Quasi-Localization of Short-Wavelength Perturbations in Softened Laplace-Lagrange Chains

We examine the long-term evolution of localized eccentricity perturbations in a softened Laplace-Lagrange chain. The secular coupling between mass elements is long-ranged, suggesting that any disturbance imposed at a single radius should disperse rapidly across the chain. We show that this expectation fails once the secular interaction is regularized by a finite softening length $\beta$: while broad eccentricity profiles propagate smoothly, as textbook secular theory anticipates, narrow profiles remain confined to their launching site for millions of secular periods. The origin of this quasi-localization lies in the dispersion relation of the softened chain. The group velocity of eccentricity waves collapses by orders of magnitude on spatial scales below the softening length, so that narrow perturbations project almost entirely onto a nearly flat sector of the band, where little differential precession is available to disperse them. Propagating the initial wavenumber spectrum through the dispersion relation reproduces the retained fraction of angular momentum deficit across the full range of perturbation widths with no adjustable parameters. The confined perturbation is flanked by an apsidally polarized (oppositely-aligned) response, which we show to be the real-space imprint of this spectral projection. The softening length therefore sets a dynamical scale below which eccentricity perturbations spread only on timescales far longer than the secular period.