Solving the quantum nonlinear Schrodinger equation with delta-type impurity

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Caudrelier, V., Mintchev, M. and Ragoucy, E. (2005). Solving the quantum

nonlinear Schrodinger equation with delta-type impurity. Journal of Mathematical Physics,

46(4), doi: 10.1063/1.1842353

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Solving the quantum nonlinear Schrödinger equation

with

␦

-type impurity

V. Caudreliera兲

Laboratoire de Physique Théorique LAPTH,b兲LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France

M. Mintchevc兲

INFN and Dipartimento di Fisica, Universitá di Pisa, Via Buonarroti 2, 56127 Pisa, Italy

E. Ragoucyd兲

Laboratoire de Physique Théorique LAPTH,b兲LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France

共Received 3 July 2004; accepted 13 October 2004; published online 18 March 2005兲

We establish the exact solution of the nonlinear Schrödinger equation with a delta-function impurity, representing a pointlike defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov–Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission 共RT兲 algebra, which captures both particle–particle and particle–impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scattering matrix of the theory is derived. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1842353兴

I. INTRODUCTION

Impurity problems arise in different areas of quantum field theory and are essential for un-derstanding a number of phenomena in condensed matter physics. At the experimental side, the recent interest in pointlike impurities 共defects兲 is triggered by the great progress in building nanoscale devices.

The interaction of quantum fields with impurities represents in general a hard and yet un-solved problem, but there are relevant achievements1–11in the case of integable systems in 1 + 1 space–time dimensions. The study12–19of the special case of purely reflecting impurities共 bound-aries兲indicates factorized scattering theory20–24as the most efficient method for dealing with this kind of problem. The method provides on-shell information about the system and allows to derive the exact scattering matrix. The goal of the present paper is to extend this framework, exploring the possibility to recover off-shell information and to reconstruct the quantum fields, generating the above scattering matrix. We test this possibility on one of the most extensively studied inte-grable systems—the nonlinear Schrödinger共NLS兲model.25–32More precisely, we are concerned below with the NLS model coupled to a delta-function impurity. The basic tool of our investiga-tion is a specific exchange algebra,6,7called reflection-transmission共RT兲algebra. The RT algebra is a generalization of the Zamolodchikov–Faddeev 共ZF兲21,23 algebra used in the case without defects. The RT algebra is originally designed for the construction of the total scattering operator from the fundamental scattering data, namely the two-body bulk scattering matrix and the

reflec-a兲_{Electronic mail: [email protected]}

b兲_{UMR 5108 du CNRS, associée à l’Université de Savoie.} c兲_{Electronic mail: [email protected]}

d兲_{Electronic mail: [email protected]}

46, 042703-1

tion and transmission amplitudes of a single particle interacting with the defect. In what follows we demonstrate that in the NLS model the same algebra allows to reconstruct the corresponding off-shell quantum field as well. Being the first exactly solvable example with nontrivial bulk scattering matrix, the NLS model sheds some light on the interplay between pointlike impurities, integrability, and symmetries. In this respect our solution clarifies a debated question about the Galilean invariance of the bulk scattering matrix.

After introducing the model in Sec. II, we establish the solution, both at the classical 共Sec. II B兲and second-quantized共Sec. III兲levels. We do this in detail, clarifying the basic properties of the solution. In Sec. IV we derive from the off-shell quantum field the total scattering matrix of the model, showing that it coincides with the one obtained directly from factorized scattering. In Sec. V we indicate some generalizations. Our conclusions and ideas about further developments are also collected there. Appendixes A and B are devoted to the proofs of some technical results.

We present below the analysis of the so-called ␦-type impurity. A wider class of defects, interacting with the NLS model and preserving its integrability, can be treated in a similar way.33 We have chosen to focus here on the particular ␦-type defect in order to keep the length of the proofs reasonable, referring to Ref. 33 for a more physically oriented treatment of the general case

共without detailed proofs兲.

II. INTRODUCING AN IMPURITY IN THE NLS MODEL

We start by recalling some well-known results about the NLS model without impurity. The reason for this is twofold: first, because this is a good guide to tackle the problem with impurity and second, because the central piece of the solution of the NLS model, the Rosales expansion,34,35can be adapted to the impurity case.

A. The model to solve

The field theoretic version of NLS is described by a classical complex field ⌽共t , x兲 whose equation of motion reads

共i⳵t+⳵x

2_兲⌽共

t,x兲= 2g兩⌽共t,x兲兩2⌽共t,x兲. 共2.1兲

The corresponding action takes the form

ANLS=

冕

R

dt

冕

R

dx共i⌽¯共t,x兲⳵_t⌽共t,x兲−兩⳵_x⌽共t,x兲兩2_{− g兩⌽共t,x兲兩}4_{兲, 共2.2兲} and, being in particular invariant under time translation, ensures the conservation of the energy

ENLS=

冕

R

dx共兩⳵x⌽共t,x兲兩2+ g兩⌽共t,x兲兩4兲. 共2.3兲

The latter is non-negative for g艌0.

It is well-known that this is a nonrelativistic integrable model36共see also Ref. 30 for a review兲 and an explicit solution for the field was given by Rosales in Ref. 34,

⌽共t,x兲=

兺

n=0

⬁

共− g兲n⌽共n兲共t,x兲, 共2.4兲

⌽共n_{兲共t,x兲=}

冕

R2n+1

兿

_i=1

j=0 n

dpi

2␲ dqj

2␲␭ ¯_共p

1兲¯¯␭共pn兲␭共qn兲¯␭共q0兲 ei兺j=0

n

共q_jx−q2_jt_兲−i_兺i=1n _共p_ix−p_i2t_兲

兿

i=1 n

共pi− qi−1兲共pi− qi兲

共2.5兲

and the overbar denotes complex conjugation.

The level n = 0 is the linear part of the field corresponding to the free Schrödinger equation. It was argued in Ref. 32 that this solution is well-defined for a large class of functions␭关containing the Schwarz space S共R兲兴 and an upper bound for g was given for the series 共2.4兲to converge uniformly in x. It also represents a physical field since it vanishes as x→±⬁. In the same paper, the authors considered NLS on the half-lineR+, which can be seen as the model on the whole line in the presence of a purely reflecting impurity sitting at the origin. Therefore, the latter represents a particular case of the model with transmitting and reflecting impurity at x = 0 we wish to contemplate in this paper. They gave the following action:

AR=

冕

R dt

冕

R+

dx共i⌽¯共t,x兲⳵_t⌽共t,x兲−兩⳵_x⌽共t,x兲兩2_{− g兩⌽共t,x兲兩}4_兲−␩

冕

R

dt兩⌽共t,0兲兩2_,

where␩苸R is the parameter controlling the boundary condition

lim

x→0+

共⳵x−␩兲⌽共t,x兲= 0. 共2.6兲

In our case, since the impurity is allowed to reflect and transmit, we must take theR− _{part into} account and we are led to work with the following action:

ART=A++A−+A0, 共2.7兲

where

A±=

冕

R

dt

冕

R±

dx共i⌽¯共t,x兲⳵t⌽共t,x兲−兩⳵x⌽共t,x兲兩2− g兩⌽共t,x兲兩4兲, 共2.8兲

A0= − 2␩

冕

R

dt兩⌽共t,0兲兩2. 共2.9兲

The form ofA_RTshows the particular status of the origin x = 0 where the impurity sits. Again, the invariance of the action under time translations ensures the conservation of the energy,

ERT=

冕

R−_丣R+

dx共兩⳵x⌽共t,x兲兩2+ g兩⌽共t,x兲兩4兲+ 2␩兩⌽共t,0兲兩2. 共2.10兲

It is positive for g艌0 ,␩艌0, which is what we assume in the rest of this paper. We will see that

␩ characterizes the transmission and reflection properties of the impurity. Using the variational principle, one deduces the equation of motion and the boundary conditions for the field:⌽共t , x兲 must be the solution of NLS onR−andR+, continuous at x = 0 and satisfy a “jump condition” at the origin. It must also vanish at infinity as a physical field.

Definition 2.1: The nonlinear Schrödinger model with a transmitting and reflecting impurity at the origin is described by the following boundary problem for the field⌽共t , x兲:

共i⳵t+⳵x

2_兲⌽共

t,x兲− 2g兩⌽共t,x兲兩2⌽共t,x兲= 0, x⫽0, 共2.11兲

lim

x→0+

lim

x→0+

兵共⳵x⌽兲共t,x兲−共⳵x⌽兲共t,− x兲其− 2␩⌽共t,0兲= 0, 共2.13兲

lim

x→±⬁⌽共

t,x兲= 0. 共2.14兲

B. Explicit solution

As announced, the Rosales solution34can be adapted suitably to solve the problem of defini-tion 2.1. Since共2.4兲is a solution of NLS onR, it is easy to devise a solution for共2.11兲. Starting from two copies of共2.4兲 and共2.5兲, one based on a function␭+ and the other on a function␭−, denoted⌽+共t , x兲and⌽−共t , x兲, respectively, we define

⌽共t,x兲=

冦

⌽+共t,x兲, x⬎0,

⌽−共t,x兲, x⬍0,

1

2共⌽+共t,0兲+⌽−共t,0兲兲, x = 0.

冧

共2.15兲

It is clearly solution of共2.11兲for x⫽0 and from the vanishing of⌽±共t , x兲as x→±⬁,共2.14兲is also satisfied. However, there is no reason why, in general,⌽共t , x兲so defined should satisfy the bound-ary conditions 共2.12兲 and 共2.13兲. In order to satisfy these conditions, we parametrize ␭+, ␭− as follows:

冉

␭+共p兲 ␭−共p兲

冊

=

冉

1 T共p兲 T共− p兲 1

冊冉

␮+共p兲

␮−共p兲

冊

+

冉

R共p兲 0 0 R共− p兲

冊冉

␮+共− p兲

␮−共− p兲

冊

, 共2.16兲

where

T共p兲= p

p + i␩, R共p兲= − i␩

p + i␩, p苸R, 共2.17兲 and␮±共p兲 are arbitrary Schwarz test functions. Then, the functions␭±共p兲 satisfy

␭±共p兲= T共± p兲␭⫿共p兲+ R共± p兲␭±共− p兲, ∀p苸R 共2.18兲

which follows from the identities

R共p兲R共− p兲+ T共p兲T共− p兲= 1 and T共p兲R共− p兲+ R共p兲T共− p兲= 0, ∀p苸R. 共2.19兲

These relations plus a particular choice for the form of ␮_± will be essential in the proof of the theorem 2.2 below.

Anticipating the quantum case, if we interpret␭₊共respectively,␭₋兲as a wave packet,共2.18兲 shows that each wave packet inR+ _{共respectively, R}−_{兲 is equivalent to the superimposition of a} transmitting part coming fromR−_{共respectively,R}+_{兲and a reflected part inR}

+共respectively,R−兲. This physical interpretation will show up in the next section when we construct a Fock represen-tation of the creation and annihilation operators.

We are now in position to state the main result of this section whose lengthy proof we defer until Appendix A.

Theorem 2.2: Let␮+,␮−be given by

␮±共k兲= ±

␮0共±k兲+共k⫿i␩兲␮1共k兲

lim

x→0+

兵⌽共t,x兲−⌽共t,− x兲其= 0,

lim

x→0+

兵共⳵x⌽兲共t,x兲−共⳵x⌽兲共t,− x兲其− 2␩⌽共t,0兲= 0.

With this result, we can say that⌽共t , x兲rewritten as

⌽共t,x兲=␪共x兲⌽+共t,x兲+␪共− x兲⌽−共t,x兲, 共2.21兲

where␪共x兲is the Heaviside function defined here to be 1₂ at x = 0, is the classical solution of the nonlinear Schrödinger model with impurity as given in definition 2.1.

We want to emphasize that these boundary conditions decouple for the nonlinear part of the field 共as shown in Appendix A兲 and this is due to the reflection-transmission property 共2.18兲 satisfied by␭+and␭−. This already gives a good hint that the construction of a local field from the quantum counterparts of␭₊,␭₋is achievable, as we now explain.

III. QUANTIZATION OF THE SYSTEM

In this section, we move on to the construction and resolution of the quantized version of NLS with impurity. As we mentioned earlier, the crucial ingredient is the RT algebra which encodes the properties of the impurity.

A. Reflection-transmission algebra

Here we rely on the constructions developed in Ref. 7 and recast them in the particular context of the scalar nonlinear Schrödinger model共no internal degrees of freedom, special form of the exchange matrix and of the generators, see also Ref. 11兲.

We consider the associative algebra with identity element 1 and two sets of generators,

兵a_␣共p兲, a_␣†共p兲; p苸R,␣= ±其and兵r共p兲, t共p兲; p苸R其, called the bulk and defect共reflection and trans-mission兲 generators. The label ␣= ± refers to the half-line R± with respect to the impurity 共in practice it will indicate where the particle is created or annihilated兲. Introducing the measurable function S :R⫻R→Cdefined by

S共p兲= p − ig

p + ig 共3.1兲

theS-matrix is defined in our context by

S=

兺

␣1,␣2=±

S␣1␣2共p1, p2兲E␣1␣1丢E␣2␣2, 共3.2兲

whereS_␣1␣2共p1, p2兲= S共␣1p1−␣2p2兲 and共E␣␤兲␴␥=␦␣␴␦␤␥. It is easy to check that S satisfies the unitarity condition and the quantum Yang–Baxter equation

S12共p1, p2兲S21共p2, p1兲=1丢1, 共3.3兲

S12共p1, p2兲S13共p1, p3兲S23共p2, p3兲=S23共p2, p3兲S13共p1, p3兲S12共p1, p2兲. 共3.4兲

Our defect generators r共p兲, t共p兲 are related to r␤_␣共p兲, t_␣␤共p兲defined in Ref. 7 by

r_␣␤共p兲=␦_␣␤r共␣p兲 and t␤_␣共p兲=⑀_␣␤t共␣p兲with⑀=

冉

0 1

1 0

冊

. 共3.5兲

All this setup gives rise to a particular RT algebra whose defining relations then read as follows.

a_␣

1共p1兲a␣2共p2兲− S共␣2p2−␣1p1兲a␣2共p2兲a␣1共p1兲= 0, 共3.6兲

a_␣1† 共p1兲a_␣2 † _共

p2兲− S共␣2p2−␣1p1兲a_␣2 † _共

p2兲a_␣1 † _共

p1兲= 0, 共3.7兲

a_␣ 1共p1兲a␣2

† _共p

2兲− S共␣1p1−␣2p2兲a_␣ 2 † _共p

2兲a␣₁共p1兲= 2␲␦共p1− p2兲关␦_␣ 1 ␣2_{1 +⑀}

␣1 ␣2_t共␣

1p1兲兴

+ 2␲␦共p₁+ p₂兲␦_␣ 1 ␣2_r共␣

1p1兲. 共3.8兲

共ii兲 Defect exchange relations,

关r共p1兲,r共p2兲兴= 0, 共3.9兲

关t共p1兲,t共p2兲兴= 0, 共3.10兲

关t共p₁兲,r共p₂兲兴= 0. 共3.11兲

共iii兲 Mixed exchange relations,

a_␣

1共p1兲r共p2兲= S共p2− p1兲S共p2+ p1兲r共p2兲a␣1共p1兲, 共3.12兲

r共p1兲a␣₂ † _共

p2兲= S共p1− p2兲S共p1+ p2兲a␣₂ † _共

p2兲r共p1兲, 共3.13兲

a_␣

1共p1兲t共p2兲= S共p2− p1兲S共p2+ p1兲t共p2兲a␣1共p1兲, 共3.14兲

t共p1兲a_␣2 † _共

p2兲= S共p1− p2兲S共p1+ p2兲a_␣2 † _共

p2兲t共p1兲. 共3.15兲

共iv兲 Finally, the defect generators are required to satisfy unitarity conditions,

t共p兲t共− p兲+ r共p兲r共− p兲= 1, 共3.16兲

t共p兲r共− p兲+ r共p兲t共− p兲= 0, 共3.17兲

which amount to implement the physical energy conservation when reflection and trans-mission occur.

Since we aim at second quantize a physical system, we now turn to the Fock representation of this algebraic setup as it is presented in Ref. 7. What we need is to represent the generators

兵a_␣共p兲, a_␣†共p兲, r共p兲, t共p兲, p苸R其as operator-valued distributions acting on a common invariant sub-space of a Hilbert sub-space,F, to be defined. We should also identify a normalizable vacuum state⍀ annihilated by a_␣ and cyclic with respect to a_␣†. Applying the general construction of Ref. 7, we know that each such Fock representation is characterized by two numerical matrices T共p兲 and

R共p兲. Here we take

T共p兲=

冉

0 T共p兲

T共− p兲 0

冊

, R共p兲=

冉

R共p兲 0

0 R共− p兲

冊

共3.18兲

with T, R given in共2.17兲. Now consider

L= 丣 ␣=±

L2共R兲 共3.19兲

具␸,␺典=

冕

R

dp

兺

␣=±

␸

¯_␣共p兲␺_␣共p兲, 共3.20兲

which makes it a Hilbert space for the associated norm denoted储·储. Then, the n-particle subspace

H共n_{兲is the subspace of the n-fold tensor productL}丢n_{defined as follows. If␸共}n_兲苸L丢n_{, we identify}

it with the column whose entries are␸_␣ 1,. . .,␣n

共n兲

. Then explicitly,H共0兲_{=Cand for n艌1,␸}共n兲_苸H共n兲_if

and only if

␸共n兲_苸L丢n_,

␸␣1¯␣n

共n兲 _共

p1, . . . , pn兲= T共␣npn兲␸␣共₁¯␣_n−1,−␣_n

n兲 _共

p1, . . . , pn−1, pn兲+ R共␣npn兲␸␣共₁¯␣_n−1␣_n n兲 _共

p1, . . . , pn−1,− pn兲,

共3.21兲

n⬎1, ␸_␣

1¯␣i␣i+1¯␣n

共n兲 _共

p1, . . . , pi, pi+1, . . . , pn兲= S共␣ipi−␣i+1pi+1兲

⫻␸␣1¯␣i+1␣i¯␣n

共n兲 _共p

1, . . . , pi+1, pi, . . . , pn兲, 1⬍i⬍n − 1. 共3.22兲

The Fock space isF=丣n=0⬁ H共

n兲_{and the common invariant subspace is the finite particle spaceD}

spanned by the linear combination of sequences ␸=共␸共0兲,␸共1兲, . . . ,␸共n兲, . . .兲 with␸共n兲苸H共n兲 and

␸共n兲_{= 0 for n large enough.Dis dense inF. We extend the scalar product, again denoted by具· , ·典,}

toF,

∀␸,␺苸F, 具␸,␺典=

兺

n=0

⬁

具␸共n兲_,␺共n兲_典

=

兺

n=0

⬁

冕

Rn

dp1¯dpn

兺

␣1,. . .,␣n=±

␸

¯_␣

1¯␣n共p1, . . . , pn兲␺␣1¯␣n共p1, . . . , pn兲.

The unit norm vacuum state is⍀=共1 , 0 , . . . , 0 , . . .兲and belongs toD.

Now, we can define the action of the smeared bulk operators兵a共f兲, a†共f兲;f苸丣␣=±C0⬁共R兲其onD as follows:

a共f兲⍀= 0, 共3.23兲 and for any␸共n兲_苸H共n兲_,

关a共f兲␸兴_␣

1¯␣n−1

共n−1兲 _共p

1, . . . , pn−1兲=

冑

n

冕

−⬁ ⬁ _dp

2␲_␣

兺

_=±f ¯

␣共p兲␸␣␣1¯␣n−1

共n兲 _{共p, p}

1, . . . , pn−1兲, 共3.24兲

关a†共f兲␸兴␣1¯␣n+1

共n+1兲 _共

p1, . . . , pn+1兲=

冑

n + 1关P共n+1兲f丢␸共n兲兴␣₁¯␣_n+1共p1, . . . , pn+1兲, 共3.25兲

where P共n兲_{is the orthogonal projector inL}丢n_{defined in Ref. 7. For completeness, the explicit form}

of共3.25兲is given in Appendix B. These operators are bounded on eachH共n兲_,

∀␸苸H共n兲_{, 储a共f兲␸储艋}

冑

_{n储f储 储␸储, 储a}†_{共f兲␸储艋}

冑

_{n + 1储f储 储␸储. 共3.26兲} In particular, they are continuous in the smearing functionf. Finally, they satisfy

关r共p兲␸兴_␣ 1¯␣n

共n_{兲 共}

p1, . . . , pn兲= S共p −␣1p1兲¯S共p −␣npn兲R共p兲S共␣npn+ p兲¯S共␣1p1+ p兲 ⫻␸␣1¯␣n

共n兲 _共p

1, . . . , pn兲, 共3.28兲

关t共p兲␸兴_␣ 1¯␣n

共n_{兲 共}

p1, . . . , pn兲= S共p −␣1p1兲¯S共p −␣npn兲T共p兲S共␣npn+ p兲¯S共␣1p1+ p兲 ⫻␸␣1¯␣n

共n兲 _共p

1, . . . , pn兲. 共3.29兲

It follows then that r and t have nonvanishing vacuum expectation values

具⍀,r共p兲⍀典= R共p兲, 具⍀,t共p兲⍀典= T共p兲. 共3.30兲

Introducing finally the operator-valued distributions a_␣共p兲, a_␣†共p兲as

a共f兲=

冕

R

dp

2␲_␣

兺

=±

f

¯

␣共p兲a␣共p兲, a†共f兲=

冕

R

dp

2␲_␣

兺

=±

a_␣†共p兲f_␣共p兲 共3.31兲 one can check that the defining relations of the RT algebra are satisfied onD. The operators a , a† will be referred to as annihilation and creation operators, respectively. Implementing the automor-phism%defined in Ref. 7 for which we know that it is realized by the identity operator for any Fock representation, we get the quantum analog of the reflection-transmission property共2.18兲

a_␣共p兲=⑀_␣␤t共␣p兲a_␤共p兲+␦_␣␤r共␣p兲a_␤共− p兲, 共3.32兲

a_␣†共p兲=⑀_␤␣a_␤†共p兲t共␤p兲+␦_␤␣a_␤†共− p兲r共−␤p兲. 共3.33兲

B. The question of operator domains

From the above it appears that the natural domain to start with isD. Actually, it is much too big for practical calculations and we would like to work on a dense subspace ofDwhich would play the role of the standard formal “state space,” a basis of which is usually denoted by

兩k1, . . . , kn典, k1⬎¯⬎kn. As a first step, we define

D0 0

=C,

D0

n_=兵a

␣1 † _共

f1兲¯a_␣

n

† _共

fn兲⍀;fi苸C0⬁共R兲, ␣i= ± ,i = 1, . . . ,n其, n艌1. 共3.34兲

One can check thatD₀n is dense inH共n兲, i.e., ⍀ is cyclic with respect to a_␣†. The corresponding domain D0, dense in D, is the linear space of sequences ␸=共␸共0兲,␸共1兲, . . . ,␸共n兲, . . .兲 with ␸共n兲 苸D0

n

and␸共n兲= 0 for n large enough.D0is stable under the action of a␣共f兲and a_␣ †_共

f兲. Finally, since T, R, and S are bounded, C⬁-functions, D₀n傺C₀⬁共Rn兲. Now in order to formulate the desired properties of the quantum field in the next paragraph, we introduce a partial ordering relation on C₀⬁共R兲by

fⱭg⇔ ∀x苸supp共f兲, ∀y苸supp共g兲, 兩x兩⬎兩y兩, 共3.35兲

which extends naturally to C₀⬁共R␣兲,␣= ±. Let us introduce

a

˜_␣†共t,x兲=

冕

R

dp

2␲a␣ †_共

a

˜_␣†_{共t, f兲=}

冕

R

dx a˜_␣†_{共t,x兲f共x兲, f苸C} 0

⬁_{共R兲. 共3.36兲}

Now, fix t苸R and␣1, . . . ,␣n and define 共vect standing for “linear span of”兲 D˜0 0₌

C and for n

艌1,

D˜ 0,␣₁¯␣_n

n _{= vect兵a˜}

␣1 † _{共t, f}

1,␣₁兲¯a˜␣_n † _{共t, f}

n,␣_n兲⍀; f1,␣₁Ɑ ¯ Ɑfn,␣_n, fi,␣_i苸C0⬁共R␣i兲,

0苸supp共fi,␣_i兲, i = 1, . . . ,n其 共3.37兲

then the following theorem holds.

Theorem 3.1:∀t苸R,∀␣1, . . . ,␣n= ±,D˜0,␣₁¯␣_n

n _{is dense inH共}n兲_.

Proof: We only need to consider n艌1. The proof relies on two known results of standard analysis. First, the Fourier transform of a C⬁-function with compact support is real analytic 共i.e., a Gevrey class 1 function兲. Second, a real analytic function vanishing on a given open subset U of an open connected set O, vanishes on the whole of O共see, e.g., Ref. 37兲.

Here, it suffices to show that D˜_0,␣ 1¯␣n

n

is dense in D₀n for any t苸R so let us consider the matrix element

A ˜

t,␸,␣₁¯␣_n共x1, . . . ,xn兲=具␸共n兲,a˜_␣

1 † _共

t,x1兲¯a˜_␣

n

† _共

t,xn兲⍀典, 共3.38兲

where␸共n兲苸D₀nis arbitrary. To prove the statement, we now have to show that A

˜

t,␸,␣₁¯␣_n共x1, . . . ,xn兲= 0, ∀兩x1兩⬎ ¯ ⬎兩xn兩⬎0, xi苸R␣i, i = 1, . . . ,n 共3.39兲

implies␸共n兲= 0. From共3.36兲, we get

A ˜

t,␸,␣1¯␣_n共x1, . . . ,xn兲=

冕

Rn

兿

_j=1

n

dp_j 2␲e

−ip_jx_j+itp2_j具␸共n兲_,a

␣1 † _共

p1兲¯a␣_n † _共

pn兲⍀典, 共3.40兲

which shows that A˜_t,␸,␣

1¯␣nis the Fourier transform of a C

⬁_{-function with compact support and is}

therefore real analytic. Condition共3.39兲amounts to saying that A˜_t,␸,␣

1¯␣nvanishes on the set

U_␣

1¯␣n=兵x苸R

n_{s . t .兩x}

1兩⬎ ¯ ⬎兩xn兩⬎0, xi苸R␣i, i = 1, . . . ,n其. 共3.41兲

U_␣

1¯␣nbeing an open subset of 共the open and connected space兲R

n_{, we conclude that A}˜ t,␸,␣₁¯␣_n vanishes onRn. This gives in turn that

具␸共n_兲,a

␣1 † _共p

1兲¯a␣_n † _共p

n兲⍀典= 0, ∀pj苸R, j = 1, . . . ,n, 共3.42兲

or, equivalently, from the cyclicity of⍀with respect to a†

␸␣1¯␣n

共n兲 _共p

1, . . . , pn兲= 0, ∀pj苸R, j = 1, . . . ,n. 共3.43兲

Now using the properties共3.21兲and共3.22兲satisfied by␸共n兲, we get

␸␣1¯␣n

共n兲 _共p

1, . . . , pn兲= 0, ∀pj苸R, ∀␣j= ± , j = 1, . . . ,n 共3.44兲

that is␸共n兲= 0. 䊏

共3.45兲 and conclude for the whole domainDby a continuity argument.

Lemma 3.2: Let f1,␣₁Ɑ¯Ɑfn,␣_n and h1,␤₁Ɑ¯Ɑhn,␤_n, then

具˜a_␣ 1 † _共

t, f1,␣1兲¯˜a␣_n † _共

t, fn,␣_n兲⍀,a˜␤₁

† _共

t,h1,␤1兲¯˜a␤_n † _共

t,hn,␤_n兲⍀典=

兿

j=1 n

␦␣j␤j具fj,␣j,hj,␤j典. 共3.46兲

In particular, for␸苸D˜_0,␣ 1¯␣n

n _{represented as}

␸=

兺

␤苸B

a ˜_␣

1 † _{共t, f}

1,␣₁ ␤ _兲¯˜a

␣n

† _{共t, f}

n,␣_n

␤ _{兲, f} 1,␣₁

␤ _{Ɑ ¯ Ɑf}

n,␣_n

␤ _{, ∀␤苸B, 共3.47兲}

where B is a finite set, one has储␸储=储兺_␤苸Bf␤1,␣₁丢¯丢fn,␤␣_n储.

Proof: To get共3.46兲, one uses an induction on n and combines共3.36兲,共3.27兲,共3.8兲, and共3.23兲 together with the support conditions on the smearing functions. Using a contour integral argument, these support conditions imply that all the contributions arising from the RT algebra vanish except for the usual ␦- term producing the right-hand side. Equation 共3.47兲 is a mere consequence of

共3.46兲. 䊏

Remark: It is important to realize that the n particle space H共n兲 _{is the central piece in this}

construction and that, on this space, any operation we have considered共scalar product, creation operator, Fourier transform兲 is continuous in the smearing functions. Since C₀⬁共R兲 is dense in

S共R兲, the Schwarz space, we can extend the above 共especially the definition ofD₀n兲to smearing functions inS共R兲.

C. Quantum field

We start by defining ⌽共t , f兲 as

⌽共t, f兲=

冕

R

dx

兺

␣=±

f¯_␣共x兲⌽_␣共t,x兲, f苸CwhereC= 丣 ␣=±

C₀⬁共R␣兲. 共3.48兲

f is viewed as a column vector f =

共

f+

f−

兲

with f␣苸C0

⬁_共R␣_{兲and 0苸supp共f}

␣兲. Following the standard argument of Ref. 29, we replace␭_␣共p兲,¯␭_␣共p兲in the Rosales expansion of the classical field共2.4兲 and共2.5兲by the operators a_␣共p兲, a_␣†共p兲in order to define

⌽␣共t,x兲=

兺

n=0

⬁

共− g兲n⌽_␣共n兲共t,x兲, g⬎0 共3.49兲

and

⌽_␣共n兲_共t,x兲=

冕

R2n+1

兿

_i=1

j=0 n

dpi

2␲ dqj

2␲a␣ †_共p

1兲¯a_␣†共pn兲a␣共qn兲¯a␣共q0兲

ei兺j=0 n

共q_jx−q2_jt_兲−i_兺i=1n _共p_ix−p_i2t_兲

兿

i=1 n

共pi− qi−1− i␣␧兲共pi− qi− i␣␧兲

,

共3.50兲

where we used an i␧ prescription depending on␣= ±.

We now have several requirements to meet for our quantum theory to be well defined. We must give a precise meaning to⌽_␣共t , x兲, show that the canonical commutation relations as well as the boundary conditions共2.12兲and共2.13兲hold in a sense we shall make precise and that⌽_␣共t , x兲 is indeed the quantum solution we look for.

共␸,␺兲哫具␸,⌽_␣共t,x兲␺典, 共3.51兲

Dcontaining only finite particle vectors, it is enough to investigate具␸,⌽_␣共n兲共t , x兲␺典for arbitrary n. Proposition 3.3:∀n艌0,∀␸,␺苸D,共t , x兲哫具␸,⌽_␣共n兲共t , x兲␺典 is a C⬁ function.

Proof: The proof is the same as in Ref. 32. 䊏 We define the conjugate⌽_␣†共t , x兲again as a quadratic form onD⫻Dby

具␸,⌽_␣†共t,x兲␺典=具⌽_␣共t,x兲␸,␺典. 共3.52兲

It has the same smoothness properties and from共3.27兲, we get

⌽_␣†_共n_兲共

t,x兲=

冕

R2n+1

兿

_i=1

j=0 n

dpi

2␲ dqj

2␲a␣ †_共

q0兲¯a_␣ †_共

qn兲a␣共pn兲¯a␣共p1兲

⫻ e

−i兺n_j=0共q_jx−q2_jt兲+i兺_i=1n 共p_ix−p_i2t兲

兿

i=1 n

共pi− qi−1+ i␣␧兲共pi− qi+ i␣␧兲

. 共3.53兲

Defining the smeared version

⌽†_{共t, f兲=}

冕

R

dx

兺

␣=±

⌽_␣†_共

t,x兲f_␣共x兲, f苸C 共3.54兲

we conclude that⌽共t , f兲and⌽†共t , f兲are understood as quadratic forms on the domainDand are related by

具␸,⌽_␣†共t, f兲␺典=具⌽_␣共t, f兲␸,␺典. 共3.55兲

To get true quantum fields, we need to show that these quadratic forms give rise to operators on

D. This requires the following two lemmas. Lemma 3.4:∀␸,␺苸D,

共i兲 For h_1,␣Ɑ¯Ɑh_n,␣,

具␸,⌽_␣共t, f_␣兲˜a_␣†_共t,h

1,␣兲¯˜a_␣†共t,hn,␣兲⍀典=

兺

j=1 n

具f_␣,hj,␣典

⫻具␸,a˜_␣†共t,h1,␣兲¯a˜␣ †_共

t,hj,␣兲

ˆ _¯˜a

␣ †_共

t,hn,␣兲⍀典,

共3.56兲 where the hatted symbol is omitted.

共ii兲 For h_␣Ɑf_␣,

具␸,⌽_␣†共t, f_␣兲˜a_␣†共t,h_␣兲␺典=具␸,a˜_␣†共t,h_␣兲⌽†_␣共t, f_␣兲␺典. 共3.57兲

共iii兲 For f_␣Ɑhj,␣, j = 1 , . . . , n,

具␸,⌽_␣†共t, f_␣兲˜a_␣†共t,h1,␣兲¯˜a_␣ †_共

t,hn,␣兲⍀典=具␸,a˜_␣

†_共

t, f_␣兲˜a_␣†共t,h1,␣兲¯˜a_␣ †_共

t,hn,␣兲⍀典.

共3.58兲

contribu-tions of⌽,⌽†onD˜₀n,␣are carried by the zeroth order corresponding to the linear problem共it is the

Fourier transform of a, a†_{兲. 䊏}

Lemma 3.5: Given␸_␣苸D˜₀n,␣,␺_␣苸D˜₀n+1,␣and f_␣苸C₀⬁共R␣兲, the quadratic form (3.51) satisfies the following boundedness condition:

兩具␸␣,⌽␣共t, f␣兲␺␣典兩艋共n + 1兲储f␣储 储␸␣储 储␺␣储. 共3.59兲

Proof: The proof is similar to that given in Ref. 32 and uses lemmas 3.2 and 3.4共i兲. 䊏 From the Riesz lemma and theorem 3.1, we conclude that ⌽_␣共t , f_␣兲:H共n+1兲_→H共n兲 _{is a}

bounded operator for any n艌0. Thus, it defines an operator on the common invariant domainD. The same holds for⌽_␣†共t , f_␣兲,H共n兲_→H共n+1兲 _{by共3.55兲. We can therefore collect our results in the}

following theorem.

Theorem 3.6:⌽共t , f兲,⌽†_{共t , f兲:D→D are Hermitian conjugate, linear operators and satisfy}

⌽共t, f兲⍀= 0, ⌽†共t, f兲⍀= a˜†_{共t, f兲⍀. 共3.60兲}

Finally, we will have a nonrelativistic quantum field if we prove the canonical commutation relations for⌽,⌽†_.

Theorem 3.7:兵⌽共t , f兲,⌽†_{共t , f兲, f苸C其realize a Fock representation of the equal time} canoni-cal commutation relations onD,

关⌽共t, f1兲,⌽共t, f2兲兴= 0 =关⌽†共t, f1兲,⌽†共t, f2兲兴, 共3.61兲

关⌽共t, f₁兲,⌽†共t, f₂兲兴=具f₁, f₂典. 共3.62兲

Proof: We know that it suffices to compute the commutators on D˜₀n,+or D˜₀n,−for arbitrary n and then extend the results by continuity toH共n兲_{and by linearity toD. From theorem 3.6, we get}

that共i兲–共iii兲of lemma 3.4 hold as operator equalities. Let us start with the first commutator. It is made out of four parts,

关⌽共t, f1兲,⌽共t, f2兲兴=关⌽+共t, f1,+兲,⌽+共t, f2,+兲兴+关⌽+共t, f1,+兲,⌽−共t, f2,−兲兴+关⌽−共t, f1,−兲,⌽+共t, f2,+兲兴 +关⌽₋共t, f_1,−兲,⌽₋共t, f_2,−兲兴. 共3.63兲

The first and fourth parts of the right-hand side are easily seen to be zero from共i兲of lemma 3.4 One has for␣= ±,

⌽␣共t, f1,␣兲⌽␣共t, f2,␣兲˜a_␣ †_共

t,h1,␣兲¯˜a_␣ †_共

t,hn,␣兲⍀=

兺

j=1 n

兺

k=1 k⫽j n

具f2,␣,hj,␣典具f1,␣,hk,␣典˜a_␣

†_共

t,h1,␣兲¯

⫻˜aˆ_␣†共t,hj,␣兲¯˜a_␣

†_共 t,hk,␣兲

ˆ _¯˜a

␣ †_共

t,hn,␣兲⍀,

共3.64兲

which is symmetric under the exchange of f1and f2implying the vanishing of the commutators. As for the mixed terms, one can check that

⌽␣共t, fi,␣兲˜a−␣ † _共t,h

1,−␣兲¯a˜−␣ † _共t,h

n,−␣兲⍀= 0, i = 1,2 共3.65兲

implying the vanishing of the second and third commutators onD˜₀n,−␣and hence on D. Now the vanishing of关⌽†_{共t , f}

1兲,⌽†共t , f2兲兴onDis obtained by Hermitian conjugation. This proves共3.61兲. Equation共3.62兲is obtained as follows. Again, we split the commutator into four parts. Now

given a state inD˜₀n,␣, we assume hk,␣Ɑf2,␣Ɑhk+1,␣ for some k and using lemma 3.4, we compute

⌽␣共t, f1,␣兲⌽_␣ †_共

t, f2,␣兲˜a_␣ †_共

t,h1,␣兲¯˜a_␣ †_共

t,hn,␣兲⍀=具f1,␣, f2,␣典a˜_␣ †_共

t,h1,␣兲¯˜a_␣ †_共

t,hn,␣兲⍀+S

共3.66兲

and

⌽_␣†_共

t, f2,␣兲⌽␣共t, f1,␣兲˜a_␣ †_共

t,h1,␣兲¯a˜_␣ †_共

t,hn,␣兲⍀=S, 共3.67兲

whereS is

兺

_j=1n 具f1,␣,hj,␣典˜a␣

†_共

t,h1,␣兲¯˜a␣ †_共

t,hj,␣兲

ˆ _¯˜a

␣ †_共

t,hk,␣兲˜a␣

†_共 t, f2,␣兲˜a␣

†_共

t,hk+1,␣兲¯a˜␣

†_共

t,hn,␣兲⍀.

This gives

关⌽+共t, f1,+兲,⌽+ †_共

t, f2,+兲兴+关⌽−共t, f1,−兲,⌽− †_共

t, f2,−兲兴=具f1,+, f2,+典+具f1,−, f2,−典=具f1, f2典, 共3.68兲

i.e., the desired contribution. It is then straightforward using共3.65兲to verify that the mixed terms do not contribute

关⌽+共t, f1,+兲,⌽− †_共

t, f2,−兲兴=关⌽−共t, f1,−兲,⌽+ †_共

t, f2,+兲兴= 0.

䊏

Now we prove that ⍀ is cyclic with respect to ⌽† _{and that ⌽共t , x兲 is the solution of the} quantum nonlinear Schrödinger equation with impurity. Extending the partial orderingⱭto func-tions inCas follows:

for f,g苸C, fⱭg⇔f_␣Ɑg_␣, ␣= ± , 共3.69兲

one can prove the following theorems.

Theorem 3.8: The space

H0共

n兲

= vect兵⌽†共t, f1兲¯⌽†共t, fn兲⍀; fi苸C, i = 1, . . . ,n, fnⱭ . . . Ɑf1其 共3.70兲 is dense inH共n兲_.

Proof: Let ␸共n兲_苸H共n兲_{and suppose}

具␸共n兲_,⌽†_{共t, f}

1兲¯⌽†共t, fn兲⍀典= 0, ∀fnⱭ ¯ Ɑf1. Then, it is true in particular for fi,−= 0, i = 1 , . . . , n but in that case, we have

⌽†_{共t, f} 1兲¯⌽

†_{共t, f}

n兲⍀= a˜+ †_{共t, f}

n,+兲¯˜a+ †_{共t, f}

1,+兲⍀

which implies␸共n兲_{= 0 sinceD}˜ 0

n,+_{is dense inH共}n兲_{. 䊏}

Theorem 3.9: The quantum field⌽is solution of the quantum nonlinear Schrödinger

equa-tion with impurity, i.e., it satisfies

共i⳵t+⳵x

2_{兲具␸,⌽共t,x兲␺典= 2g具␸,:⌽⌽}†_{⌽:共t,x兲␺典 共3.71兲}

and the following boundary conditions:

lim

x_→0+

具␸,兵⌽₊共t,x兲−⌽₋共t,− x兲其␺典= 0, 共3.72兲

lim

x→0+

⳵x具␸,兵⌽+共t,x兲+⌽−共t,− x兲其␺典= 2␩lim

x→0具␸

,⌽共t,x兲␺典, 共3.73兲

lim

x→±⬁具␸

for any␸,␺苸D.

Proof: Inspired by the classical case, we split the field as follows:

⌽共t,x兲=␪共x兲⌽₊共t,x兲+␪共− x兲⌽₋共t,x兲. 共3.75兲

The main difficulty here is to specify a normal ordering prescription for the analog of the cubic term. We adopt the prescription detailed in Ref. 32 for the normal ordering denoted :¯: and apply it to⌽_␣,␣= ±. Then following Ref. 32共theorem 5兲, one gets that the quantum field⌽_␣is solution of the nonlinear Schrödinger equation on the half-lineR␣: for all␸,␺苸D,

共i⳵t+⳵x

2_{兲具␸,⌽}

␣共t,x兲␺典= 2g具␸,:⌽␣⌽␣†⌽␣:共t,x兲␺典. 共3.76兲

The situation is now similar to the classical case and we have to check the quantum analog of

共2.12兲–共2.14兲. The idea lies again in realizing that Eqs.共3.72兲–共3.74兲 can be cast into a zeroth-order/linear problem. Following the line of argument of Ref. 32共theorem 6兲, one shows that given

␸,␺苸D, there exists␹苸H共1兲 _{such that 具␸,⌽共t , f兲␺典=具⍀,⌽共t , f兲␹典 and ␹ is independent of f.} This gives in particular具␸,⌽_␣共t , f_␣兲␺典=具⍀,⌽_␣共t , f_␣兲␹典,␣= ± and we can compute

具␸,⌽_␣共t,x兲␺典=具a˜_␣†共t,x兲⍀,␹典=

冕

R

dp

2␲e

ipx−ip2t_␹

␣共p兲. 共3.77兲

Then, Eqs.共3.72兲and共3.73兲are easily obtained using the property共3.21兲satisfied by␹. Finally, since ␹_␣苸L2_{共R兲, 具␸,⌽}

␣共t , x兲␺典 as a function of x is also in L2共R兲 and therefore vanishes at infinity. Noting that limx_→±_⬁具␸,⌽共t , x兲␺典= limx_→±_⬁具␸,⌽±共t , x兲␺典, we get共3.74兲. 䊏 We have finally achieved the goal of this section: we have explicitly constructed off-shell local fields for the quantum nonlinear Schrödinger system on the line in the presence of a transmitting and reflecting impurity. As mentioned in Ref. 7, this remained a challenging open problem for which we brought an answer here. In other words, the quantum inverse scattering method remains valid in the presence of an impurity provided that the ZF algebra is replaced by the RT algebra.

IV. SCATTERING THEORY

Scattering theory in the presence of an impurity was studied on general grounds in Ref. 7 by introducing the RT algebra which, being a generalization of the ZF and boundary algebras, is believed to prove fundamental also in the study of off-shell correlations functions and symmetries for 1 + 1-dimensional integrable systems with impurity.

In this section, we aim at giving some credit to this in the context of the nonlinear Schrödinger model. Indeed from the above results, we can get some insight in the correlations functions of the theory. The correlations functions vanish unless they involve the same number of⌽and⌽†_and for a given 2n-point function, we need at most the first共n − 1兲-order terms in the Rosales expan-sion of the field. This reads

具⍀,⌽共t1,x1兲¯⌽共tn,xn兲⌽†共tn+1,xn+1兲¯⌽†共t2n,x2n兲⍀典

=

兺

K艋n−1 L艋n−1

gK+L具⍀,⌽共k1兲共_t

1,x1兲¯⌽共kn兲共tn,xn兲⌽†共l1兲共tn+1,xn+1兲¯⌽†共ln兲共t2n,x2n兲⍀典, 共4.1兲

where K =兺_i=1n kiand L =兺i=1 n

liand the sum runs over all n-uplets共k1, . . . , kn兲,共l1, . . . , ln兲苸Z+

n

such that K , L艋n − 1.

具⍀,⌽共t1,x1兲⌽*共t2,x2兲⍀典=

冕

−⬁

+_⬁ dp

2␲e

−ip2t_12兵␪共x

1兲␪共x2兲关eipx12+ R共p兲eipx˜12兴+␪共− x1兲␪共− x2兲关eipx12 + R¯共p兲eipx˜_{12兴+␪共x}

1兲␪共− x2兲T共p兲eipx12+␪共− x1兲␪共x2兲¯T共p兲eipx12其.

共4.2兲

More importantly, using the Haag–Ruelle approach suitably, we can relate off-shell and asymptotic theories and, doing so, fill the gap of our quantum field theory. Indeed, on the one hand, we know from Ref. 7 that the Fock representation of the RT algebra generates the asymptotic states of a general integrable theory with impurity with corresponding S-matrix. On the other hand, in this paper we constructed off-shell local time-dependent fields whose behavior as t→±⬁we would like to know.

A. Asymptotic theory

The first step is to characterize wave packets for the free Schrödinger equation which take into account the presence of the impurity at x = 0. We adopt the following setup. For f苸C₀⬁共R兲, we define

ft共x兲=

冕

R

dp

2␲f共p兲e

ipx−ip2t_{. 共4.3兲}

We transpose the partial ordering共3.35兲to functions of the variable p. Definition 4.1: Given n , m艌1, consider two sets of functions

Hn=兵hi,␣_i苸C0⬁共R␣i兲, i = 1, . . . ,n其 andGm=兵gi,␤_i苸C0⬁共R

−␤_{i兲, i = 1, . . . ,m其, 共4.4兲} where the functions obey the following order prescriptions:

h1,␣₁Ɑ ¯ Ɑhn,␣_n, gm,␤_mⱭ ¯ Ɑg1,␤₁. 共4.5兲 We also define

h_i,␣

i

␪ _{共x兲=␪共␣}

ix兲hi,␣_i t _共

x兲, g_i,␤

i

␪ _{共x兲=␪共␤}

ix兲gi,␤_i t _共

x兲. 共4.6兲

By construction, h_i,␣

i

␪ _{共x兲 represent wave packets in R}␣i _{moving away from the impurity towards}

␣i⬁ while gi,␪␤_i共x兲 represent wave packets in R␤i moving towards the impurity. One already

un-derstands that they will be relevant for the so-called “out” and “in” states, respectively. In fact, this is the main theorem of this section for which we need some preliminary results.

From the preceding section, we know the exchange and commutation properties of⌽†and a˜† smeared with ordered functions in the variable x. Here, our wave packets were constructed from ordered functions in p but we made no assumption as to their ordering in x. Therefore, we must include all the possibilities and this requires the use of the permutation group of n elementsSn.

For␴苸Sn,␲苸Sm, n , m艌2, we introduce

␪h␴共␣1x1, . . . ,␣nxn兲=

兿

i,j=1 i⬍j

n

␪共␣␴ix␴i−␣␴jx␴j兲, 共4.7兲

␪g␲共␤1x1, . . . ,␤mxm兲=

兿

i,j=1 i⬎j m

␪共␤␲ix␲i−␤␲jx␲j兲, 共4.8兲

兺

␴苸S_n

␪h␴共␣1x1, . . . ,␣nxn兲= 1 =

兺

␲苸S_m

␪g␲共␤1x1, . . . ,␤mxm兲. 共4.9兲

Lemma 4.2: Given any two sets of functions inHn andGm,

共i兲 The following limits hold:

lim

t→+⬁储 h_1,␣

1

␪ _{丢 ¯ 丢h}

n,␣_n

␪ _{− h} 1,␣₁

t

丢 ¯ 丢h_n,␣

n

t _{储= 0,}

lim

t→−⬁储 g_1,␤

1

␪ _{丢 ¯ 丢g}

m,␤_m

␪ _{− g} 1,␤₁

t

丢 ¯ 丢g_m,␤

m

t _储

= 0. 共4.10兲

共ii兲 Let enbe the identity ofSnand let us define

H_␣ 1¯␣n

␴ _共x

1, . . . ,xn兲= h1,␪␣₁共x1兲¯hn,␪␣_n共xn兲␪h␴共␣1x1, . . . ,␣nxn兲,

G_␤ 1¯␤m

␲ _共x

1, . . . ,xm兲= g1,␪␤₁共x1兲¯gm,␪␤_m共xm兲␪g␲共␤1x1, . . . ,␤mxm兲. 共4.11兲

Then

lim

t→+⬁储 H_␣1¯␣

n

␴ _{储= 0, lim}

t→−⬁储 G_␤1¯␤

m

␲ _{储= 0 for all␴⫽e}

n, ␲⫽em. 共4.12兲

共iii兲 The following estimate is valid for any F苸L2_共Rn_兲,

冐

冕

Rn

dx1¯dxnF共x1, . . . ,xn兲˜a_␣

1 † _共

t,x1兲¯˜a_␣

n

† _共

t,xn兲

冐

艋

冑

n!储F储. 共4.13兲

Proof: The ideas are the same as those detailed in Ref. 32 from theorem 7 onwards and rest especially on the use of the weak limit

lim

t→±⬁ eitk

k ± i␧= 0. 共4.14兲

We just stress again that in our case all the above holds thanks to the use of the RT algebra and by paying careful attention to the support conditions encoded in共4.5兲. 䊏

We are now in position to identify the asymptotic behavior of the field as t→±⬁.

Theorem 4.3: The following limits hold in the strong sense in the Fock space F:

lim

t→+⬁⌽ †_共t,h

1,␣₁

␪ _兲¯⌽†_共t,h

n,␣_n

␪ _{兲⍀= a} ␣1 † _共

h1,␣₁兲¯a␣_n † _共

hn,␣_n兲⍀, 共4.15兲

lim

t→−⬁⌽ †_共t,g

1,␤₁

␪ _兲¯⌽†_共t,g

m,␤_m

␪ _{兲⍀= a} ␤1 † _共

g1,␤₁兲¯a_␤

m

† _共

gm,␤_m兲⍀. 共4.16兲

Proof: We note first that from 共3.75兲 one gets ⌽†共t , h_i,␣

i

␪ _兲=⌽ ␣i

†_{共t , h}

i,␣_i

␪ _{兲 and ⌽}†_{共t , g}

i,␤_i

␪ _兲

=⌽_␤

i

†_共 t , g_i,␤

i

␪ _{兲so that}

⌽†_共t,h

i,␣_i

␪ _{兲⍀= a˜} ␣i

†_共t,h

i,␣_i

␪ _{兲⍀ and⌽}†_共t,g

i,␤_i

␪ _{兲⍀= a˜} ␤i

†_共t,g

i,␤_i

␪ _{兲⍀. 共4.17兲}

Moreover, forf_␣苸C₀⬁共R␣兲, one has

储⌽†_{共t, f} ␣ ␪_{兲⍀− a}

␣ †_共

f_␣兲⍀储=储˜a_␣†共t, f_␣␪兲⍀− a˜†_␣共t, f_␣t兲⍀储艋储f_␣␪− f_␣t储, 共4.19兲

fplaying the role ofhorg. Now we want to compute the left-hand sides of Eqs.共4.15兲and共4.16兲 for n , m艌2. We give details for Eq. 共4.15兲,

⌽†_共t,h 1,␣₁

␪ _兲¯⌽†_共t,h

n,␣_n

␪ _兲⍀=

兺

␴苸Sn

冕

Rn

dx1¯dxnH_␣

1¯␣n

␴ _共x

1, . . . ,xn兲⌽_␣

1 † _共

t,x1兲¯⌽_␣

n

† _共 t,xn兲⍀

=

兺

␴苸Sn

冕

Rn

dx1¯dxnH␴␣₁¯␣_n共x1, . . . ,xn兲˜a␣_␴

1

† _共 t,x_␣

␴1兲¯

a ˜_␣

␴n

† _共 t,x_␣

␴n兲⍀

= a˜_␣ 1 † _共t,h

1,␣₁ ␪ _兲¯˜a

␣n

† _共t,h

n,␣_n

␪ _兲⍀+

兺

␴苸Sn

␴⫽e_n

冕

Rn

dx1¯dxnH_␣

1¯␣n

␴ _共x

1, . . . ,xn兲

⫻兵˜a_␣

␴1

† _共 t,x_␣

␴1兲¯

a ˜_␣

␴n

† _共 t,x_␣

␴n兲⍀

− a˜_␣ 1 † _共

t,h_1,␣ 1 ␪ _兲¯˜a

␣n

† _共 t,h_n,␣

n

␪ _兲⍀其,

共4.20兲

where we used point共iii兲of lemma 3.4 and共3.61兲for⌽†in the second equality. Applying共4.13兲 then gives

储⌽†_共t,h 1,␣₁

␪ _兲¯⌽†_共t,h

n,␣_n

␪ _{兲⍀− a} ␣1 † _共

h1,␣₁兲¯a␣_n † _共

hn,␣_n兲⍀储艋

冑

n!储h␪1,␣₁丢 ¯ 丢hn,␪␣_n− h1,␣₁

t

丢 ¯

丢h_n,␣

n

t _储

+ 2

冑

n!

兺

␴苸Sn

␴⫽e_n

储H_␣ 1¯␣n

␴ _{储, 共4.21兲}

implying共4.15兲by points共i兲–共ii兲of lemma 4.2 Similar computations give

储⌽†_共t,g 1,␤₁

␪ _兲¯⌽†_共t,g

m,␤_m

␪ _{兲⍀− a} ␤1 † _共

g1,␤₁兲¯a_␤

m

† _共

gm,␤_m兲⍀储艋

冑

m!储g␪1,␤₁丢 ¯ 丢gm,␪␤_m− g1,␤₁

t

丢 ¯ 丢g_m,␤

m

t _{储+ 2}

冑

_m!

兺

␲苸Sm

⫽e_m

储G_␤ 1¯␤m

␲ _{储, 共4.22兲}

proving共4.16兲. 䊏

B. Scattering matrix

Now that we have identified the natural “free” dynamics approached by our interacting field as t→±⬁, we are left with the verification of asymptotic completeness allowing the construction of a unitary S-matrix. We emphasize here that our “in” and “out” spaces are slightly different from those exhibited in Ref. 7 because of our ordering involving absolute values, so that we must recheck their properties.

Proposition 4.4: Let

Fin_{= vect兵⍀,a} ␤1 † _共

g1,␤₁兲¯a␤_m † _共

gm,␤_m兲⍀, ␤i= ± , i = 1, . . . ,m, m艌1其, 共4.23兲

Fout_{= vect兵⍀,a} ␣1 † _共

h1,␣₁兲¯a_␣

n

† _共

hn,␣_n兲⍀,␣i= ± , i = 1, . . . ,n, n艌1其, 共4.24兲

wherehi,␣_iand gj,␤_j run overHn andGm.

Then,Finand Foutare separately dense in F.