Divergence Without Transition in Adiabatic Theory: Exact Cancellation in Reflectionless Potentials

Abstract

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.