Heun Spectrum, Threshold Hierarchy, and One-Loop Quantum Mass of the Christ--Lee Kink

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 ${\epsilon }_1=\sqrt{\sqrt{3}-1}$. For small deformation, the kink separates into two widely spaced ${\phi }^6$-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 ${\phi }^4$ 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 ${\phi }^4$ 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.