LINEAR SYSTEMS, HANKEL PRODUCTS AND THE SINH-GORDON EQUATION

. Let ( − A, B, C ) be a linear system in continuous time t > 0 with input and output space C 2 and state space H . The scattering functions φ ( x ) ( t ) = Ce − ( t +2 x ) A B determines a Hankel integral operator Γ φ ( x ) ; if Γ φ ( x ) is trace class, then the Fredholm determinant τ ( x ) = det( I + Γ φ ( x ) ) determines the tau function of ( − A, B, C ). The paper establishes properties of algebras including R x = (cid:82) ∞ x e − tA BCe − tA dt on H . Thus the paper obtains solutions of the sinh-Gordon PDE. The tau function for sinh-Gordon satisﬁes a particular Painl´eve III (cid:48) nonlinear ODE and describes a random matrix model, with asymptotic distribution found by the Coulomb ﬂuid method to be the solution of an electrostatic variational problem on an interval.


Introduction
This paper is concerned with the Fredholm determinants of operators that are introduced via linear systems and with their applications to the sinh-Gordon equations.The study has application in Tracy and Widom's approach [41] to random matrix theory.
We begin by fixing some notation concerning linear systems.Let H be a separable complex Hilbert space with orthonormal basis (e j ) ∞ j=0 , and let L(H) denote the algebra of bounded linear operators on H.We shall denote the adjoint of B ∈ L(H) by B † .Let H 0 be a separable complex Hilbert space which serves as the input and output space; let B : H 0 → H and C : H → H 0 be bounded linear operators.On the state space H, let (T t ) t>0 be a strongly continuous and bounded semigroup with infinitesimal generator The scattering function of (−A, B, C) is φ(t) = Ce −tA B, which is a bounded and weakly continuous function φ : (0, ∞) → L(H 0 ).
Suppose that φ ∈ L 2 ((0, ∞); L(H 0 )).Then the Hankel integral operator with scattering function φ is the operator Such Hankel operators do not themselves form an algebra, although they have an algebraic structure which has been exploited in [35] and section 3.5 of [28].An important fact is that every bounded self-adjoint Hankel integral operator on L 2 (0, ∞) can be realised as the Hankel operator associated with a linear system (−A, B, C) in continuous time with state space H, and such that φ(t) = Ce −tA B where (e −tA ) is a strongly continuous semigroup on H (see [34] and [32]).
This tau function is analogous to the tau function introduced by Jimbo, Miwa and Ueno [25] to describe the isomonodromy of rational differential equations, and generalizes the classical theta function.Mumford [33] constructed solutions of the cubic nonlinear Schrödinger, KdV , mKdV and sine-Gordon equations in terms of classical theta functions on abelian varieties.Ercolani and McKean [14] extended the analysis of KdV to infinitedimensional abelian varieties in which case the scattering function φ is of rapid decay.
In [6], we reinterpreted the results of [35] and [28] in terms of an algebra of operators on H with a special associative product, and showed how nonlinear differential equations such as KdV emerge from algebraic identities in this product.In the present paper, we continue this analysis by addressing the sinh-Gordon equation in section 5.The simplest Darboux transformation of a linear systems is (−A, B, C) → (−A, B, −C) which takes R → −R.In section 5, we introduce a 2 × 2 block matrix system which enables use to introduce a tau function for the sinh-Gordon equation in the form det In [13], the authors consider the algebra of integrable operators of the form λI + K where K is an integral operator on L 2 (L) for a curve L in C with kernel where N j=1 f j (z)g j (z) = 0.They achieve several results using Riemann-Hilbert theory and make applications to some integrable operators in random matrix theory.Integrable operators of this form include the Christoffel-Darboux kernel from orthogonal polynomials when L is a real subinterval.In [41], the authors systematically consider these kernels and their scaling limits for classical orthogonal polynomials and obtain integrable operators that are fundamental to random matrix theory.In this application, the Fredholm determinant det(I + λK) is of primary interest.In particular, they show that the Airy and Bessel kernels can be expressed as products of Hankel integral operators on L 2 (0, ∞).
In section 3 we give sufficient conditions T ∈ A to be expressed as a product of Hankel integral operators.

Differential rings and Darboux addition
Let u ∈ C ∞ (R; C) be the potential in Schrödinger's equation.Gelfand and Dikii [18] considered the algebra A = C[u, u ′ , . . .] of differential polynomials generated by the potential and its derivatives.In this section we consider a generalization of this relating to linear systems.In [32] there is an existence theory covering the self-adjoint case.
We suppose that φ is a scalar scattering function that can be realized from a linear system (−A, B, C) from (1.1) as φ(t) = Ce −tA B, where A : H → H, B : C → H and is trace class, and satisfies the Lyapunov equation With We introduce the vector space of linear operators and introduce the associative product and the derivation and the bracket operation ) gives a homomorphism of differential rings.
(ii) Let the potential of the linear system (−A, B, C) be Proof.See [6] Lemma 4.2, and a related approach is developed in [12].Definition 2.3.[19] (i) For a potential u ∈ C ∞ (R; C) the stationary KdV hierarchy for a sequence (f ℓ ) is the recurrence relation f 0 = 1, (ii) The stationary hierarchy for the sine-Gordon and modified KdV equation is similar, with u replaced by (2.10) The solutions sequence for these hierarchies are differential polynomials in u by [19] Remark 2.2, and by Lemma 2.2 there exists X j ∈ A σ such that f ℓ = ⌊X ℓ ⌋.For the stationary KdV hierarchy, we showed in [6] that when f 0 = (1/2)u, the terms f ℓ = (−1) ℓ 2⌊A 2ℓ−1 ⌋ give a solution.In the physical models discussed in [23], KdV has a partition function which is the square of a tau function; whereas mKdV has a partition function that is the product of tau functions; for sinh-Gordon, one considers quotients of tau functions.
Suppose that u ∈ C ∞ (R; R) is bounded and let λ 0 be the bottom of the spectrum of L = −d 2 /dx 2 + u.For λ < min{λ 0 , 0} and signs {±}, we consider solutions: (2.12) Let G(x, y; λ) be the Greens integral kernel that implements (λI . (2.13) In the case where u is a real continuous and 2π-periodic potential, the differential equation −h ′′ +uh = λh is known as Hill's equation, and is associated with a hyperelliptic spectral curve Σ with points {(λ, ±) : λ ∈ C} giving a two-sheeted cover of C as in [31].
The notion of Darboux addition refers to the operation on potentials that corresponds to the addition rule for pole divisors of Baker-Akhiezer functions on Σ, as discussed in [30].In our case, we have a scattering potential u, and we consider the corresponding addition rule on the potentials, without seeking to interpret directly the notion of the spectral curve.Nevertheless, there is a simple operation on the linear system (−A, B, C) that gives rise to Darboux addition on the associated potential from Lemma 2.2 (ii).
Then the infinitesimal addition satisfies or equivalently Proof.We begin by recalling McKean's calculation.By the composition formula (2.14), the infinitesimal addition satisfies where from the differential equation Combining (2.17), (2.18) and (2.13), and taking δλ → 0, one can express the infinitesimal transformation in terms of the diagonal of the Greens function as in (2.15).In [6] Theorem 5.4, we computed this diagonal Greens function, and we deduce where the summands of this series are as in (5.4) of [6].Hence by summing the resulting geometric series, we have

Products of Hankel operators
In what follows we shall require some of the concepts of abstract differential calculus as developed in [11].Let S be an associative and unital complex algebra.There is a natural multiplication map µ : S ⊗ S → S, µ a j ⊗ b j = a j b j .We shall denote the nullspace of this map as Ω 1 S and so we have an exact sequence In this section, we shall mainly be interested in the case where S = C(x), the space of rational functions.In these cases, the tensor product S ⊗S may be regarded as consisting of functions of the form f j (x)g j (y) acting on a product space.There are several natural derivation maps that act on these spaces.
See (6.13) for a significant example.
Products of Hankel operators arise from rational differential equations, involving poles.
As is common in control theory, we let S be the unital complex algebra of stable rational functions, namely the space of f (s)/g(s) where f (s), g(s) ∈ C[s] satisfy degree f (s) ≤ degree g(s), and all the zeroes of g(s) satisfy Re s < 0. We allow poles at ∞ by adjoining s to form S[s]. Now let p 1 , . . ., p r ∈ C be distinct points satisfying Re p j > 0, and for P = {p 1 , . . ., p r , ∞}, introduce the algebra by formally adjoining the inverses of (s−p j )/(s+1) ∈ S. The possible poles of elements of and define Ω(s) ∈ M n×n (S p ) by The residues are Ω ∞ (∞), Ω 1 (p 1 ), . . ., Ω r (p r ).We assume further that Ω(s) ⊤ = Ω(s), where Ω ⊤ denotes the transpose.Let We also let , (w j ) m j=1 = m j=1 z j w j be the standard bilinear pairing, and • the operator norm on M m×m (C).Given a solution of the differential equation Let Ω be as in (3.3) and suppose that which is a finite-rank M m×m (C)-valued kernel.Here the entries of the difference quotient (Ω(z) − Ω(w))/(z − w) may be expressed as a sum of products of rational functions in z and rational functions in w, which are bounded for z, w > 0 by the partial fraction decomposition of Ω(s) in (3.3).Note that in which the matrix R is constant on cross diagonals, and is real symmetric and satisfies R 2 = I r×r , with trace R = 0 for even r and 1 for odd r, hence R is similar via a real orthogonal matrix to a real diagonal matrix with signature 0 or 1, respectively.From the right-hand side of (3.6) we extract coefficients depending on w or z such that Also, k(z, w) → 0 as z → ∞ or w → ∞.Likewise the right-hand side of (1.2) converges to 0 as z → ∞ or w → ∞, so we have the identity (3.6).Recall that if Theorem 3.3.Let k be a trace-class kernel as in Proposition 3.2, and let k x (u, v) = k(x + u, x + v).Then for all λ ∈ C, there exists a matrix function T (x, y) ∈ M N ×N (C) Proof.Using Proposition 3.2, write K in the form N j=1 Γ ψ j Γ φ j with ψ j , φ j ∈ L 2 ((0, ∞); R), or in matrix form Consider N × N matrix functions First we factorize K as a product of matrix Hankel operators.By Proposition 3.2 K is a complex-linear combination of Hankel products Γ ψ Γ φ , where ψ, φ ∈ L 2 ((0, ∞); R).
dt is finite, the Hankel operator with scattering function φ is self-adjoint and Hilbert-Schmidt, hence can be realised by a linear system (−A, B, C) in continuous time with input and output space C; this follows from [32].We can apply this to the real and imaginary parts of the entries of Φ 1 and Φ 2 , and introduce for j = 1, 2 the matrices We can suppose that there are linear systems (−A 1 , B 1 , C 1 ) and (−A 2 , B 2 , C 2 ) with state space H and input and output spaces Let − Â, B, Ĉ be the linear system which has scattering function The operator function Also, T is a solution of Theorem 3.3 applies in particular to the sinh-Gordon equation, as we discuss in section 5.
Example 3.4.For the weight x α e −x on (0, ∞), Laguerre's polynomials n (x) and w α (x) = u ′ α (x); then these give a solution of Laguerre's differential equation so one can apply Proposition 3.2.In particular, for α = 1, we obtain a Hankel product The Laguerre system is associated with orthogonal polynomials for the weight x α e −x on (0, ∞), which is the limit of the semi-classical weights x α (1−x/n) n from [26].In section 6, we consider the singular weight w α,s (s) = x α e −x−s/x , which has log w α,s (x) not integrable over (0, 1), hence lies beyond the scope of Szegö's theory on asymptotic formulas for orthogonal polynomials.

A linear system for Darboux addition
The pair of linear systems (−A, B, C) giving I + R x and (−A, B, −C) giving I − R x are related by a Darboux transformation, as discussed in Theorem 3.4 of [6].As there, we introduce the matricial system which has scattering function so that when I ± R x are invertible, we have We can then define where (ii) The Fredholm determinant satisfies (iii) and with q(x) = −4⌊A⌋, the function Proof.(i) From Lyapunov's equation, we have (ii) Then so by Lemma 2.5 (4.12) and likewise We have x), so as in Riccati's equation Example 4.2.With φ(x) = Ai(x/2), we have This example is considered in detail by Hastings and McLeod [22].

Solutions of the sinh-Gordon equation
Howland [24] observed that Hankel matrices are analogous to one-dimensional Schrödinger operators with the role of the Laplacian played by Carleman's operator (5.1) Power [36] gives several unitarily equivalent forms of this operator.To obtain such operators in terms of linear systems, we recall Proposition 2.1 from [3].Let H = L 2 ((0, ∞); C) which has dense linear subspace Suppose that h ∈ L ∞ ((0, ∞); R) and h(y)/ √ y ∈ L 2 (0, ∞).Then we let where the input and output operators depend upon the real parameter t > 0. Then we introduce the scattering function which satisfies the linear PDE ∂ 2 φ ∂x∂t = 2φ, (5.4) which may be interpreted as a linear counterpart of the sinh-Gordon equation.We also introduce In the following result, the pair (x, t) may be regarded as light-cone co-ordinates, rather than space and time.
Theorem 5.1.For the linear system (5.2),let S(x; t) = log det(I + R (x;t) ) − log det(I − R (x;t) ). (5.5) Then S gives a solution of the sinh-Gordon equation Proof.As an integral operator on L 2 (0, ∞), R (x;t) has a kernel which has the form of a Howland operator.This R (x;t) evidently defines a Hilbert-Schmidt linear operator on L 2 (0, ∞).Suppose for simplicity that h is real-valued.Then R (x;t) is the Schur product of the Carleman operator Γ from (5.1) with kernel 1/(y + z) and the tensor product of h(y)e −xy−t/y with itself.Power [36] showed that Γ has spectrum [0, π], hence R (x;t) is self-adjoint and positive with trace trace So R (x;t) is trace class and there exists x 0 > 0 such that for all x > x 0 , we have R (x;t) < 1; hence I ± R (x;t) are invertible.(Similar results hold for complex h by polarization.) Note that due to the special form of the linear system gives an operator of rank one so We write 2S(x; t) = ∞ x ψ(s; t)ds, where ψ(x; t) = −4V (x, x) as in (4.10).By (4.12), we have so by integrating with respect to x we deduce which implies when we differentiate with respect to t and multiply by ψ, that We also have and from (5.8), Also, from (5.9) we have We need to combine the equations (5.10) and (5.13) and thereby obtain the sinh-Gordon equation.The expression (5.11) involves factors I −R; whereas (5.12) involves only I +R; we reconcile these by multiplying by ψ = 4⌊(I +R)/(I −R)⌋, which cancels out the factors of I − R. From (5.8), we have hence from the * product (2.6)we obtain hence we obtain so we need to compare the integral in (5.18) with the sinh 2S term in (5.6).
We introduce the integration operator F ψ : L 2 (0, ∞) → L 2 (0, ∞), which depends upon t > 0 by Choosing f = 1, we obtain by induction the formula Now we express the right-hand side of (5.6) as a composition of operators Hence the equation (5.18) gives which is the sinh-Gordon equation.
(ii) By Cauchy-Schwarz inequality where the inner integral is bounded and converges to zero as Re z → ∞, so Γ φ (z;t)

2
L 2 → 0 by the dominated convergence theorem, hence the result.
(iii) The conclusion of Corollary 5.2(ii) holds, but the hypotheses are not quite satisfied, so we provide a special argument.Let be the modified Bessel function of the third kind of order 1, also known as MacDonald's function of the second kind, which is holomorphic on {z Re z > 0} with |K 1 (z)| ≤ K 1 (Re z).This also satisfies by simple estimates following from cosh u = 1 + 2 sinh 2 (u/2).Then for α = 0, we have dx is finite for all x 0 , s > 0 and we can deduce Γ φ (x 0 ) L 2 → 0 as x 0 → ∞, as in Corollary 5.2(ii).

Hankel determinants
The Wishart ensemble is a standard model in random matrix theory [16, page 91] which produces the Laguerre ensemble of random eigenvalues on (0, ∞).The Laguerre ensemble is in turn used as a model in the theory of wireless transmission, and in an integral model of quantum field theory at finite temperature [8].The KP , KdV and sinh-Gordon differential equation can be interpreted as aspects of a single hierarchy.
Let α = 0.By Andréief's identity [15], we have e −t cosh u j cosh u j du j (6.3) with K 1 as in (5.21).With the change of variable x j = 2/(1 + cosh u j ), this may be written as This is associated with the generalized unitary ensemble for the scalar potential with and Dyson introduced a technique for funding the asymptotics of such determinants, which has been refined by Chen [10, page 4603] and others into the Coulomb fluid method.In this, the point distribution n −1 n j=1 δ x j of the x j is approximated in the weak topology on probability measures as n → ∞ by a continuous distribution ρ which is found by potential theory.We write Fix ξ ∈ (0, 1) and write t for 2nξ.The zeros of u ′ n (x) may be approximated by the zeros of h is positive for all 0 < ξ < 1/2 and all sufficiently large n, so u n is convex near to this minimizer.For a continuous V : [0, 1] → R, we introduce the energy functional where V indicates an electric field and the log |x − y| term involves electrostatic interaction between points x, y ∈ [0, 1], where the charge distribution is ρ.The equilibrium distribution is defined to be the minimizer of this energy functional; see [38].
We consider the approximate scaled potential that is given by the first two summands in (6.4), namely with which has a local minimum at x = 2ξ.Proposition 6.2.
(i) The equilibrium distribution for u 0 is σ 0 , where (ii) For 1/4 < ξ < 1/2, the free logarithmic Sobolev inequality (iii) Let ρ n be the minimizer of E un/n (ρ) over all continuous ρ such that ρ ≥ 0 and 1 0 ρ(x)dx = 1.Suppose that ρ n has a continuous density which is supported in a single interval.Then ρ n converges weakly as n → ∞ to σ 0 .
Proof.(i) By standard results [38] Theorem 1.3 and p. 215, there exists a unique continuous probability density function on [0, 1] that attains the minimum of E V (ρ) for continuous V .We aim to solve the singular integral problem u ′ 0 (x) = p.v. by a similar argument to (6.10).We need a > 0 to ensure that the resulting ρ is integrable.
The relevant integrals arise from (248) in the appendix to [10].Let (P j,N ) ∞ j=0 be the monic polynomials that are orthogonal for the weight e −N v(z) , which gives rise to the integral equation (6.14) multiplied through by N, with the normalizations preserved.Then using the results of [9], we observe that N for z ∈ C \ R.This type of double scaling is standard in random matrix theory and addresses the singularity of the weight; see section 4 of [9] for more details.
dy = 2π H σ 0 (x) for the equilibrium density σ 0 subject to the constraint b a σ 0 (y) dy = 1.Since u 0 is convex, the solution is continuous and positive on a single interval (a, b), where 0 < a < b < 1, and we haveσ 0 (x) = (b − x)(x − a) 2π 2 b a u ′ 0 (x) − u ′ 0 (y) x − y dy (b − y)(y − a) = (b − x)(x − a)step follows from the substitution y = a+(b−a) sin 2 φ and some elementary integrals.The initial factor has the form of a semicircular distribution, which is modulated by a rational function with poles at x = 0 outside of the support interval (a, b).The endpoints of this interval are subject to the constraints 0 = b a u ′ 0 (x) dx (b − x)(x − a) We have u ′′ 0 (x) = (8xξ − 2x)/x 3 > 8ξ − 2 > 0. The result follows from Theorem 3.1 of [1].(iii) Returning to the original potential, we have u n = nu 0 + f where the correction term f (x) = (1/2) log(1 − x) − log(2 − x).There is a corresponding correction to the equilibrium density, ρ n = σ 0 + ρ/n where ρ(x) = 1 2π 2 (b − x)(x − a) p.v. b a (b − y)(y − a) y − x f ′ (y) dy a sum of products of rational functions of z and y, and there exist b > a > 0, such that the integral equation v(z) = 2 b a log |z − y|ρ(y)dy + C (z ∈ (a, b)) (6.14)for some C ∈ R with the normalization ρ(y) ≥ 0 and b a ρ(x) dx gives the number of zeros of P N,N (z) and have an asymptotic formula log P N,N (z) ∼ N b a log(z − y)ρ(y)dy (N → ∞)