Kouichi Takemura
We review several results on the finite-gap potential and Heun's differential equation, and we discuss relationships among the finite-gap potential, the WKB analysis and Heun's differential equation.
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The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding to Heun's differential equation and the ultradiscrete limit $q\to 0$ are considered.
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.
The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one more singularity. In this paper we compare these equations with linear differential equations isomonodromy deformations of which are described by the Painlevé equations $P_{II}-P_{VI}$.
We derive the most general families of differential operators of first and second degree semi-commuting with the differential operators of the Heun class. Among these families we classify all those families commuting with the Heun class. In particular, we discover that a certain generalized Heun equation commutes with the Heun differential operator allowing us to construct the general solution to a complicated fourth order linear differential equation with variable coefficients which Maple 16 cannot solve.
Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.
We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our results to exceptional Jacobi polynomials.
We relate two parameter solutions of the sixth Painlevé equation and finite-gap solutions of the Heun equation by considering monodromy on a certain class of Fuchsian differential equations. In the appendix, we present formulae on differentials of elliptic modular functions, and obtain the ellitic form of the sixth Painlevé equation directly.
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency (non-branching) of a singularity. For the elliptical repr…
The generating function of Stieltjes-Carlitz polynomials is a solution of Heun's differential equation and using this relation Carlitz was the first to get exact closed forms for some Heun functions. Similarly the associated Stieltjes-Carlitz polynomials lead to a new differential equation which we call associated Heun. Thanks to the link with orthogonal polynomials we are able to deduce two integral relations connecting associated Heun functions with different parameters and to exhibit the set of associated Heun functions which generalize Carlitz's. Part of these results were used by the auth…
Distributional Solution and Spectral Shift Function of Heun Differential Equation
该研究将Heun算子纳入SL(2,C)李代数的泛包络代数框架，并利用群上Fourier变换求其精确可解算子的Green函数与谱移位函数。
On zeros of polynomials associated with Heun class equations
本文研究Heun类方程局部解系数中附加参数多项式的零点，通过数值计算获得相关谱数据。
In the present article we introduce and study a novel type of solutions of the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form of the Fuchsian equations with an arbitrary number $N\geq 4$ of regular singular points. We derive the symmetry group of these equations which turns to be a proper extension of the Mobius group. We also introduce and study new series solution of symmetric form of the general Heun's differential equation (N=4) which treats simultaneously and on an equal foot…
The Painlevé equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a ``time variable'' $t$. From ladder operators one finds second order linear ordinary differential equations for associated orthogonal polynomials with coefficients being rational functions. The Painlevé and related functions appear as the residues of these rational functions. We will be interested in the situation when $n$, the order of the Hankel matrix and also t…
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 introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are well-known polynomials while others are either new or modified versions of known ones.
The present article discusses the two point connection problem for Heun's differential equation. We employ a contour integral method to derive connection matrices for a sub-class of the Heun equation containing 3 free parameters. Explicit expressions for the connection coefficients are found.
Degenerations of $q$-Heun equation
本文通过若干q-Painlevé方程相关的线性q-差分方程，系统构造了q-Heun方程的合流、双合流与二重合流退化形式，并研究了它们到相应微分方程的极限过程。
Normalization Integrals of Orthogonal Heun Functions
本文提出一种基于微分方程极限过程的方法，利用局部幂级数解及其导数显式计算正交Heun函数的归一化积分，为构建标准正交基提供高效替代方案。
We review Darboux-Crum transformation of Heun's differential equation. By rewriting an integral transformation of Heun's differential equation into a form of elliptic functions, we see that the integral representation is a generalization of Darboux-Crum transformation. We also consider conservation of monodromy with respect to the transformations.
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide explicit integral representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly-confluent and triconfluent, with integrals involving only rational functions and exponential integrands. All the series are illustrated with concrete examples of us…
Accessory Parameter of Confluent Heun Equations, Voros Periods and classical irregular conformal blocks
本文通过Voros周期推导连通Heun方程的附加参数形式级数展开，并将其与Bonelli等人的经典共形块进行比较，探讨Zamolodchikov型猜想，扩展了Mironov等人的工作，给出了选择谱曲线循环以获得共形块的详细方法。
It is known that the Painleve VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painleve-Heun correspondence.
The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form $-d^2/dx^2+V(g;x)$, where the potential is an elliptic function depending on a coupling vector $g\in{\mathbb R}^4$. Alternatively, this operator arises from the $BC_1$ specialization of the $BC_N$ elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on $g$, we associate to this operator a self-adjoint operator $H(g)$ on the Hilbert space ${\mathcal H}=L^2([0,ω_1],dx)$, where $2ω_1$ is the real period of …
The confluent Heun equation is one of 4 confluent forms of Heun's differential equation in which is the Fuchsian equation of second order with four regular singularities. A confluent Heun function is applicable to diverse areas such as theory of rotating/non-rotating black hole, the gauge theories on thick brane words, Schr$\ddot{\mbox{o}}$dinger equation for hydrogen molecule ion in Stark effect and etc. The confluent Heun function consists of the three term recurrence relation in its power series, and we show that the function is divergent on the boundary of the disc of convergence.
Van Vleck spectra of high-order Heun operators:\ finite-band universality and exterior asymptotics
研究Fuchs指标为1的经典Heun算子的高阶类似物，通过线性Van Vleck多项式形成谱多项式，得到有限带行列式表示和普遍性定理，结合相关定理加强外部收敛结论，证明确定性准则等。
Les Houches Lectures on Exact WKB Analysis and Painlevé Equations
本文通过介绍精确 WKB 分析理论及其在 Painlevé 方程研究中的应用，探讨了线性微分方程的单值性问题和 resurgent 结构分析。
Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent forms: (1) Confluent Heun (two regular and one irregular singularities), (2) Doubly confluent Heun (two irregular singularities), (3) Biconfluent Heun (one regular and one irregular singularities), (4) Triconfluent Heun equations (one irregular singularity). Mathieu equation in algebraic forms is also derived from the Confluent Heun equation by changing all …
The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Moebius automorphisms and F-homotopies, and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of 3F2 type is examined, and Pfaff and Euler transformations of 3F2(a1,a2,e+1;b1,e;x) are derived. They ex…
A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D_n. Each of the 192 expressions is labeled by an element of D_4. Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that t…
Three schemes, whose expressions are not too complex, are selected for the numerical integration of a system of stochastic differential equations in the Stratonovich interpretation: the integration methods of Heun, Milstein, and derivative-free Milstein. The strong (path-wise) convergence is studied for each method by comparing the final points after integrating with $2^n$ and $2^{n-1}$ time steps. We also compare the time that the computer takes to carry out the integration with each scheme. Putting both things together, we conclude that, at least for our system, the Heun method is by far the…
A class of Schrödinger-type second-order linear differential equations with a large parameter $u$ is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of $u$. These formal series solutions are called the WKB solutions. We show that under mild conditions on the potential function of the equation, the WKB solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. It is established that the formal series expansions are the asymptotic expansions, uniform with resp…
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.
We study the system of partial differential equations which characterizes the Pearcey integral from the viewpoint of the exact WKB analysis. It is shown that the Borel transform of the WKB solutions to the system can be written as a linear combination of the branches of an algebraic function of degree $4$. Moreover, some connection formulas for the system are given.
On the Convergence of WKB Approximations of the Damped Mathieu Equation
本文研究阻尼Mathieu方程在质量参数趋于零时WKB近似的收敛性，证明基本解集的周期部分和指数部分均收敛，并给出特征指数和周期部分的渐近误差阶。
We consider the difference Schr{ö}dinger equation $ψ$(z + h) + $ψ$(z -- h) + v(z)$ψ$(z) = 0 where z is a complex variable, h > 0 is a parameter, and v is an analytic function. As h $\rightarrow$ 0 analytic solutions to this equation have a standard quasiclassical behavior near the points where v(z) = $\pm$2. We study analytic solutions near the points z 0 satisfying v(z 0) = $\pm$2 and v (z 0) = 0. For the finite difference equation, these points are the natural analogues of the simple turning points defined for the differential equation --$ψ$ (z) + v(z)$ψ$(z) = 0. In an h-independent neighbor…
Orthogonal polynomials: from Heun equations to Painlevé equations
本文研究了四种权函数对应的正交多项式，给出了其满足的二阶线性微分方程及Heun方程，利用等单值形变方法将其转化为Painlevé方程，并讨论了Hankel行列式的渐近行为。
In this paper, we study the weak asymptotic in the plane of some wave functions resulting from the WKB techniques applied to a Shrodinger equation with quartic oscillator and having some boundary condition. In first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions .Especially , we investigate the properties of the Cauchy transform of the root counting measure of a re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) …
This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconflue…
We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard-Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous d…
Another approach to WKB analysis
本文提出一种新的方法研究WKB方程，不依赖传统假设，仅使用高级微积分和微分方程理论，无需假设拐点为简单或存在连接两个拐点的Stokes曲线。
In my joint papers with Iwaki and Koike ([IKoT1, IKoT2]) we found an intriguing relation between the Voros coefficients in the exact WKB analysis and the free energy in the topological recursion introduced by Eynard and Orantin in the case of the confluent family of the Gauss hypergeometric differential equations. In this paper we discuss its generalization to the case of the hypergeometric differential equations associated with $2$-dimensional degenerate Garnier systems.
We develop a $D-$module approach to various kinds of solutions to several classes of important differential equations by long divisions of different differential operators. The zeros of remainder maps of such long divisions are handled by an analogue of Hensel's lemma established recently from valuation theory. In particular, this explains the common origin of some classically known special function series solutions of Heun equations and usual Frobenius series solutions. Moreover, these remainder maps also generate eigenvalue problems that lead to non-trivial factorizations of certain generali…
Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight
研究不连续半经典Laguerre权函数，推导出与Painlevé IV方程相关的递推系数，并揭示正交多项式满足双退化Heun方程。
The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti's transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace …
Semiclassical expansion for exactly solvable differential operators
研究线性微分方程\(\MM (v(z,\eta))=\eta^M{v(z,\eta)}\)，证明其WKB解在斯托克斯区域的波莱尔可和性，对\(M = 3\)给出斯托克斯复形描述，还表明欧拉 - 柯西方程的WKB解在通常意义下收敛。
We use some properties of solutions of Riccati equation for establishing boundedness and stability criteria for solutions of second order linear ordinary differential equations. We show that the conditions on coefficients of the equations, appearing in the proven criteria, do not follow from the conditions, which ensure the application of the WKB approximation to the second order linear equations. On these examples we compare the obtained results wit the results obtained by the Liapunov and Bogdanov methods, by a method involving estimates of solutions in the Lozinski's logarithmic norms, and …
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