Homogenization of point interactions

TL;DR

This paper studies the effective behavior of a quantum particle interacting with many point-like potentials, showing that as the number of points increases, the system converges to a Schrödinger operator with a smooth potential.

Contribution

It proves the homogenization of a quantum system with many singular point interactions, establishing strong resolvent convergence to a regular potential operator.

Findings

Operators converge strongly to a Schrödinger operator with a regular potential.

Homogenization holds under uniform distribution and negative scattering lengths.

Convergence extends to uniform resolvent sense with trapping potentials.

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.