A Generalized Hypergeometric Function II. Asymptotics and D4 ...

Commun. Math. Phys. 243, 389–412 (2003) Digital Object Identifier (DOI) 10.1007/s00220-003-0969-3

Communications in

Mathematical Physics

A Generalized Hypergeometric Function II. Asymptotics and D4 Symmetry S.N.M. Ruijsenaars Centre for Mathematics and Computer Science, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands Received: 28 November 2002 / Accepted: 22 May 2003 Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: In previous work we introduced and studied a function R(a+ , a− , c; v, v) ˆ that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function E(a+ , a− , γ ; v, v), ˆ with parameters γ ∈ C4 related to the couplings 4 c ∈ C by a shift depending on a+ , a− . We show that the E-function is invariant under all maps γ → w(γ ), with w in the Weyl group of type D4 . Choosing a+ , a− positive and γ , vˆ real, we obtain detailed information on the |Re v| → ∞ asymptotics of the E-function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R → E. Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . D4 -Invariance: First Steps . . . Asymptotics: The Key Results Proofs of Theorems 1.1 and 1.2

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389 395 399 406

1. Introduction This paper may be viewed as a continuation of our previous paper Ref. [1]. The “relativistic” hypergeometric function R(a+ , a− , c0 , c1 , c2 , c3 ; v, v) ˆ at issue here was first introduced in Ref. [2], as an eigenfunction of a relativistic Hamiltonian of Calogero-Moser type. This generalizes the fact that in suitable variables the 2 F1 -function is an eigenfunction of a nonrelativistic Hamiltonian of Calogero-Moser type. Likewise, the discretization of 2 F1 yielding the Jacobi polynomials generalizes to discretizations of R yielding the Askey-Wilson polynomials [3, 4]. Our lecture notes [2] and Ref. [1] (henceforth referred to as I) contain extensive background material and references to related work. More recent surveys pertinent to our R-function include Refs. [5, 6].

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We have occasion to invoke various results from I, some of which we begin by summarizing. This summary also serves to collect some definitions and notation from I, with an eye on making this paper and its sequel Ref. [7] somewhat more self-contained. We begin by recalling how R(a+ , a− , c; v, v) ˆ is defined via an integral involving our “hyperbolic gamma function” G(a+ , a− ; z) as a building block. First, we introduce quantities as ≡ min(a+ , a− ), al ≡ max(a+ , a− ), a ≡ (a+ + a− )/2, α ≡ 2π/a+ a− , (1.1) s1 ≡ c0 + c1 − a− /2, s2 ≡ c0 + c2 − a+ /2, s3 ≡ c0 + c3 ,

(1.2)

cˆ0 ≡ (c0 + c1 + c2 + c3 )/2.

(1.3)

Second, we take a+ , a− > 0 and c ∈ R4 unless explicitly stated otherwise. To ease the exposition, we also choose at first Re v and Re vˆ positive and s1 , s2 , s3 ∈ (−a, a). Then we can define the R-function by the contour integral 1 F (c0 ; v, z)K(c; z)F (cˆ0 ; v, ˆ z)dz, (1.4) R(c; v, v) ˆ = (a+ a− )1/2 C where F (b; y, z) ≡

G(z + y + ib − ia) G(z − y + ib − ia) , G(y + ib − ia) G(−y + ib − ia)

K(c; z) ≡

3 G(isj ) 1 . G(z + ia) G(z + isj )

(1.5)

(1.6)

j =1

(To unburden the notation, we often suppress the dependence on the parameters a+ , a− .) As concerns the G-function occurring in these formulas, we recall from I Appendix A that it can be written as G(a+ , a− ; z) = E(a+ , a− ; z)/E(a+ , a− ; −z),

(1.7)

where E(z) is an entire function (closely related to Barnes’ double gamma function) vanishing solely in the points + ≡ ia + zkl , zkl

k, l ∈ N,

(1.8)

where zkl ≡ ika+ + ila− .

(1.9)

Thus G has the same zeros as E and poles located only at − + ≡ −zkl , zkl

k, l ∈ N.

(1.10)

As a consequence of these pole/zero features, the function K has four upward pole sequences for z ∈ i[0, ∞), whereas F has two downward pole sequences starting at z = ±y − ib.

Generalized Hypergeometric Function II

391

Finally, the contour C is given by a horizontal line Im z = d, indented (if need be) so that it passes above the points −v − ic0 , −vˆ − i cˆ0 in the left half plane and the points v −ic0 , vˆ −i cˆ0 in the right half plane, and so that it passes below 0. Thus the four upward pole sequences of the integrand lie above C and the four downward ones lie below C. The integrand has exponential decay as |Re z| → ∞, uniformly for Im z in compact subsets of R, so that the choice of d is immaterial. The analyticity properties of the R-function are known in considerable detail. In particular, it extends to a meromorphic function in all of its eight arguments, provided a+ , a− stay in the right half plane (RHP). Moreover, the (eventual) pole locations are explicitly known. Specifically, introducing new parameters γ0 ≡ c0 − a+ /2 − a− /2, γ1 ≡ c1 − a− /2, γ2 ≡ c2 − a+ /2, γ3 ≡ c3 ,

(1.11)

and defining γˆ0 , . . . , γˆ3 by (1.16) below, the function ˆ Rren (a+ , a− , c; v, v)

3

E(a+ , a− ; δv + iγµ )E(a+ , a− ; δ vˆ + i γˆµ ),

(1.12)

G(a+ , a− ; isj )−1 · R(a+ , a− , c; v, v), ˆ

(1.13)

δ=+,− µ=0

where ˆ ≡ Rren (a+ , a− , c; v, v)

3 j =1

is analytic in RHP2 × C6 , cf. I Theorem 2.2. Hence Rren can only have poles for + + δv = −iγµ + zkl , δ vˆ = −i γˆµ + zkl , δ = +, −, µ = 0, 1, 2, 3, k, l ∈ N. (1.14)

The factors G(isj ) in K are convenient for normalization purposes. We are taking them out in the renormalized function Rren , however, since they give rise to poles (and zeros) that are independent of the variables v and v. ˆ These factors are also absent in the E-function, which is the main object of study in this paper. We now proceed to define this function. To this end we introduce the c-function c(a+ , a− , p; y) ≡

3 1 G(a+ , a− ; y − ipµ ), p ∈ C4 , G(a+ , a− ; 2y + ia)

(1.15)

µ=0

and dual parameters pˆ ≡ Jp, where J is the self-adjoint and orthogonal matrix   1 1 1 1 1  1 1 −1 −1  J ≡  . 2 1 −1 1 −1 1 −1 −1 1

(1.16)

(1.17)

Notice that this entails cˆ and γˆ are again related by (1.11). Introducing the function 2 2 χ (a+ , a− , p) ≡ exp(iα[p · p/4 − (a+ + a− + a+ a− )/8]), p ∈ C4 ,

(1.18)

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we now set E(a+ , a− , γ ; v, v) ˆ ≡ χ (a+ , a− , γ )Rren (a+ , a− , c; v, v)/c(a ˆ ˆ + , a− , γ ; v)c(a+ , a− , γˆ ; v).

(1.19)

The E-function just defined has various symmetry properties that are readily established from its definition. These include (cf. the paragraph in I containing Eq. (2.7)): ˆ = E(a+ , a− , γ ; v, v), ˆ λ > 0, (scale invariance), (1.20) E(λa+ , λa− , λγ ; λv, λv) ˆ = E(a+ , a− , γ ; v, v), ˆ E(a− , a+ , γ ; v, v)

(parameter symmetry),

ˆ v) = E(a+ , a− , γ ; v, v), ˆ E(a+ , a− , γˆ ; v,

(self − duality),

ˆ = −u(a+ , a− , γ ; v)E(a+ , a− , γ ; v, v), ˆ E(a+ , a− , γ ; −v, v)

(1.21) (1.22)

(reflection symmetry), (1.23)

where the u-function is defined by u(a+ , a− , p; y) ≡ −c(a+ , a− , p; y)/c(a+ , a− , p; −y).

(1.24)

Using the relation (cf. (1.2), (1.11)) sj = γ0 + γj + a,

j = 1, 2, 3,

(1.25)

it is also straightforward to check that E is invariant under any permutation of γ1 , γ2 , γ3 . In fact, however, E has a far stronger “hidden” γ -symmetry. Specifically, E is invariant not only under all permutations of γ0 , γ1 , γ2 , γ3 , but also under flipping the sign of any pair of γµ ’s. As is well known, the resulting invariance group, which we denote by W , is the Weyl group of the Lie algebra D4 . We state the symmetry just explained in the following theorem, which is a principal result of this paper. Theorem 1.1 (D4 symmetry). The E-function satisfies ˆ = E(a+ , a− , γ ; v, v), ˆ ∀w ∈ W. E(a+ , a− , w(γ ); v, v)

(1.26)

Morally speaking, the D4 invariance of E follows from its being a joint eigenfunction of four independent AOs (analytic difference operators) that are W-invariant. These AOs arise by similarity transforming the four Askey-Wilson type AOs from I with the above c-function, and their W-invariance is an obvious corollary of Lemmas 2.1 and 2.2. We would like to stress that the Askey-Wilson polynomials do not exhibit D4 -invariance for any choice of normalization. The three-term recurrence obtained by taking vˆ = i cˆ0 + ina− , n ∈ N,

(1.27)

in the pertinent R-function AO is not even S4 -invariant, cf. Eqs. (3.33)–(3.39) in I. (More precisely, it cannot be rendered S4 -invariant via any change of parameters such as (1.11).) For the similarity-transformed AO, however, this discretization of vˆ gives rise to a three-term recurrence of the form a0 c1 P1 + b0 P0 = 2 cos(2rv)P0 , an cn+1 Pn+1 + bn Pn + Pn−1 = 2 cos(2rv)Pn , n > 0, (1.28)


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instead of I (3.34), with an , bn , cn given by I (3.37)–(3.39). The S4 -invariance of the new recurrence coefficients (hence of the resulting polynomials with P0 ≡ 1) now follows from Lemmas 2.1 and 2.2. To be sure, S4 -invariance with this normalization is not a new result. Indeed, Askey and Wilson already pointed out S4 -invariance for their polynomials with a normalization that is connected to the above normalization by a manifestly S4 -invariant similarity, cf. Eqs. (1.24)–(1.27) in Ref. [3]. From our viewpoint, the reason that the D4 -symmetry of the E-function breaks down to S4 -symmetry of the corresponding Askey-Wilson polynomials is the non-invariance of the v-discretization ˆ (1.27) under any sign changes of γ0 , γ1 , γ2 and γ3 . (Recall that cˆ0 equals µ γµ /2 + a.) Even so, within the polynomial context the matrix J (1.17) is already important to understand self-duality properties, and this fact generalizes to the many-variable AskeyWilson (or Koornwinder [8]) polynomials, cf. Ref. [9]. This may be viewed as a hint for the existence of a D4 -symmetric interpolation (which for the case of one variable does exist, as transpires from our results). To appreciate this, recall that D4 admits outer automorphisms connecting the defining and spinor representations of SO(8) (“triality”). In this picture, the matrix J connects the weights of the defining representation and the even spinor representation, as is readily verified. It is therefore a natural question whether other interpolations of the Askey-Wilson polynomials can also be made D4 -symmetric by a suitable similarity. Specifically, we are thinking of the interpolations of Grünbaum and Haine [10], and the special linear combination of the Ismail-Rahman functions [11] studied by Suslov [12, 13] and by Koelink/Stokman [14]. (See Ref. [5] for a comparison of the latter interpolations to our R-function.) The same question can be asked for the |q| = 1 Askey-Wilson function recently introduced by Stokman [15]. We proceed to comment on our proof of Theorem 1.1. As already indicated, the D4 invariance of the four AOs for which E is an eigenfunction “explains” why E is itself D4 -invariant. For a complete proof, though, we cannot extract enough information from the joint eigenfunction property. For one thing, very little is known in general about joint eigenspaces of commuting AOs. For another, even for the case at hand, where we know that E(γ ; v, v) ˆ and E(w(γ ); v, v), ˆ w ∈ W , satisfy the same four AEs (analytic difference equations), there is no general result yielding proportionality. (See Sect. 1 of Ref. [16] for an appraisal of the general situation.) As will become clear below, the main problem to prove Theorem 1.1 consists in showing that a Casorati determinant of E(γ ; v, v) ˆ and E(w(γ ); v, v) ˆ vanishes identically. The pertinent result of Sect. 2 is that a certain (a priori unknown) periodic multiplier in the Casorati determinant has no poles (Lemma 2.3). The known pole locations (1.14) of Rren are the key to obtain this result. However, we are not able to prove within the “algebraic” context of Sect. 2 that the multiplier actually vanishes. We can only show this in Sect. 4, after establishing (in Sect. 3) the asymptotic behavior of the E-function as Re v → ∞ in a suitable strip. The relevant result (Lemma 3.2) involves a restriction on the parameters, which is however solely a consequence of our proof strategy. Indeed, after using Lemma 3.2 to complete the proof of Theorem 1.1, the resulting W -invariance of E can be exploited to extend the domain of validity of our asymptotics results to arbitrary parameters. More precisely, defining the parameter set ≡ (0, ∞)2 × R4 , the restricted parameter set

(1.29)

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∗ ≡ \ {(a, a, 0, 0, 0, 0) | a > 0},

(1.30)

and the leading asymptotics function ˆ ≡ exp(iαv v) ˆ − u(a+ , a− , γˆ ; −v) ˆ exp(−iαv v), ˆ α = 2π/a+ a− , Eas (a+ , a− , γ ; v, v) (1.31) we obtain the following theorem. (We only detail the Re v → ∞ asymptotics of E; thanks to the reflection symmetry (1.23) and the u-asymptotics (3.6), the Re v → −∞ asymptotics is immediate from this.) Theorem 1.2 (Asymptotics). Letting (σ, a+ , a− , γ , δ, v) ˆ ∈ [1/2, 1) × × (0, ∞)2 , we have ˆ < C(σ, a+ , a− , γ , δ, v) ˆ exp(−σ αas v), |(E − Eas )(a+ , a− , γ ; v, v)|

(1.32)

for all v > δ. Here, C is a positive continuous function on [1/2, 1) × × (0, ∞)2 . Next, let (a+ , a− , γ , δ, v) ˆ ∈ ∗ × (0, ∞)2 . Then we have ˆ < C(a+ , a− , γ , δ, Im v, v) ˆ exp(−ρ(a+ , a− , γ )Re v), |(E − Eas )(a+ , a− , γ ; v, v)| (1.33) for all v ∈ C satisfying Re v > δ. Here, C is a positive continuous function on ∗ × (0, ∞) × R × (0, ∞) and ρ is a positive continuous function on ∗ . Finally, let a+ = a− = a, γ = 0 and (σ, a, δ, τ, v) ˆ ∈ [1/2, 1) × (0, ∞)2 × [0, 1) × (0, ∞). Then we have ˆ < C(σ, a, δ, τ, v) ˆ exp(−σ (1 − τ )αaRe v), |(E − Eas )(a, a, 0; v, v)|

(1.34)

for all v ∈ C satisfying Re v > δ, |Im v| ≤ τ a,

(1.35)

with C continuous on [1/2, 1) × (0, ∞)2 × [0, 1) × (0, ∞); moreover, |E(a, a, 0; v, v)| ˆ < C(a, δ, Im v, v), ˆ

(1.36)

for all v ∈ C satisfying Re v > δ, with C continuous on (0, ∞)2 × R × (0, ∞). Admittedly, our proofs of Theorems 1.1 and 1.2 are not exactly straightforward. Of course, we cannot exclude the existence of more direct proofs, avoiding the entanglement of W-invariance and asymptotics that characterizes our proof strategy. In any event, we have tried to render our reasoning more accessible (and possibly more amenable to shortcuts) by opting for an exposition that isolates readily understood key features before turning to detailed technical arguments. At the end of Sect. 4 we also indicate why a direct proof of Theorem 1.2 for arbitrary γ ∈ R4 and Im v ∈ R seems intractable. To conclude this introduction, we would like to mention that the result of Theorem 1.1 is by its nature “best possible”, whereas it is likely that the bounds in Theorem 1.2 are not optimal. (For example, there is evidence that (1.33) holds true with ρ = σ αas , σ ∈ [1/2, 1), for all (a+ , a− , γ ) ∈ , but we are only able to prove this for Im v = 0, cf. (1.32).) On the other hand, the results encoded in Theorem 1.2 are sufficiently strong to handle problems arising in the Hilbert space context of our next paper (Ref. [7]) in this series.


395

2. D4 -Invariance: First Steps We begin by recalling from I that R(a+ , a− , c; v, v) ˆ is a joint eigenfunction of four Askey-Wilson type AOs, two acting on v and two on v, ˆ cf. I (3.1)–(3.5). It transpires from I (3.3) why the parameters cµ , µ = 0, 1, 2, 3, can be viewed as coupling constants: When they all vanish, the coefficients of the AOs are constant. On the other hand, this parametrization breaks symmetry properties that only become visible in terms of the shifted parameters γµ . Therefore, we switch to AOs with coefficients depending on γ , obtained from their counterparts in loc. cit. via the parameter change (1.11). To be specific, we first define the coefficient function

4 3µ=0 cosh(π[y − ipµ − ia− /2]/a+ ) C(a+ , a− , p; y) ≡ − , (2.1) sinh(2πy/a+ ) sinh(2π [y − ia− /2]/a+ ) which is manifestly invariant under permutations of the four parameters pµ . Now we define the AO A(a+ , a− , p; y) ≡ C(a+ , a− , p; y) exp(−ia− ∂/∂y) + (y → −y) + Vb (a+ , a− , p; y), (2.2) where Vb (a+ , a− , p; y)

  3 π  pµ + a −  . ≡ −C(a+ , a− , p; y) − C(a+ , a− , p; −y) − 2 cos a+

(2.3)

µ=0

Then the fourAOs I (3.2) amount to A(a+ , a− , γ ; v), A(a− , a+ , γ ; v), A(a+ , a− , γˆ ; v) ˆ and A(a− , a+ , γˆ ; v). ˆ Furthermore, their action on the meromorphic function R(a+ , a− , c(γ ); v, v) ˆ yields eigenvalues 2 cosh(2π v/a ˆ + ), 2 cosh(2π v/a ˆ − ), 2 cosh(2π v/a+ ) and 2 cosh(2π v/a− ), respectively. It is plain from the above definitions that the “external field” Vb (p; y) (2.3) is S4 -invariant in p. It is not obvious, but true that it is actually D4 -invariant. This stronger symmetry is manifest from a second formula for Vb , obtained in the next lemma. Lemma 2.1. We have Vb (a+ , a− , p; y) =

d− (p/a+ ) cosh(2πy/a+ ) + d+ (p/a+ ) cos(π a− /a+ ) , (2.4) sinh(2π[y − ia− /2]/a+ ) sinh(2π [y + ia− /2]/a+ )

with d± (p) ≡ C(p) ± S(p),

S(p) ≡ 4

3

(2.5)

sin(πpµ ),

(2.6)

cos(πpµ ).

(2.7)

µ=0

C(p) ≡ 4

3 µ=0

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Proof. Clearly, the functions on the right-hand sides of (2.3) and (2.4) both have period ia+ and limit 0 as |Re y| → ∞. Thus we need only show equality of residues at their poles in a period strip, choosing a+ /a− irrational to ensure the poles are simple. To this end, we note first that the poles due to the factors sinh(±2πy/a+ ) in (2.3) cancel by virtue of evenness. Likewise, by evenness it suffices to compare residues at y = ia− /2 and y = ia. Their equality amounts to two equations that are solved by (2.5)–(2.7).

Obviously, A(a+ , a− , p; y) is S4 -invariant, but not D4 -invariant. (Indeed, the coefficients of the shifts are not invariant under any sign flips, cf. (2.1).) We now define a similarity-transformed AO that is D4 -invariant. Specifically, we set A(a+ , a− , p; y) ≡ c(a+ , a− , p; y)−1 A(a+ , a− , p; y)c(a+ , a− , p; y).

(2.8)

Then we have the following explicit formula for A. Lemma 2.2. The AO (2.8) can be written A(a+ , a− , p; y) = exp(−ia− ∂/∂y) + Va (a+ , a− , p; y) exp(ia− ∂/∂y) +Vb (a+ , a− , p; y),

(2.9)

with Va (a+ , a− , p; y)

16 3µ=0 cosh(π[y + ipµ + ia− /2]/a+ ) cosh(π [y − ipµ + ia− /2]/a+ ) . (2.10) ≡ sinh(2πy/a+ ) sinh(2π[y + ia− /2]/a+ )2 sinh(2π [y + ia− ]/a+ ) Proof. We recall the G-function satisfies the AEs G(a+ , a− ; z + iaδ /2) = 2 cosh(π z/a−δ ), G(a+ , a− ; z − iaδ /2)

δ = +, −.

(2.11)

Using the δ = − AE and the definition (1.15) of the c-function, we obtain C(a+ , a− , p; y) = c(a+ , a− , p; y)/c(a+ , a− , p; y − ia− ).

(2.12)

Thus the coefficient of the shift y → y − ia− in (2.8) equals 1, as asserted in (2.9). Moreover, for the coefficient of the y-shift over ia− we obtain Va (y) = C(−y)C(y + ia− ), and hence (2.10) follows from (2.1).

(2.13)

It is immediate from Lemmas 2.1 and 2.2 that A(a+ , a− , p; y) is invariant under taking p → w(p) for all w ∈ W . Therefore, E(a+ , a− , w(γ ); v, v), ˆ w ∈ W , is a joint eigenfunction of the four AOs A(a+ , a− , γ ; v), A(a− , a+ , γ ; v), A(a+ , a− , γˆ , v), ˆ A(a− , a+ , γˆ ; v), ˆ denoted briefly as A+ , A− , Aˆ + , Aˆ − , resp. For later purposes we note that due to (2.12)–(2.13) we have Va (a+ , a− , p; y) = u(a+ , a− , p; y + ia− )/u(a+ , a− , p; y),

(2.14)

with u defined by (1.24). We also observe that (2.10) entails invariance of Va under arbitrary sign flips of pµ . (This is not true for Vb , since S(p) (2.6) changes sign under an odd number of flips.)


397

Next, we point out the crucial relation J W J = W,

(2.15)

which is readily verified. More specifically, it is useful to note that a permutation of γ1 , γ2 , γ3 yields the same permutation of γˆ1 , γˆ2 , γˆ3 , whereas the transpositions γ0 ↔ γj transform as follows: γ → (γ1 , γ0 , γ2 , γ3 ) ⇒ γˆ → (γˆ0 , γˆ1 , −γˆ3 , −γˆ2 ),

(2.16)

γ → (γ2 , γ1 , γ0 , γ3 ) ⇒ γˆ → (γˆ0 , −γˆ3 , γˆ2 , −γˆ1 ),

(2.17)

γ → (γ3 , γ1 , γ2 , γ0 ) ⇒ γˆ → (γˆ0 , −γˆ2 , −γˆ1 , γˆ3 ).

(2.18)

As a consequence, the dual c-function c(γˆ ; v) ˆ occurring in the definition (1.19) of the E-function is invariant under permutations leaving γ0 fixed, but not under arbitrary permutations, in contrast to the c-function c(γ ; v). Before turning to the W-invariance of E announced above, it is expedient to define a fundamental domain DW for the W-action on R4 . This domain may be viewed as the closure of a Weyl chamber, and it is particularly suited for our later purposes. Specifically, we set DW ≡ {γ ∈ R4 | γ0 ≤ γ1 ≤ γ2 ≤ 0, γ3 ∈ [γ2 , −γ2 ]}.

(2.19)

It is readily verified that this entails J DW = DW .

(2.20)

For the remainder of this section we fix vˆ ∈ (0, ∞) and parameters satisfying a+ , a− , γ0 , γ1 , γ2 , γ3 linearly independent over Q.

(2.21)

γ (j ) ≡ wj (γ ), wj ∈ W, j = 1, 2, w1 = w2 ,

(2.22)

Ej (v) ≡ E(a+ , a− , γ (j ) ; v, v), ˆ j = 1, 2.

(2.23)

Next, we set

and introduce

We are now going to study the Casorati determinant D(v) ≡ E1 (v + ia− /2)E2 (v − ia− /2) − (i → −i),

(2.24)

pertinent to the eigenvalue AEs A+ Ej (v) = 2 cosh(2π v/a ˆ + )Ej (v), j = 1, 2.

(2.25)

We need some well-known features of Casorati determinants, detailed for example in Sect. 1 of Ref. [16]. Our goal is to show that D(v) vanishes identically. We are only able to prove this by obtaining a contradiction from the assumption that this is not the case. Furthermore, we can only arrive at this contradiction in Sect. 4, after determining the |Re v| → ∞ asymptotics of E in Sect. 3.

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In this section, however, we derive a key feature of D(v), assuming from now on D(v) is not identically 0. For a start, this assumption together with (2.25) entails that D(v) satisfies the first order AE, D(v + ia− /2) 1 = . D(v − ia− /2) Va (a+ , a− , γ ; v)

(2.26)

(Cf. Eqs. (1.1)–(1.6) in Ref. [16].) By (2.14), it now follows that the function m(v) ≡ −D(v)u(a+ , a− , γ ; v + ia− /2)

(2.27)

is a meromorphic ia− -periodic function. Since we explicitly know the (eventual) poles of D(v) and the poles of the G-function, we are able to show that m(v) cannot have poles. This is the gist of the next lemma and its proof, with which we conclude this section. Lemma 2.3. Assume D(v) does not vanish identically. Then m(v) is an entire ia− -periodic function that does not vanish identically. Proof. We have already seen that m(v) is an ia− -periodic meromorphic function. To show that m(v) is pole-free, we first note that from (1.24), (1.15) and the reflection equation G(−z) = 1/G(z) (cf. (1.7)), we have −u(a+ , a− , γ ; v + ia− /2) = G (v)/G(2v + ia− + ia)G(2v + ia− − ia), (2.28) with G (v) ≡

3

G(v + ia− /2 + iδγµ ).

(2.29)

δ=+,− µ=0

We now introduce E˜j (v) ≡ Ej (v)/G(2v + ia), j = 1, 2,

(2.30)

˜ D(v) ≡ E˜1 (v + ia− /2)E˜2 (v − ia− /2) − (i → −i),

(2.31)

and rewrite m(v) as ˜ m(v) = G (v)D(v)G(2v − ia− + ia)/G(2v + ia− − ia) ˜ sinh(2πv/a− )/ sinh(2π v/a+ ), = G (v)D(v)

(2.32)

where we used the G-AEs (2.11). ˜ Next, we study the poles of G (v) and D(v). Clearly, the poles of G (v) are located at v = iδγµ − ia+ /2 − ika+ − ila− , δ = +, −, µ = 0, 1, 2, 3, k ∈ N, l ∈ N∗ , (2.33) ˜ cf. (1.7)–(1.10). Turning to D(v), we deduce using (2.23), (1.19), (1.15) and (1.7) that we have E˜j (v) = χ (γ )c(J γ (j ) ; v) ˆ −1 Rren (c(γ (j ) ); v, v) ˆ

3 (j ) E(−v + iγµ ) (j )

µ=0

E(v − iγµ )

.

(2.34)


399

Now the function v → Rren (c(γ (j ) ); v, v) ˆ

3

E(−v + iγµ(j ) )E(v + iγµ(j ) )

(2.35)

µ=0

is entire, cf. the paragraph containing (1.11). Therefore, E˜j (v) can only have poles at the zero locations of the function 3

3

E(v − iδγµ(j ) ) =

δ=+,− µ=0

E(v − iδγµ ).

(2.36)

δ=+,− µ=0

˜ These are given by iδγµ + ia + zkl , cf. (1.8), so we finally conclude that D(v) can only have poles at v = iδγµ + ia+ /2 + ika+ + ila− , δ = +, −, µ = 0, 1, 2, 3, k, l ∈ N.

(2.37)

The upshot is that eventual poles of m(v) sinh(2π v/a+ ) must be located at the points (2.33) and (2.37). Let us now assume that m(v) has a pole at v = v0 , so as to derive a contradiction. To begin with, by ia− -periodicity our assumption entails that m(v) has poles at all points v = v0 + ij a− ,

j ∈ Z.

(2.38)

Only one of these poles can be matched by a pole of the factor 1/ sinh(2π v/a+ ) in (2.32), since we have a+ /a− ∈ / Q. Save for at most one pole, therefore, all poles (2.38) must be located at (2.33) or (2.37). Furthermore, since (2.37) consists of upward pole sequences and (2.33) of downward ones, the poles (2.38) must be located at (2.37) for j → ∞ and at (2.33) for j → −∞. From this we see that iv0 can be written in two ways as a Q-linear combination of a+ , a− , γ , with distinct coefficients of a+ . In view of our requirement (2.21), this yields the desired contradiction.

3. Asymptotics: The Key Results In this paper we focus on the asymptotic behavior of E(a+ , a− , γ ; v, v) ˆ for |Re v| → ∞ with the parameters a+ , a− positive. In this case the |Re z| → ∞ asymptotics of G(a+ , a− ; z) obtained in I Theorem A.1 simplifies considerably. Indeed, in I (A.23)– (A.24) we have φ± = φmax = φmin = 0, and the asymptotics domain I (A.32) reduces to A ≡ {z ∈ C | Re z > al }, al = max(a+ , a− ).

(3.1)

Setting 2 2 + a− )/48 + f (a+ , a− ; z)]), z ∈ ±A, (3.2) G(a+ , a− ; z) = exp(∓iα[z2 /4 + (a+

it now follows from I Theorem A.1 that for σ ∈ [1/2, 1) we have |f (a+ , a− ; z)| < C(σ, a+ , a− , Im z) exp(−σ αas |Re z|), as = min(a+ , a− ), z ∈ ±A, (3.3)

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S.N.M. Ruijsenaars

where C is continuous on [1/2, 1)×(0, ∞)2 ×R. (Here and below, we find it convenient to formulate decay bounds that are uniform on compact subsets of a given set S in terms of positive continuous functions on S, generically denoted by C.) Next, we recall that for a+ , a− positive, G(a+ , a− ; z) has no poles and zeros for |Re z| > 0. It readily follows that there exist functions C± (a+ , a− , δ, Im z) that are continuous on (0, ∞)3 × R and such that 0 < C− < |G(a+ , a− ; z)| exp(−αIm z|Re z|/2) < C+ ,

(3.4)

for all z ∈ C satisfying |Re z| > δ. In the sequel, we will frequently invoke these estimates. The asymptotic behaviors of the c-function (1.15) and u-function (1.24) as |Re y| → ∞ easily follow from (3.1)–(3.3). Specifically, we obtain

∓1 c(p; ±y) exp αy < Cc exp(−σ αas Re y), (3.5) p /2 + a χ (p) − 1 µ µ

|u(p; ±y)χ (p)∓2 − 1| < Cu exp(−σ αas Re y),

(3.6)

for all y ∈ C satisfying Re y > al ; the functions Cs = Cs (σ, a+ , a− , p, Im y) with s = c, u are continuous on [1/2, 1) × × R, with defined by (1.29). At this point it is expedient to add two more estimates whose verification is routine. Specifically, we see from (2.4) and (2.10) that there exist functions Cs (a+ , a− , p, δ) with s = a, b that are continuous on × (0, ∞) and such that |Va (a+ , a− , p; ±y) − 1| < Ca exp(−αa− Re y),

(3.7)

|Vb (a+ , a− , p; ±y)| < Cb exp(−αa− Re y),

(3.8)

for all y ∈ C satisfying Re y > δ. For the remainder of this section we choose vˆ ∈ [r− , r+ ], 0 < r− < r+ ,

(3.9)

Re v > r+ + al .

(3.10)

This ensures that the two downward pole sequences due to F (cˆ0 ; v, ˆ z) are at a distance at least r− from the imaginary axis and the ones due to F (c0 ; v, z) at a distance larger than vˆ + al . Moreover, the integrand Iren (γ ; v, v, ˆ z) ≡ (a+ a− )−1/2

F (γ0 + a; v, z)F (γˆ0 + a; v, ˆ z) ,

3 G(z + ia) j =1 G(z + ia + itj )

tj ≡ γ0 + γj = γˆ0 + γˆj , j = 1, 2, 3,

(3.11)

(3.12)

of the function Rren (cf. (1.13) and (1.2)–(1.6), (1.11)) has simple poles in the z-plane at ±vˆ − i cˆ0 . We continue to study the residues of Iren at these points. To do so, we recall that the residue of G(z) at z = −ia is given by Res G(z)|z=−ia = i(a+ a− )1/2 /2π,

(3.13)


401

cf. I (A.20). The residue at z = vˆ − i cˆ0 is therefore given by i F (c0 ; v, vˆ − i cˆ0 )G(2vˆ − ia) i = c(γˆ ; −v)F ˆ (c0 ; v, vˆ − i cˆ0 ),

3 2π 2π µ=0 G(vˆ + i γˆµ )

(z = vˆ − i cˆ0 ). (3.14)

Likewise, the residue at −vˆ − i cˆ0 yields i c(γˆ ; v)F ˆ (c0 ; v, −vˆ − i cˆ0 ), 2π

(z = −vˆ − i cˆ0 ).

(3.15)

Let us now obtain the Re v → ∞ asymptotics of these residues multiplied by the factor −2π iχ (γ )/c(γ ; v)c(γˆ ; v), ˆ

(3.16)

in anticipation of their contribution to the E-function asymptotics (recall (1.19)). For (3.14) we obtain R+ (γ ; v, v) ˆ = −u(γˆ ; −v)χ ˆ (γ )F (c0 ; v, vˆ − i cˆ0 )/c(γ ; v),

(3.17)

for which we have by (3.5) and (3.1)–(3.3), ˆ + u(γˆ ; −v) ˆ exp(−iαv v)| ˆ < C(σ, a+ , a− , γ , Im v, v) ˆ exp(−σ αas Re v), |R+ (γ ; v, v) (3.18) for all v ∈ C satisfying Re v > vˆ + al ; the function C is continuous on [1/2, 1) × × (0, ∞)2 . Likewise, for the residue (3.15) we get R− (γ ; v, v) ˆ = χ (γ )F (c0 ; v, −vˆ − i cˆ0 )/c(γ ; v),

(3.19)

and |R− (γ ; v, v) ˆ − exp(iαv v)| ˆ < C(σ, a+ , a− , γ , Im v, v) ˆ exp(−σ αas Re v).

(3.20)

Clearly, (3.18) and (3.20) reveal the plane wave terms featuring in Eas (1.31). To exploit this, however, we must bound the contour integral shifted across the poles at z = ±vˆ − i cˆ0 . We now turn to this task. We begin by defining the shifted contour. Save for three eventual indentations, it is the horizontal line Im z = −cˆ0 − η,

η ∈ (0, as ).

(3.21)

The η-restriction ensures that all poles of the downward sequences starting at ±v−i ˆ cˆ0 are below the contour, except those at ±v−i ˆ cˆ0 . (Later on, we impose further case-dependent restrictions on η.) The eventual indentations are defined as follows. Setting m ≡ min(0, a − s1 , a − s2 , a − s3 ),

(3.22)

we indent the contour downwards at the imaginary axis, so that its distance to i[m, ∞) is bounded below by d ≡ min(r− /2, as /2).

(3.23)

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S.N.M. Ruijsenaars

(Hence no indentation is needed for m ≥ −cˆ0 + as /2, for instance.) Likewise, if need be we indent the contour upwards at Re z = ±Re v, so that it stays at a distance as /2 from ±Re v + i(−∞, −c0 + as ]. (We will let Im v vary over [−as , as ], so this ensures that the pole sequences starting at ±v − ic0 stay below the contour.) We have depicted the situation in Fig. 1 for a choice of parameters where the three indentations are needed, choosing them rectangular for convenience. The poles of Iren (3.11) lie at ±vˆ − i cˆ0 and on the half lines symbolized by the vertical arrows. We have also chosen Im v = −as . Denoting the shifted contour just defined by Cs , we deduce from the above that we have (E − R+ − R− )(γ ; v, v) ˆ = χ (γ )[c(γ ; v)c(γˆ ; v)] ˆ −1 Iren (γ ; v, v, ˆ z)dz. (3.24) Cs

We may and will put the four z-independent G-functions in Iren in front of the integral. This yields a v-independent prefactor χ (γ )[c(γˆ ; v)G( ˆ vˆ + i γˆ0 )G(−vˆ + i γˆ0 )]−1

(3.25)

that is bounded as (a+ , a− , γ ) varies over -compacts and vˆ varies over [r− , r+ ]. Thus we may and shall omit it in estimating E − R+ − R− . By (3.5) and (3.4), the v-dependent prefactor P (γ ; v) ≡ [c(γ ; v)G(v + iγ0 )G(−v + iγ0 )]−1

(3.26)

obeys |P (a+ , a− , γ ; v)| < CP (a+ , a− , γ , δ, Im v) exp(α(γˆ0 − γ0 + a)Re v),

(3.27)

for all v ∈ C satisfying Re v > δ, with CP continuous on × (0, ∞) × R. Therefore, we are left with estimating the integral IL (γ ; v, v, ˆ z)dz, (3.28) L(γ ; v, v) ˆ = Cs

where IL (γ ; v, v, ˆ z) ≡

G(z + v + iγ0 )G(z − v + iγ0 )G(z + vˆ + i γˆ0 )G(z − vˆ + i γˆ0 ) .

G(z + ia) 3j =1 G(z + ia + itj ) (3.29)

To this end we begin by observing that (3.4) entails |IL | < Ct (a+ , a− , γ , Im v, v, ˆ Im z) exp(∓α(2aRe z−Im vRe v)), ±Re z > Re v+as /2, (3.30) with Ct continuous on × R × [r− , r+ ] × R. Therefore the integrals over the tails of Cs are majorized by C exp(−α(2a ∓ Im v)Re v).

(3.31)

Here and from now on, we use the symbol C to denote a positive function of the parameters a+ , a− , γ and η that can be chosen independent of vˆ ∈ [r− , r+ ], Im v ∈ [−as , as ] and Re v ∈ [r+ + al , ∞), and that is continuous for (a+ , a− , γ ) in the parameter set at issue.


403

In connection with this convention, we add three remarks. First, though the parameter set on which (3.31) holds true is all of (1.29), it will be necessary to restrict the parameters in various ways later on. Second, we repeat that we are going to fix the shift parameter η in (3.21) in a case-dependent way. (Of course, the total contour integral is η-independent, but since we can only estimate it piecemeal, the pieces do have dependence on η.) Third, at this stage there seems to be no reason to restrict Im v to [−as , as ], but the need for this restriction will become apparent shortly. Next, we combine these estimates with the bound (3.27) on the prefactor to obtain upper bounds C exp(α(γˆ0 − γ0 − a ± Im v)Re v)

(3.32)

for the tail contributions to E − R+ − R− . Clearly, for these contributions to converge to 0 as Re v → ∞, it suffices that we have |Im v| < γ0 − γˆ0 + a.

(3.33)

At this point we should stress that we do not know whether (3.33) is necessary for convergence to 0. This restriction is however essential for our analysis to yield the desired convergence. (See also our remarks at the end of Sect. 4 in this connection.) Indeed, for later purposes we need to let Im v vary at least over [−as , as ]. Thus we see that (3.33) can only hold on all of the latter interval when the parameters satisfy γ0 − γˆ0 + a − as > 0.

(3.34)

Obviously, this condition is not valid for arbitrary parameters. In Sect. 4, we will see that for a+ = a− it does hold true on a suitable fundamental domain for the W -action. But when a+ equals a− , (3.34) is plainly false for γ = 0. It should be stressed that these arguments apply irrespective of the choice of shift parameter η. (Recall that (3.30) holds true for all Im z ∈ R.) Therefore, we proceed in two stages. For the remainder of this section, we choose the parameters in a subset r of defined by the restrictions as ∈ (0, al /8],

(3.35)

γ ≤ as /2,

(3.36)

where · denotes the euclidean norm on R4 . This entails |γµ |, |γˆµ | ≤ as /2 (recall J is orthogonal), so that we have γ0 − γˆ0 + a − as > al /4.

(3.37)

Moreover, the requirement (3.36) is clearly W -invariant. Accordingly, we can use the asymptotics results obtained in this section to complete the proof of Theorem 1.1 in the next one. Once we have shown W -invariance of E, we can proceed to the second stage, where we choose γ in a fundamental domain for which γ0 − γˆ0 ≥ 0 with equality only for γ = 0. Due to W -invariance, we can then deduce the asymptotic behavior for arbitrary γ , obtaining Theorem 1.2. After this sketch of our proof strategy, we turn to estimating the contour integral L(γ ; v, v) ˆ (3.28). The pertinent result is the following lemma.

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S.N.M. Ruijsenaars

Lemma 3.1. Assume the parameters a+ , a− and γ are restricted by (3.35) and (3.36), and the variables vˆ and v by (3.9), (3.10) and Im v ∈ [−as , as ].

(3.38)

|L(γ ; v, v)| ˆ < C exp(−α(γˆ0 − γ0 + a + η)Re v),

(3.39)

Then we have

with C defined below (3.31). Proof. We have already seen that the integrals over the right and left tail are bounded above by (3.31). Since we have |Im v| ≤ as , we obtain an upper bound C exp(−αal Re v)

(3.40)

for both tail integrals. Next, we examine the integral over the right indentation. On this piece of the contour, the G-function G(z − v + iγ0 ) in (3.29) is bounded independently of Re v, and so we deduce from (3.4), |IL | < C exp(α[−4aRe z + (Im z + γ0 )(Re v − Re z) + Im v(Re v + Re z)]/2). (3.41) Since |Re z−Re v| ≤ as /2 on the indentation, and since Im v ≤ as , this can be majorized by |IL | < C exp(−αal Re v).

(3.42)

As the length of the indentation is bounded, its contribution is bounded by (3.40). Proceeding analogously for the left indentation, we see that its contribution is once more bounded by (3.40). Next, we study the contribution of the middle indentation. We need only consider the two v-dependent G-functions in (3.29), since the remaining ones stay bounded. Thus we get from (3.4) |IL | < C exp(α[(Im z + γ0 )Re v + Im vRe z]).

(3.43)

Now Im v and Re z stay bounded and we have Im z ≤ −γˆ0 − a − η on the middle indentation. Hence we obtain |IL | < C exp(−α(γˆ0 − γ0 + a + η)Re v),

(3.44)

and since the indentation length is bounded, its contribution to L is majorized by C exp(−α(γˆ0 − γ0 + a + η)Re v).

(3.45)

We proceed with the horizontal piece of the contour where Re z varies from d (3.23) ˆ we can proceed just as for the midto Re v − as /2, cf. Fig. 1. On the part from d to v, dle indentation, from which we deduce its contribution is bounded by (3.45). On the remainder, however, we must take all G-functions into account. Doing so, we obtain from (3.4), |IL | < C exp(α[(Im z + γ0 )Re v + (Im v − Im z − γ0 − 2a)Re z]), vˆ < Re z < Re v − as /2.

(3.46)


405

Fig. 1. The shifted contour Cs in the z-plane

Consider now the coefficient A of Re z. Since Im z = −cˆ0 − η and Im v ≤ as , we have A ≡ Im v − Im z − γ0 − 2a ≤ B ≡ −al /2 + as /2 + γˆ0 − γ0 + η.

(3.47)

Using (3.35), (3.36) and η < as , we obtain B < −al /8.

(3.48)

Hence the contribution of the horizontal part Re z ∈ [v, ˆ Re v − as /2] is again majorized by (3.45). Finally, repeating this analysis for the horizontal part Re z ∈ [−Re v + as /2, −d], we obtain once more an upper bound (3.45) for its contribution. The upshot is that the contribution to L for |Re z| ≤ Re v − as /2 is bounded above by (3.45), and the remaining contribution by (3.40). The difference of the Re v-coefficients equals −αB, so by (3.48) it is positive. Therefore, the estimate (3.39) follows.

From (3.39) we obtain an upper bound on the shifted contour integral L(γ ; v, v) ˆ for Re v → ∞, so we can deduce information on the asymptotics of E from our previous analysis. In order to obtain the same type of decay bound from the two residue contributions and from the shifted contour integral, we choose η = σ as .

(3.49)

Now we are prepared for our next lemma, which concludes this section. Lemma 3.2. Letting (σ, a+ , a− , γ , v) ˆ ∈ [1/2, 1) × r × [r− , r+ ], we have |(E − Eas )(γ ; v, v)| ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas Re v),

(3.50)

for all v ∈ C satisfying Re v > r+ + al , |Im v| ≤ as , with C continuous on [1/2, 1) × r × [r− , r+ ]. Proof. In view of (3.18) and (3.20), it suffices to show |(E − R+ − R− )(γ ; v, v)| ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas Re v).

(3.51)

Recalling (3.24)–(3.28), we need only multiply (3.27) and (3.39) to see that (3.51) follows from our choice (3.49).

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S.N.M. Ruijsenaars

4. Proofs of Theorems 1.1 and 1.2 With Lemmas 2.3 and 3.2 at our disposal, it is not difficult to prove the W -invariance of the E-function, as will now be detailed. Proof of Theorem 1.1. We recall that for vˆ > 0 and parameters restricted by (2.21), we have already obtained information on the Casorati determinant D(v) (2.24), cf. (2.27) and Lemma 2.3. In addition to (2.21), we at first restrict the parameters by requiring (3.35)–(3.36). For notational convenience, we also choose a− = as .

(4.1)

(On account of the parameter symmetry (1.21), this choice is not a restriction.) Now (3.36) is a W -invariant restriction, so that Lemma 3.2 applies to Ej (v), j = 1, 2, cf. (2.22)–(2.23). In particular, it follows from Lemma 3.2 that Ej (v) remains bounded as Re v → ∞ for vˆ ∈ [r− , r+ ] and |Im v| ≤ as . Next, we recall from (3.6) that u(γ ; v) remains bounded as |Re v| → ∞, uniformly for Im v in R-compacts. Therefore, both factors on the rhs of (2.27) remain bounded as Re v → ∞, uniformly for |Im v| ≤ a− /2. In view of the reflection symmetry (1.23), this is also the case for Re v → −∞. It follows that the entire ia− -periodic function m(v) remains bounded as |Re v| → ∞, uniformly for vˆ ∈ [r− , r+ ] and |Im v| ≤ a− /2. From Liouville’s theorem we then infer that m(v) is constant. We continue to invoke Lemma 3.2 once more, this time to show the constant equals 0. First, we note that we may write (cf. (1.15) and (1.24))

3 µ=0 G(y − ipµ )G(y + ipµ ) u(p; y) = − . (4.2) G(2y + ia)G(2y − ia) From this we obtain u(w(p); y) = u(p; y), ∀w ∈ W.

(4.3)

Recalling (2.15), we deduce that the u-functions in the asymptotics (3.50) of Ej (v) satisfy u(J γ (j ) ; −v) ˆ = u(J γ ; −v), ˆ j = 1, 2.

(4.4)

As a consequence, the finite part of the asymptotics of Ej (v) (as Re v → ∞ with vˆ ∈ [r− , r+ ] and |Im v| ≤ a− ) is the same for j = 1 and j = 2. But then we have lim

Re v→∞

D(v) = 0, |Im v| ≤ a− /2,

(4.5)

and so the constant vanishes, as announced. The upshot is that the function m(v) vanishes identically. Hence the assumption of Lemma 2.3 is false, i.e., D(v) vanishes identically. This entails that the quotient Q(a+ , a− , γ , w1 , w2 , v; ˆ v) ≡ E(a+ , a− , w1 (γ ); v, v)/E(a ˆ ˆ (4.6) + , a− , w2 (γ ); v, v), satisfies Q(a+ , a− , γ , w1 , w2 , v; ˆ v + ia− /2) = Q(a+ , a− , γ , w1 , w2 , v; ˆ v − ia− /2), cf. (2.24).

(4.7)


407

We have now proved ia− -periodicity of Q(v) for vˆ ∈ [r− , r+ ] and parameters a+ , a− , γ restricted by (2.21), (3.35), (3.36) and (4.1). But for Re v, Re vˆ > 0 (say), the functions E1 and E2 are real-analytic in a+ , a− , γ for a+ , a− > 0 and γ ∈ R4 , and for fixed parameters they are meromorphic in v and vˆ (as follows from I). Therefore, (4.7) holds true for parameters a+ , a− > 0, γ ∈ R4 , and variables v, vˆ ∈ C. Now E1 and E2 are also invariant under a+ ↔ a− , cf. (1.21). Hence the same is true for Q. But then we have Q(a+ , a− ; v + ia+ ) = Q(a− , a+ ; v + ia+ ) = Q(a− , a+ ; v) = Q(a+ , a− ; v), (4.8) so Q(v) has both period ia− and period ia+ . Choosing a+ , a− > 0 with a+ /a− ∈ / Q, it follows that Q(v) is constant. By denseness and real-analyticity in a+ , a− , we see that Q(v) is constant for all a+ , a− > 0. A priori, the constant could depend on a+ , a− , γ , w1 , w2 and v, ˆ however. To show that this is not so, we reconsider (4.6), with vˆ and the parameters restricted so that Lemma 3.2 applies. Then E1 (v) and E2 (v) have equal asymptotics as v → ∞, so that Q(v) equals 1. Invoking once again analyticity, we deduce Q(v) = 1 for arbitrary parameters and variables. Hence Theorem 1.1 follows.

Now that we have proved W -invariance of E, we need only determine the Re v → ∞ asymptotics of E for γ in a fundamental domain to handle γ ∈ R4 . (As already became clear from the analysis leading to (3.34), our proof strategy cannot be directly applied to arbitrary γ .) Specifically, we choose γ ∈ diag(−1, 1, 1, −1)DW ,

(4.9)

with DW defined by (2.19). Hence we have 0 ≤ |γ3 | ≤ −γ2 ≤ −γ1 ≤ γ0 .

(4.10)

γˆ0 ≤ γ0 /2,

(4.11)

γ0 − γˆ0 + a − as ≥ γ0 /2 + (al − as )/2.

(4.12)

This entails

so that

If we now reconsider the contribution of the tail integrals for this γ -choice (cf. the paragraph containing (3.34)), we are led to a dichotomy. To be specific, for γ0 > 0 or al > as we see that it converges exponentially to 0 as Re v → ∞, with a rate that can be chosen uniformly for |Im v| ≤ as . But when we have both γ0 = 0 (which entails γ = 0) and al = as = a, then we only get exponential convergence to 0 for |Im v| ≤ a − , > 0, with a rate that goes to 0 as → 0. On the other hand, the tail contribution does remain bounded as Re v → ∞, uniformly for |Im v| ≤ a, cf. (3.31)–(3.32). After these introductory observations, we are prepared to complete the proof of Theorem 1.2. Proof of Theorem 1.2. Since Eas (a+ , a− , γ ; v, v) ˆ (1.31) is manifestly W -invariant and E(a+ , a− , γ ; v, v) ˆ is also W -invariant (as proved in Theorem 1.1), we may and will restrict γ by (4.9). Moreover, both functions are uniformly bounded on any compact subset of the set × {Re v > 0} × (0, ∞), since they are real-analytic in a+ , a− , γ , vˆ

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and analytic in v on the latter set. Therefore, we need only prove the bounds in the theorem for vˆ varying over an arbitrary interval [r− , r+ ] with 0 < r− < r+ , and for δ equal to r+ + al . Hence we can follow the reasoning in Sect. 3. Specifically, recalling the analysis leading to (3.18) and (3.20), we deduce that to obtain (1.32) , it suffices to show ˆ < C(σ, a+ , a− , γ , v) ˆ exp(−σ αas v), |(E − R+ − R− )(a+ , a− , γ ; v, v)|

(4.13)

for all v > r+ + al , with C continuous on [1/2, 1) × × [r− , r+ ]. Adapting the arguments below (3.24), we first obtain (3.31) and (3.32) with Im v = 0. Due to our γ -choice (4.9), we have γˆ0 − γ0 ≤ 0, so the tail contributions to E − R+ − R− are majorized by C exp(−αav).

(4.14)

On the right indentation we may invoke (3.41) with Im v = 0. Arguing as before, we see that its contribution to E − R+ − R− is bounded by C exp(α(γˆ0 − γ0 − a)v).

(4.15)

Repeating the reasoning for the left indentation, we obtain again an upper bound (4.15) for its contribution. Since γˆ0 − γ0 ≤ 0, the bound (4.15) is majorized by (4.14). Turning to the middle indentation, we are once more led to (3.45), so its contribution to E − R+ − R− is bounded by C exp(−αηv).

(4.16)

(Recall we need to multiply by (3.27).) More generally, we obtain this bound on the part of the contour where we have −vˆ < Re z < v. ˆ Next, we invoke (3.46) with Im v = 0. The coefficient of Re z equals cˆ0 + η − γ0 − 2a = γˆ0 − γ0 + η − a ≤ η − a < 0,

(4.17)

since η < as . Choosing η = σ as from now on, so that Im z + γ0 = −γˆ0 + γ0 − a − σ as

(4.18)

on this part of the contour, we deduce that its contribution to E − R+ − R− is bounded by C exp(−σ αas v).

(4.19)

Likewise, the part Re z ∈ [−v + as /2, −d] leads to (4.19). Putting the pieces together, we obtain (4.13) and hence (1.32). We proceed with the special case γ = 0, a+ = a− = as = al = a. Thus we have c0 = cˆ0 = a and m = 0 (cf. (3.22)), while (3.27) reduces to |P (a, a, 0; v)| < CP (a, a, 0, δ, Im v) exp(αaRe v).

(4.20)

As we have already pointed out, in this case our estimates on the contribution of the tail integrals to the rhs of (3.24) yield an upper bound C exp(α(−a + |Im v|)Re v),

(4.21)


409

cf. (3.32). For |Im v| ≤ τ a with τ ∈ [0, 1), this is bounded by a multiple of exp((τ − 1)αaRe v), but for |Im v| ≤ a we can only deduce boundedness. Proceeding as in the proof of Lemma 3.1, we reach the same conclusion for the right and left indentations. Since m = 0 in this special case, no middle indentation occurs. For Re z ∈ [−v, ˆ v] ˆ we obtain as before from (3.4), |IL | < C exp(−α(a + η)Re v).

(4.22)

Combining this with (4.20), we see that the contribution of this line segment to the rhs of (3.24) is majorized by C exp(−αηRe v).

(4.23)

Turning to the bound (3.46), we see it reduces to |IL | < C exp(α[(−a − η)Re v + (Im v − a + η)Re z]), vˆ < Re z < Re v − a/2. (4.24) Taking Im v = a, we infer from (4.20) that the contribution of this line segment cannot be bounded by (4.23). (Recall C is by definition independent of Im v ∈ [−a, a].) On the other hand, fixing η ∈ (0, a), we get for Im v ≤ a: Re v−a/2−ia−iη Re v | dzIL (γ ; v, v, ˆ z)| < C exp(α(−a − η)Re v) dxeαηx v−ia−iη ˆ

0

< C exp(−αaRe v).

(4.25)

Combined with (4.20), this yields boundedness for |Im v| ≤ a. Fixing τ ∈ [0, 1) and σ ∈ [1/2, 1) and taking Im v ≤ τ a, η = σ (1 − τ )a, we have Im v − a + η ≤ (σ − 1)(1 − τ )a < 0. Then the contribution of the pertinent integral to (3.24) is bounded above by C(σ, a, τ, v) ˆ exp(−αηRe v), η = σ (1 − τ )a,

(4.26)

where C is continuous on [1/2, 1) × (0, ∞) × [0, 1) × [r− , r+ ]. Repeating these arguments for the line segment −Re v + a/2 < Re z < −v, ˆ we obtain the same conclusion. The upshot is that the rhs of (3.24) is bounded uniformly for |Im v| ≤ a and vˆ ∈ [r− , r+ ], whereas for |Im v| ≤ τ a with τ ∈ [0, 1), this can be improved to (4.26). Recalling our previous results (3.18) and (3.20), we see that we have now proved the decay assertion (1.34), whereas (1.36) has only been shown to hold for |Im v| ≤ a. To extend (1.36) to Im v ∈ R, we invoke the AE E(v − ia, v) ˆ + Va (v)E(v + ia, v) ˆ + Vb (v)E(v, v) ˆ = 2 cosh(2π v/a)E(v, ˆ v), ˆ (4.27) and the bounds (3.7)–(3.8) on Va and Vb . Specifically, taking v → v + ia in (4.27), we obtain (1.36) for Im v ∈ [−2a, −a]. Clearly, we can now proceed in a recursive, strip-by-strip fashion to deduce (1.36) for Im v ∈ (−∞, −a]. Multiplying (4.27) by Va (v)−1 , we can take v → v − ia to obtain (1.36) for Im v ∈ [a, 2a], and then recursively for Im v ∈ [a, ∞). Thus we have now proved Theorem 1.2 for the special case γ = 0, a+ = a− . It remains to prove (1.33) for vˆ ∈ [r− , r+ ], Re v > r+ + al and γ satisfying (4.9), with a+ = a− in case γ0 = 0. Setting r ≡ γ0 − γˆ0 + a − as ,

(4.28)

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we have r > 0 (cf. (4.12)), and the tail contributions to E − R+ − R− are bounded above by C exp(−αrRe v)),

(4.29)

cf. (3.32). Using the bound (3.41) for the right indentation and its counterpart for the left one, we deduce that their contribution is also majorized by (4.29). Turning to the middle indentation, we conclude as before that its contribution to L is bounded above by (3.45). Combining this with (3.27), we see that its contribution to E − R+ − R− is majorized by C exp(−αηRe v).

(4.30)

Again, we obtain the same result for the line segments d ≤ |Re z| ≤ v. ˆ Next, consider the bound (3.46). In the present case, the coefficient A of Re z satisfies A ≤ as + γˆ0 + η − γ0 − a = η − r.

(4.31)

Thus we can ensure that it is negative by choosing η in (0, min(as , r)). Doing so, the contribution of this piece of Cs to E − R+ − R− is again bounded above by (4.30). For −Re v + as /2 < Re z < −v, ˆ we reach the same conclusion. In summary, we have |(E − R+ − R− )(γ ; v, v)| ˆ < C exp(−αηRe v), η ∈ (0, min(as , r)),

(4.32)

with r given by (4.28). Recalling (3.18) and (3.20), we deduce |(E − Eas )(γ ; v, v)| ˆ < C exp(−ρRe v), ρ ≡ α min(σ as , r).

(4.33)

As a consequence, (1.33) follows with Im v ∈ [−as , as ]; for γ restricted by (4.9), we can choose for instance ρ(a+ , a− , γ ) = α min(as /2, γ0 − γˆ0 + a − as ),

(4.34)

and this choice can be extended to arbitrary (a+ , a− , γ ) ∈ ∗ by requiring that ρ be W -invariant. Finally, to extend (1.33) to Im v ∈ R, we exploit the eigenvalue AE for E that involves v-shifts over ±ias . To avoid clumsy formulas, we assume from now on as = a− . (The case as = a+ reduces to obvious notation changes. Alternatively, we can invoke (1.21).) Accordingly, we start from the AE, F (v − ia− ) + Va (a+ , a− , γ ; v)F (v + ia− ) + Vb (a+ , a− , γ ; v)F (v) = 2 cosh(αa− v)F ˆ (v),

(4.35)

ˆ Just as in the special case already handled, this can be obeyed by E(a+ , a− , γ ; v, v). done recursively, so we only detail the first step. Specifically, taking v → v + ia− in the AE (4.35), we see that we may write (E −Eas )(v, v) ˆ = − exp(iαv v)+u( ˆ γˆ ; −v) ˆ exp(−iαv v)−V ˆ ˆ a (v+ia− )E(v+2ia− , v) +[exp(αa− v) ˆ + exp(−αa− v) ˆ − Vb (v + ia− )]E(v + ia− , v). ˆ (4.36) Letting Im v ∈ [−2a− , −a− ], we are entitled to use (1.33) for the E-functions on the rhs. Doing so, and using also (3.7)–(3.8), we see that the eight terms staying away from 0 as Re v → ∞ cancel pairwise. Noting ρ < αa− , the remaining terms manifestly have decay O(exp(−ρRe v)). Therefore, (1.33) holds for Im v ∈ [−2a− , −a− ].


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With Theorem 1.2 proved, we can shed more light on the asymptotics of the shifted contour integral L(γ ; v, v) ˆ (3.28), both for γ that do not satisfy (3.34) and for |Im v| > as . Indeed, from (3.18), (3.20) and Theorem 1.2 we deduce ˆ = O(exp(−ρRe v)), Re v → ∞, (E − R+ − R− )(γ ; v, v)

(4.37)

where we have ρ > 0 for arbitrary (fixed) γ ∈ R4 \ {0} and uniformly for Im v in arbitrary R-compacts. Choosing in particular Im v ∈ [−as , as ], we may invoke (3.24), and all of the arguments leading to (3.33) apply as well. The point is now that (3.4) also yields lower bounds on the prefactors (3.25), (3.26) of the same type as the upper bounds. (That is, (3.25) is bounded away from 0, and we may also take |P | → |P |−1 in (3.27).) From this we see that for arbitrary a+ , a− and γ = 0 we have L(γ ; v, v) ˆ = O(exp([α(γ0 − γˆ0 − a) − ρ]Re v)), ρ > 0, Re v → ∞,

(4.38)

uniformly for |Im v| ≤ as ; provided we enlarge the right and left indentations of Cs (so that the pole sequences starting at ±v − ic0 stay below Cs ), we obtain (4.38) even for Im v in an arbitrary R-compact. We now explain why this estimate is remarkable. To this end we point out that when we use (3.2)–(3.3) to bound IL (3.29) for Re z > Re v + al , it becomes clear that we have not only |IL | ∈ exp(−α(2aRe z − Im vRe v))(C− , C+ ), 0 < C− < C+ ,

Re z > Re v + al , (4.39)

but also that we are estimating away only one diverging phase (as Re v → ∞), viz., the factor exp(−iα(Re v)2 /2). Since this factor is z-independent, it is quite plausible (though it does not rigorously follow) that the modulus of the integral over Re z > Re v + al is bounded below by K exp(α(Im v − 2a)Re v), with K > 0. Assuming this is indeed the case, we see from (4.38) that whenever we have Im v−a ≥ γ0 − γˆ0 , the quotient of the integral over the contour tail Re z > Re v +al and the integral L over the whole contour Cs diverges exponentially as Re v → ∞. This comparison reveals why our piecemeal reasoning cannot be directly applied to arbitrary Im v and γ . In this connection we should also draw the reader’s attention to the phase factor exp(iα(Re z)2 /2) that is estimated away in (3.46). Since it is divergent on the pertinent interval, it can (and apparently does) supply the cancellations that must occur for Im v − a ≥ γ0 − γˆ0 . In view of this state of affairs, it may be regarded as a felicitous circumstance that there does exist an (Im v)- and γ -window that is not only accessible via piecewise estimates, but large enough to open up the asymptotics of E for arbitrary Im v and γ . Acknowledgements. We would like to thank the referee for some valuable suggestions.

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