Quantum Painlevé Equations: from Continuous to Discrete

On a q-Deformation of the Discrete Painlevé I Equation and q-Orthogonal Polynomials

I present a $q$-analog of the discrete Painlevé I equation, and a special realization of it in terms of $q$-orthogonal polynomials.

Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation

Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as "symmetries" or the Bäcklund transformations in Painlevé equations. We thereby propose a quantization of the discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of $W(D_4^{(1)})$.

Affine Weyl groups, discrete dynamical systems and Painleve equations

A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.

Quantizing the discrete Painlevé VI equation : The Lax formalism

A discretization of Painlevé VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point of view. It turns out that the second approach is also successful and gives the same quantization as in the first approach.

非平稳差分方程与仿射Laumon空间：离散Painlevé方程的量子化

Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation

本研究建立了非平稳差分方程与量子化离散Painlevé VI方程的关联，证明原方程可分解为耦合系统，并推测仿射Laumon空间的瞬子配分函数是其解。

Quantum Painlevé systems of type $A^{(1)}_l$

We propose quantum Painlevé systems of type $A_l^{(1)}$. These systems, for $l=1$ and $l\ge 2$, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type $A_l^{(1)}$ studied by M. Noumi and Y. Yamada \cite{NYhigherorder}, respectively. These quantizations enjoy the affine Weyl group symmetries of type $A_l^{(1)}$ as well as the Lax representations. The quantized systems of type $A_1^{(1)}$ and type $A_l^{(1)}$ ($l= 2n$) can be obtained as the continuous limits of the discrete systems constructed from the affin…