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.
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.
In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlevé equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra.
We study first the supersymmetric quantum mechanics (SUSY QM), specially the cases of the harmonic and radial oscillators. Then, we obtain a new Wronskian formula for the confluent SUSY transformation and apply the SUSY QM to the inverted oscillator. After that, we present the polynomial Heisenberg algebras (PHA). We study the general systems described by PHA: for zeroth- and first-order we obtain the harmonic and radial oscillators, respectively; for second- and third-order PHA, the potential is determined in terms of solutions to Painlevé IV and V equations ($P_{IV}$ and $P_{V}$), respective…
In these lecture notes we shall study first the supersymmetric quantum mechanics (SUSY QM), specially when applied to the harmonic and radial oscillators. In addition, we will define the polynomial Heisenberg algebras (PHA), and we will study the general systems ruled by them: for zero and first order we obtain the harmonic and radial oscillators, respectively; for second and third order PHA the potential is determined by solutions to Painleve IV (PIV) and Painleve V (PV) equations. Taking advantage of this connection, later on we will find solutions to PIV and PV equations expressed in terms …
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…
In this paper we will explicitly work out the complex first-order SUSY transformation for the harmonic oscillator in order to obtain both real and complex new exactly-solvable potentials. Furthermore, we will show that this systems lead us to exact complex solutions of the Painlevé IV equation with complex parameters. We present some concrete examples of such solutions.
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.
In this paper we shall use the algebraic method known as supersymmetric quantum mechanics (SUSY QM) to obtain solutions to the Painlevé V (PV) equation, a second-order non-linear ordinary differential equation. For this purpose, we will apply first the SUSY QM treatment to the radial oscillator. In addition, we will revisit the polynomial Heisenberg algebras (PHAs) and we will study the general systems ruled by them: for first-order PHAs we obtain the radial oscillator, while for third-order PHAs the potential will be determined by solutions to the PV equation. This connection allows us to int…
In this work, supersymmetric quantum mechanics will be used to obtain complex solutions to Painleve IV equation with real parameters. We will also focus on the properties of the associated Hamiltonians, i.e. the algebraic structure, the eigenfunctions and the energy spectra.
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $ε$ in the interior of the Whitham oscillatory zone, it is known to be only of order $ε^{1/3}$ near the leading edge of this zone. To obtain a…
In this article we will obtain real and complex solutions to the Painleve IV equation through supersymmetric quantum mechanics. Then we will classify them into real solution hierarchies and also the complex solution hierarchies, which are hardly studied.
We rigorously compute the integrable system for the limiting $(N\rightarrow\infty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $n\in\mathbb{Z}_{\geq 1}$ at positive temperature. More precisely, the edge momentum statistics in the harmonic trap $n=1$ are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlevé-II integro-differential transcendent, cf. \cite{LW,ACQ}. For general $n\geq 2$, a novel higher order finit…
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 consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlevé equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data $s\equiv(s_1,s_2,s_3)\subset\mathbb{C}^3$ of the associated Flaschka-Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$, between the oscillatory power-like decay asymptotics for $|s_1|<1$ (Ablowitz-Segur) to the power-like growth behavior for $|s_1|=1$ (Hastings-McLeod) and from the latter to the singular oscillatory power-like growth for $|s_1|>1$ (Kapaev). It is shown t…
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…
A generalisation of the Stieltjes relations for the Painlevé-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher Painlevé-IV equation. The approach is based on the interpretation of the Stieltjes relations as local trivial monodromy conditions for certain Schrödinger equations in the complex domain. As a corollary a new class of the Schrödinger operators with trivial monodromy is constructed in terms of the Painlevé-IV transcendents…
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real…
Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painlevé IV equation, several solutions of this non-linear second-order differential equation will be simply found.
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.
This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlevé system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic M…
The type III Hermite $X_m$ exceptional orthogonal polynomial family is generalized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$ odd such that $m_2 > m_1$) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also…
These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlevé hierarchy. The construction is based on dressing ch…
Poles of solutions to the Painleve-I equation are intimately related to the theory of the cubic anharmonic oscillator. In particular, poles of integrale tritronquee are in 1-1 correspondence with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We analyze this pair of quantization conditions by means of a suitable version of the complex WKB method.
We extend Painlevé's determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general 'multiple-valued' Cauchy's problems. We study $C^0-$continuability (near singularities) of solutions.
We consider a perturbed version of the second Painlevé equation ($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings-McLeod and tritronquée solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type solution of the perturbed equation is holomorphic, real-valued and positive on the whole real-line, while the tritronquée-type solution is holomorphic in a large sector of the complex plane. These properties also characterise the corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising because th…
We study expectations of powers and correlation functions for characteristic polynomials of $N \times N$ non-Hermitian random matrices. For the $1$-point and $2$-point correlation function, we obtain several characterizations in terms of Painlevé transcendents, both at finite-$N$ and asymptotically as $N \to \infty$. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginib…
We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine t…
We study deformations of the harmonic oscillator algebra known as polynomial Heisenberg algebras (PHAs), and establish a connection between them and extended affine Weyl groups of type $A^{(1)}_m$, where $m$ is the degree of the PHA. To establish this connection, we employ supersymmetric quantum mechanics to first connect a polynomial Heisenberg algebra to symmetric systems of differential equations. This connection has been previously used to relate quantum systems to non-linear differential equations; most notably, the fourth and fifth Painlevé equations. Once this is done, we use previous s…
In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to expressible in terms of the third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.
We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region $\arg x \in [-\fracπ{3},\fracπ{3}]\cup [\frac{2π}{3},\frac{ 4 π}{3}]$, which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane, and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlev…
Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$ \frac{d}{dλ}Y= \left(\frac t2σ_3 + \frac{A_0}λ+\frac{A_1}{λ-1}\right)Y. $$ The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.
Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each `integrable system' increases. Certain intermediate systems are constructed and also tested for the Painlevé property. The Lie symmetries are also computed for completeness.
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine therelationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé Property in the parent second order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion …
In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonloga…
We consider the asymptotic behaviour of solutions of the first $q$-difference Painlevé equation in the limits $|q|\rightarrow 1$ and $n\rightarrow\infty$. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis…
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…
In this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of the Painlevé differential equations. Since all Painlevé differential equations (apart from the sixth one) exhibit Stokes phenomenon, it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The decorated character is defined as complexification of the bordered cusped Teichmüller space introduced in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n>1 bordered cusps is a Poisson manifold of dimens…
Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation
Modified Painlevé systems with meromorphic solutions for polynomial Hamiltonians of all degrees
Faculty of Mathematics Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学与机械学院）；School of Mathematics and Physics, University of Portsmouth（朴茨茅斯大学数学与物理学院）
A family of random variables $\mathbf{X}(s)$, depending on a real parameter $s>-\frac{1}{2}$, appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives, in the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative. Our first main result establishes a connection between the characteristic function of $\mathbf{X}(s)$ and the $σ$-Painlevé III' equation in the full range of parameter values $s>-\frac{1}{2}$. Our second main result gives …
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We fou…
In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its polynomial algebra. We will show how this polynomial algebra can be directly realized as a deformed oscillator algebra. This particular algebraic structure allows to find the unitary representations and the corresponding energy spectrum. We apply this construction to a family of caged anisotropic oscillators. The method can be used to generate new superinteg…
The classical Painlevé equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymp…
In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials in the real line, understood as those whose matrix of w…
This is a review of recent developments in the theory of beta ensembles of random matrices and their relations with conformal filed theory (CFT). There are (almost) no new results here. This article can serve as a guide on appearances and studies of quantum Painlevé and more general multidimensional linear equations of Belavin-Polyakov-Zamolodchikov (BPZ) type in literature. We demonstrate how BPZ equations of CFT arise from $β$-ensemble eigenvalue integrals. Quantum Painlevé equations are relatively simple instances of BPZ or confluent BPZ equations, they are PDEs in two independent variables…
Second degree polynomial Heisenberg algebras are realized through the harmonic oscillator Hamiltonian, together with two deformed ladder operators chosen as the third powers of the standard annihilation and creation operators. The corresponding solutions to the Painleve IV equation are easily found. Moreover, three different sets of eigenstates of the deformed annihilation operator are constructed, called the good, the bad and the ugly coherent states. Some physical properties of such states will be as well studied.
We show that the new third-order complex nonlinear wave equation, introduced recently by Müller-Hoissen [arXiv:2202.04512], does not pass the Painlevé test for integrability. We find two reductions of this equation, one integrable and one non-integrable, whose solutions jointly cover all solutions of the original equation.
