Mahouton Norbert Hounkonnou, André Ronveaux
This paper addresses new results on the factorization of the general Heun's operator, extending the investigations performed in previous works [{\it Applied Mathematics and Computation} {\bf 141} (2003), 177 - 184 and {\bf 189} (2007), 816 - 820]. Both generalized Heun and generalized Lamé equations are considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erdélyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.
By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lamé equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic solutions of Heun's equation. Each of the quasi-periodic solutions is doubly degenerate.
An analysis of the generalized confluent Heun equation $(α_2r^2+α_1r)\,y''+(β_2r^2+β_1r+β_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{α_i, β_i, \varepsilon_i\}$ are real parameters, is presented. With the aid of these general results, the quasi exact solvability of the Schrödinger eigenproblem generated by the softcore Coulomb potential $V(r)=-e^2Z/(r+b),\, b>0$, is explicitly resolved. Necessary and sufficient conditions for polynomial solvability are given. A three-term recurrence relation is provided to generate the coefficients of polynomial solutions exp…
We consider methods for constructing explicit solutions of the non-stationary Lamé equation, which is a generalization of the classical Lamé equation, that has appeared in works on integrable models, conformal field theory, high energy physics and representation theory. We also present a general method for constructing integral representations of solutions to the non-stationary Lamé equation by a recursive scheme. Explicit integral representations, for special values of the model parameters, are also presented. Our approach is based on kernel function methods which can be naturally generalized…
New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials for which the stationary one-dimensional Schrödinger equation is reduced to t…
The present paper analyze the constraints on the confluent Heun type-equation, $(a_{3,1}r^2+a_{3,2}r)y"+(a_{2,0}r^2+a_{2,1}r+a_{2,2})y'-(τ_{1,0}r+τ_{1,1})y=0,$ where $|a_{3,1}|^2+|a_{3,2}|^2\neq 0, $ and $a_{i,j},i=3,2,1, j=0,1,2$ are real parameters, to admit polynomial solutions. The necessary and sufficient conditions for the existence of these polynomials are given. A three-term recurrence relation is provided to generate the polynomial solutions explicitly. We, then, prove that these polynomial solutions are a source of finite sequences of orthogonal polynomials. Several properties, such …
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 …
Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlevé equations. There are important examples, however, where one has to use higher order equations. Heun equation is…
We study the factorizations of Ising low-temperature correlations C(M,N) for $ν=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlevé VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlevé V…
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…
Blowing-up the edge: connection formulae and stability chart of the Lamé equation
本文通过超对称规范理论与二维共形场论研究周期谱问题，提出用爆破方程极限重求和半经典 Virasoro 块的新方法，并精确刻画了 Lamé 方程等量子可积系统的稳定性图与连接问题。
We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lamé potentials $V^{PT}(x)=-a(a+1)m \sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2 (y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}$, where $y \equiv ix+β$, and there are four parameters $a,b,f,g$. This work is a substantial generalization of previous work with the associated Lamé potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)}$ and their corresponding PT-invariant counterparts $V^{PT}(x)=-V(ix+β)$, both of which involving just two parameters $a,b$. We show that f…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。