经典和量子可积系统若干前沿问题  

Several frontier problems in classical and quantum integrable systems

作　　者：楼森岳[1] LOU Senyue(School of Physical Science and Technology,Ningbo University,Ningbo 315211,China)

机构地区：[1]宁波大学物理科学与技术学院,浙江宁波315211

出　　处：《宁波大学学报(理工版)》2025年第1期1-13,共13页Journal of Ningbo University:Natural Science and Engineering Edition

基　　金：国家自然科学基金重点项目(12235007)。

摘　　要：对经典和量子可积性的前沿研究方面提出一些有意义的探索问题:(1)玻色(对易)可积系统、费米(反对易)可积系统、任意子(q对易)可积系统及其混合系统的相关问题;(2)量子可积性的算子变换理论;(3)线性量子薛定谔方程的无限自由度引起的有限自由度不可积性;(4)与潘勒韦类型的非解析种子解对应的达布变换和反散射理论;(5)非局域对称和局域对称的一般对偶理论;(6)从互反变换推广理论引入的高维可积系统的求解;(7)如何从达布变换、反散射方法、黎曼-希尔伯特方法等传统可积系统理论中获得多线性分离变量解;(8)如何用传统方法推导超可积、超对称可积、任可积和任对称可积系统的玻色化方法的严格解;(9)对称性可求解可积系统多波解的猜想;(10)对称性求解完全可积和部分可积系统的一般猜想;(11)从1+1维双线性可积系统推广到高维可积系统、离散可积系统、超对称可积系统和任对称可积系统的猜想.This paper proposes some significant exploration problems in the frontier research on classical and quantum integrability,including:(1)Problems related to bosonic(commuting)integrable systems,fermionic(anti-commuting)integrable systems,anyonic(q-commuting)integrable systems,and their mixed systems;(2)The theory of operator transformations in quantum integrability;(3)The non-integrability of linear quantum Schrödinger equations due to infinite freedoms;(4)Darboux transformations and inverse scattering theory corresponding to non-analytic seed solutions like Painlevétype solutions;(5)General dual theory between nonlocal symmetries and local symmetries;(6)The solution of high-dimensional integrable systems from the theory of reciprocal transformation extension;(7)How to obtain multilinearly separated variable solutions from traditional theories such as the Darboux transformations,the inverse scattering methods,and the Riemann-Hilbert approaches;(8)How to derive exact solutions obtained from the bosonization approach for supersymmetric integrable systems,super-integrable systems,ren-integrable systems,and ren-symmetric integrable systems by using traditional methods;(9)Conjectures on multiwave solutions of integrable systems via symmetries;(10)General conjectures on all exact solutions via symmetry approach for completely and partially integrable systems;(11)Conjectures to extend 1+1-dimensional bilinear integrable systems to higher-dimensional integrable systems,discrete integrable systems,supersymmetric integrable systems,and ren-symmetric integrable systems.

关 键 词：经典和量子可积性 任对称和超对称可积性 对称性猜想 高维可积形变猜想 

分 类 号：O4[理学—物理] O1