Heun Spectrum, Threshold Hierarchy, and One-Loop Quantum Mass of the Christ--Lee Kink
Abstract
We study the fluctuation spectrum and renormalized one-loop quantum mass of the Christ--Lee kink over the full positive range of its deformation parameter. The fluctuation equation is reduced exactly to general-Heun form, yielding parity-resolved continuum solutions and implicit quantization conditions for the discrete spectrum. At the first nontrivial continuum crossing, the odd Heun solution truncates to a polynomial, giving the exact threshold ε1=√(√3−1). For small deformation, the kink separates into two widely spaced ϕ6-like interfaces and the fluctuation operator develops a long central cavity. This geometry explains the logarithmic growth of the bound-state count, the asymptotically geometric hierarchy of continuum-threshold crossings, and the emergence of a soft relative-translation mode. In the opposite limit, the spectrum approaches the ϕ4 Pöschl--Teller problem, while its even threshold resonance becomes a shallow bound state at finite deformation. The one-loop correction is evaluated in a fixed vacuum-normal-ordering prescription using a common-regulator spectral trace and independently through an imaginary-frequency functional determinant. The two calculations agree throughout representative finite deformations and reproduce the exact ϕ4 limit. In the small-deformation regime, the extended central region produces a logarithmically enhanced negative quantum correction. These results provide a unified analytic and numerical description of how the spectral reorganization of the Christ--Lee kink controls its semiclassical quantum mass.
Related
- On pointwise convergence of singular integrals of Christ-Journé type
- A Sum-of-Squares Hierarchy with Quadratic Convergence for Quantum Channel Coding
- Quantum hypothesis testing of non-mixed-unitarity: A multifaceted hierarchy of quantum channel discrimination
- From Ehrenfest to Heisenberg scales: a hierarchy of mixing times for a phase-randomized quantum baker walk
- Shimura curves of discriminant 14 and 15 and associated Heun Functions
BibTeX
@misc{mulyanto2026heun,
title = {{Heun Spectrum, Threshold Hierarchy, and One-Loop Quantum Mass of the Christ--Lee Kink}},
author = {Mulyanto and Afif Jalu Pradipta and Ardian Nata Atmaja and Bobby Eka Gunara},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27178},
eprint = {2609.27178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27178},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Heun Spectrum, Threshold Hierarchy, and One-Loop Quantum Mass of the Christ--Lee Kink
AU - Mulyanto
AU - Afif Jalu Pradipta
AU - Ardian Nata Atmaja
AU - Bobby Eka Gunara
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27178
UR - https://arxiv.org/abs/2609.27178
AB - We study the fluctuation spectrum and renormalized one-loop quantum mass of the Christ--Lee kink over the full positive range of its deformation parameter. The fluctuation equation is reduced exactly to general-Heun form, yielding parity-resolved continuum solutions and implicit quantization conditions for the discrete spectrum. At the first nontrivial continuum crossing, the odd Heun solution truncates to a polynomial, giving the exact threshold ε_1=√(√3−1). For small deformation, the kink separates into two widely spaced ϕ^6-like interfaces and the fluctuation operator develops a long central cavity. This geometry explains the logarithmic growth of the bound-state count, the asymptotically geometric hierarchy of continuum-threshold crossings, and the emergence of a soft relative-translation mode. In the opposite limit, the spectrum approaches the ϕ^4 Pöschl--Teller problem, while its even threshold resonance becomes a shallow bound state at finite deformation. The one-loop correction is evaluated in a fixed vacuum-normal-ordering prescription using a common-regulator spectral trace and independently through an imaginary-frequency functional determinant. The two calculations agree throughout representative finite deformations and reproduce the exact ϕ^4 limit. In the small-deformation regime, the extended central region produces a logarithmically enhanced negative quantum correction. These results provide a unified analytic and numerical description of how the spectral reorganization of the Christ--Lee kink controls its semiclassical quantum mass.
ER -