Non-Harmonic Effective Potentials from Thin-Shell Dynamics and Their Spectral Implications

We show that non-harmonic effective potentials arise naturally from the nonlinear dynamics of thin-shell configurations. Expanding the shell potential around a characteristic radius associated with a bounce, a minimal quadratic-plus-quartic structure is obtained without introducing additional assumptions. After coarse-graining, this induces an effective internal potential for bound configurations that deviates from the harmonic approximation.

The resulting excitation spectrum is not uniformly spaced: low-lying modes remain approximately harmonic, while higher excitations increasingly reflect the nonlinear structure of the underlying geometry. This provides a natural mechanism for generating non-equidistant spectra without introducing additional fundamental degrees of freedom.

We argue that this behavior captures a minimal structural feature of composite systems, such as quarkonium families, where bound states exhibit hierarchies of excitations with non-uniform spacing. In this framework, spectral deformation can be interpreted as a residual signature of nonlinear geometric dynamics after coarse-graining.