P. Deift, X. Zhou
The authors use steepest descent ideas to obtain a priori $L^p$ estimates for solutions of Riemann-Hilbert Problems. Such estimates play a crucial role, in particular, in analyzing the long-time behavior of solutions of the perturbed nonlinear Schrödinger equation on the line.
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In this paper we present a general scheme to generate constructive solutions to the Riemann--Hilbert problem via middle convolution and illustrate this approach for a Fuchsian system with four singular points.
In this paper we give the asymptotic behavior of type I multiple orthogonal polynomials for a Nikishin system of order two with two disjoint intervals. We use the Riemann-Hilbert problem for multiple orthogonal polynomials and the steepest descent analysis for oscillatory Riemann-Hilbert problems to obtain the asymptotic behavior in all relevant regions of the complex plane.
$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{ρ(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where $ρ(x)$ is a positive $m$ times continuously differentiable function on $[-1,1]$, $m\geq3$.
We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the limit as the degree of the polynomial approaches infinity.
We develop the theory of Riemann-Hilbert problems necessary for the results in part one of this series of papers. In particular, we obtain solutions for a family of non-linear Riemann-Hilbert problems through classical contraction principles and saddle-point estimates. We use compactness arguments to obtain the required smoothness property on solutions. We also consider limit cases of these Riemann-Hilbert problems where the jump function develops discontinuities of the first kind together with zeroes of a specific order at isolated points in the contour. Solutions through Cauchy integrals are…
In this paper we study an analogue of the classical Riemann-Hilbert problem stated for the classes of difference and $q$-difference systems. The Birkhoff's existence theorem was generalized in this paper.
In this paper we describe various applications of the Riemann-Hilbert method to the theory of orthogonal polynomials on the line and on the circle.
In the paper we give an upper estimate of the number of apparent singularities that are sufficient for construction of a system of linear differential equations on a Riemann surface with given fuchsian singularities and monodromy.
In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $\mathcal{H}_L:L^2([b_L,0])\to L^2([0,b_R])$ and $\mathcal{H}_R:L^2([0,b_R])\to L^2([b_L,0])$. These operators arise when one studies the interior problem of tomography. The diagonalization of $\mathcal{H}_R,\mathcal{H}_L$ has been previously obtained, but only asymptotically when $b_L\neq-b_R$. We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes $\mathcal{H}_R,\mathcal{H}_L$ explicitly. We also find the asymptotics …
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.
In this article, we set up a method of reconstructing to polylogarithms $\mathrm{Li}_k(z)$ from zeta values $ζ(k)$ via the Riemann-Hilbert problem. This is referred to as "a recursive Riemann-Hilbert problem of additive type." Moreover, we suggest a framework of interpreting the connection problem of the Knizhnik-Zamolodochikov equation of one variable as a Riemann-Hilbert problem.
We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-PIV equations are obtained for the three term recurrence relation coefficients.
In the present work we use the Levelt's valuation theory to describe all monodromy representations that can be realized by Riemann equation. Also we show that if the monodromy of Riemann equation lies in $SL(2,\mathbb{C})$, then such a monodromy can be realized by a more special Riemann-Sturm-Liouville equation as well. After all the criterion for hypergeometric equation to have monodromy in $SL(2,\mathbb{Z})$ is presented.
We generalize some classical results for the Schlesinger system of partial differential equations and give the explicit form of its solution, associated with rational matrix functions in general position.
In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hilbert problem.
Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann-Hilbert perspective
该研究从黎曼-希尔伯特视角出发，推导单位圆上带权正交多项式及相关函数的微分方程，分析递推系数性质，并以广义修正Jacobi、Bessel权为案例验证方法。
We describe the close connection between the linear system for the sixth Painlevé equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise logarithmic modifications of the classical Röhrl-Deligne vector bundle. We show that the solutions of the Riemann-Hilbert problem are in bijection with some families of local filtrations which are stable under the prescribed monodromy maps. We introduce the notion of Birkhoff-Grothendieck trivialisation, and show that its computation corresponds to geodesic paths in some local affine Bruhat-Tits building. We use this to compute how the type of a bundle changes under stalk modifications, and give several corresponding alg…
Matrix Orthogonal Polynomials: A Riemann--Hilbert approach
本文提出利用 Riemann–Hilbert 分析研究矩阵正交多项式，从权重矩阵的微分性质推导递推系数性质，并以 Hermite、Laguerre 和 Jacobi 型矩阵多项式为例进行验证。
We give some additions to the article "On the generalized Riemann-Hilbert problem with irregular singularities" by Bolibruch, Malek, Mitschi (math/0410483). In particular, a weak GRH-problem and the GRH-problem for scalar differential equations are discussed.
In this work, some theorems are established for orthogonal trigonometric polynomials (OTP) including Favard, Baxter, Geronimus, Rakhmanov, Szegö and the strong Szegö theorems which are important in the theory of orthogonal polynomials on the unit circle (OPUC). All these results are based on the mutual representation theorem of OPUC and OTP which deduced by a Riemann-Hilbert problem simultaneously characterizing them. In addition, Szegö recursions, four-term recurrences and some new identities for OPUC and OTP are also obtained by using their Riemann-Hilbert characterizations respectively.
In a larger size Riemann-Hilbert problem matching the local parametrices with the global parametrix is often a major issue. In this article we present a result that should tackle this problem in natural situations. We prove that, in a general setting, it is possible to obtain a double matching, that is, a matching condition on two circles instead of one circle. We discuss how this matching approach can be used to obtain local scaling limits of correlation kernels and apply our result to several examples from the existing literature.
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance dependent correlation functions of integrable models described by a six-vertex R-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann--Hilbert based analysis.
In this paper we study those polynomials orthogonal with respect to a particular weight over the union of disjoint intervals first introduced by N.I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a previous paper, that involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann-Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchs…
We show how the Riemann-Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete Bessel kernel are computed as examples.
In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.
There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas, culminating in a beautiful refinement due to Baez-Duarte. The purpose of this little survey is to dis-entangle the resulting web of complications, and reveal the essential simplicity of the main results.
We give sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincare ranks at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover in particular the irreducibility condition on the monodromy given by Bolibrukh and Kostov in the classical case. We apply the above criteria to solve the inverse problem in differential Galois theory with a better control of the singularities.
A Riemann-Hilbert approach to the computation of transform pairs
提出一种统一方法，结合谱理论、Riemann-Hilbert问题和逆散射理论，构建并数值评估线性变系数偏微分方程的变换对，通过混合解析-数值策略实现精确计算，尤其适用于间断系数情形。
A novel approach to Riemann--Hilbert problems of particular class is introduced. The approach is applicable to problems in which the multiplicative jump is set on a half-line. Such problems are linked to some Wiener--Hopf problems motivated by diffraction theory. The new approach is based on ordinary differential equations: the Riemann--Hilbert problem is reduced to finding a coefficient of an ordinary differential equation and solving this equation. The new method leads to an efficient numerical algorithm and opens a road to new asymptotical and analytical advances.
For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with arbitrary polynomial potentials.
We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and fu…
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.
This is an expanded version of the talk given at the conference ``Constructive Functions Tech-04''. We survey some recent results on canonical representation and asymptotic behavior of polynomials orthogonal on the unit circle with respect to an analytic weight. These results are obtained using the steepest descent method based on the Riemann-Hilbert characterization of these polynomials.
In this article we present the hexagon equations for dilogarithms which come from the analytic continuation of the dilogarithm $\mathrm{Li}_2(z)$ to ${\mathbf P}^1 \setminus {0,1,\infty}$. The hexagon equations are equivalent to the coboundary relations for a certain 1-cocycle of holomorphic functions on ${\mathbf P}^1$, and are solved by the Riemann-Hilbert problem of additive type. They uniquely characterize the dilogarithm under the normalization condition.
On the Riemann-Hilbert approach to asymptotics of tronquée solutions of Painlevé I
本文利用Riemann-Hilbert方法与最速下降法，通过构造含互补误差函数的局部参数化子，给出Painlevé I方程tronquée解超出庞加莱展开的非微扰贡献细节。
The tacnode Riemann-Hilbert problem is a 4 x 4 matrix valued RH problem that appears in the description of the local behavior of two touching groups of non-intersecting Brownian motions. The same RH problem was also found by Duits and Geudens to describe a new critical regime in the two-matrix model. Delvaux gave integral representations for some of the entries of the 4 x 4 matrix. We complement this work by presenting integral representations for all of the entries. As a consequence we give an explicit formula for the Duits-Geudens critical kernel.
We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analy…
In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights ---which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann-Hilbert problem are derived. An explicit and general example is presented to illustrate the theoretical results of the work. Related matrix eigenvalue problem…
A Riemann-Hilbert approach to Fredholm determinants of Hankel composition operators: scalar-valued kernels
本文通过矩阵值Riemann-Hilbert问题刻画一类Hankel复合算子的Fredholm行列式，在核无非可积结构的条件下仅利用Hankel复合性质给出描述，并展示了方法效率、计算秩一扰动行列式及推导Akhiezer-Kac渐近定理。
Quantum Stokes matrices and quantum Riemann-Hilbert-Birkhoff maps
本文引入非交换亚纯线性常微分方程组（极点阶数为p+1）的量子斯托克斯矩阵，证明其满足量子交换关系，并解释为Riemann-Hilbert-Birkhoff映射的形变量子化。
In this paper, we show that every regular singular $\mathscr{D}$-module in $\mathbb{C}^n$ whose singular locus is a normal crossing is isomorphic to a quiver $\mathscr{D}$-module - a $\mathscr{D}$-module whose definition is based on certain representations of the hypercube quiver. To be more precise we give an equivalence of the respective categories. Our definition of quiver $\mathscr{D}$-modules is based on the one of Khoroshkin and Varchenko. To prove the equivalence, we use an equivalence by Galligo, Granger and Maisonobe for regular singular $\mathscr{D}$-modules whose singular locus is a…
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