Curvature-Induced Shifts of Higgs and Goldstone Modes in 2D Superconducting Shells

We present an analytical theory of collective excitations in a two-dimensional superconductor confined to a spherical shell. Using an undamped, second-order time-dependent Ginzburg-Landau (TDGL) framework, we demonstrate that the uniform spatial curvature of the spherical manifold S 2 ensures a homogeneous ground state, mathematically guaranteeing the complete linear decoupling of the amplitude (Higgs) and phase (Goldstone) fluctuations. By expanding these fields in a basis of spherical harmonics, we derive closed-form dispersion relations for both modes. We show that finite-size geometric confinement on a closed surface explicitly discretizes the excitation spectra into quantized multipole ladders. Furthermore, we demonstrate that the constant positive spatial curvature of the sphere explicitly dictates the non-linear l(l +1) mode spacing and the 2l +1 degeneracy of these ladders. For the Higgs mode, the finite-size quantization of the Laplace-Beltrami spectrum imposes an angular-momentum-dependent gradient penalty that blue-shifts higher-order multipoles. This geometric mass shift is controlled entirely by the geometric ratio ξ GL /R, while leaving the fundamental uniform breathing mode invariant. Furthermore, by incorporating 3D long-range Coulomb interactions via Pearl electrodynamics under a quasistatic approximation, we demonstrate that the acoustic Goldstone branch is converted into a highly stiffened, discrete spherical plasmon spectrum in the terahertz domain. Our analytical scaling laws offer direct spectroscopic signatures for probing spatial curvature in core-shell superconducting nanoparticles and sub-micron shell structures.