Generalized definition of time delay in scattering theory
Christian G´erard and Rafael Tiedra de Aldecoa
D´epartement de math´ematiques, Universit´e de Paris XI, 91 405 Orsay Cedex France
E-mails: christian.gerard@math.u-psud.fr and rafael.tiedra@math.u-psud.fr
Abstract
We advocate for the systematic use of a symmetrized definition of time delay in scattering theory. In two-body scattering processes, we show that the symmetrized time delay exists for arbitrary dilated spatial regions symmetric with respect to the origin. It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions. We also prove that the symmetrized time delay is invariant under an appropriate mapping of time reversal. These results are also discussed in the context of classical scattering theory.
1 Introduction
This paper is devoted to the definition of time delay (in terms of sojourn times) in two- body scattering theory. Its purpose is to advocate for the use of a symmetrized definition of time delay. Our main arguments supporting this point of view are the following:
(A) In two-body scattering processes, symmetrized time delay does exist for arbitrary dilated spatial regions symmetric with respect to the origin (usual time delay does exist only for spherical spatial regions [12]). It is equal to the usual time delay plus a new contribution, which vanishes in the case of spherical spatial regions.
(B) Symmetrized time delay is invariant under an appropriate mapping of time reversal.
Usual time delay is not.
(C) Symmetrized time delay generalizes to multichannel-type scattering processes. Usual time delay does not.
Let us recall the usual definition of time delay for a two-body scattering process in Rd,d ≥1. Consider a bounded open setΣinRd containing the origin and the dilated spatial regionsΣr := {rx | x ∈ Σ},r > 0. LetH0 := −12∆ be the kinetic energy operator inH:=L2(Rd)and letHbe a selfadjoint perturbation ofH0such that the wave
operatorsW± := s-limt→±∞eitHe−itH0 exist and are complete (so that the scattering operatorS:= (W+)∗W−is unitary). Then one defines for some statesϕ∈ Handr >0 two sojourn times, namely:
Tr0(ϕ) :=
Z ∞
−∞
dt Z
x∈Σr
ddx(e−itH0ϕ)(x)2 and
Tr(ϕ) :=
Z ∞
−∞
dt Z
x∈Σr
ddx(e−itHW−ϕ)(x)2.
If the stateϕis normalized the first number is interpreted as the time spent by the freely evolving statee−itH0ϕinside the setΣr, whereas the second one is interpreted as the time spent by the associated scattering statee−itHW−ϕwithin the same region. The (usual) time delay of the scattering process with incoming stateϕforΣris defined as
τrin(ϕ) :=Tr(ϕ)−Tr0(ϕ). (1.1) For a suitable initial stateϕ, a sufficiently short-ranged interaction andΣspherical, the limit ofτrin(ϕ)asr→ ∞exists and is equal to the expectation value in the stateϕof the Eisenbud-Wigner time delay operator [2, 3].
However it is shown in [12] that this limit does not exist ifΣis not spherical. There are other situations where the usual definition of time delay is inappropriate. Examples are multichannel-type scattering processes such asN-body scattering [13, 6, 10], scattering with dissipative interactions [9], step potential scattering [4] and scattering in waveg- uides [14]. In such cases time delay of the form (1.1) does not admit a limit due to the “non conservation” of the kinetic energy. Therefore one has to modify the defini- tion (1.1) by replacing the free sojourn timeTr0(ϕ)with the symmetrized free sojourn time12
Tr0(ϕ) +Tr0(Sϕ)
(see e.g. [10, Sec. V.(a)] or [14, Sec. 1]). The associated sym- metrized time delay forΣrtakes the form:
τr(ϕ) :=Tr(ϕ)−12
Tr0(ϕ) +Tr0(Sϕ)
. (1.2)
Our purpose in this paper is to show that the limit ofτr(ϕ)as r→ ∞exists ifΣ is an arbitrary region symmetric w.r.t. the origin and to derive an appropriate stationary formula for it.
Let us finally give a more precise description of this paper. In section 2 we intro- duce a natural condition on the setΣ(see Assumption 2.1) under which the existence of symmetrized time delay will be proved. We also give some auxilliary results on aver- aged characteristic functions. Section 3 is devoted to symmetrized time delay in classical scattering; its existence under Assumption 2.1 and its invariance under an appropriate mapping of time reversal is shown. In Section 4 we prove the same results in quantum scattering and we derive a stationary formula for the symmetrized time delay.
2 Averaged characteristic functions
LetΣbe a bounded open set inRd containing0. For eachr > 0 we setΣr := {rx | x∈Σ}. We shall simply say thatΣis star-shaped (resp. symmetric) wheneverΣis star- shaped (resp. symmetric) with respect to0. ClearlyΣis star-shaped iffΣr
1 ⊂ Σr
2 for 0< r1≤r2. Moreover to each open star-shaped setΣwe can associate a strictly positive continuous function`ΣonSd−1defined by
`Σ(ω) := sup{µ≥0|µω∈Σ}. (2.3) Conversely to each strictly positive continuous function`onSd−1 one can associate a unique open star-shaped setΣsuch that`=`Σ.
We shall also consider the following class of spatial regionsΣ(1lΣstands for the characteristic function forΣ):
Assumption 2.1. Σis a bounded open set inRdcontaining0and satisfying the condition Z +∞
0
dµ[1lΣ(µx)−1lΣ(−µx)] = 0, ∀x∈Rd.
Ifp ∈ Rd, then the numberR+∞
0 dt1lΣ(tp)is the sojourn time inΣof a free classical particle moving along the trajectoryt7→x(t) :=tp,t≥0. Clearly ifΣ =−Σ(i.e. ifΣ is symmetric), thenΣsatisfies Assumption 2.1. Moreover ifΣis star-shaped and satisfies Assumption 2.1, thenΣ =−Σ.
Lemma 2.2. LetΣbe a bounded open set inRdcontaining0. Then (a) The limit
RΣ(x) := lim
ε&0
Z +∞ ε
dµ
µ 1lΣ(µx) + lnε exists for eachx∈Rd\ {0}.
(b) The (even) functionGΣ:Rd\ {0} →Rgiven by
GΣ(x) := 12[RΣ(x) +RΣ(−x)]
satisfies
GΣ(x) =GΣ x
|x|
−ln|x|.
(c) IfΣis star-shaped, then
GΣ(ω) = 12[ln(`Σ(ω)) + ln(`Σ(−ω))] (2.4) for eachω∈Sd−1.
Proof. Letx∈Rd\ {0}. Then point (a) follows from the equalities lim
ε&0
Z +∞ ε
dµ
µ 1lΣ(µx) + lnε
= Z +∞
1
dµ
µ 1lΣ(µx) + lim
ε&0
Z 1 ε
dµ
µ [1lΣ(µx)−1]
= Z +∞
1
dµ
µ 1lΣ(µx) + Z 1
0
dµ
µ [1lΣ(µx)−1]. Furthermore we have forλ >0
RΣ(λx) = lim
ε&0
Z +∞ ε
dµ
µ 1lΣ(µλx) + lnε
= lim
ε&0
Z +∞ λε
dµ
µ 1lΣ(µx) + ln(λε)−lnλ
=RΣ(x)−lnλ,
which proves point (b). Finally point (c) follows from a direct computation.
We give now some properties of the functionsRΣandGΣ, which follow easily from Lemma 2.2.
Remark 2.3. (a) Let us considerR∗+ :=]0,+∞[endowed with the multiplication as a Lie group with Haar measure dµµ. ThenRΣis the (renormalized) average of1lΣ with respect to the action ofR∗+onRd.
(b) IfΣis equal to the unit open ballB:={x∈Rd| |x|<1}, then we have
GB(x) =−ln|x|, x∈Rd\ {0}. (2.5) (c) To each setΣone can associate a unique symmetric star-shaped setΣesuch that
GΣ=GΣe.
Indeed it suffices to take the symmetric star-shaped setΣedefined by the even, strictly positive, continuous function`onSd−1given by`(ω) := exp(GΣ(ω)).
3 Symmetrized time delay in classical scattering
In this section we study the symmetrized time delay in classical scattering defined in terms of sojourn times in the setsΣr. For simplicity we restrict ourselves to scattering by compactly supported potentials. Under a natural assumption on the potential we show that Assumption 2.1 is a necessary condition for the existence of the symmetrized time delay.
We also prove the existence of the symmetrized time delay for symmetric star-shaped regions withC2-boundary and show that the symmetrized time delay and the usual time delay are equal if the setsΣrare spherical.
LetΣbe a bounded open set containing 0and letV be a realC2-potential with compact support. A scattering trajectory forV is a mapΦ :R3t7→(x(t), p(t))∈R2d solution ofx˙(t) = p(t),p˙(t) = −∇V(x(t))satisfying|x(t)| → ∞ ast → ±∞ and E := 12p2(t) +V(x(t)) >0. SinceV has compact support there exist timest− ≤t+, positionsx±, and momentap±with|p±|=√2Esuch that
x(t) =x±+ (t−t±)p± if ±t≥ ±t±.
Thus the incoming and outgoing free trajectoriesΦ±associated to the scattering trajectory Φare given by
x±(t) :=x±+ (t−t±)p±, p±(t) :=p±. For eachr∈R\ {0}we define the characteristic function
χr(x) := 1lΣ(r−1x), x∈Rd,
and for each suitable trajectoryΨ :R3t7→(y(t), v(t))∈R2dwe set Tr(Ψ) :=
Z +∞
−∞
dt χr(y(t)).
Then the sojourn time inΣrof the scattering trajectoryΦisTr(Φ)and the sojourn times inΣrof the incoming and outgoing free trajectoriesΦ±areTr(Φ±). The usual time delay of the scattering process forΣris defined as
τrin(Φ) :=Tr(Φ)−Tr(Φ−).
It is known [12, Sec. 2] thatlimr→∞τrin(Φ) exists only ifΣis spherical. In this case τrin(Φ) converges to the classical analogue τcl(Φ) of the Eisenbud-Wigner time delay [11].
One can also define the symmetrized time delay:
τr(Φ) :=Tr(Φ)−12[Tr(Φ+) +Tr(Φ−)].
Remark 3.1. Consider the mappingf(of full time reversal)f : Φ7→Φrev, whereΦrev : t7→(x(−t),−p(−t)), and set
τrout(Φ) := (τrin◦f)(Φ). SinceTr(Φrev) =Tr(Φ)andTr((Φrev)−) =Tr(Φ+), we have
τr(Φ) = 1
2
τrin(Φ) +τrout(Φ) .
Thusτris the mean value of the usual time delayτrin and of the time delayτroutcorre- sponding to the time reversed scattering process. In particularτris invariant under full time reversal, namely one hasτr◦f =τrsincefis an involution.
The following assumption concerns the potentialV. Givenω ∈ Sd−1,S(ω) ⊂ Sd−1stands for the set of all outgoing directions of scattering trajectories with incoming directionω.
Assumption 3.2. For allω∈Sd−1,S(ω)contains a neighborhood ofω.
We believe that Assumption 3.2 should be satisfied by typical nonzero potentials.
A possible way to prove it would be to show that at high energies the differential of the outgoing direction with respect to the incoming impact parameter is non zero.
Theorem 3.3. (a) Assume thatV satisfies Assumption 3.2. Thenlimr→∞τrexists and is finite for all scattering trajectories only ifΣsatisfies Assumption 2.1.
(b) IfΣis symmetric and star-shaped with aC2-boundary, thenlimr→∞τrexists and is finite for all scattering trajectories.
(c) IfΣis spherical, thenlimr→∞[τr(Φ)−τrin(Φ)] = 0. Proof. (a) A direct calculation gives
Tr(Φ) = 1
√2E Z 0
−∞
dt χr(x−+tω−)+
Z t+
t−
dt χr(x(t))+ 1
√2E Z +∞
0
dt χr(x++tω+)
and
Tr(Φ±) = 1
√2E Z +∞
−∞
dt χr(x±+tω±),
whereω±:=p±/|p±|,|p±|=√
2E. It follows that
τr(Φ) = Z t+
t−
dt χr(x(t)) + 1
√8E[I(x+, ω+, r)−I(x−, ω−, r)], (3.6) where
I(x, ω, r) :=
Z +∞
−∞
dt sgn(t)χr(x+tω), x∈Rd, ω∈Sd−1.
The first term in the r.h.s. of Equation (3.6) tends tot+−t−, so it is sufficient to consider the convergence of the second term. SinceI(x, ω, r) =rI(r−1x, ω,1)andlimx→0I(x, ω,1) = I(0, ω,1), we get
I(x, ω, r) =rI(0, ω,1) +o(r), and
I(x+, ω+, r)−I(x−, ω−, r) =r[I(0, ω+,1)−I(0, ω−,1)] +o(r). Thuslimr→∞τr(Φ)exists only if
I(0, ω+,1) =I(0, ω−,1).
Varying the scattering trajectory keepingω−fixed, we deduce from Assumption 3.2 that I(0,·,1)is locally constant onSd−1, and thus constant by connexity. SinceI(0,·,1)is
an odd function we get the equalityI(0, ·,1) = 0, which is equivalent to Assumption 2.1.
(b) Suppose that Σ is symmetric star-shaped with a C2-boundary, and let `Σ : Sd−1 →(0,∞)be the function associated toΣas in Section 2. We extend`ΣtoRdby homogeneity. Givenx∈Rdorthogonal toω ∈Sd−1, we denote bys± ≡s±(r−1x, ω) the entrance and exit times of the free trajectoryt7→(r−1x+tω, ω)inΣ(s±exist ifris large enough). Since`Σ(ω) =`Σ(−ω), the timess±satisfys±(0, ω) =±`Σ(ω)and are solutions of the equations
s±=± `Σ(r−1x+s±ω)2−r−2x21/2 . Doing a Taylor expansion aroundr−1= 0, we get
s±(r−1x, ω) =±`Σ(ω)±r−1`Σ(ω)−1x·(∇`Σ)(±ω) +O(r−2). (3.7) Thus, for eachx∈Rd,ω∈Sd−1, andx⊥:=x−(x·ω)ω, we get
I(r−1x⊥, ω,1) =s+(r−1x⊥, ω) +s−(r−1x⊥, ω) =O(r−1). This implies that
I(x, ω, r) =rI(r−1x, ω,1) =r Z +∞
−∞
dtsgn t−r−1(x·ω)
1lΣ(r−1x⊥+tω)
=rI(r−1x⊥, ω,1) +O(1)
=O(1),
and the existence oflimr→∞τr(Φ)follows from Equation (3.6).
(c) IfΣis spherical, then Equation (3.7) gives Tr(Φ±) = r
√2E +O(r−1).
This implies thatlimr→∞[τr(Φ)−τrin(Φ)] = 0.
4 Symmetrized time delay in quantum scattering
4.1 Sojourn times
In this section we gather some properties of the (quantum) sojourn times associated to the free HamiltonianH0=−12∆and the full HamitonianH inH=L2(Rd). We first recall some definitions.
Σis a bounded open set inRdcontaining0, andΣr ={rx | x ∈ Σ}. We write 1lH
0(·)for the spectral measure ofH0andQfor the (vector) position operator inH. We seth·i:=p
1 +| · |2, and we always assume that:
Assumption 4.1. The wave operatorsW±exist and are complete. The projectionsχr(Q) are locallyH-smooth on]0,+∞[\σpp(H).
For latter use we introduce the following definition:
Definition 4.2. Lets≥0, then Ds:={ϕ∈ D(hQis)|1lH
0(J)ϕ=ϕfor some compact setJ in]0,+∞[\σpp(H)}. It is clear thatDsis dense inHand thatDs1 ⊂ Ds2 ifs1≥s2.
Forr >0and an appropriate scattering stateϕ∈ H, we define the free sojourn time Tr0(ϕ) :=
Z +∞
−∞
dtχr(Q) e−itH0ϕ2
and the full sojourn time Tr(ϕ) :=
Z +∞
−∞
dtχr(Q) e−itHW−ϕ2.
Due to Assumption 4.1, one shows easily that these times are finite ifϕ∈ D0. The time delay of the scattering process with incoming stateϕ∈ D0forΣris then defined as
τrin(ϕ) :=Tr(ϕ)−Tr0(ϕ).
SinceSD0⊂ D0one can also define the symmetrized time delay of the scattering process with incoming stateϕ∈ D0:
τr(ϕ) :=Tr(ϕ)−12
Tr0(ϕ) +Tr0(Sϕ) .
Finally we define for eachr >0the auxiliary sojourn timeτrfree(ϕ)(see [14, Sec. 2.1]) τrfree(ϕ) := 1
2
Z 0
−∞
dt kχr(Q) e−itH0ϕk2− kχr(Q) e−itH0Sϕk2
(4.8) +1
2
Z +∞ 0
dt kχr(Q) e−itH0Sϕk2− kχr(Q) e−itH0ϕk2 , which is also finite ifϕ∈ D0.
Lemma 4.3. Suppose that Assumption 4.1 holds and letϕ∈ D0be such that
(W−−1l) e−itH0ϕ∈L1(R−,dt) (4.9) and (W+−1l) e−itH0Sϕ∈L1(R+,dt). (4.10) Then
lim
r→∞
τr(ϕ)−τrfree(ϕ)= 0.
Proof. Fort∈R, set
f−(t) :=e−itHW−ϕ−e−itH0ϕ and f+(t) :=e−itHW+ϕ−e−itH0Sϕ. We know from Hypotheses (4.9) and (4.10) thatf±∈L1(R±). Using the inequality
kuk2− kvk2≤ ku−vk(kuk+kvk), u, v∈ H, (4.11) we obtain the estimates
kχr(Q) e−itHW−ϕk2− kχr(Q) e−itH0ϕk2≤2f−(t)kϕk, kχr(Q) e−itHW−ϕk2− kχr(Q) e−itH0Sϕk2≤2f+(t)kϕk.
Since s-limr→+∞χr(Q) = 1l, then the scalars on the l.h.s. above converge to0asr→ +∞. Thus the claim follows from (4.8) and Lebesgue’s dominated convergence theorem.
4.2 Time reversal
We now collect some elementary remarks related to time reversal for the (complete) scat- tering system{H0, H}.
Time reversal is implemented by the antiunitary involution H 3ϕ7→ϕ.
The HamiltonianHis invariant under time reversal if Hϕ=Hϕ, ϕ∈ D(H).
In such a case one has the identitiese−itHϕ= eitHϕ,W±ϕ=W∓ϕand
Sϕ=S−1ϕ (4.12)
for eachϕ∈ H. Consider the bijection
f :H → H, ϕ7→Sϕ,
which we call full time reversal. The mapfcorresponds to time reversal for the full scat- tering process, i.e. it interchanges past and future scattering data and reverses the direction of time. Furthermore one sees easily from (4.12) thatfis an antiunitary involution.
In order to give a rigourous interpretation of full time reversal we introduce the spaceEof scattering trajectories, i.e. the space of continuous maps
R3t7→Φ(t)∈ H, such that
i(∂tΦ)(t) =HΦ(t) ∀t∈R (in the weak sense) and w-limt→±∞Φ(t) = 0.
The spaceEis invariant under the involution
R:E → E, (RΦ)(t) := Φ(−t).
One can associate to a trajectoryΦ∈ Ea vectorϕ:=T(Φ)∈ Hdefined by the constraint s-limt→−∞ Φ(t)−e−itH0ϕ= 0.
Due to the completeness of the wave operators we know thatT :E → His bijective, and we have
f(ϕ) = T RT−1(ϕ), ϕ∈ H. (4.13) Equation (4.13) provides a rigourous meaning to full time reversal as a map interchanging past and future scattering data and reversing the direction of time.
Lemma 4.4. Assume thatHis invariant under time reversal, and set τrout(ϕ) := (τrin◦f)(ϕ).
Then one has the equalities τr(ϕ) =12
τrin(ϕ) +τrout(ϕ) and τr(ϕ) = (τr◦f)(ϕ). (4.14) Thusτr(ϕ)is the mean value of the usual time delay τrin(ϕ)and of the time de- layτrout(ϕ)corresponding to the time reversed scattering process. In particularτr(ϕ)is invariant under full time reversal.
Proof. SinceH0is invariant under time reversal, one gets Tr0(ϕ) =Tr0(ϕ). This together with time reversal invariance ofHyields
Tr(Sϕ) =Tr(ϕ). Thus
τrout(ϕ) =τrin(Sϕ) =Tr(Sϕ) +Tr0(Sϕ) =Tr(ϕ) +Tr0(Sϕ),
which implies the first identity in (4.14). The second identity follows from the fact thatf is an involution.
4.3 Time delay
In the present section we shall give the proof of the existence of the symmetrized time delay. We first fix some notation. IfA, B are two symmetric operators, then we set for eachϕ∈ D(A)∩ D(B):
(ϕ,[A, B]ϕ) := (Aϕ, Bϕ)−(Bϕ, Aϕ).
Ifqis a quadratic form with domainD(q), andS is unitary, then we set for eachϕ ∈ D(q)∩S−1D(q):
(ϕ, S∗[q, S]ϕ) :=q(Sϕ)−q(ϕ).
If A is an operator with domain D(A) andS is unitary, then we define the operator S∗[A, S]with domainD(A)∩S−1D(A)by
S∗[A, S] :=S∗AS−A.
We also recall that the functionGΣwas introduced in Section 2 and thatD2⊂ D(Q2)∩ D(GΣ(P)). Therefore the quadratic formi[Q2, GΣ(P)]is well defined onD2.
The proof of the next proposition can be found in the appendix.
Proposition 4.5. LetΣsatisfy Assumption 2.1. Suppose that Assumption 4.1 is verified.
Then we have for allϕ∈ D2the equality lim
r→∞
Z +∞ 0
dt ϕ, eitP2/2χr(Q) e−itP2/2−e−itP2/2χr(Q) eitP2/2 ϕ
=− ϕ,i[Q2, GΣ(P)]ϕ
. (4.15)
We are now in a position to give the proof of our main theorem. It involves the operator
A0:= 12 P
P2 ·Q+Q· PP2
, which is well-defined and symmetric onD1.
Theorem 4.6. LetΣsatisfy Assumption 2.1. Suppose that Assumption 4.1 is verified. Let ϕ∈ D2satisfy (4.9), (4.10) andSϕ∈ D2. Then the limitτΣ(ϕ) = limr→∞τr(ϕ)exists, and one has
τΣ(ϕ) =−12 ϕ, S∗i
Q2, GΣ P
|P|
, S ϕ
−(ϕ, S∗[A0, S]ϕ). (4.16) The quadratic formi[Q2, GΣ(P
|P|)]and the operatorA0are well defined onD2, so all the commutators in the above formula are well defined sinceϕ, Sϕ∈ D2.
Proof. The expression (4.8) forτrfree(ϕ)can be rewritten as τrfree(ϕ) =−12
Z +∞ 0
dt ϕ, eitP2/2
χr(Q) e−itP2/2−e−itP2/2
χr(Q) eitP2/2 ϕ +1
2
Z +∞ 0
dt Sϕ, eitP2/2χr(Q) e−itP2/2−e−itP2/2χr(Q) eitP2/2 Sϕ
. Applying Proposition 4.5, we get
r→lim+∞τrfree(ϕ) =12 ϕ,i[Q2, GΣ(P)]ϕ
−12 Sϕ,i[Q2, GΣ(P)]Sϕ
=−12 ϕ, S∗i[Q2, GΣ(P)], S ϕ
.
By Lemma 2.2.(b), we haveGΣ(P) =GΣ |P|P
−ln|P|, and we know from [2, Sec. 2]
that
i
2[Q2,−ln|P|] =A0, as quadratic forms onD2. This yields
limr→+∞τrfree(ϕ) =−12 ϕ, S∗i
Q2, GΣ |P|P , S
ϕ
−(ϕ, S∗[A0, S]ϕ). We conclude by using Lemma 4.3.
Remark 4.7. The second term in Formula (4.16) coincides with the usual value of time delay; it is equal to the limit (forΣspherical) ofτrin(ϕ)asr→+∞(see [2, Prop. 1]).
The first term is a new contribution to time delay determined by the shape of the setΣ. If Σis spherical, this contribution vanishes due to Remark 2.3.(b), and then one gets (under the hypotheses of Theorem 4.6) the equality
r→∞lim τr(ϕ) = lim
r→∞τrin(ϕ).
Remark 4.8. Under the hypotheses of Theorem 4.6, the two following facts are true wheneverΣis an open bounded set containing the origin (see [12, Sec. 3]):
(a) The equalitylimr→∞
τr(ϕ)−τrfree(ϕ)= 0holds.
(b) The difference Z +∞
0
dt Sϕ,eitP2χr(Q) e−itP2−e−itP2χr(Q) eitP2, S ϕ
−r Z +∞
0
du Sϕ,h
|H0|−1/2
1lΣ u|PP|
−1lΣ −u|P|P , Si
ϕ
remains bounded asr→ ∞.
The integrand in the second term in (b) can be written as Sϕ,|H0|−1/2[M(P), S]ϕ
, where
M(x) =|x|
Z +∞ 0
dµ[1lΣ(µx)−1lΣ(−µx)], x∈Rd.
The combination of facts (a) and (b) shows thatτr(ϕ)can have a limit forϕin a dense setE ⊂ Honly if
Sϕ,|H0|−1/2[M(P), S]ϕ= 0, ∀ϕ∈ E,
which implies that[M(P), S] = 0. Therefore, if the scattering operatorS has no other symmetry than[S, P2] = 0, one hasM(P) =F(P2)for some functionF, and it follows thatM ≡0sinceM(x) =−M(−x). In consequenceτr(ϕ)can have a limit forϕin a dense setEonly ifΣsatisfies Assumption 2.1.
Remark 4.9. One could also consider time delay for setsΣrtranslated by a vectora∈ Rd. Obviously this is equivalent to determining the time delay (4.16) when the origin of the spatial coordinate system is translated to the pointa. In this case one has
ϕ7→ϕa:= eiP·aϕ, S7→Sa:= eiP·aSe−iP·a, andτΣ(ϕ)becomes
τΣa(ϕ) :=−12 ϕa, Sa∗i
Q2, GΣ P
|P|
, Sa ϕa
−(ϕa, Sa∗[A0, Sa]ϕa). Using the formulas
e−iP·a
Q2, GΣ |PP|eiP·a =
Q2, GΣ |P|P
−2a·
Q, GΣ P
|P|
,
e−iP·aA0eiP·a =A0−a·PP2, one gets
τΣa(ϕ) =τΣ(ϕ) +a· ϕ, S∗i
Q, GΣ P
|P|
, S
ϕ+a· ϕ, S∗[P P2, S]ϕ
. (4.17) Due to its very definition time delay given by Formula (4.17) is clearly covariant under spatial translations.
4.4 Stationary formulas
In the sequel we derive stationary formulas for the symmetrized time delay in the case of scattering by a short-ranged potential. We refer to [5] for the treatment of this issue in the case of the usual time delay.
Suppose thatΣsatisfy Assumption 2.1. Then we know from Remark 2.3.(c) that there exists a symmetric star-shaped setΣesuch that
τΣ(ϕ) =τΣe(ϕ),
forϕsatisfying the hypotheses of Theorem 4.6. Thus with no loss of generality we may assume thatΣis symmetric and star-shaped. We also assume that the boundary∂Σof Σis aC2 hypersurface, so that the functions`ΣandGΣ(see Formulas (2.3) and (2.4)) associated toΣareC2. In such a case one has
i 2
Q2, GΣ |P|P =−12
Q· ∇GΣ |P|P +∇GΣ |P|P
·Q=:BΣ,
as quadratic forms onD2 (note that BΣ is a well-defined symmetric operator onD1).
Thus we can rewriteτΣ(ϕ)as
τΣ(ϕ) =−(ϕ, S∗[BΣ, S]ϕ)−(ϕ, S∗[A0, S]ϕ). (4.18)
LetU :L2(Rd)→R⊕
R+dλL2(Sd−1)be the spectral transformation forH0, i.e. the unitary mapping defined by
(Uϕ)(λ, ω) = (2λ)(d−2)/4(Fϕ)(√ 2λω), whereF denotes the Fourier transform. One has
UH0U−1= Z ⊕
R+
dλ λ and USU−1= Z ⊕
R+
dλ S(λ),
where {S(λ)}λ≥0 ⊂ B L2(Sd−1)
is the scattering matrix for the pair{H0, H}. For shortness we shall setϕ(λ) := (T ϕ)(λ,·)∈L2(Sd−1).
If the interactionV :=H−H0is a potential that decays faster than|x|−2at infinity, then there exists a dense setE ⊂ Hsuch that the hypotheses of Theorem 4.6 are satisfied for anyϕ ∈ E (a precise definition of V andE can be found in [3, Prop. 3], see also [7, 8] for related results). Furthermore the functionλ7→ S(λ)is strongly continuously differentiable onE, and the second term in (4.18) is equal to the Eisenbud-Wigner time delay for anyϕ∈ E:
−(ϕ, S∗[A0, S]ϕ) =−i Z ∞
0
dλ
ϕ(λ), S(λ)∗d
S(λ) dλ
ϕ(λ)
L2(Sd−1)≡(ϕ, τE-Wϕ). Let us now consider the first term in (4.18). Since the functionx7→GΣ |x|x
is homoge- neous of degree0, one has
x·(∇GΣ) x
|x|
= 0,
namely the vector field(∇GΣ) x
|x|
is orthogonal to the radial direction. In fact a direct calculation shows that
UBΣU−1= Z ⊕
R+
dλ λ−1bΣ(ω, ∂ω),
where bΣ(ω, ∂ω)is a symmetric first order differential operator on Sd−1 with C1 co- efficients. Therefore the operatorUBΣU−1is essentially selfadjoint onUD1, and its closure is decomposable in the spectral representation ofH0, i.e.
UBΣU−1=UBΣU−1≡ Z ⊕
R+
dλ BΣ(λ).
This yields the equality τΣ(ϕ) =−
Z +∞ 0
dλ(ϕ(λ), S∗(λ)[BΣ(λ), S(λ)]ϕ(λ))
L2(Sd−1)
−i Z +∞
0
dλ
ϕ(λ), S(λ)∗d
S(λ) dλ
ϕ(λ)
L2(Sd−1).
In consequence the time delay (4.18) is the sum of two contributions, each of these being the expectation value of an operator decomposable in the spectral representation ofH0.
Appendix
Proof of Proposition 4.5. (i) For anyF ∈L∞(Rd)ands∈Rone has eisP2/2F(Q) e−isP2/2=F(Q+sP), e−isQ2/2
F(P) eisQ2/2=F(P+sQ), (4.19) which imply the identity
eitP2/2F(Q) e−itP2/2=Z−1/tF(tP)Z1/t, (4.20) wheret∈R∗andZτ := eiτQ2/2. Formula (4.20) and the change of variablesµ=rt−1, ν=r−1, lead to the equalities
Z +∞ 0
dt ϕ, eitP2/2χr(Q) e−itP2/2−e−itP2/2χr(Q) eitP2/2 ϕ
= Z +∞
0
dµ
νµ2 (ϕ,(Z−νµχµ(P)Zνµ−Zνµχ−µ(P)Z−νµ)ϕ). One has also
Z +∞ 0
dµ
µ2 [χµ(P)−χ−µ(P)] = Z +∞
0
ds[χ(sP)−χ(−sP)] = 0
due to Assumption 2.1. Hence the l.h.s of (4.15) can be written as K∞(ϕ) := lim
ν&0
Z +∞ 0
dµ Kν,µ(ϕ), (4.21)
where
Kν,µ(ϕ) := 1
νµ2(ϕ,[Z−νµχµ(P)Zνµ−χµ(P)]ϕ)
−νµ12 (ϕ,[Zνµχ−µ(P)Z−νµ−χ−µ(P)]ϕ).
(ii) To prove the statement, we shall show that one may interchange the limit and the integral in (4.21), by invoking Lebesgue’s dominated convergence theorem. This will be done in (iii) below. If one assumes that this interchange is justified for the moment, then direct calculations give
K∞(ϕ) = Z +∞
0
dµ d
dν Kν,µ(ϕ)
ν=0
=−12
Z +∞ 0
dµ
µ ϕ,i [Q2, χµ(P)] + [Q2, χ−µ(P)]
ϕ
. (4.22)
