Jost solutions and direct scattering for the continuum Calogero-Moser equation
Rupert L. Frank, Larry Read
TL;DR
This work constructs a rigorous direct scattering theory for the continuum Calogero-Moser equation by analyzing the Lax operator $L_q=-i\partial_x-qC_+\overline{q}$ on the Hardy space $L^2_+(\mathbb{R})$, introducing generalized Jost functions and a distorted Fourier transform to diagonalize the absolutely continuous spectrum. It establishes a limiting absorption principle, defines the scattering data $(\beta, \Gamma, \lambda_j, \gamma_j)$, and proves trace formulas that relate these data to the potential $q$, including Birman–Krein type identities. The paper also develops two inverse-scattering reconstruction schemes for $q$ from the spectral data, and derives high-energy and eigenvalue expansions for the Jost functions, as well as a detailed time evolution of the scattering data under the CM flow, highlighting conserved quantities and their role in the integrable structure. Overall, the results lay a comprehensive spectral-analytic foundation for solving the continuum Calogero-Moser equation via inverse scattering and for understanding its conserved quantities through trace identities.
Abstract
We propose an inverse scattering transform for the continuum Calogero-Moser equation. We give a rigorous treatment of the direct scattering problem by constructing the associated Jost solutions and introducing a distorted Fourier transform, as well as deriving trace formulas for the eigenvalues of the Lax operator.