2011/11/02 by Chang-Shou Lin, Chang‐Shou Lin, Dong Ye +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Black Holes and Theoretical Physics #Combinatorics #Degenerate energy levels #Diagonal #Diagonal matrix #Geometry #Invertible matrix #Mathematical analysis #Mathematical physics #Mathematics #Measure (data warehouse) #Nonlinear Waves and Solitons #Order (exchange) #Physics #Pure mathematics #Quantization (signal processing) #Quantum mechanics #Toda lattice #Triangular matrix #math.AP #math.DG #msc:35B20
paper · pdf · doi:10.1007/s00222-012-0378-3
28 pages
arxiv created 2011/11/02 · openalex publication_date 2012/01/17 · arxiv updated 2015/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the following Toda system Δui + \D ∑j = 1n aijeuj = 4πγiδ0 in\mathbb R2, ∫\mathbb R2eui dx < ∞, ∀ 1≤ i ≤ n, where γi > -1, δ0 is Dirac measure at 0, and the coefficients aij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: ∑j=1n aij∫\R2euj dx = 4π(2+γi+γn+1-i), ∀ 1≤ i ≤ n. This generalizes the classification result by Jost and Wang for γi=0, ∀ 1≤ i≤ n. (ii) We prove that if γi+γi+1+...+γj ∉ \mathbb Z for all 1≤ i≤ j≤ n, then any solution ui is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.