经典和量子可积系统若干前沿问题
PDF下载 (524)楼森岳.经典和量子可积系统若干前沿问题[J].宁波大学学报(理工版),2025,38(1):1-13.DOI:10.20098/j.cnki.1001-5132.2024.1013
LOU Senyue.Several frontier problems in classical and quantum integrable systems[J].Journal of Ningbo University(Natural Science & Engineering Edition),2025,38(1):1-13.DOI:10.20098/j.cnki.1001-5132.2024.1013
| Title: | Several frontier problems in classical and quantum integrable systems |
| 作者: | 楼森岳 |
| Author(s): | LOU Senyue |
| 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 |
| Keywords: | classical and quantum integrability; ren-symmetric and supersymmetric integrability; symmetry conjuecture; higher dimensional integrable conjuecture |
| 分类号: | O4; O1 |
| DOI: | 10.20098/j.cnki.1001-5132.2024.1013 |
| 文献标识码: | A |
| 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. |
| Abstract: | 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 Painlevé type 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. |
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| 备注/Memo: | 收稿日期: 2024−10−28. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/ 基金项目: 国家自然科学基金重点项目（12235007）. 第一作者: 楼森岳, 教授, 主要研究方向: 非线性物理、应用数学、粒子物理和量子场论、非线性大气和海洋动力学. E-mail: lousenyue@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
