发表机构
Catholic Kwandong University（天主教江原大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Bruhat-Tits树的算术商，建立了Eisenstein变换与Jacobi算子散射变换的显式酉对应，推导出矩阵值Maass-Selberg公式，并利用共振计算区分特征值与散射极点。
AI 中文摘要
对于具有有限多个尖点的Bruhat-Tits树的算术商，我们建立了球Eisenstein变换（其Eisenstein级数通过常数项归一化）与具有有限核的关联Jacobi算子的散射变换之间的显式酉对应关系。追踪Haar测度、稳定子权重、高度坐标和尖点宽度，我们得到Plancherel测度，并表明绝对连续谱的重数等于尖点的个数。由离散Green恒等式，我们推导出关于Hermite矩阵$iS(\theta)^*\partial_\theta S(\theta)$的矩阵值Maass-Selberg公式，其中$S(\theta)$是散射矩阵。其迹由$\det S(\theta)$决定，而完整矩阵保留了额外的尖点间信息。在相应的归一化改变后，通过消去尖点射线得到的有限Schur补与Arends-Peterson-Weich的共振矩阵一致。利用他们的共振计算作为输入，我们区分完全支撑在有限核中的特征值与散射矩阵的极点。Nagao商和$\Gamma_0(T)$商，以及由$\mathbb F_3$上的椭圆曲线产生的四尖点商，使归一化和矩阵值结论变得明确。
英文摘要
For arithmetic quotients of Bruhat--Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass--Selberg formula for the Hermitian matrix $iS(θ)^*\partial_θS(θ)$, where $S(θ)$ is the scattering matrix. Its trace is determined by $\det S(θ)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $Γ_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.
Comments39 pages, 4 figures
