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 \(ε_1=\sqrt{\sqrt{3}-1}\). 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.