发表机构
ETH Zurich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
通过Bargmann-Fock表示将线性化硬球Boltzmann算子精确径向化，揭示su(1,1)结构并构造全局Weyl序列，实现谱的精确控制。
AI 中文摘要
我们研究三维硬球的线性化Boltzmann碰撞算子，并在任何Sonine截断之前重新组织完整算子，使其角度和径向结构显式化。从Carleman表示出发，碰撞积分成为正交平移的叠加；在Hermite/Fock和Bargmann形式下，这些平移为所有Hermite碰撞括号生成一个解析相干态核。旋转协变性分离出固定角动量扇区，每个扇区精确约化为具有$\mathfrak{su}(1,1)$结构的一变量径向Bargmann空间。径向算子为$\mathfrak A_\ell=\kappa\pi G(\mathcal J_\ell)-\mathfrak C_\ell$，其中$\mathcal J_\ell$是显式的Laguerre-Jacobi算子，$\mathfrak C_\ell$是紧的。一个酉交织子将该实现与经典Burnett/Sonine坐标等同，恢复Burnett矩阵以及标准的应力和热流修正。对于未截断的算子，紧致增益仅给出紧致修正，而损失项设定公共本质谱$[\nu_{\min},\infty)$。径向压缩的$\mathfrak{su}(1,1)$相干包在每个固定角度扇区中产生阈值Weyl序列。随着$\ell\to\infty$且压缩尺度为$\ell^{-1}$，它们在整个本质谱带内部提供全空间Weyl序列，揭示$(\ell,j)$平面中的连续谱走廊。在固定$\ell$且$j\to\infty$时，矩阵尾部趋近于一个通用Toeplitz算子，其在$\theta=\pi$处的零点标志着同一阈值逃逸的局部特征。因此，Fock重构提供了精确的径向化以及对全局谱几何的构造性控制。
英文摘要
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 $\mathfrak A_\ell=κπG(\mathcal J_\ell)-\mathfrak C_\ell$, 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 $[ν_{\min},\infty)$. 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 $\langle j\rangle\sim\ell^2$, 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 $θ=π$ is the local signature of the same threshold escape. Thus the Fock reformulation provides exact radialization and constructive control of the global spectral geometry.
