We consider a family of tronquée solutions of the Painelvé II equation \begin{equation*} q''(s)=2q(s)^3+sq(s)-(2α+\frac12), \qquad α> -\frac12, \end{equation*} which is characterized by the Stokes multipliers $$s_1=-e^{-2απi },\quad s_2=ω, \quad s_3=-e^{2 απi} $$ with $ω$ being a free parameter. These solutions include the well-known generalized Hastings-McLeod solution as a special case if $ω=0$. We derive asymptotics of integrals of the tronquée solutions and the associated Hamiltonians over the real axis for $α> -1/2$ and $ω\geq 0$, with the constant terms evaluated explicitly. Our results agree with those already known in the literature if the parameters $α$ and $ω$ are chosen to be special values. Some applications of our results in random matrix theory are also discussed.
We express discrete Painlevé equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painlevé equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlevé equation and present explicit birational transformations relating them.
Recently, a quantum version of Painleve equations from the point of view of their symmetries was proposed by H. Nagoya. These quantum Painleve equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian. We give a characterization of the quantum Painleve equations by certain holomorphic properties. Namely, we introduce canonical transformations such that the Painleve Hamiltonian system is again transformed into a polynomial Hamiltonian system, and we show that the Hamiltonian can be uniquely characterized through this holomorphic property.
We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann-Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painleve V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles…
It is demonstrated that a certain integral equation can be solved using the Painleve equation of third kind. Inversely, a special solution of this Painleve equation can be expressed as the ratio of two infinite series of spheroidal functions with known coefficients.
We find and study a two-parameter family of coupled Painlevé II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables $t,s$. Setting $s=0$, we can obtain an autonomous version of the coupled Painlevé II systems. We also show its symmetry and holomorphy conditions.
A $q$-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear $q$-difference equation associated with the $q$-Painlevé VI equation. Then we obtain integral transformations. We investigate the $q$-middle convolution in terms of the affine Weyl group symmetry of the $q$-Painlevé VI equation. We deduce an integral transformation on the $q$-Heun equation.
This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic represent…
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three variables. We give a relation between it and the second member of the second Painlevé hierarchy. Moreover, we give some relations between an autonomous version of its polynomial Hamiltonian system in two variables and the mKdV hierarchies.
In this letter we will use higher-order supersymmetric quantum mechanics to obtain several families of complex solutions of the Painleve IV equation with real parameters. We shall also study the algebraic structure, the eigenfunctions and the energy spectra of the corresponding non-hermitian Hamiltonians.
The 8-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ 8-parameter 'relativistic' Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé-Calogero correspondence to the highest level in the two hierarchies.
We present an new system of ordinary differential equations with affine Weyl group symmetry of type E_6^{(1)}. This system is expressed as a Hamiltonian system of sixth order with a coupled Painleve VI Hamiltonian.
We construct a solution of an analog of the Schrödinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of variables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schrödinger equation corresponding to the Hamiltonian $ H_{II} (z, t, q_1, q_2, p_1, p_2) $ of Hamiltonian system with respect to $t$ which is compatible with $P_…
Explicit Hamiltonian structure of the Flaschka-Newell Painlevé II hierarchy via symmetry reduction of the Painlevé IV hierarchy
Université Jean Monnet（让·莫内大学）；CNRS, Institut Camille Jordan UMR 5208（法国国家科学研究中心，卡米尔·若尔当研究所）；Institut Universitaire de France（法兰西学院）；Université de Genève（日内瓦大学）
We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painleve transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.
This is the third article in our series of articles exploring connections between dynamical systems of Stäckel-type and of Painlevé-type. In this article we present a method of deforming of minimally quantized quasi-Stäckel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schrödinger equations posses common, multi-time solutions. As in the classical case, we obtain here both magnetic and non-magnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps betwe…
We study a Hamiltonian system without the Painlevé property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the Painlevé differential equations. The system in question was obtained by Takasaki as part of the Painlevé-Calogero correspondence and possesses the algebro-Painlevé property, being related by an algebraic transformation to the fourth Painlevé equation. We provide an atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Ham…
We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.
The extension of the Painlevé-Calogero coorespondence for n-particle Inozemtsev systems raises to the multi-particle generalisations of the Painlevé equations which may be obtained by the procedure of Hamiltonian reduction applied to the matrix or non-commutative Painlevé systems, which also gives isomonodromic formulation for these non-autonomous Hamiltonian systems. We provide here dual systems for the rational multi-particle Painlevé systems (PI,PII and PIV) by reduction from another intersection a coadjoint orbit of GL(n) action with the level set of moment map. We describe this duality in…
This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so call…
The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Wid…
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standa…
Symmetries and solutions of the Painleve IV equation are presented in an alternative framework which provides the bridge between the Hamiltonian formalism and the symmetric Painleve IV equation. This approach originates from a method developed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy with the Darboux-Backlund and Miura transformations. In the Hamiltonian formalism the Darboux-Backlund transformations are introduced as maps between solutions of the Hamilton equations corresponding to two allowed values of Hamiltonian's discrete parameter. The action of the g…
The Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries. A canonical quantization of the Painlevé equations preserving the affine Weyl group symmetries has been studied. While, the Painlevé equations are isomonodromic equations for certain second-order linear differential equations. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and construct symmetries of the quantum Lax equations. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.
We consider Schrödinger equations for the quantum Painlevé equations. We present hypergeometric solutions of the Schrödinger equations for the quantum Painlevé equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schrödinger equations for the quantum Painlevé equations and those of the KZ equation or the confluent KZ equations.
We discuss a family of Hamiltonians given by particular rational extensions of the singular oscillator in two-dimensions. The wave functions of these Hamiltonians can be expressed in terms of products of Laguerre and exceptional Jacobi polynomials. We show that these systems are superintegrable and admit an integral of motion that is of fourth-order. As such systems have been classified, we see that these potential satisfy a non-linear equation related to Painlevé VI. We begin by demonstrating the process with the simpler example of rational extensions of the harmonic oscillator and use the cl…
We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) is the isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbert problem. The approach allows us to gain control of the analysis of the convergence as $n\to\infty$. By an appropriate choice of the external source matrix in Kontsevich's integral, we show that the limit produces the isomonodromic tau function of a special tronquée solution of the first Painlevé hierarchy, and we identify the solution in terms of the Stokes' data of the associated linear problem. We also show that there are several tau functi…
Hamiltonian representation of isomonodromic deformations of twisted rational connections: The Painlevé $1$ hierarchy
Université de Lyon（里昂大学）；Université Jean Monnet Saint-Étienne（圣艾蒂安让·莫内大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心 5208 联合实验室，卡米尔·若当研究所）；Institut Universitaire de France（法兰西学院）
Unstable separatrix solutions for the first and second Painlevé transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise to separatrix solutions. Similarly, for a fixed initial slope, say $y'(0)= 0$, there is a discrete set of initial values $y(0)=c_n$ that give rise to separatrix solutions. For Painlevé I the large-$n$ asymptotic behavior of $b_n$ is $b_n\sim B_{\rm I}n^{3/5}$ and that of $c_n$ is $c_n\sim C_{\rm I}n^{2/ 5}$, and for Painlevé II the large-$n$ asymptotic behavi…
In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant. Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the…
We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.
We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.
We investigate the structure of $τ$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $τ$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $τ$-functions associated with the configuration of generic nine points in the projective plane.
Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed.
In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defin…
Quadratic Bureau-Guillot systems with the first and second Painlevé transcendents in the coefficients. Part I: geometric approach and birational equivalence
We prove a set of general theorems that provide new nonlocal constants and first integrals for nonlinear Jacobi-type ordinary differential equations. Applications include equations of the Painleve-Gambier classification.
A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the third-order ladder op…
We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials…
We consider two Lax systems for the homogeneous Painlevé II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the early 2010's. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka-Newell Lax system: the tacnode…
In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlevé IV and Painlevé V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the Bäcklund transformations relating them.
Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second- order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.
The procedure of the "quantum" linearization of the Hamiltonian ordinary differential equations with one degree of freedom is introduced. It is offered to be used for the classification of integrable equations of the Painleve type. By this procedure and all natural numbers $n$ we construct the solutions $Ψ(\hbar,t,x,n)$ to the non-stationary Shrödinger equation with the Hamiltonian $H = (p^2+q^2)/2$ which tend to zero as $x\to\pm\infty$. On the curves $x=q_n (\hbar, t) $ defined by the old Bohr-Sommerfeld quantization rule the solutions satisfy the relation $i\hbar Ψ'_x\equiv p_n (\hbar, t) Ψ$…
A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $τ$-function for certain Painlevé transcendents. The coefficients in this expansion are given in terms of conformal blocks of a two-dimensional conformal field theory, which can be written as infinite sums over pairs of partitions. In this note a discrete matrix model is proposed on a lattice whose partition function can be used to obtain a multiple integral representation for the length restricted partial sums of the Painlevé conformal blocks. This leads to expr…
As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a simple technique for generating solutions of the Painlevé IV equation. Finally, we classify these solutions into three relevant hierarchies.
