A delta-shell interaction is the idealization of a thin spherical barrier: infinitely thin, with a finite integrated strength. Place several concentric delta shells around the origin, and a quantum particle scatters off a layered onion of singular potentials.
The paper on scattering matrices for Schrödinger operators with concentric delta shells (arXiv: 2603.24028) shows that the scattering matrix in each angular momentum channel is determined by the determinant of a finite-dimensional boundary matrix.
The formula is S_l(k) = det K_l(k² − i0) / det K_l(k² + i0), where K_l is an N×N matrix constructed from the shell positions and strengths. The same matrix appears in the resolvent formula that defines the self-adjoint operator. The infinite-dimensional scattering problem collapses to a finite-dimensional determinant.
For two concentric shells in the s-wave channel, the interaction between shells produces nontrivial threshold effects. Generically, the scattering length (the zero-energy scattering parameter) is finite and explicit. But at a critical configuration — when a zero-energy solution exists whose exterior constant term vanishes — the scattering length diverges and S₀(k) → −1 as k → 0. The shells conspire to create a resonance at zero energy.
The through-claim: singular potentials have finite-dimensional scattering. Each delta shell contributes one dimension to the boundary matrix. The scattering of a wave off N concentric shells is entirely encoded in an N×N determinant — the channels don't interact, and within each channel, the problem is matrix algebra.
2603.24028. Mathematical physics / scattering theory / delta-shell interactions / Schrödinger operators / threshold effects.