﬀerently,”

text: thus

text: necessitating

text: retrospective

text: reasoning.

text: Counterfactuals

text: are

text: placed

text: at

text: the

text: top

text: of

text: the

text: hierarchy

text: because

text: they

text: subsume

text: interventional

text: and

text: associational

text: questions.

text: If

text: we

text: have

text: a

text: model

text: that

text: can

text: answer

text: counterfactual

text: queries,

text: we

text: can

text: also

text: answer

text: questions

text: about

text: interventions

text: and

text: observations.

text: For

text: example,

text: the

text: interventional

text: question,

text: What

text: will

text: happen

text: if

text: we

text: double

text: the

text: price?

text: can

text: be

text: answered

text: by

text: asking

text: the

text: counterfactual

text: question:

text: What

text: would

text: happen

text: had

text: the

text: price

text: been

text: twice

text: its

text: current

text: value?

text: Likewise,

text: associational

text: questions

text: can

text: be

text: answered

text: once

text: we

text: can

text: answer

text: interventional

text: questions;

text: we

text: simply

text: ignore

text: the

text: action

text: part

text: and

text: let

text: observations

text: take

text: over.

text: The

text: translation

text: does

text: not

text: work

text: in

text: the

text: opposite

text: direction.

text: Interventional

text: questions

text: cannot

text: be

text: answered

text: from

text: purely

text: observational

text: information

text: (i.e.,

text: from

text: statistical

text: data

text: alone).

text: No

text: counterfactual

text: question

text: involving

text: retrospection

text: can

text: be

text: answered

text: from

text: purely

text: interventional

text: information,

text: such

text: as

text: that

text: acquired

text: from

text: controlled

text: experiments;

text: we

text: cannot

text: re-run

text: an

text: experiment

text: on

text: subjects

text: who

text: were

text: treated

text: with

text: a

text: drug

text: and

text: see

text: how

text: they

text: behave

text: had

text: they

text: not

text: given

text: the

text: drug.

text: The

text: hierarchy

text: is

text: therefore

text: directional,

text: with

text: the

text: top

text: level

text: being

text: the

text: most

text: powerful

text: one.

text: names

text: used

text: for

text: inferences

text: at

text: this

text: layer

text: are:

text: “model-free,”

text: “model-blind,”

text: “black-box,”

text: or

text: “data-centric.”

text: Darwiche

text: used

text: “function-ﬁtting,”

text: for

text: it

text: amounts

text: to

text: ﬁtting

text: data

text: by

text: a

text: complex

text: function

text: deﬁned

text: by

text: the

text: neural

text: network

text: architecture.

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Unfortunately, in the massless case, neither nor are isolated points in the spec- e0 e1 trum of the free Hamiltonian which is why the interacting ground-state and the reso- nance ( and ) cannot be constructed using standard results from regular pertur- λ0 λ1 bation theory. Several technologies were developed to overcome this diﬃculty. Two succesful methods that recently received a lot of attention are: The so-called multi- scale method (see e.g. [21, 22, 5, 4]) and the renormalization group method (see e.g. [7, 9, 8, 6, 10, 2, 15, 16, 25, 14, 12]). In both cases, a family of spectrally dilated Hamilto- nians is analyzed since this allows for complex eigenvalues. In this paper, we will employ the multiscale technique. This technique invokes an infrared cut-oﬀ which is then removed using an inductive scheme. In each step of the induction, lower and lower boson momenta are added to the interaction and regular perturbation theory is used to construct the re- spective ground-state and resonance. In order to reach the limit of no infrared cut-oﬀ good control over the closing gap is essential. Note that such a procedure has been introduced for the construction of resonances in the Pauli-Fierz model [4, 5]. In this work we also construct resonances (and ground-state eigenvalues) but this is not our main purpose. Our main purpose is to prove that resonances are analytic with respect to the dilation parameter and coupling constant, and furthermore, to provide certain spectral and resol- vent estimates that allow for the control of the dynamics including the scattering regime. In a forthcoming paper, these estimates are employed to address scattering theory for the model at hand. We want to remark that in [11] the time evolution of this model is studied using the spectral renormalization method. Some results derived therein are similar to some of ours, however, utilizing diﬀerent methods, respectively. What we call resonances and ground-states multiscale analysis is an inductive con- struction of a sequence of Hamiltonians that enjoy infrared cutoﬀs and satisfy certain properties. As the parameter of the sequence tends to inﬁnity these cutoﬀs are removed. Our mutliscale analysis is the content of Theorem 4.5. Its basic scheme is presented in Section and the proof of Theorem 4.5 is carried out in Section . Our proofs of analyticity and resolvent and spectral estimates are not part of our multiscale analysis employed in the contruction of the resonance and ground-state energy, they only use it as mathematical input. The latter results are presented in Section (spectral and resolvent estimates) and Section (analyticity). Theorem 4.5 is only an intermediate but necessary step. As we mention above, the method of multiscale analysis for resonances is introduced in [4, 5]. However, the results in [4, 5] cannot be used directly to prove analyticity because many of the estimations therein consider the dilation parameter, , to be purely imaginary whereas θ analyticity requires estimates that are uniform for in an open set. Thus, although it θ does not involve major obstacles, for the sake of analyticity, many of the given calculations and some of the proofs need to be redone. For the convenience of the reader and in order to keep this work self-contained we provide them in the proof of Theorem 4.5. Note that Theorem 4.5 is applied to the Spin-Boson model while [4, 5] address the Pauli-Fierz model. This gives us the opportunity to review the multiscale technique for a non-trivial but more tractable model. In Section (in particular, in Subsection ) we introduce a new inductive scheme that is used to study resolvent and spectral estimates. This scheme is independent and diﬀerent from the scheme used to construct resonances. It allows to localize the spectrum in two cones with vertices at the location of the resonance and ground-state energy, respectively, and allows for arbitrary small apex angles provided the coupling constant is suﬃciently small. We want to emphasize that such a result requires a more subtle analysis than localizing the spectrum in cuspidal domains. Additionally, we provide estimates for the resolvent operator in the vicinity of the cones. The study of analytic properties of resonances and ground sate eigenvalues in the con- 2

text: Unfortunately,

text: in

text: the

text: massless

text: case,

text: neither

text: nor

text: are

text: isolated

text: points

text: in

text: the

text: spec-

text: trum

text: of

text: the

text: free

text: Hamiltonian

text: which

text: is

text: why

text: the

text: interacting

text: ground-state

text: and

text: the

text: reso-

text: nance

text: and

text: cannot

text: be

text: constructed

text: using

text: standard

text: results

text: from

text: regular

text: pertur-

text: λ0

text: λ1

text: bation

text: theory.

text: Several

text: technologies

text: were

text: developed

text: to

text: overcome

text: this

text: diﬃculty.

text: Two

text: succesful

text: methods

text: that

text: recently

text: received

text: a

text: lot

text: of

text: attention

text: are:

text: The

text: so-called

text: multi-

text: scale

text: method

text: (see

text: e.g.

text: and

text: the

text: renormalization

text: group

text: method

text: (see

text: e.g.

text: In

text: both

text: cases,

text: a

text: family

text: of

text: spectrally

text: dilated

text: Hamilto-

text: nians

text: is

text: analyzed

text: since

text: this

text: allows

text: for

text: complex

text: eigenvalues.

text: In

text: this

text: paper,

text: we

text: will

text: employ

text: the

text: multiscale

text: technique.

text: This

text: technique

text: invokes

text: an

text: infrared

text: cut-oﬀ

text: which

text: is

text: then

text: removed

text: using

text: an

text: inductive

text: scheme.

text: In

text: each

text: step

text: of

text: the

text: induction,

text: lower

text: and

text: lower

text: boson

text: momenta

text: are

text: added

text: to

text: the

text: interaction

text: and

text: regular

text: perturbation

text: theory

text: is

text: used

text: to

text: construct

text: the

text: re-

text: spective

text: ground-state

text: and

text: resonance.

text: In

text: order

text: to

text: reach

text: the

text: limit

text: of

text: no

text: infrared

text: cut-oﬀ

text: good

text: control

text: over

text: the

text: closing

text: gap

text: is

text: essential.

text: Note

text: that

text: such

text: a

text: procedure

text: has

text: been

text: introduced

text: for

text: the

text: construction

text: of

text: resonances

text: in

text: the

text: Pauli-Fierz

text: model

text: In

text: this

text: work

text: we

text: also

text: construct

text: resonances

text: (and

text: ground-state

text: eigenvalues)

text: but

text: this

text: is

text: not

text: our

text: main

text: purpose.

text: Our

text: main

text: purpose

text: is

text: to

text: prove

text: that

text: resonances

text: are

text: analytic

text: with

text: respect

text: to

text: the

text: dilation

text: parameter

text: and

text: coupling

text: constant,

text: and

text: furthermore,

text: to

text: provide

text: certain

text: spectral

text: and

text: resol-

text: vent

text: estimates

text: that

text: allow

text: for

text: the

text: control

text: of

text: the

text: dynamics

text: including

text: the

text: scattering

text: regime.

text: In

text: a

text: forthcoming

text: paper,

text: these

text: estimates

text: are

text: employed

text: to

text: address

text: scattering

text: theory

text: for

text: the

text: model

text: at

text: hand.

text: We

text: want

text: to

text: remark

text: that

text: in

text: the

text: time

text: evolution

text: of

text: this

text: model

text: is

text: studied

text: using

text: the

text: spectral

text: renormalization

text: method.

text: Some

text: results

text: derived

text: therein

text: are

text: similar

text: to

text: some

text: of

text: ours,

text: however,

text: utilizing

text: diﬀerent

text: methods,

text: respectively.

text: What

text: we

text: call

text: resonances

text: and

text: ground-states

text: multiscale

text: analysis

text: is

text: an

text: inductive

text: con-

text: struction

text: of

text: a

text: sequence

text: of

text: Hamiltonians

text: that

text: enjoy

text: infrared

text: cutoﬀs

text: and

text: satisfy

text: certain

text: properties.

text: As

text: the

text: parameter

text: of

text: the

text: sequence

text: tends

text: to

text: inﬁnity

text: these

text: cutoﬀs

text: are

text: removed.

text: Our

text: mutliscale

text: analysis

text: is

text: the

text: content

text: of

text: Theorem

text: Its

text: basic

text: scheme

text: is

text: presented

text: in

text: Section

text: and

text: the

text: proof

text: of

text: Theorem

text: is

text: carried

text: out

text: in

text: Section

text: Our

text: proofs

text: of

text: analyticity

text: and

text: resolvent

text: and

text: spectral

text: estimates

text: are

text: not

text: part

text: of

text: our

text: multiscale

text: analysis

text: employed

text: in

text: the

text: contruction

text: of

text: the

text: resonance

text: and

text: ground-state

text: energy,

text: they

text: only

text: use

text: it

text: as

text: mathematical

text: input.

text: The

text: latter

text: results

text: are

text: presented

text: in

text: Section

text: (spectral

text: and

text: resolvent

text: estimates)

text: and

text: Section

text: (analyticity).

text: Theorem

text: is

text: only

text: an

text: intermediate

text: but

text: necessary

text: step.

text: As

text: we

text: mention

text: above,

text: the

text: method

text: of

text: multiscale

text: analysis

text: for

text: resonances

text: is

text: introduced

text: in

text: However,

text: the

text: results

text: in

text: cannot

text: be

text: used

text: directly

text: to

text: prove

text: analyticity

text: because

text: many

text: of

text: the

text: estimations

text: therein

text: consider

text: the

text: dilation

text: parameter,

text: to

text: be

text: purely

text: imaginary

text: whereas

text: θ

text: analyticity

text: requires

text: estimates

text: that

text: are

text: uniform

text: for

text: in

text: an

text: open

text: set.

text: Thus,

text: although

text: it

text: θ

text: does

text: not

text: involve

text: major

text: obstacles,

text: for

text: the

text: sake

text: of

text: analyticity,

text: many

text: of

text: the

text: given

text: calculations

text: and

text: some

text: of

text: the

text: proofs

text: need

text: to

text: be

text: redone.

text: For

text: the

text: convenience

text: of

text: the

text: reader

text: and

text: in

text: order

text: to

text: keep

text: this

text: work

text: self-contained

text: we

text: provide

text: them

text: in

text: the

text: proof

text: of

text: Theorem

text: Note

text: that

text: Theorem

text: is

text: applied

text: to

text: the

text: Spin-Boson

text: model

text: while

text: address

text: the

text: Pauli-Fierz

text: model.

text: This

text: gives

text: us

text: the

text: opportunity

text: to

text: review

text: the

text: multiscale

text: technique

text: for

text: a

text: non-trivial

text: but

text: more

text: tractable

text: model.

text: In

text: Section

text: (in

text: particular,

text: in

text: Subsection

text: we

text: introduce

text: a

text: new

text: inductive

text: scheme

text: that

text: is

text: used

text: to

text: study

text: resolvent

text: and

text: spectral

text: estimates.

text: This

text: scheme

text: is

text: independent

text: and

text: diﬀerent

text: from

text: the

text: scheme

text: used

text: to

text: construct

text: resonances.

text: It

text: allows

text: to

text: localize

text: the

text: spectrum

text: in

text: two

text: cones

text: with

text: vertices

text: at

text: the

text: location

text: of

text: the

text: resonance

text: and

text: ground-state

text: energy,

text: respectively,

text: and

text: allows

text: for

text: arbitrary

text: small

text: apex

text: angles

text: provided

text: the

text: coupling

text: constant

text: is

text: suﬃciently

text: small.

text: We

text: want

text: to

text: emphasize

text: that

text: such

text: a

text: result

text: requires

text: a

text: more

text: subtle

text: analysis

text: than

text: localizing

text: the

text: spectrum

text: in

text: cuspidal

text: domains.

text: Additionally,

text: we

text: provide

text: estimates

text: for

text: the

text: resolvent

text: operator

text: in

text: the

text: vicinity

text: of

text: the

text: cones.

text: The

text: study

text: of

text: analytic

text: properties

text: of

text: resonances

text: and

text: ground

text: sate

text: eigenvalues

text: in

text: the

text: con-


text: ∈ S | | Lemma 3.10. Let (see (1.24)) and let be small enough (depending on and θ g ν ). Then, there is a constant 0 (independent of , , , , and ) such that ρ1 C(3.37) > θ n g ρ0 ρ ν ∈ { } for every 0 1 i , (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (1) − ≤ | | (3.37) λi ei g C(3.37). (cid:12) D E (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Proof. It follows from Lemma 3.6 that ⊗ Ω (1) ⊗ Ω 1 for | | small enough ϕi , Pi ϕi > 2 g (depending on and ). We calculate D ν ρ1 E D E D E D E ϕ ⊗ Ω(1), H(1),θP (1)ϕ ⊗ Ω(1) V (1),θϕ ⊗ Ω(1), P (1)ϕ ⊗ Ω(1) (1) = i i i = + i i i (3.38) λi ⊗ Ω(1) (1) ⊗ Ω(1) ei g ⊗ Ω(1) (1) ⊗ Ω(1) . ϕ , P ϕ ϕ , P ϕ i i i i i i Let now ∈ C such that | − | = 1 sin( ). Eq. (3.26) (which requires | | to be z ei z 4ρ1 ν g small enough -depending on and ) then allows to obtain ν ρ1 (cid:13) (cid:13) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13) (cid:13) 1 (cid:13) λ(1) − e ≤ 4 gV (1),θϕ ⊗ Ω(1) ≤ 4 |e − z| gV (1),θ ≤ |g|c, (3.39) i i i i (1) − ,θ H0 z for some constant (independent of , , , , and ) . Here, we have used (3.17) c θ n g ρ0 ρ ν from Lemma 3.3 in the last step. Notice that in this work we assume that the imaginary part of , , is positive. Then, strictly speaking, we do not have the right to use our results θ ν for . However, the restriction we impose by assuming that is not negative is irrelevant. θ ν This is assumed only for convenience in order to simplify our notation. Of course, the − − same results hold true if we take 16 . π/ < ν < ν ∈ S | | Remark 3.11. Let (see (1.24)) and let be small enough (depending on and θ g ν ). Then ρ1 Z 1 1 (1) = − d (3.40) Pi 2πi (1) z H(1),θ − z , γ i becawuhsee,refoγri(1s)m:al[0l ,e2nπo]ug→h |C|,, t(1)7→ˆ(λ1)(in⊂) + (141ρ)1\s{in((ν1))}e,its.eeTLheismfmolalo3w.s10f.rom Remark 3.7, g γ , γ B λ i i i i ∈ S | | Lemma 3.12. Let (see (1.24)) and let be small enough (depending on and θ g ν ∈ { } ). Take 0 1 . Then, there is a constant 0 (independent of , , , , ρ1 i , C(3.41) > θ n g ρ0 ρ and ) such that ν (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) 1 (cid:12) 1 (cid:12) (1) ≤ C(3.41) ∀ ∈ (1) (3.41) H(1),θ − z Pi sin(ν) ρ1 + λ(1) − z , z Bi . i 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: Lemma

text: Let

text: (see

text: and

text: let

text: be

text: small

text: enough

text: (depending

text: on

text: and

text: θ

text: g

text: ν

text: Then,

text: there

text: is

text: a

text: constant

text: (independent

text: of

text: and

text: such

text: that

text: ρ1

text: θ

text: n

text: g

text: ρ0

text: ρ

text: ν

text: for

text: every

text: i

label: equation

text: λi

text: ei

text: g

text: D

text: E

text: Proof.

text: It

text: follows

text: from

text: Lemma

text: that

text: Ω

text: Ω

text: for

text: small

text: enough

text: ϕi

text: Pi

text: ϕi

text: g

text: (depending

text: on

text: and

text: We

text: calculate

text: D

text: ν

text: ρ1

text: E

text: D

text: E

text: D

text: E

text: D

text: E

text: ϕ

text: V

text: P

text: i

text: i

text: i

text: i

text: i

text: i

text: λi

text: ei

text: g

text: ϕ

text: P

text: ϕ

text: ϕ

text: P

text: ϕ

text: i

text: i

text: i

text: i

text: i

text: i

text: Let

text: now

text: C

text: such

text: that

text: sin(

text: Eq.

text: (which

text: requires

text: to

text: be

text: z

text: ei

text: z

text: ν

text: g

text: small

text: enough

text: -depending

text: on

text: and

text: then

text: allows

text: to

text: obtain

text: ν

text: ρ1

text: e

text: gV

text: |e

text: z|

text: gV

text: |g|c,

text: i

text: i

text: i

text: i

text: ,θ

text: z

text: for

text: some

text: constant

text: (independent

text: of

text: and

text: Here,

text: we

text: have

text: used

text: c

text: θ

text: n

text: g

text: ρ0

text: ρ

text: ν

text: from

text: Lemma

text: in

text: the

text: last

text: step.

text: Notice

text: that

text: in

text: this

text: work

text: we

text: assume

text: that

text: the

text: imaginary

text: part

text: of

text: is

text: positive.

text: Then,

text: strictly

text: speaking,

text: we

text: do

text: not

text: have

text: the

text: right

text: to

text: use

text: our

text: results

text: θ

text: ν

text: for

text: However,

text: the

text: restriction

text: we

text: impose

text: by

text: assuming

text: that

text: is

text: not

text: negative

text: is

text: irrelevant.

text: θ

text: ν

text: This

text: is

text: assumed

text: only

text: for

text: convenience

text: in

text: order

text: to

text: simplify

text: our

text: notation.

text: Of

text: course,

text: the

text: same

text: results

text: hold

text: true

text: if

text: we

text: take

text: π/

text: ν

text: ν

text: S

text: Remark

text: Let

text: (see

text: and

text: let

text: be

text: small

text: enough

text: (depending

text: on

text: and

text: θ

text: g

text: ν

text: Then

text: ρ1

text: Z

text: d

text: Pi

text: z

text: z

text: γ

text: i

text: |C|,,

text: Remark

text: g

text: γ

text: γ

text: B

text: λ

text: i

text: i

text: i

text: i

text: S

text: Lemma

text: Let

text: (see

text: and

text: let

text: be

text: small

text: enough

text: (depending

text: on

text: and

text: θ

text: g

text: ν

text: Take

text: Then,

text: there

text: is

text: a

text: constant

text: (independent

text: of

text: ρ1

text: i

text: θ

text: n

text: g

text: ρ0

text: ρ

text: and

text: such

text: that

text: ν

text: z

text: Pi

text: sin(ν)

text: ρ1

text: z

text: z

text: Bi

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: H(n,n+1),θ := e−θH(n,n+1),0. (4.13) f f ∈ For every function h h(n,n+1), we denote the creation and annihilation operators, an,n+1(h), a∈∗n,n+1(h), on F [h(n,n+1)] according to Eq. (1.10). We use the same notation for functions h h but then understand the argument as h restricted to h(n,n+1). K ⊗ F Furthermore, we ﬁx the following operator (deﬁned on [h(n,n+1)], and hence, on H (n+1) - see Remark 1.2) (cid:16) (cid:17) V (n,n+1),θ := σ1 ⊗ a +1(fθ) + a +1(fθ)∗ . (4.14) n,n n,n In this notation we obtain (see Remark 1.2): H(n+1),θ = H(n),θ + H(n,n+1),θ + gV (n,n+1),θ. (4.15) f Lemma 4.7. Suppose that (P1)-(P4) hold true for all ∈ N such that ≤ . Then, m m n (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) | | 2 (n) ≤ 2 + g ≤ 3 = 0 1 (4.16) P , i , i ρ | | ≤ (notice that 2, see Deﬁnition 4.3- recall Remark 4.4) and g ρ/ | (n) − (1)| ≤ 2| | (4.17) λ λ g . i i P Proof. Eq. (4.17) is a consequence of Property ( 1). We estimate (cid:13) (cid:13) (cid:13) (cid:13) X (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (n) ≤ (1) + n (j) − (j−1) ⊗ Pi Pi X (cid:18) P(cid:19)i Pi PΩ(j−1,j) =2 j ≤ 2 + |g| n−1 1 j ≤ 2 + 2|g| (4.18) 2 , ρ ρ =0 j P ≤ where apply the induction hypothesis ( 3) for and use Deﬁnition 3.14 and j n Deﬁnition 4.3. Deﬁnition 4.8. Let ∈ N and ∈ {0 1}. We deﬁne the region n i , (cid:26) (cid:18) (cid:19)(cid:27) 2 (n) := (n) \ ∈ C : Im( ) ∈ −∞ Im( (n)) − sin( ) (4.19) Mi Bi z z , λi 5ρn+1 ν . 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: f

text: f

text: For

text: every

text: function

text: h

text: we

text: denote

text: the

text: creation

text: and

text: annihilation

text: operators,

text: on

text: F

text: according

text: to

text: Eq.

text: We

text: use

text: the

text: same

text: notation

text: for

text: functions

text: h

text: h

text: but

text: then

text: understand

text: the

text: argument

text: as

text: h

text: restricted

text: to

text: K

text: F

text: Furthermore,

text: we

text: ﬁx

text: the

text: following

text: operator

text: (deﬁned

text: on

text: and

text: hence,

text: on

text: H

text: see

text: Remark

text: V

text: σ1

text: a

text: a

text: n,n

text: n,n

text: In

text: this

text: notation

text: we

text: obtain

text: (see

text: Remark

text: H(n),θ

text: gV

text: f

text: Lemma

text: Suppose

text: that

text: hold

text: true

text: for

text: all

text: N

text: such

text: that

text: Then,

text: m

text: m

text: n

text: (n)

text: g

text: P

text: i

text: i

text: ρ

text: (notice

text: that

text: see

text: Deﬁnition

text: recall

text: Remark

text: and

text: g

text: ρ/

text: (n)

text: λ

text: λ

text: g

text: i

text: i

text: P

text: Proof.

text: Eq.

text: is

text: a

text: consequence

text: of

text: Property

text: We

text: estimate

text: X

text: (n)

text: n

text: (j)

text: Pi

text: Pi

text: X

text: Pi

text: j

text: |g|

text: j

text: ρ

text: ρ

text: j

text: P

text: where

text: apply

text: the

text: induction

text: hypothesis

text: for

text: and

text: use

text: Deﬁnition

text: and

text: j

text: n

text: Deﬁnition

text: Deﬁnition

text: Let

text: N

text: and

text: We

text: deﬁne

text: the

text: region

text: n

text: i

text: (n)

text: (n)

text: C

text: Im(

text: Im(

text: (n))

text: sin(

text: Mi

text: Bi

text: z

text: z

text: λi

text: ν

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof. We calculate: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Hf(n,n+1) + r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (cid:12) Hf(n,n+1),θ − (z − λ(in)) (cid:12)(cid:12)(cid:12)(cid:12) y∈{0}∪[ρn+1,∞) (cid:12)(cid:12)(cid:12)(cid:12)e−θy + λ(in) − z (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) ≤ |eθ| + |eθ| y∈{0}∪su[ρpn+1,∞) (cid:12)ee−−θθyr −+ λλ(i(inn)) +− zz (cid:12) ≤ |eθ| + (1 + |eθ|) y∈{0}∪su[ρpn+1,∞) (cid:12)(cid:12)e−θyz +−λλ(i(inn)) − z(cid:12)(cid:12) | | ≤ |eθ| + 4(1sin+( e)θ ) ≤ sin1(0 ), (4.40) ν ν where we have used (4.34) in the second last step. Lemma 4.11. Suppose that (P1)-(P4) hold true for all ∈ N such that ≤ . Then, (cid:13) (cid:13) m m n (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 1 (cid:13) 2500 V (n,n+1),θ H(n),θ + H(cid:12)(cid:12)(n,n+1),θ (cid:12)(cid:12)− z ≤ ρ sin(ν)2 Cn+1ρµn (4.41) (cid:12)f (cid:12) for all z ∈ M(cid:12)(cid:12)(cid:12) i(n) \ {λ(cid:12)(cid:12)(cid:12)(in)} such that z − λ(in) ≥ 110ρn+1 sin(ν) and for all i ∈ {0, 1}. Proof. Set = − (n) ≥ 1 sin( ). We observe that (cid:13)(cid:13) r z λi 10ρn+1 (cid:13)(cid:13) ν (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (cid:13)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)V (n,n+1),θ H(n),θ + H1(n,n+1(cid:13)(cid:13)(cid:13)),(cid:16)θ − z (cid:13) ≤ (cid:13)V (n,n+1),θ H(n(cid:17),n+11) + r(cid:13) (cid:13)H(n,nH+1f()n,,θn−+1(cid:13)(cid:13)(cid:13))(z+−rλ(n))(cid:13) f (cid:13) f f (cid:13) i (cid:13) (cid:13) (cid:13) 1 (cid:13) · (n,n+1),θ − ( − (n)) (4.42) Hf z λi H(n),θ + H(n,n+1),θ − z . f Lemma 4.10 yields (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) H(n,n+1) + r (cid:13) ≤ 10 f (4.43) H(n,n+1),θ − (z − λ(n)) sin(ν), f i and furthermore, we obtain from functional calculus that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16)H(cid:13)(cid:13)(cid:13) f(n,n+(cid:13)(cid:13)(cid:13)1),θ − (z − λ(in))(cid:17)(cid:13)(cid:13)(cid:13)(cid:13)H(n),θ + Hf1(n,n+1),θ − z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)≤ y∈{0}∪su[ρpn+1,∞) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)He(−nθ)y,θ ++λe(i−nθ)y−−zz (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ Pi(n) + ∈{0}∪su[ p ∞) (cid:13)Hλ((inn)),θ−−(z(z−−ee−−θyθy))Pi(n)(cid:13) ≤ 3 + Cn+1 ≤ 4Cn+1. (4.44) y ρn+1, In the last step, we use Lemma 4.7 for the ﬁrst term. For the second term, we utilize P Lemma 3.13, Deﬁnition 3.14 and induction hypothesis ( 4), together with Lemma 4.7 ≥ and the Deﬁnition of in Deﬁnition 4.1 - see Remark 4.4 (notice that C C sin(4νc/2) ≥ s4iCn((3ν./432)) and kPi(n)k ≤ 4). 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: We

text: calculate:

text: r

text: sup

text: y

text: r

text: (z

text: λ(in))

text: λ(in)

text: z

text: |eθ|

text: |eθ|

text: λλ(i(inn))

text: zz

text: |eθ|

text: |eθ|)

text: +−λλ(i(inn))

text: |eθ|

text: e)θ

text: ν

text: ν

text: where

text: we

text: have

text: used

text: in

text: the

text: second

text: last

text: step.

text: Lemma

text: Suppose

text: that

text: hold

text: true

text: for

text: all

text: N

text: such

text: that

text: Then,

text: m

text: m

text: n

text: V

text: H(n),θ

text: z

text: ρ

text: for

text: all

text: z

text: i(n)

text: such

text: that

text: z

text: λ(in)

text: sin(ν)

text: and

text: for

text: all

text: i

text: Proof.

text: Set

text: (n)

text: sin(

text: We

text: observe

text: that

text: r

text: z

text: λi

text: ν

text: H(n),θ

text: z

text: f

text: f

text: f

text: i

text: (n))

text: Hf

text: z

text: λi

text: H(n),θ

text: z

text: f

text: Lemma

text: yields

text: r

text: f

text: (z

text: λ(n))

text: sin(ν),

text: f

text: i

text: and

text: furthermore,

text: we

text: obtain

text: from

text: functional

text: calculus

text: that

text: (z

text: z

text: ++λe(i−nθ)y−−zz

text: Pi(n)

text: p

text: y

text: In

text: the

text: last

text: step,

text: we

text: use

text: Lemma

text: for

text: the

text: ﬁrst

text: term.

text: For

text: the

text: second

text: term,

text: we

text: utilize

text: P

text: Lemma

text: Deﬁnition

text: and

text: induction

text: hypothesis

text: together

text: with

text: Lemma

text: and

text: the

text: Deﬁnition

text: of

text: in

text: Deﬁnition

text: see

text: Remark

text: (notice

text: that

text: C

text: C

text: and

text: kPi(n)k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

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text: Proof. In this proof we explicitly emphasize the dependence of (n) on and write (n) ≡ P θ P P (n),θ. We deﬁne Ψ(n),θ := P (n),θϕ ⊗ Ω(n+1), see Remark 1.2. Piroposition 4.14, Proiperty Pi i i i 3, Deﬁnition 3.14 (see Remark 3.11) and the restrictions for in Deﬁnition 4.3 g imply that kΨ(i(cid:13)n),θ − ϕ(cid:13)i ⊗ Ω(n+1)k ≤ 1102 .(cid:12) (cid:28)This guarantees th(cid:29)at(cid:12) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) 1 Ψ(n),θ ≤ 2 and Ψ(n),θ (n+1),θΨ(n),θ ≥ (4.68) i i , Pi i 2. Notice that in this work we assume that the imaginary part of , , is positive. Then, θ ν strictly speaking, we do not have the right to use the symbol Ψ(n),θ := P (n),θϕ ⊗ Ω(n+1). i i i However, the restriction we impose by assuming that is not negative is irrelevant. This ν is assumed only for convenience in order to simplify our notation. Of course, the same (cid:0) (cid:1) − − results hold true if we take 16 (we use this fact in the present proof as well π/ < ν < ν as (n),θ = (n),θ ∗). Then, we obtain Pi Pi (cid:28) (cid:29) (cid:28) (cid:29) (cid:28) (cid:29) (cid:28) (cid:29) Ψ(n),θ, H(n+1),θP (n+1),θΨ(n),θ H(n+1),θΨ(n),θ, P (n+1),θΨ(n),θ (n+1) = i i i = i i i λ i Ψ((cid:28)n),θ (n+1),θΨ(n),θ (cid:29)Ψ(n),θ (n+1),θΨ(n),θ , P , P i i i i i i (cid:28) (cid:29) V (n,n+1),θΨ(n),θ, P (n+1),θΨ(n),θ = (n) + i i i (4.69) λ g . i Ψ(n),θ (n+1),θΨ(n),θ , P i i i Now(cid:12)(cid:12)(cid:12) we choose z (cid:12)(cid:12)(cid:12)∈ C s(cid:13)(cid:13)(cid:13)(cid:13)uch that |z − λ(in)| = ρn+11s0in(ν(cid:13)(cid:13)(cid:13)(cid:13))(cid:13)(cid:13)(cid:13). We get that (cid:13)(cid:13)(cid:13) λ(in+1(cid:12)(cid:12)(cid:12)) − λ(in) (cid:12)(cid:12)(cid:12)≤(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) 2 gV (n,n+1),θPi(n),θϕi ⊗ Ω(n+1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Pi(n+1),θΨ(in),θ (cid:13) 1 (cid:13) ≤ 54 z − λ(n) gV (n,n+1),θ (cid:18) (cid:19) i H(n),θ + H(n,n+1),θ − z f ≤ g54ρn+11s0in(ν) ρ s2in50(ν0)2 Cn+1ρµn ≤ |g|C(n+1)+1ρ1n+µ ≤ |g| 12 n ρn, (4.70) where we use Lemmas 4.11 and 4.7 and the deﬁnition of in Deﬁnition 4.1 (it C implies that 54 2500 ≤ , see also Remark 4.4). C sin( )2 ν P Proof of Property ( 4): Lemma 4.16. Suppose that (P1)-(P4) hold true for all ∈ N with ≤ . Then, for m m n ∈ { } 0 1 : i , (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) 1 (n+1) ≤ 3168 + 8Cn+1 (cid:12)1 (cid:12) ∀ ∈ (m) (4.71) H(n+1),θ − z Pi sin(ν) ρ +1 + z − λ(n) , z Mi , n i and hence, for all ∈ (n+1) (recall (n+1) ⊂ (n) from the proof of Proposition z B B M i i i 4.14). 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: In

text: this

text: proof

text: we

text: explicitly

text: emphasize

text: the

text: dependence

text: of

text: (n)

text: on

text: and

text: write

text: (n)

text: P

text: θ

text: P

text: P

text: (n),θ.

text: We

text: deﬁne

text: Ψ(n),θ

text: P

text: (n),θϕ

text: see

text: Remark

text: Piroposition

text: Proiperty

text: Pi

text: i

text: i

text: i

text: Deﬁnition

text: (see

text: Remark

text: and

text: the

text: restrictions

text: for

text: in

text: Deﬁnition

text: g

text: imply

text: that

text: guarantees

text: Ψ(n),θ

text: and

text: Ψ(n),θ

text: i

text: i

text: Pi

text: i

text: Notice

text: that

text: in

text: this

text: work

text: we

text: assume

text: that

text: the

text: imaginary

text: part

text: of

text: is

text: positive.

text: Then,

text: θ

text: ν

text: strictly

text: speaking,

text: we

text: do

text: not

text: have

text: the

text: right

text: to

text: use

text: the

text: symbol

text: Ψ(n),θ

text: P

text: (n),θϕ

text: i

text: i

text: i

text: However,

text: the

text: restriction

text: we

text: impose

text: by

text: assuming

text: that

text: is

text: not

text: negative

text: is

text: irrelevant.

text: This

text: ν

text: is

text: assumed

text: only

text: for

text: convenience

text: in

text: order

text: to

text: simplify

text: our

text: notation.

text: Of

text: course,

text: the

text: same

text: results

text: hold

text: true

text: if

text: we

text: take

text: (we

text: use

text: this

text: fact

text: in

text: the

text: present

text: proof

text: as

text: well

text: π/

text: ν

text: ν

text: as

text: (n),θ

text: (n),θ

text: Then,

text: we

text: obtain

text: Pi

text: Pi

text: Ψ(n),θ,

text: P

text: i

text: i

text: i

text: i

text: i

text: i

text: λ

text: i

text: P

text: P

text: i

text: i

text: i

text: i

text: i

text: i

text: V

text: P

text: (n)

text: i

text: i

text: i

text: λ

text: g

text: i

text: Ψ(n),θ

text: P

text: i

text: i

text: i

text: we

text: choose

text: z

text: C

text: that

text: |z

text: λ(in)|

text: We

text: get

text: that

text: λ(in)

text: gV

text: z

text: λ(n)

text: gV

text: i

text: H(n),θ

text: z

text: f

text: ρ

text: |g|

text: n

text: ρn,

text: where

text: we

text: use

text: Lemmas

text: and

text: and

text: the

text: deﬁnition

text: of

text: in

text: Deﬁnition

text: (it

text: C

text: implies

text: that

text: see

text: also

text: Remark

text: C

text: sin(

text: ν

text: P

text: Proof

text: of

text: Property

text: Lemma

text: Suppose

text: that

text: hold

text: true

text: for

text: all

text: N

text: with

text: Then,

text: for

text: m

text: m

text: n

text: i

text: (m)

text: z

text: Pi

text: sin(ν)

text: ρ

text: z

text: λ(n)

text: z

text: Mi

text: n

text: i

text: and

text: hence,

text: for

text: all

text: (recall

text: (n)

text: from

text: the

text: proof

text: of

text: Proposition

text: z

text: B

text: B

text: M

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 Proof of Proposition 2.1 We deﬁne the sequence of vectors (see Remark 1.2) Ψ(n) := (n) ⊗ Ω ∈ N (6.1) P ϕ , n . λi i i P Due to the Property ( 3) in Theorem 4.5, we know that the sequence above converges ⊗ 6 to the non-zero limit Ψ := Ω = 0 (see the discussion above Eq. (4.68)). Note that P ϕ (see Remark 1.2) λi i i Hθ = H(n),θ + H(n,∞),θ + gV (n,∞),θ = H˜ (n),θ + gV (n,∞),θ (6.2) f and set z = λ(n) − 10ρ e−iν. Then, i n 1 HθΨ(n) = λ(n)Ψ(n) + g(λ(n) − z)V (n,∞),θ ˜ − Ψ(n). (6.3) λi i λi i H(n),θ z λi concLleumdemtaha5t.8 implies that V (n,∞),θ H˜(n)1,θ−z tends to zero as n tends to inﬁnity. We nl→im∞ HθΨ(λni) = λiΨλi. (6.4) As Hθ is a closed operator, Ψ belongs to its domain and is an eigenvector of Hθ corresponding to the eigenvalue .λiFurthermore, as is rank-one, Ψ spans its range. λ P 7 Analyticity i i λi In this section we assume that Deﬁnitions 4.1, 4.2, 4.3 hold true. We recall that we use the symbol to represent any generic (indeterminate) constant that does not depend c on , , , and dilation parameters (here do not only use , but also and ). In this n g ρ ρ0 θ η λ section we prove Theorems 2.3 and 2.5. Lemma 7.1. For every 0 and ∈ (0 16) (and every ∈ N): r > λ, η D , π/ n (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) H0(n),λ + r ≤10 (7.1) . (n),η + H0 r Moreover, for large enough (independent of , , , and and ) and every in r n g ρ ρ0 η λ z the resolvent set of H(n),η: (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 1 1 (n),λ ≤ 20 + 20(| | + 2) (7.2) H0 ( ) − z r/ ( ) − . H n ,η z H n ,η z 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof

text: of

text: Proposition

text: We

text: deﬁne

text: the

text: sequence

text: of

text: vectors

text: (see

text: Remark

text: Ψ(n)

text: (n)

text: Ω

text: N

text: P

text: ϕ

text: n

text: λi

text: i

text: i

text: P

text: Due

text: to

text: the

text: Property

text: in

text: Theorem

text: we

text: know

text: that

text: the

text: sequence

text: above

text: converges

font: default

text: to

text: the

text: non-zero

text: limit

text: Ψ

text: Ω

text: (see

text: the

text: discussion

text: above

text: Eq.

text: Note

text: that

text: P

text: ϕ

text: (see

text: Remark

text: λi

text: i

text: i

text: Hθ

text: H(n),θ

text: H(n,∞),θ

text: gV

text: (n,∞),θ

text: H˜

text: (n),θ

text: gV

text: (n,∞),θ

text: f

text: and

text: set

text: z

text: λ(n)

text: e−iν.

text: Then,

text: i

text: n

text: HθΨ(n)

text: λ(n)Ψ(n)

text: g(λ(n)

text: z)V

text: (n,∞),θ

text: Ψ(n).

text: λi

text: i

text: λi

text: i

text: H(n),θ

text: z

text: λi

text: implies

text: that

text: V

text: (n,∞),θ

text: tends

text: to

text: zero

text: as

text: n

text: tends

text: to

text: inﬁnity.

text: We

text: nl→im∞

text: HθΨ(λni)

text: λiΨλi.

text: As

text: Hθ

text: is

text: a

text: closed

text: operator,

text: Ψ

text: belongs

text: to

text: its

text: domain

text: and

text: is

text: an

text: eigenvector

text: of

text: Hθ

text: corresponding

text: to

text: the

text: eigenvalue

text: .λiFurthermore,

text: as

text: is

text: rank-one,

text: Ψ

text: spans

text: its

text: range.

text: λ

text: P

text: Analyticity

text: i

text: i

text: λi

text: In

text: this

text: section

text: we

text: assume

text: that

text: Deﬁnitions

text: hold

text: true.

text: We

text: recall

text: that

text: we

text: use

text: the

text: symbol

text: to

text: represent

text: any

text: generic

text: (indeterminate)

text: constant

text: that

text: does

text: not

text: depend

text: c

text: on

text: and

text: dilation

text: parameters

text: (here

text: do

text: not

text: only

text: use

text: but

text: also

text: and

text: In

text: this

text: n

text: g

text: ρ

text: ρ0

text: θ

text: η

text: λ

text: section

text: we

text: prove

text: Theorems

text: and

text: Lemma

text: For

text: every

text: and

text: (and

text: every

text: N):

text: r

text: λ,

text: η

text: D

text: π/

text: n

text: r

text: (n),η

text: r

text: Moreover,

text: for

text: large

text: enough

text: (independent

text: of

text: and

text: and

text: and

text: every

text: in

text: r

text: n

text: g

text: ρ

text: ρ0

text: η

text: λ

text: z

text: the

text: resolvent

text: set

text: of

text: H(n),η:

text: (n),λ

text: z

text: r/

text: H

text: n

text: ,η

text: z

text: H

text: n

text: ,η

text: z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: IEEE COMMUNICATIONS MAGAZINE (ACCEPTED FROM OPEN CALL) 2 long-term evolution (3GPP-LTE) and layered division multiplexing (LDM) in digital TV standard ATSC 3.0 [7]. It has also been known that more efﬁcient use of wireless spectrum can be achieved by cognitive radio (CR), where secondary users (SUs) intelligently adapt their operating parameters to access a spectrum band occupied by primary users (PUs), in an opportunistic or collaborative manner [8], [9]. To date, exploiting CR technology to support emerging applications has received considerable attention, where novel architectures for CR networks based on full-duplex, device- to-device, and multiple-input multiple-output (MIMO) have been studied to further increase spectrum efﬁciency. More particularly, existing research on the combination of NOMA and CR (i.e., [10]–[14]) has shown the possibility to meet 5G requirements of high throughput, massive connectivity, as well as low latency. Despite these potential beneﬁts, building efﬁcient cognitive NOMA is a challenging issue in practice. This is because both NOMA and CR are interference-limited, and thus, coexistence of inter-network interference between the primary and secondary networks and intra-network interference (also called co-channel interference) caused by power domain multiplexing of NOMA undoubtedly results in severe performance degradation of reception reliability. Therefore, it is necessary to combine NOMA with CR in an appropriate manner for minimizing the interference and better utilizing the underlying spectrum resource. In this article, we integrate NOMA capabilities into CR concepts to achieve more intelligent spectrum sharing. We begin our attempt with a concise introduction of NOMA and CR paradigms, along with motivations of cognitive NOMA as well as cooperative strategies to further improve performance. Then we provide an in-depth overview of three cognitive NOMA architectures, namely underlay NOMA networks, overlay NOMA networks, and CR-inspired NOMA networks (termed CR-NOMA networks), and discuss their key operating principles. Note that both NOMA and CR are interference-limited, which might undermine reception reliability. To address this challenging issue, for each cognitive NOMA architecture, we propose a cooperative relaying strategy for reliability enhancement. Since studies of cognitive NOMA networks are still in a nascent stage, several potential challenges in this research ﬁeld are also outlined, many of which are promising avenues for future works. Finally, we make concluding remarks to this article. II. RATIONALES OF COGNITIVE NOMA NETWORKS This section ﬁrst introduces basics of NOMA and CR. Then the motivations of cognitive NOMA networks and cooperative strategies are discussed.

text: IEEE

text: COMMUNICATIONS

text: MAGAZINE

text: (ACCEPTED

text: FROM

text: OPEN

text: CALL)

text: long-term

text: evolution

text: and

text: layered

text: division

text: multiplexing

text: (LDM)

text: in

text: digital

text: TV

text: standard

text: ATSC

text: It

text: has

text: also

text: been

text: known

text: that

text: more

text: efﬁcient

text: use

text: of

text: wireless

text: spectrum

text: can

text: be

text: achieved

text: by

text: cognitive

text: radio

text: (CR),

text: where

text: secondary

text: users

text: (SUs)

text: intelligently

text: adapt

text: their

text: operating

text: parameters

text: to

text: access

text: a

text: spectrum

text: band

text: occupied

text: by

text: primary

text: users

text: (PUs),

text: in

text: an

text: opportunistic

text: or

text: collaborative

text: manner

text: To

text: date,

text: exploiting

text: CR

text: technology

text: to

text: support

text: emerging

text: applications

text: has

text: received

text: considerable

text: attention,

text: where

text: novel

text: architectures

text: for

text: CR

text: networks

text: based

text: on

text: full-duplex,

text: device-

text: to-device,

text: and

text: multiple-input

text: multiple-output

text: (MIMO)

text: have

text: been

text: studied

text: to

text: further

text: increase

text: spectrum

text: efﬁciency.

text: More

text: particularly,

text: existing

text: research

text: on

text: the

text: combination

text: of

text: NOMA

text: and

text: CR

text: (i.e.,

text: has

text: shown

text: the

text: possibility

text: to

text: meet

text: requirements

text: of

text: high

text: throughput,

text: massive

text: connectivity,

text: as

text: well

text: as

text: low

text: latency.

text: Despite

text: these

text: potential

text: beneﬁts,

text: building

text: efﬁcient

text: cognitive

text: NOMA

text: is

text: a

text: challenging

text: issue

text: in

text: practice.

text: This

text: is

text: because

text: both

text: NOMA

text: and

text: CR

text: are

text: interference-limited,

text: and

text: thus,

text: coexistence

text: of

text: inter-network

text: interference

text: between

text: the

text: primary

text: and

text: secondary

text: networks

text: and

text: intra-network

text: interference

text: (also

text: called

text: co-channel

text: interference)

text: caused

text: by

text: power

text: domain

text: multiplexing

text: of

text: NOMA

text: undoubtedly

text: results

text: in

text: severe

text: performance

text: degradation

text: of

text: reception

text: reliability.

text: Therefore,

text: it

text: is

text: necessary

text: to

text: combine

text: NOMA

text: with

text: CR

text: in

text: an

text: appropriate

text: manner

text: for

text: minimizing

text: the

text: interference

text: and

text: better

text: utilizing

text: the

text: underlying

text: spectrum

text: resource.

text: In

text: this

text: article,

text: we

text: integrate

text: NOMA

text: capabilities

text: into

text: CR

text: concepts

text: to

text: achieve

text: more

text: intelligent

text: spectrum

text: sharing.

text: We

text: begin

text: our

text: attempt

text: with

text: a

text: concise

text: introduction

text: of

text: NOMA

text: and

text: CR

text: paradigms,

text: along

text: with

text: motivations

text: of

text: cognitive

text: NOMA

text: as

text: well

text: as

text: cooperative

text: strategies

text: to

text: further

text: improve

text: performance.

text: Then

text: we

text: provide

text: an

text: in-depth

text: overview

text: of

text: three

text: cognitive

text: NOMA

text: architectures,

text: namely

text: underlay

text: NOMA

text: networks,

text: overlay

text: NOMA

text: networks,

text: and

text: CR-inspired

text: NOMA

text: networks

text: (termed

text: CR-NOMA

text: networks),

text: and

text: discuss

text: their

text: key

text: operating

text: principles.

text: Note

text: that

text: both

text: NOMA

text: and

text: CR

text: are

text: interference-limited,

text: which

text: might

text: undermine

text: reception

text: reliability.

text: To

text: address

text: this

text: challenging

text: issue,

text: for

text: each

text: cognitive

text: NOMA

text: architecture,

text: we

text: propose

text: a

text: cooperative

text: relaying

text: strategy

text: for

text: reliability

text: enhancement.

text: Since

text: studies

text: of

text: cognitive

text: NOMA

text: networks

text: are

text: still

text: in

text: a

text: nascent

text: stage,

text: several

text: potential

text: challenges

text: in

text: this

text: research

text: ﬁeld

text: are

text: also

text: outlined,

text: many

text: of

text: which

text: are

text: promising

text: avenues

text: for

text: future

text: works.

text: Finally,

text: we

text: make

text: concluding

text: remarks

text: to

text: this

text: article.

text: II.

text: RATIONALES

text: OF

text: COGNITIVE

text: NOMA

text: NETWORKS

text: This

text: section

text: ﬁrst

text: introduces

text: basics

text: of

text: NOMA

text: and

text: CR.

text: Then

text: the

text: motivations

text: of

text: cognitive

text: NOMA

text: networks

text: and

text: cooperative

text: strategies

text: are

text: discussed.