绝热理论中无跃迁的散度：无反射势中的精确抵消
Divergence Without Transition in Adiabatic Theory: Exact Cancellation in Reflectionless Potentials
AI总结：
研究绝热理论中无跃迁的散度，以与科特韦格 - 德弗里斯孤子相关的无反射势层级为例，通过达布分解等方法，揭示虽构造发散但反向波系数为零等情况，表明散度诊断局部对角化失败，全局辛连接决定反射是否存在。
AI中文摘要：
绝热不变量在等离子体物理中起着核心作用，其微扰构造常是渐近的且呈现阶乘增长。我们表明这种散度本身并不意味着非绝热跃迁。对于与科特韦格 - 德弗里斯孤子相关的无反射势层级，尽管贝里相位积分迭代和相应的李变换构造发散，但精确的反向波系数为零。达布分解明确给出了透射波。其模和相位定义了一种范式，其中移动的正则框架在相互作用区域内变形但回到其原始渐近形式，仅留下相移且无作用变化。精确的相位积分是导数迭代的不稳定不动点。对于单孤子对称点，即使精确的非对角连接系数为零，在单个李变换族中显式的博雷尔计算也显示出非零奇点。精确连接数据的解析性要求这些依赖于表示的模糊性在完整的连接矩阵中抵消。因此，散度诊断局部对角化的失败，而全局辛连接决定反射是否存在。
英文摘要：
Adiabatic invariants play a central role in plasma physics, from magnetic moment and bounce action to wave action in slowly varying media. Their perturbative constructions are often asymptotic, and exhibit factorial growth. We show that such divergence does not by itself imply non-adiabatic transitions. For the reflectionless potential hierarchy associated with Korteweg--de Vries solitons, the exact backward-wave coefficient vanishes, although Berry's phase-integral iteration and the corresponding Lie-transform construction are divergent. Darboux factorisation gives the transmitted wave explicitly. Its modulus and phase define a normal form in which the moving canonical frame is distorted inside the interaction region but returns to its original asymptotic form, leaving only a phase shift and no action change. The exact phase integral is nevertheless an unstable fixed point of the derivative iteration. For the one-soliton symmetry point, an explicit Borel calculation exhibits nonzero singularities in an individual Lie-transform family even though the exact off-diagonal connection coefficient vanishes. Analyticity of the exact connection data then requires these representation-dependent ambiguities to cancel in the completed connection matrix. Thus divergence diagnoses failure of local diagonalisation, whereas the global symplectic connection determines whether reflection survives.
