AI 中文总结
本研究在单组分四次基准模型中解析O(3)临界泡的局域标量模，发现ℓ=1、2分波连续谱在特定过冷度合并，ℓ=0分波则保持束缚至几何壁交叉，揭示了平面形状模在有限过冷下的两种不同命运。
AI 中文摘要
在相共存条件下，简并的四次标量势允许存在一个精确的平面扭结，其法向涨落算子为修正的Pöschl–Teller算子，平移态和正形状态均处于以Λ=4为起点的连续谱之下。我们在平滑的单组分四次基准模型中，将两个相连的光谱带延拓至有限过冷状态，并解析得到了ℓ=0、1、2的情况。O(3) bounce通过奇异配置法得到，而径向欧几里得海森矩阵则通过有限差分离散化和独立阈值射击法进行分析。正的ℓ=2和ℓ=1分支分别在δ_merge,2=0.0900472和δ_merge,1=0.1410162处达到共同的假真空连续谱阈值。在每个端点处，u_ℓ∝ρ^(-ℓ)是平方可积的，定义了一个阈值本征态；然而，在端点之外，对应分支不存在可归一化本征态的延拓。ℓ=0形状态保持束缚直至几何壁交叉。其平面模重叠、径向质心以及不同的内部和外部衰减长度，揭示了不对称的波函数形变。因此，角光谱溶解和随后类真真空核的几何损失是两个不同的现象，展现了平面形状模式在有限过冷下的两种不同命运。
英文摘要
At phase coexistence, a degenerate quartic scalar potential admits an exact planar kink whose normal fluctuation operator is the modified Pöschl--Teller operator, with translational and positive shape states below a continuum beginning at $Λ=4$. We continue the two connected spectral bands through finite supercooling in a smooth one-component quartic benchmark and resolve $\ell=0,1,2$. The $O(3)$ bounce is obtained by singular collocation, while the radial Euclidean Hessian is analyzed by finite-difference diagonalization and independent threshold shooting. The positive $\ell=2$ and $\ell=1$ branches reach the common false-vacuum continuum threshold at $δ_{{\rm merge},2}=0.0900472$ and $δ_{{\rm merge},1}=0.1410162$, respectively. At each endpoint, $u_\ell\proptoρ^{-\ell}$ is square integrable and defines a threshold eigenstate; beyond the endpoint, however, no normalizable eigenstate continuation exists for the corresponding branch. The $\ell=0$ shape state remains bound up to the geometric wall crossover. Its planar-mode overlap, radial centroid, and distinct interior and exterior decay lengths reveal asymmetric wave-function deformation. Thus, angular spectral dissolution and the later geometric loss of a true-vacuum-like core are separate phenomena, revealing two distinct finite-supercooling fates of the planar shape mode.
Comments30 pages, 14 figures
