Universal low-energy behavior in a quantum Lorentz gas with Gross-Pitaevskii potentials

TL;DR

This paper demonstrates that a quantum Lorentz gas with Gross-Pitaevskii potentials exhibits universal low-energy behavior as the number of obstacles grows large, with the effective potential depending only on scattering length and obstacle density.

Contribution

It establishes the convergence of the quantum Hamiltonian to a universal limiting operator and characterizes the fluctuations around this limit, simplifying the scattering model for slow neutrons.

Findings

Convergence of the Hamiltonian to a universal limit

Explicit characterization of fluctuations

Modeling of neutron scattering in condensed matter

Abstract

We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.