The propagator of the attractive delta-Bose gas in one dimension
aa r X i v : . [ m a t h - ph ] J a n The propagator of the attractive delta-Bose gas inone dimension
Sylvain Prolhac ‡ and Herbert Spohn § Zentrum Mathematik and Physik Department,Technische Universit¨at M¨unchen,D-85747 Garching, Germany
Abstract.
We consider the quantum δ -Bose gas on the infinite line. For repulsiveinteractions, Tracy and Widom have obtained an exact formula for the quantumpropagator. In our contribution we explicitly perform its analytic continuationto attractive interactions. We also study the connection to the expansion of thepropagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide anindependent proof of their completeness.PACS numbers: 02.30.Ik 05.30.Jp Keywords: delta-Bose gas, quantum propagator, Bethe ansatz, completeness, analyticcontinuation. ‡ [email protected] § [email protected] he propagator of the attractive delta-Bose gas in one dimension
1. Introduction
Quantum particles on the real line interacting through a δ -potential are governed bythe Hamiltonian H κ = − n X j =1 ∂ ∂x j − κ n X j 0, has beenstudied in great detail and we refer to [1, 2, 3, 4, 5, 6, 7]. The attractive case, κ > 0, hasreceived less attention. One reason is that the structure of the Bethe equations is morecomplicated. On top, physical applications are not obviously in reach. In the recentyears, there has been renewed interest. We have now available a detailed study of theeigenfunctions [8, 9, 10, 11] and, as argued by Calabrese and Caux [12], applicationsto real materials are in sight. A further motivation comes from the one-dimensionalKadar-Parisi-Zhang (KPZ) equation [13]. Its replica solution is given in terms of thepropagator of the attractive δ -Bose gas [14, 15] which can be used to obtain exactsolutions for some special initial conditions [16, 17, 18, 11, 19, 20, 21, 22, 23, 24, 25].In the KPZ context, and also in other cases, one is actually interested in thequantum propagator h x | e − tH κ | y i , t ≥ 0. In principle, e − tH κ can be expanded in a sum(integral) over eigenfunctions. But one might hope to have at disposal more conciseexpressions for the propagator. In the repulsive case, Tracy and Widom [26] carriedout such a program. The resulting expression we refer to as TW formula, which willbe discussed below, including its relation to the expansion in eigenfunctions. A naturalissue is to extend such a program to the attractive case, which is the topic of ourcontribution.By symmetry the propagator h x | e − tH κ | y i can be restricted to the domain Λ = { x | x ≤ . . . ≤ x n } ⊂ R n . Using the Bose symmetry, H κ of (1.1) is then defined by H κ ψ ( x , . . . , x n ) = − n X j =1 ∂ ∂x j ψ ( x , . . . , x n ) , x ∈ Λ ◦ , (1.2)with the boundary conditions (cid:18) ∂∂x j +1 − ∂∂x j + κ (cid:19) ψ ( x , . . . , x n ) | x j +1 = x j = 0 , (1.3)where the limit x j +1 = x j is taken from the interior, Λ ◦ , of Λ. The Hamiltonian H κ isa self-adjoint operator and h x | e − tH κ | y i is continuous in x, y ∈ Λ. In particular,lim t → h x | e − tH κ | y i = 1 n ! n Y j =1 δ ( x j − y j ) , x, y ∈ Λ . (1.4)Throughout the paper x, y ∈ Λ, hence the position of the particles are orderedincreasingly. As will be proved in Appendix A, h x | e − tH κ | y i is analytic in κ for otherwise he propagator of the attractive delta-Bose gas in one dimension κ ≤ 0, to κ > 0. As written, the TW formula becomes singular at κ = 0. Thereforethe main task is to understand the structure of the analytic continuation in κ . As aresult, we will arrive at various formulas for the propagator. One formula will be justthe expansion in Bethe ansatz eigenfunctions, which thus implies their completeness.The issue of completeness for the attractive δ -Bose gas on the line has beenstudied before. In his thesis, Stephen Oxford [27] proves completeness of thegeneralized eigenfunctions defined as bounded Bethe ansatz eigenfunctions. He usesfunctional analytic methods to construct the Hilbert space isometry from the generalizedeigenfunctions and thereby the spectral representation of H κ . A similar strategy is usedby Babbitt and Thomas [28] for the ground state representation of the ferromagneticHeisenberg model on Z . Heckman and Opdam [29] exploit the fact that the δ -Bose gasturns up in the representation theory of graded Hecke algebras. (We are grateful toBal´azs Pozsgay for pointing out this reference.) They have results for the case when theinteraction strength is allowed to be pair dependent. But only for H κ their expressionsimplifies and they arrive at a Plancherel formula, which is the completeness relation.For the system on the line, studied here, the set of admissible wave numbersis known explicitly. For a bounded system, in particular with periodic boundaryconditions, the discrete set of wave numbers are the solutions to the Bethe equations,a coupled system of n transcendental equations. Completeness becomes more difficultto establish and to our knowledge only for the repulsive case a completeness proof isavailable [30].The article is organized as follows. In Section 2, we recall the Tracy and Widomformula for the propagator in the repulsive case, and rewrite it in terms of Betheeigenstates. In Section 3, we summarize our main results on the propagator withattractive interactions. These results are proved in Section 4, by performing explicitlythe analytic continuation to κ > 0. A further rewriting represents the propagator interms of the known Bethe eigenstates. The special case of the propagator with allparticles starting and ending at 0 is handled in Section 5. In Appendix A, we provethat the (imaginary time) propagator is an analytic function of the coupling. δ -Bose gas with repulsive interaction ( κ < ) Let S n be the set of all n ! permutations of the integers between 1 and n . In the following,we use the notations n Y j 0. By analytic continuation of (2.2) from κ < κ > 0, we will derive in Section3 an exact expression for the propagator in the attractive case κ > D n,M the set of the M -tuples ~n = ( n , . . . , n M ) such that n j ≥ j = 1 , . . . , M and n + . . . + n M = n . For ~n ∈ D n,M , the clusters Ω j ( ~n ) ≡ Ω j , j = 1 , . . . , M are defined byΩ j = { n + . . . + n j − + 1 , . . . , n + . . . + n j } . (3.1)From the Bethe ansatz point of view, the clusters will correspond to bound states of theparticles. The function r ~n ≡ r , acting on { , . . . , n } , is defined by r ( a ) = s for a = n + . . . + n j − + s ∈ Ω j . (3.2)More visually, one has a . . . n n + 1 . . . n + n . . . n − n M + 1 . . . n Cluster Ω Ω . . . Ω M r ( a ) 1 . . . n . . . n . . . . . . n M . (3.3)Finally, we call S ′ n ( ~n ) (respectively S ′′ n ( ~n )) the subset of S n containing only thepermutations σ (resp. τ ) such that for all j = 1 , . . . , M and a, b ∈ Ω j with a < b one has σ − ( a ) < σ − ( b ) (resp. τ − ( a ) > τ − ( b )).In the attractive case, the following expression for the propagator is proved inSection 4. Theorem 1. For fixed ~n , let µ j , j = 1 , . . . , M , be arbitrary real numbers satisfying theconstraint − n j < µ j ≤ . (3.4) For κ > and x, y ∈ Λ , one has h x | e − tH κ | y i = n X M =1 κ n − M n ! M !(2 π ) M X ~n ∈ D n,M M Y j =1 ( n j !( n j − Z R M d q . . . d q M X σ ∈ S ′ n ( ~n ) X τ ∈ S ′′ n ( ~n ) M Y j =1 Y a ∈ Ω j (cid:16) e i( q j +i κ ( µ j + r ( a ) − x σ − a ) − y τ − a ) ) e − t ( q j +i κ ( µ j + r ( a ) − (cid:17) × M Y j, k = 1 j = k Y a ∈ Ω j b ∈ Ω k σ − ( a ) > σ − ( b ) τ − ( a ) < τ − ( b ) (cid:18) ( q j + i κ ( µ j + r ( a ))) − ( q k + i κ ( µ k + r ( b ))) + i κ ( q j + i κ ( µ j + r ( a ))) − ( q k + i κ ( µ k + r ( b ))) − i κ (cid:19) . (3.5) All the apparent poles in the integrand cancel except for simple poles at q j + i κµ j = q k + i κ ( µ k + n k ) and q j + i κ ( µ j + n j ) = q k + i κµ k , j < k . The integrand vanishes at q j + i κµ j = q k + i κµ k and q j + i κ ( µ j + n j ) = q k + i κ ( µ k + n k ) .he propagator of the attractive delta-Bose gas in one dimension n particles are arranged in M clusters of size n , . . . , n M , hence the extra summationsover M and ~n . Furthermore, Eq. (3.5) contains a summation over two permutations σ and τ instead of only one for the TW formula (2.2). In the special case x = y = 0discussed in Section 5, both summations over σ and τ can be eliminated.For the attractive case, the propagator can also be written in terms of a summationover the eigenstates of the Hamiltonian (1.1). The Bethe eigenfunctions for attractiveinteraction are (see [11], Eq. (B.26) and (B.48); in [11], r ( a ) is equal to r ( σ ( a )) withour notations, and Ω j to σ − (Ω j )) ψ ( x ; M, ~n, q ) = κ n − M √ n ! M Y j =1 q n j !( n j − X σ ∈ S ′ n ( ~n ) M Y j =1 Y a ∈ Ω j (cid:16) e i ( q j +i κ ( r ( a ) − nj − )) x σ − a ) (cid:17) × M Y j 12 ( n j − n j ) (cid:19) . (3.7)The relation of the propagator with the Bethe eigenfunctions is stated as next theoremin terms of (3.6) and (3.7). Theorem 2. For κ > and x, y ∈ Λ , one has h x | e − tH κ | y i = n X M =1 M !(2 π ) M X ~n ∈ D n,M Z R M d q . . . d q M ψ ( x ; M, ~n, q ) ψ ( y ; M, ~n, q ) e − tE ( M,~n,q ) . (3.8)As in the case of repulsive interaction discussed in Section 2, taking t = 0 yields thecompleteness relation for the Bethe eigenstates (3.6). Their orthonormality is proved in[11], Appendix B. 4. Analytic continuation from κ < to κ > κ > 0. Theorem 1 and Theorem 2 are proved. The contours of integration in (2.2) can be moved freely as long as the denominators q j − q k − i κ keep a strictly positive imaginary part. In particular, if the integration is he propagator of the attractive delta-Bose gas in one dimension q j ∈ R + i λ ( n − j ), j = 1 , . . . , n , with λ > 0, we obtain a formula valid for all κ such that ℑ ( q j − q k − i κ ) = ( k − j ) λ − κ > j < k , i.e. for all κ < λ . One obtains h x | e − tH κ | y i = 1 n !(2 π ) n n Y a =1 (cid:18)Z R +i λ ( n − a ) d q a (cid:19)X σ ∈ S n n Y a < bσ − ( a ) > σ − ( b ) q a − q b + i κq a − q b − i κ n Y a =1 (cid:16) e i q a ( x σ − a ) − y a ) e − tq a (cid:17) . (4.1)In the following, we want to further move the contours of integration, but this timethe contours will have to cross poles of the integrand, which will add several new termsresulting from the residues at these poles, symbolically, ✲ × = ✲ × ✚✙✛✘ ✛ (4.2)If one denotes by j → k the action of taking the residue at q j = q k + i κr with r ∈ Z ,the terms, obtained after moving the contours of integration, correspond to collectionsof j → k such that, for each ℓ = 1 , . . . , n , ℓ → . . . appears only once in the collection(since after taking the residue at q ℓ = q m + i κr , the integrand no longer contains q ℓ ).Each term thus corresponds to a forest (a set of trees), for example { → , → , → , → } ⇔ 12 7 4 35 6 ❄❆❆❯ ✁✁☛ ❄ (4.3)The particular trees obtained in this fashion depend on the order in which the contoursare moved. Here, we choose to move first the contour for q n − in such a way that itcrosses only the pole at q n − = q n + i κ . Then, we move the contour for q n − in such away that it crosses only the poles at q n − = q n + i κ and q n − = q n − + i κ (in which casewe still have an integration over both q n − and q n ), or only the pole at q n − = q n + 2i κ (in which case the residue at q n − = q n + i κ has been taken). We continue in this fashionuntil in the final step the contour for q is moved.In principle, after moving the contours, the propagator h x | e − tH κ | y i will be expressedas a sum over forests. In fact, it turns out that during this procedure there are manycancellations which remove all the forests which contain trees with “branches”: in otherwords, only the forests with merely “branchless” trees (like a , a → b , a → b → c , a → b → c → d , . . . ) remain after these cancellations. Instead of a sum overforests, we end up with a sum over partitions of { , . . . , n } (each element of the partitioncorresponding to one of the branchless trees of the forest).In the context of the distribution of the leftmost particle in the asymmetricsimple exclusion process, the procedure described here bears some similarity with the he propagator of the attractive delta-Bose gas in one dimension { , . . . , n } and not just over subsets of { , . . . , n } . Weexpect that in the case of the full transition probability for the asymmetric exclusionprocess an expression with an integration over large circles would require summing overall partitions of { , . . . , n } .A proof of the previous statements is based on induction w.r.t. an integer ℓ suchthat all the contours for q ℓ +1 , . . . , q n − have already been moved.We introduce a few notations. For a boolean condition c , { c } is defined to beequal to 1 if c is true and 0 otherwise. For ~n ∈ D n,M , the set P n ( ~n ) contains all thepartitions ~A = { A , . . . , A M } of { , . . . , n } with | A j | = n j , j = 1 , . . . , M . The partition ~A verifies A ∪ . . . ∪ A M = { , . . . , n } and for j = k A j ∩ A k = ∅ . The partitions are notordered, i.e. the partition ~B = { A R (1) , . . . , A R ( M ) } is considered to be the same elementof P n ( ~n ) as ~A for all R ∈ S M . Each A j is called a cluster , and will correspond to abound state of particles in the Bethe ansatz point of view. For a partition ~A , we define d ~A ( a ), a = 1 , . . . , n , (abbreviated as d ( a ) to lighten the notation) to be the rank of a inits cluster A j , starting with rank 0 for the largest element of the cluster, rank 1 for thesecond largest, . . . , and rank | A j | − A j .With these notation, the following lemma can be stated. Lemma 1. Let ℓ be an integer between and n − . For fixed M = 1 , . . . , n , let ǫ j , j = ℓ + 1 , . . . , M be distinct numbers with ≤ ǫ j < . Then, for < κ < λ one has h x | e − tH κ | y i = n X M =1 κ n − M n !(2 π ) M X ~n ∈ D n,M M Y j =1 ( n j !( n j − ℓ Y j =1 (cid:18)Z R +i λ ( n − j ) d q j (cid:19) × M Y j = ℓ +1 Z R − i κǫ j d q j ! X ~A ∈ P n ( ~n ) X σ ∈ S n ℓ Y j =1 { A j = { j }} M Y j =1 Y a, b ∈ A j a < b { σ − ( a ) >σ − ( b ) } × M Y j =1 Y a ∈ A j (cid:16) e i( q j +i κd ( a ))( x σ − a ) − y a ) e − t ( q j +i κd ( a )) (cid:17) × M Y j, k = 1 j = k Y a ∈ A j b ∈ A k a < bσ − ( a ) > σ − ( b ) (cid:18) ( q j + i κd ( a )) − ( q k + i κd ( b )) + i κ ( q j + i κd ( a )) − ( q k + i κd ( b )) − i κ (cid:19) . (4.4) Proof. The constraint on the ǫ j , j = ℓ + 1 , M , implies that (4.4) is well defined since allthe poles are at q j = q k + i δ with ℑ ( q j ) = ℑ ( q k ) + δ .For ℓ = n − 1, the identity between the expressions (4.1) and (4.4) of h x | e − tH κ | y i is immediate. All the clusters must have size 1, and only M = n contributes. Since thepoles for q n are at q n = q j − i κ , j = 1 , . . . , n − 1, the contour for q n can be moved freelyfrom R to R − i κǫ n , provided 0 ≤ ǫ n < κ > ℓ : we assume that he propagator of the attractive delta-Bose gas in one dimension h x | e − tH κ | y i is valid for ℓ ≥ ℓ replaced by ℓ − ~n , we want to move the contour of integration for q ℓ from R + i λ ( n − ℓ ) to R − i κǫ ℓ with 0 ≤ ǫ ℓ < ǫ ℓ different from all the other ǫ k , k = ℓ + 1 , . . . , M . In orderto accomplish this, one needs to take into account the residues of the poles at q ℓ = z for − κǫ ℓ < ℑ ( z ) < λ ( n − ℓ ). The only poles for q ℓ are at z = q j − i κ , j = 1 , . . . , ℓ − 1, andat z = q m + i κ ( d ( c ) + 1), m = ℓ + 1 , . . . , M , c ∈ A m . In the first case, using κ < λ and j ≤ ℓ − 1, one finds ℑ ( z ) > λ ( n − ℓ ), which implies that these poles do not contributewhen moving the contour for q ℓ . In the second case, using 0 < κ < λ , 0 ≤ ǫ m < ℓ + d ( c ) + 1 ≤ n , one has − κǫ ℓ ≤ < ℑ ( z ) < λ ( n − ℓ ), which implies that all thesepoles contribute a residue (with a factor − π corresponding to a clockwise contourintegration).Moving the contour for q ℓ produces several terms: one term corresponding to theintegration over q ℓ ∈ R − i κǫ ℓ , for which the integrand still depends on q ℓ , and one termfor each c ∈ A m , m = ℓ + 1 , . . . , M , for which the residue at q ℓ = q m + i κ ( d ( c ) + 1)has been taken. The latter term corresponds to merging the cluster A ℓ = { ℓ } and thecluster A m . Assuming σ − ( ℓ ) > σ − ( c ) (otherwise, the pole vanishes), this term is equalto ( − π ) κ n − M n !(2 π ) M M Y j =1 ( n j !( n j − ℓ − Y j =1 (cid:18)Z R +i λ ( n − j ) d q j (cid:19) M Y j = ℓ +1 Z R − i κǫ j d q j !X ~A ∈ P n ( ~n ) X σ ∈ S n ℓ Y j =1 { A j = { j }} M Y j =1 Y a, b ∈ A j a < b { σ − ( a ) >σ − ( b ) } × (cid:16) e i( q m +i κ ( d ( c )+1))( x σ − ℓ ) − y ℓ ) e − t ( q m +i κ ( d ( c )+1)) (cid:17) × M Y j = 1 j = ℓ Y a ∈ A j (cid:16) e i( q j +i κd ( a ))( x σ − a ) − y a ) e − t ( q j +i κd ( a )) (cid:17) × M Y j, k = 1 j = kj, k = ℓ Y a ∈ A j b ∈ A k a < bσ − ( a ) > σ − ( b ) (cid:18) ( q j + i κd ( a )) − ( q k + i κd ( b )) + i κ ( q j + i κd ( a )) − ( q k + i κd ( b )) − i κ (cid:19) × M Y j, k = 1 j = kj = ℓ Y a ∈ A j a < ℓσ − ( a ) > σ − ( ℓ ) (cid:18) ( q j + i κd ( a )) − ( q m + i κ ( d ( c ) + 1)) + i κ ( q j + i κd ( a )) − ( q m + i κ ( d ( c ) + 1)) − i κ (cid:19) × M Y j, k = 1 j = kk = ℓ Y b ∈ A k ℓ < bσ − ( ℓ ) > σ − ( b ) (cid:18) ( q m + i κ ( d ( c ) + 1)) − ( q k + i κd ( b )) + i κ ( q m + i κ ( d ( c ) + 1)) − ( q k + i κd ( b )) − i κ (cid:19) he propagator of the attractive delta-Bose gas in one dimension × Y b ∈ A m b = cσ − ( ℓ ) > σ − ( b ) (cid:18) ( q m + i κ ( d ( c ) + 1)) − ( q m + i κd ( b )) + i κ ( q m + i κ ( d ( c ) + 1)) − ( q m + i κd ( b )) − i κ (cid:19) × (( q m + i κ ( d ( c ) + 1)) − ( q m + i κd ( c )) + i κ ) . (4.5)The last line of (4.5) contributes a factor 2i κ and the line before contributes Y b ∈ A m b = cσ − ( ℓ ) > σ − ( b ) (cid:18) d ( c ) − d ( b ) + 2 d ( c ) − d ( b ) (cid:19) . (4.6)Let us first assume that c = min( A m ). Then, for all b ∈ A m , b = c one has σ − ( b ) < σ − ( c ). Together with σ − ( c ) < σ − ( ℓ ), it implies that all the elements ofthe cluster A m (except c ) contribute in (4.6). This results in a factor ( n m + 1) n m / − π ) and 2i κ , we obtain a factor 2 πκ ( n k + 1) n k . The term with c = min( A m ) thus corresponds exactly to the term of (4.4) with ℓ replaced by ℓ − ~A replaced by ~B , obtained from ~A by merging the cluster { ℓ } with A m (after a renaming of the q j , n j , ǫ j to q j − , n j − , ǫ j − for ℓ + 1 ≤ j ≤ M ).It remains to show that for c = min( A m ), the residues cancel each other. Sincethe σ − ( b ) are ordered in the same way as the d ( b ) for b ∈ A m , there exists a uniquenumber f ∈ A m such that for b ∈ A m , if b ≥ f then σ − ( b ) < σ − ( ℓ ), and if b < f then σ − ( b ) > σ − ( ℓ ). Since σ − ( c ) < σ − ( ℓ ), one has necessarily f ≤ c (or equivalently d ( f ) ≥ d ( c )). Then, (4.6) rewrites Y b ∈ A m b = cb ≥ f (cid:18) d ( c ) − d ( b ) + 2 d ( c ) − d ( b ) (cid:19) . (4.7)The rest of the argument depends on the relative values of d ( c ) and d ( f ). If d ( f ) ≥ d ( c ) + 2, then, there exists b ∈ A m such that b ≥ f and d ( b ) = d ( c ) + 2, thus (4.7) isequal to zero. Since d ( f ) ≥ d ( c ), the only cases left are d ( f ) = d ( c ) + 1 and d ( f ) = d ( c ),for which (4.7) rewrites respectively d ( c ) − d ( f ) + 2 d ( c ) − d ( f ) Y b ∈ A m b > c (cid:18) d ( c ) − d ( b ) + 2 d ( c ) − d ( b ) (cid:19) = − ( d ( c ) + 1)( d ( c ) + 2)2 , (4.8)and Y b ∈ A m b > c (cid:18) d ( c ) − d ( b ) + 2 d ( c ) − d ( b ) (cid:19) = ( d ( c ) + 1)( d ( c ) + 2)2 . (4.9)Let us call c ′ the element of A m such that d ( c ′ ) = d ( c ) + 1 ( c ′ is the smallest element of A m larger that c ). One notes that the two previous cases are exchanged when replacing σ − by σ − ◦ θ ℓ,c ′ , with θ ℓ,c ′ the permutation exchanging ℓ and c ′ . Thus, summing overall permutations σ , the residues at q ℓ = q m + i κ ( d ( c ) + 1) cancel. he propagator of the attractive delta-Bose gas in one dimension < κ < λ and ℓ between 0 and n − 1. In particular, for ℓ = 0, one has h x | e − tH κ | y i = n X M =1 κ n − M n !(2 π ) M X ~n ∈ D n,M M Y j =1 ( n j !( n j − × M Y j =1 Z R − i κǫ j d q j ! X ~A ∈ P n ( ~n ) X σ ∈ S n M Y j =1 Y a, b ∈ A j a < b { σ − ( a ) >σ − ( b ) } × M Y j =1 Y a ∈ A j (cid:16) e i( q j +i κd ( a ))( x σ − a ) − y a ) e − t ( q j +i κd ( a )) (cid:17) × M Y j, k = 1 j = k Y a ∈ A j b ∈ A k a < bσ − ( a ) > σ − ( b ) (cid:18) ( q j + i κd ( a )) − ( q k + i κd ( b )) + i κ ( q j + i κd ( a )) − ( q k + i κd ( b )) − i κ (cid:19) . (4.10)This expression no longer depends on λ . Hence, it is valid in the entire range κ > Exchanging A j and A k in (4.10) is the same as exchanging q j and q k , or ǫ j and ǫ k . Sincethe ǫ j are arbitrary numbers satisfying a constraint (0 ≤ ǫ j < ǫ j different)which is the same for all j , it is possible to add an extra sum over all permutations ofthe A j , compensated by a global factor 1 /M !. This is equivalent to summing now over ordered partitions ~A = ( A , . . . , A n ), such that for all R ∈ S M different from the identitypermutation, the ordered partition ~B = ( A R (1) , . . . , A R ( M ) ) is distinct from ~A .There exists a bijection between ordered partitions ~A such that | A j | = n j , j = 1 , . . . , M , and permutations τ ∈ S ′′ n ( ~n ) ( S ′′ n ( ~n ) is defined after Eq. (3.5)). Bythis bijection, the cluster A j is equal to { τ − ( a ) , a ∈ Ω j ( ~n ) } ≡ τ − (Ω j ) and one has1 + d ~A ( a ) = r ~n ( τ ( a )), using the definitions (3.1) and (3.2). Eq. (4.10) becomes h x | e − tH κ | y i = n X M =1 κ n − M n ! M !(2 π ) M X ~n ∈ D n,M M Y j =1 ( n j !( n j − × M Y j =1 Z R − i κǫ j d q j ! X σ ∈ S n X τ ∈ S ′′ n ( ~n ) M Y j =1 Y a, b ∈ τ − (Ω j ) a < b { σ − ( a ) >σ − ( b ) } × M Y j =1 Y a ∈ τ − (Ω j ) (cid:16) e i( q j +i κ ( r ( τ ( a )) − x σ − a ) − y a ) e − t ( q j +i κ ( r ( τ ( a )) − (cid:17) × M Y j, k = 1 j = k Y a ∈ τ − (Ω j ) b ∈ τ − (Ω k ) a < bσ − ( a ) > σ − ( b ) (cid:18) ( q j + i κr ( τ ( a ))) − ( q k + i κr ( τ ( b ))) + i κ ( q j + i κr ( τ ( a ))) − ( q k + i κr ( τ ( b ))) − i κ (cid:19) . (4.11) he propagator of the attractive delta-Bose gas in one dimension a and b by τ − ( a ) and τ − ( b ). We also replace σ by τ − ◦ σ . Because of the definition of S ′′ n ( ~n ), if a, b ∈ Ω j with τ − ( a ) < τ − ( b ) then a > b .This implies that the constraint with the { ... } is equivalent to σ ∈ S ′ n ( ~n ) (defined afterEq. (3.5)). We obtain h x | e − tH κ | y i = n X M =1 κ n − M n ! M !(2 π ) M X ~n ∈ D n,M M Y j =1 ( n j !( n j − M Y j =1 Z R − i κǫ j d q j !X σ ∈ S ′ n ( ~n ) X τ ∈ S ′′ n ( ~n ) M Y j =1 Y a ∈ Ω j (cid:16) e i( q j +i κ ( r ( a ) − x σ − a ) − y τ − a ) ) e − t ( q j +i κ ( r ( a ) − (cid:17) × M Y j, k = 1 j = k Y a ∈ Ω j b ∈ Ω k σ − ( a ) > σ − ( b ) τ − ( a ) < τ − ( b ) (cid:18) ( q j + i κr ( a )) − ( q k + i κr ( b )) + i κ ( q j + i κr ( a )) − ( q k + i κr ( b )) − i κ (cid:19) . (4.12) One has the factorization M Y j, k = 1 j = k Y a ∈ Ω j b ∈ Ω k σ − ( a ) > σ − ( b ) τ − ( a ) < τ − ( b ) (cid:18) ( q j + i κr ( a )) − ( q k + i κr ( b )) + i κ ( q j + i κr ( a )) − ( q k + i κr ( b )) − i κ (cid:19) = M Y j The expression ϕ κ ( x ; M, ~n, q ) M Y j 12 ( n j − n j ) − κ n j (cid:18) µ j + ( n j − (cid:19) . (4.28)The choice µ j = − ( n j − / ϕ κ ( x ; M, ~n, q + i κ~µ ) and ϕ κ ( y ; M, ~n, q + i κ~µ + i κ~n − i κ ) are complex conjugates of eachother. Introducing the Bethe eigenfunctions [11] ψ ( x ; M, ~n, q ) = ϕ κ (cid:18) x ; M, ~n, q − i κ ~n + i κ (cid:19) , (4.29)one finds for µ j = − ( n j − / h x | e − tH κ | y i = n X M =1 M !(2 π ) M X ~n ∈ D n,M Z R M d q . . . d q M ψ ( x ; M, ~n, q ) ψ ( y ; M, ~n, q ) M Y j =1 e − tn j q j + tκ ( n j − n j ) . (4.30)This is the result (3.8) of Theorem 2. he propagator of the attractive delta-Bose gas in one dimension 5. The special case x = y = 0In the special case x = y = 0, Eq. (4.30) simplifies. One has X σ ∈ S ′ n ( ~n ) M Y j Let f ( a, b ) be arbitrary complex coefficients. Then X σ ∈ S n sign( σ ) n Y a
6. Conclusions Exact formulas for the transition probability of the one-dimensional asymmetric simpleexclusion process, a non-equilibrium exactly solvable model, have been derived a fewyears ago [32, 33, 31]. Subsequently the method was adapted to obtain an exact formulafor the propagator of the quantum δ -Bose gas with repulsive interaction [26]. Here weanalytically continued this formula to the case of attractive interaction.An advantage of our approach, compared to the usual Bethe ansatz, is that thequestion of the completeness of the Bethe eigenfunctions can be completely avoided.In fact, such kind of exact expressions for the propagator can be used to prove thecompleteness of the Bethe ansatz, at least on the infinite line. In principle, it mightalso be possible to use the same kind of approach for the case of periodic boundaryconditions, see e.g. [33, 34].The exact expression for the propagator of the repulsive δ -Bose gas, derived in[26] and used here, bears some formal similarity with the coordinate Bethe ansatz,which is the original ansatz introduced by Bethe to diagonalize the Hamiltonian ofHeisenberg spin chain. Since then, other descriptions of eigenstates have been developed,in particular algebraic Bethe ansatz, which makes clearer the structures underlying thequantum integrability of such type of models. It would be of interest to understandwhether it is possible to write down the propagator using an approach closer to thealgebraic Bethe ansatz. Acknowledgments We thank Tomohiro Sasamoto for most instructive discussions. S.P. acknowledges thesupport through a DFG research project (SP181-24). he propagator of the attractive delta-Bose gas in one dimension Appendix A. Analyticity in the coupling of the propagator of the δ -Bose gas In the case of a standard Schr¨odinger operator of the form − ∆ + λV one can use Kato’stheory to establish that e − t ( − ∆+ λV ) is analytic in λ . The δ -potential corresponds toa boundary condition and we are not aware of a functional analytic argument for theholomorphic dependence on κ . Instead, we will use the Feynman-Kac representation. Proposition 1. For fixed x, y ∈ R n , t > , the function κ 7→ h x | e − tH κ | y i is holomorphicon C .Proof. By the Feynman-Kac formula one has the representation h x | e − tH κ | y i = E x,y (cid:0) e κX ( t ) (cid:1) p t ( x − y ) . (A.1)Here p t ( x − y ) is the Brownian motion transition kernel. The expectation E x,y is overthe standard Brownian bridge, b ( t ), starting at x and ending at y at time t . Let L j,k ( t ), j < k , be the local time at 0 for { b i ( s ) − b j ( s ) , ≤ s ≤ t } , i.e. L j,k ( t ) = Z t d s δ ( b j ( s ) − b k ( s )) . 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