Integrable Systems and Topology of Isospectral Manifolds
Alexei V. Penskoi
备注：
5 pages, LaTeX, Talk given at the Workshop "Classical and Quantum Integrable Systems-2007" (CQIS-2007), Dubna, Russia. This is an extended version of the short communication math-ph/0701061. v2: minor changes
发表信息：
Teor. Mat. Fiz. 155 (2008), no 1, 140-146 (Russian). Translation in Theor. Math. Phys. 155 (2008), no 1, 627-632.
The well known Liouville-Arnold theorem says that if a level surface of integrals of an integrable system is compact and connected, then it is a torus. However, in some important examples of integrable systems the topology of a level surface of integrals is quite complicated. This is due to the fact that in these examples the phase space has points where either the Hamiltonian is singular or the symplectic form is singular or degenerate. In such situations the Liouville-Arnold theorem does not apply. However, sometimes it is possible to define the corresponding flow on the whole level surface of integrals and use this flow to investigate the topology. Tomei (1982) and Fried (1986) used the Toda lattice to study the topology of the isospectral variety of Jacobi matrices. We recall these results and we also expose new results concerning the topology of the isospectral variety of zero-diagonal Jacobi matrices. This topology is studied using the Volterra system.
Generation of higher-dimensional isospectral-nonisospectral integrable hierarchies associated with a new class of higher-dimensional column-vector loop algebras
Cetraro lectures on integrable systems and moduli spaces of curves
National Research University Higher School of Economics（国立高等经济大学）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Università degli Studi di Padova（帕多瓦大学）
Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems
我们研究了一种特殊的超对称性的对无反射量子力学系统所描述的n-孤子的一般形式，取决于n标度和n平移参数的潜力。证明了如果子系统的离散能级不同，则超代数对平移参数不敏感，是由两个2n阶微分超荷、两个2n+1阶超荷和两个由子系统的Lax积分组成的2n+1阶玻色积分生成的.当一个子系统的r个离散能级与另一个子系统的任意r个离散能级重合时，奇异超对称性发生了约化，两个相互独立的纠缠生成元的总阶数降为4 n-2 r +1，非线性超代数结构获得了对r个相对平移的依赖性.对于一个完整的成对符合的比例参数控制的束缚态的能量和传输散射振幅，新兴的isospectrality检测到的一个蜕变的Lax积分到玻色子中心荷。在等谱类，我们揭示了一个特殊的情况下，给一个新的家庭的有限间隙一阶Bogoliubov-德Gennes系统相关的AKNS可积族。
