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Homogenization of point interactions

Domenico Cafiero, Michele Correggi, Davide Fermi

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 Schrödinger operator with a regular electrostatic potential. The result is obtained via $Γ$-converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense.

Homogenization of point interactions

Abstract

We consider a non-relativistic quantum particle in , or , 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 of the family of operators to a Schrödinger operator with a regular electrostatic potential. The result is obtained via -converge of the associated quadratic forms. Moreover, in presence of an external trapping potential, the convergence is lifted to uniform resolvent sense.
Paper Structure (11 sections, 17 theorems, 111 equations, 1 figure)

This paper contains 11 sections, 17 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and Main Results
  3. Preliminaries
  4. $\Gamma-$convergence
  5. Measures and distribution of centers
  6. Green functions
  7. Hilbert scales associated to $H_{0}$
  8. Proofs
  9. Proof of the $\Gamma-\liminf$ inequality
  10. Proof of the $\Gamma-\limsup$ (in)equality
  11. Proof of the uniform resolvent convergence

Key Result

Theorem 2.1

Let ass:UUass:xixjass:a hold. Then, the sequence of quadratic forms $\left\{ Q_{N} \right\}_{N \in \mathbb{N}}$$\Gamma$-converges to $Q_{\infty}$ with respect to both the weak and strong topology of $L^2(\mathbb{R}^d)$. More precisely, the following two conditions are fulfilled:

Figures (1)

  • Figure 1: A schematic illustration of \ref{['ass:UU']} and \ref{['ass:xixj']}. The points $\mathbf{x}_j$ cluster within a fixed region following the density distribution $U$, while maintaining a minimal mutual distance $r >\ell\,N^{-1/d}$, see \ref{['eq:infxixj']}.

Theorems & Definitions (32)

  • Remark 2.1: Regularity of $\bm{\mathrm{A}}$ and $V$
  • Remark 2.2: Well-posedness of $Q_{N}$ and $H_{N}$
  • Remark 2.3: Assumptions
  • Theorem 2.1: $\Gamma-$convergence
  • Corollary 2.1: Operator convergence
  • Remark 2.4: Bound states and dynamics
  • Remark 2.5: Relaxing the assumptions
  • Remark 2.6: Domains with boundaries and curved geometries
  • Theorem 3.1: $\Gamma-$covergence and operator convergence 1
  • Theorem 3.2: $\Gamma-$covergence and operator convergence 2
  • ...and 22 more