2013/09/20 by Dario Bambusi, Bambusi, Dario, Thomas Kappeler +3 · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.DS #math.MP
paper · pdf · doi:10.48550/arxiv.1309.5324
openalex publication_date 2013/09/20 · arxiv created 2015/05/22 · arxiv updated 2015/05/25 · openalex created_date 2022/12/14 · openalex updated_date 2026/07/28
For periodic Toda chains with a large number N of particles we consider states which are N-2-close to the equilibrium and constructed by discretizing arbitrary given C2-functions with mesh size N-1. Our aim is to describe the spectrum of the Jacobi matrices L_N appearing in the Lax pair formulation of the dynamics of these states as N → ∞. To this end we construct two Hill operators H_± -- such operators come up in the Lax pair formulation of the Korteweg-de Vries equation -- and prove by methods of semiclassical analysis that the asymptotics as N → ∞ of the eigenvalues at the edges of the spectrum of L_N are of the form ± (2-(2N)-2 λ^± _n + ⋯ ) where (λ^± _n)_n ≥ 0 are the eigenvalues of H_± . In the bulk of the spectrum, the eigenvalues are o(N-2)-close to the ones of the equilibrium matrix. As an application we obtain asymptotics of a similar type of the discriminant, associated to L_N.