Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients
Abstract
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)*∂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 Γ0(T) quotients, together with a four-cusp quotient arising from an elliptic curve over 𝔽3, make the normalizations and matrix-valued conclusions explicit.
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BibTeX
@misc{hong2026eisenstein,
title = {{Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients}},
author = {Soonki Hong and Sanghoon Kwon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25925},
eprint = {2609.25925},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25925},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients
AU - Soonki Hong
AU - Sanghoon Kwon
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25925
UR - https://arxiv.org/abs/2609.25925
AB - 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)^*∂_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 𝔽_3, make the normalizations and matrix-valued conclusions explicit.
ER -