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The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto's theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric. [ABSTRACT FROM AUTHOR]

We prove that each member of the non-commutative nonlinear Schrödinger and modified Korteweg–de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine–Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Pöppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Pöppe product—the product rule for Hankel operators pioneered by Ch. Pöppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a 'Pöppe polynomial' expansion for basic compositions in terms of 'linear signature expansions' representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order. • Non-commutative NLS and mKdV hierarchies. • Establish abstract Poppe algebra for this hierarchy. • Poppe algebra establishes the existence and uniqueness of the hierarchy. • Non-commutative sine–Gordon equation is the first negative order member. [ABSTRACT FROM AUTHOR]

Let w be a semiclassical weight that is generic in Magnus's sense, and (p n) n = 0 ∞ ({p_n})_{n = 0}^\infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes's integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise. We include examples where the Wiener–Hopf factors are found explicitly. [ABSTRACT FROM AUTHOR]

The periodic Benjamin–Ono equation is an autonomous Hamiltonian system with a Gibbs measure on L 2 (T). This paper shows that the Gibbs measures on bounded balls of L2 satisfy some logarithmic Sobolev inequalities. The space of n-soliton solutions of the periodic Benjamin–Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As n → ∞, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterized by atomic measures in the complex plane. The limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the probability density function, P (c , α , β , n) d c , of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑ j = 1 n f ( x j ) ≔ ∑ j = 1 n x j , denoted by M f (λ , α , β , n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β, x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the time-evolved Jacobi weight, namely, w (x; λ, α, β) = xα(1 − x)β e−λx, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + ⋯ + Pn(0, λ), orthogonal with respect to w (x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w (x; λ, α, β)/x. It is shown that rn(2iez) satisfies a Chazy II equation. There is another auxiliary variable, denoted as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w (x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: PV(α2/2, − β2/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function M f (λ , α , β , n) when n is sufficiently large. Furthermore, we deduce a new expression of M f (λ , α , β , n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the Paley-Wiener theorem, one deduces that P (c , α , β , n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1]. [ABSTRACT FROM AUTHOR]

Let A be the generator of a strongly continuous cosine family (cos(tA))tϵR on a complex Banach space E. The paper develops an operational calculus for integral transforms and functions of A using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on E. Examples include the Mellin transform and theMehler-Fock transform. The paper uses functional calculus for the cosine family cos(t√Δ)which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm-Liouville hypergroups with Laplace representation as well as analogues to the Kunze-Stein phenomenon in the hypergroup convolution setting. [ABSTRACT FROM AUTHOR]

This paper analyses the periodic spectrum of Schrödinger's equation − f ″ + q f = λ f when the potential is real, periodic, random and subject to the invariant measure ν N β of the periodic KdV equation. This ν N β is the modified canonical ensemble, as given by Bourgain (1994) [7] , and ν N β satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues ( λ n ) . For β , N > 0 small, there exists a set of positive ν N β measure such that ( ± 2 ( λ 2 n + λ 2 n − 1 ) ) n = 0 ∞ gives a sampling sequence for Paley–Wiener space PW ( π ) and the reproducing kernels give a Riesz basis. Let ( μ j ) j = 1 ∞ be the tied spectrum; then ( 2 μ j − j ) belongs to a Hilbert cube in ℓ 2 and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence ( μ j ) j = 1 ∞ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics ∑ j g ( λ 2 j ) with test function g ∈ PW ( π ) satisfy Gaussian concentration inequalities. [ABSTRACT FROM AUTHOR]

This article considers Whittaker's confluent hypergeometric function Wκ,μ where κ is real and μ is real or purely imaginary. Then ϕ(x) = x−μ−1/2Wκ,μ(x) arises as the scattering function of a continuous time linear system with state space L²(1/2, ∞) and input and output spaces C. The Hankel operator Γϕ on L²(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w0(x) = xb(1 − x)a. The operation of translating ϕ is equivalent to deforming w0 to give wt(x) = e−t/xxb(1 − x)a. The determinant of the Hankel matrix of moments of wε satisfies the σ form of Painlevé's transcendental differential equation PV. It is shown that Γϕ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski [Commun. Math. Phys. 211, 335-358 (2000)]. Whittaker kernels are closely related to systems of orthogonal polynomials for a Pollaczek-Jacobi type weight lying outside the usual Szegö class. [ABSTRACT FROM AUTHOR]

This paper concerns Gibbs measures ν for some nonlinear PDE over the D -torus T D . The Hamiltonian H = ∫ T D ‖ ∇ u ‖ 2 − ∫ T D | u | p has canonical equations with solutions in Ω N = { u ∈ L 2 ( T D ) : ∫ | u | 2 ≤ N } ; this N is a parameter in quantum field theory analogous to the number of particles in a classical system. For D = 1 and 2 ≤ p < 6 , Ω N supports the Gibbs measure ν ( d u ) = Z − 1 e − H ( u ) ∏ x ∈ T d u ( x ) which is normalized and formally invariant under the flow generated by the PDE. The paper proves that ( Ω N , ‖ ⋅ ‖ L 2 , ν ) is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic KdV , the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of Ω N , a logarithmic Sobolev inequality also holds in the critical case p = 6 . For D = 2 , the Gross–Piatevskii equation has H = ∫ T 2 ‖ ∇ u ‖ 2 − ∫ T 2 ( V ⁎ | u | 2 ) | u | 2 , for a suitable bounded interaction potential V and the Gibbs measure ν lies on a metric probability space ( Ω , ‖ ⋅ ‖ H − s , ν ) which satisfies LSI . In the above cases, ( Ω , d , ν ) is the limit in L 2 transportation distance of finite-dimensional ( Ω n , ‖ ⋅ ‖ , ν n ) given by Fourier sums. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation,, arises from a Hamiltonian on infinite-dimensional phase space. For, Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measureon ballsin phase space such that the Cauchy problem foris well posed on the support of, and thatis invariant under the flow. This paper shows thatsatisfies a logarithmic Sobolev inequality (LSI) for the focusing caseandonfor allN>0; alsosatisfies a restricted LSI foron compact subsets ofdetermined by Hölder norms. Hence forp = 4, the spectral data of the periodic Dirac operator inwith random potentialsubject toare concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries. [ABSTRACT FROM PUBLISHER]

The periodic Korteweg–de Vries (KdV) equation arises from a Hamiltonian system with infinite-dimensional phase space . Bourgain has shown that there exists a Gibbs probability measure ν on balls in the phase space such that the Cauchy problem for KdV is well posed on the support of ν, and ν is invariant under the KdV flow. This paper shows that ν satisfies a logarithmic Sobolev inequality. This paper also presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schrödinger equation, for small values of N. [ABSTRACT FROM AUTHOR]

Abstract: Painlevé''s transcendental differential equation may be expressed as the consistency condition for a pair of linear differential equations with matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let be the orthogonal projection; then the Fredholm determinant defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand–Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in ; so can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé''s equation with . [Copyright &y& Elsevier]

Abstract: In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems with one-dimensional input and output spaces, there exists a Hankel operator Γ with kernel such that is the generating function of a determinantal random point field on . The inverse scattering transform for the Zakharov–Shabat system involves a Gelfand–Levitan integral equation such that the trace of the diagonal of the solution gives . When is a finite matrix and , there exists a determinantal random point field such that the largest point has a generalised logistic distribution. [Copyright &y& Elsevier]

Abstract: Suppose that Γ is a continuous and self-adjoint Hankel operator on with kernel and that with . If a and b are both quadratic, hyperbolic or trigonometric functions, and ϕ satisfies a suitable form of Gauss''s hypergeometric differential equation, or the confluent hypergeometric equation, then . The paper catalogues the commuting pairs Γ and L, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half-plane. [Copyright &y& Elsevier]

Abstract: Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper gives sufficient conditions for an integrable operator to be the square of a Hankel operator, and applies the condition to the Airy, associated Laguerre, modified Bessel and Whittaker functions. [Copyright &y& Elsevier]

Abstract: Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy–Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill''s equation give a new family of Tracy–Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann–Weyl anti-commutation relations and leave invariant the subspaces of that are the ranges of projections given by the Tracy–Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble. [Copyright &y& Elsevier]

Let X be a closed linear subspace of the Lebesgue space Lp(Ω; μ) for some 1 < p < ∞, and let − A be an invertible operator that is the generator of a bounded holomorphic semigroup Tt on X. Then for each 0 < α < 1 the maximal function supt>0 |Ttf(x)| belongs to Lp(Ω; μ) for each f in the domain of Aα. If moreover iA generates a bounded C0-group and A has spectrum contained in (0, ∞), then A has a bounded H∞ functional calculus. [ABSTRACT FROM PUBLISHER]

The generalized orthogonal ensemble of n × n real symmetric matrices X has probability measure νn(dX) = Zn-1exp{ -ntrace v(X)}dX where dX is the product of Lebesgue measure on the matrix entries and v(x)≥ (2+δ)log|x| with δ>0. The eigenvalue distribution is concentrated on [-A/2, A/2] for some A<∞. This paper establishes concentration and transportation inequalities for the distribution of eigenvalues of X under νn when v is twice differentiable with v′′(x)≥ -κ where 3A2κ <1. If v′′(x)≥ κ0 > 0, or if the variance of the trace is O(1/n2), then the empirical distribution of eigenvalues converges weakly almost surely to some non-random probability measure on [-A/2, A/2] as $n\backslash rightarrow\backslash infty$. These conditions are satisfied for certain polynomial potentials. The logarithmic energy is displacement convex as a functional on charge distributions, with fixed mean, along the real line. When the trace distribution satisfies a logarithmic Sobolev inequality, or equivalently a quadratic transportation inequality, the joint eigenvalue distributions and the limiting equilibrium measure likewise satisfy quadratic transportation inequalities in the sense of Talagrand.(24) [ABSTRACT FROM AUTHOR]

The circular ensembles of Dyson satisfy isoperimetric inequalities and concentration of measure phenomena for large particle numbers analogous to the isoperimetric inequality for surface measure on the sphere in Euclidean space of high dimension. This leads to a geometrical proof of a result of Johansson [ Bull. Sci. Math. (2) 112 , (1988), 257-304] that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as n →∞. [ABSTRACT FROM AUTHOR]

We consider operators on the matrix-valued disc algebra. We show that any bounded linear operator T: A—5’X to a Banach space X of cotype 2 induces a bounded operator G½’X defined on some weighted Bergman space G½ on the unit disc. We give sufficient conditions on the weight ½ for the formal inclusion A—5’G½ to be 2C*-summing. Decompositions of T with respect to operator-valued measures are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a particular Painlevé V (denoted PV) that arises from multiinput- multi-output wireless communication systems. Such PV appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the gamma density xαe-x, x > 0, for α > -1 by (1 + x/t)λ with t > 0 a scaling parameter. Here the λ parameter "generates" the Shannon capacity; see Chen, Y. and McKay, M. R. [IEEE Trans. Inf. Theory 58, 4594-4634 (2012)]. It was found that the MGF has an integral representation as a functional of y(t) and y0(t), where y(t) satisfies the "classical form" of PV. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity and the signal-to-noise ratio, P, tends to infinity such that s = 4n²/P is finite. Under such double scaling, the MGF, effectively an infinite determinant, has an integral representation in terms of a "lesser" PIII. We also consider the situations where α = k + 1=2, k ∊ N, and α 2 {0, 1, 2, . . . }, α 2 {1, 2, . . . }, linking the relevant quantity to a solution of the two-dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, which appears naturally, we obtain the double scaled MGF for small and large s, together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian. [ABSTRACT FROM AUTHOR]