Quantum scattering by intersecting δ-potential barriers: from Gaudin's kaleidoscope to quantum Galperin billiards
Abstract
We develop a general framework for the scattering of a plane wave by a class of Gaudin kaleidoscope models: straight δ-potential barriers intersecting at a common point. The projected Lippmann-Schwinger equations split into a singular part, a finite pole closure generated by a geometric moving rule, and a regular remainder, the outgoing state being an atomic measure on the circle. At the special angles π/N, where the intersecting barriers generate the dihedral group DN, the number of channels stays fixed, whereas at generic angles channels are created and destroyed, opening smoothly from zero weight as a critical direction is crossed. This leaves no singular trace in the probabilities, being carried instead by the phase shifts, a quantum-classical correspondence beyond the reach of the coordinate Bethe ansatz. The same equations solve the quantum Galperin billiards exactly, the method of images reducing them to a single linear system whose phase shifts follow in closed form.
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BibTeX
@misc{yu2026quantum,
title = {{Quantum scattering by intersecting δ-potential barriers: from Gaudin's kaleidoscope to quantum Galperin billiards}},
author = {Yi-Cong Yu and Wen-Jie Qiu and Xiaoming Cai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23406},
eprint = {2609.23406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23406},
}
RIS
TY - UNPB
TI - Quantum scattering by intersecting δ-potential barriers: from Gaudin's kaleidoscope to quantum Galperin billiards
AU - Yi-Cong Yu
AU - Wen-Jie Qiu
AU - Xiaoming Cai
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23406
UR - https://arxiv.org/abs/2609.23406
AB - We develop a general framework for the scattering of a plane wave by a class of Gaudin kaleidoscope models: straight δ-potential barriers intersecting at a common point. The projected Lippmann-Schwinger equations split into a singular part, a finite pole closure generated by a geometric moving rule, and a regular remainder, the outgoing state being an atomic measure on the circle. At the special angles π/N, where the intersecting barriers generate the dihedral group D_N, the number of channels stays fixed, whereas at generic angles channels are created and destroyed, opening smoothly from zero weight as a critical direction is crossed. This leaves no singular trace in the probabilities, being carried instead by the phase shifts, a quantum-classical correspondence beyond the reach of the coordinate Bethe ansatz. The same equations solve the quantum Galperin billiards exactly, the method of images reducing them to a single linear system whose phase shifts follow in closed form.
ER -