2002/02/28 by K. Vaninsky, K.L. Vaninsky
Mathematics · Physics and Astronomy · #Connection (principal bundle) #Elliptic function #Holomorphic and Operator Theory #Homogeneous space #Inverse #Isospectral #Lattice (music) #Poisson bracket #Poisson manifold #Quantum Mechanics and Non-Hermitian Physics #Quotient #Spectral Theory in Mathematical Physics #Toda lattice #math-ph #math.MP
paper · pdf · doi:10.1016/s0393-0440(02)00135-3
published as Jour. Geometry and Physics 46 (2003) 283-307 · 26 pages, 2 figures
arxiv created 2002/02/28 · openalex publication_date 2003/04/30 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three--diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one--to--one correspondence with their Weyl functions. These are rational functions mapping the upper half--plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic--geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear--fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.