Bargmann-Fock Representation and Global Estimates for the Linearized Hard-Sphere Boltzmann Operator
Abstract
We study the linearized Boltzmann collision operator for three-dimensional hard spheres and reorganize the complete operator so that its angular and radial structures are explicit before any Sonine truncation. From the Carleman representation, the collision integral becomes a superposition of orthogonal translations; in Hermite/Fock and Bargmann form these generate a single analytic coherent-state kernel for all Hermite collision brackets. Rotational covariance separates fixed angular-momentum sectors, each reducing exactly to a one-variable radial Bargmann space with an su(1,1) structure. The radial operator is Aℓ=κπG(𝒥ℓ)−Cℓ, where 𝒥ℓ is an explicit Laguerre--Jacobi operator and Cℓ is compact. A unitary intertwiner identifies this realization with the classical Burnett/Sonine coordinate, recovering the Burnett matrix and standard stress and heat-flux corrections. For the untruncated operator, the compact gain gives only compact corrections, while the loss sets the common essential spectrum [νmin,∞). Radially squeezed su(1,1) coherent packets yield threshold Weyl sequences in every fixed angular sector. With ℓ→∞ and squeezing scaled as ℓ−1, they furnish full-space Weyl sequences throughout the essential band interior, with ⟨ j⟩∼ℓ2, revealing quadratic continuum corridors in the (ℓ,j) plane. At fixed ℓ and j→∞, the matrix tail approaches a universal Toeplitz operator whose zero at θ=π is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.
Related
- Operator-theoretic and analytic properties of a driven single-mode Kerr cavity in Bargmann space
- Schauder estimates for the linearized very fast diffusion equations in bounded domains
- Geometric Viewpoint of L∞ Estimates for Linearized Monge-Ampère Equations
- Eigenvalues of the linearized Navier-Stokes operator near locally Couette laminar flows
- Global Hölder estimates for linearized Monge-Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization
BibTeX
@misc{karlin2026bargmannfock,
title = {{Bargmann-Fock Representation and Global Estimates for the Linearized Hard-Sphere Boltzmann Operator}},
author = {Ilya Karlin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09548},
eprint = {2609.09548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09548},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bargmann-Fock Representation and Global Estimates for the Linearized Hard-Sphere Boltzmann Operator
AU - Ilya Karlin
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09548
UR - https://arxiv.org/abs/2609.09548
AB - We study the linearized Boltzmann collision operator for three-dimensional hard spheres and reorganize the complete operator so that its angular and radial structures are explicit before any Sonine truncation. From the Carleman representation, the collision integral becomes a superposition of orthogonal translations; in Hermite/Fock and Bargmann form these generate a single analytic coherent-state kernel for all Hermite collision brackets. Rotational covariance separates fixed angular-momentum sectors, each reducing exactly to a one-variable radial Bargmann space with an su(1,1) structure. The radial operator is A_ℓ=κπG(𝒥_ℓ)−C_ℓ, where 𝒥_ℓ is an explicit Laguerre--Jacobi operator and C_ℓ is compact. A unitary intertwiner identifies this realization with the classical Burnett/Sonine coordinate, recovering the Burnett matrix and standard stress and heat-flux corrections. For the untruncated operator, the compact gain gives only compact corrections, while the loss sets the common essential spectrum [ν_min,∞). Radially squeezed su(1,1) coherent packets yield threshold Weyl sequences in every fixed angular sector. With ℓ→∞ and squeezing scaled as ℓ^−1, they furnish full-space Weyl sequences throughout the essential band interior, with ⟨ j⟩∼ℓ^2, revealing quadratic continuum corridors in the (ℓ,j) plane. At fixed ℓ and j→∞, the matrix tail approaches a universal Toeplitz operator whose zero at θ=π is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.
ER -