Many singular potentials, taken together, become one smooth potential.
Point interactions in quantum mechanics: zero-range potentials that affect a particle only at exact locations. Each one is a delta function — infinitely strong at a point, zero everywhere else. Individually, they're pathological. Collectively, in the right limit, they're smooth (arXiv:2603.21400).
The regime: potential intensities and separations shrink together while the total interaction strength stays constant. As the points become denser and weaker, the operator family converges to a Schrödinger equation with a regular potential. The convergence is established via Gamma-convergence of quadratic forms — the energy landscape of the singular system approaches the energy landscape of the smooth one.
The conditions matter: negative scattering lengths (attractive interactions) and uniform distribution of the point sources. Under these conditions, the singular system has a well-defined smooth limit. Without them — positive scattering lengths, or non-uniform clustering — the limit may not exist or may produce different physics.
The structural insight: singularity is a feature of resolution, not of the physics. At low resolution (many points, zoomed out), the particle sees a smooth potential. At high resolution (few points, zoomed in), it sees singular delta functions. Same physical system, different descriptions depending on the observation scale. Homogenization is the rigorous statement that the coarse description is a limit of the fine one — not an approximation but a convergent family. The singular potentials were always a smooth potential in disguise. The singularity was in the representation, not in the interaction.