All Research

Abstract

We determine how the spectrum of a concentrating signed potential form splits between escaping eigenvalues and a bounded remainder as its centers coalesce. Zero profile integrals alone do not rule out escape, and relative leading-order estimates need not determine logarithmic shifts to bounded error. For two unequal asymmetric zero-mean profiles separated by more than their width, we obtain logarithmic formulas with bounded additive remainders on both escaping scales throughout the mesoscopic collision regime. Exact primitive correlations and Green geometry, together with a cancellation in a quadratic spectral pencil, control mixed trace terms that can exceed the slower scale. For a noncancelling cluster at bounded separation-to-width ratios and zero background, we identify the operator remaining after removal of the actual escaping pair. Aggregate inverse-square tails determine its essential and absolutely continuous channels; vanishing aggregate tails give trace-norm convergence. Fixed-threshold eigenvalue counts and signed spectral masses quantify how a bounded bulk balances the escaping pair's logarithmic trace drift. At exact cancellation, a primitive estimate controls the complete operator for measurable profiles. An anchored cusp family exhibits regularity-dependent transition scales and a logarithmic endpoint, with a vanishing rank-two approximation error along every joint path. Identical even profiles supply explicit scalar matching through all collision ratios.