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 $\mathfrak{su}(1,1)$ structure. The radial operator is ${\mathfrak{A}}_{\ell }=\kappa \pi G({\mathcal{J}}_{\ell })-{\mathfrak{C}}_{\ell }$, where ${\mathcal{J}}_{\ell }$ is an explicit Laguerre--Jacobi operator and ${\mathfrak{C}}_{\ell }$ 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 $[{\nu }_{\min },\infty )$. Radially squeezed $\mathfrak{su}(1,1)$ coherent packets yield threshold Weyl sequences in every fixed angular sector. With $\ell \to \infty$ and squeezing scaled as ${\ell }^{-1}$, they furnish full-space Weyl sequences throughout the essential band interior, with $⟨j⟩\sim {\ell }^2$, revealing quadratic continuum corridors in the $(\ell ,j)$ plane. At fixed $\ell$ and $j\to \infty$, the matrix tail approaches a universal Toeplitz operator whose zero at $\theta =\pi$ is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.