Homogenization of point interactions

Abstract

We consider a non-relativistic quantum particle in $\mathbb{R}^d$, $d=2$ or $d = 3$, interacting with singular zero-range potentials concentrated on a large collection of points. We analyze the homogenization regime where the intensities of the singular potentials and the distances between the points simultaneously go to zero as their number grows, while the total interaction strength remains finite. Assuming that the singular potentials have negative scattering lengths and are uniformly distributed, we prove the strong resolvent convergence as $N \to \infty$ of the family of operators to a Schr\"{o}dinger operator with a regular electrostatic potential. The result is obtained via $\Gamma$-converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense.