Closed-Orbit Theory of Spatial Density Oscillations in Finite Fermion Systems

TL;DR

This paper develops a semi-classical approach using Gutzwiller's Green function to analyze spatial density oscillations in finite fermion systems, deriving universal relations and analytical formulas applicable across various potentials.

Contribution

It introduces a novel semi-classical framework linking classical orbits to quantum density oscillations, including universal relations and a local virial theorem for arbitrary potentials.

Findings

Derived universal relations for density oscillations in spherical potentials.

Established a local virial theorem valid for non-integrable potentials.

Provided analytical formulas for density oscillations in one-dimensional systems.

Abstract

We investigate the particle and kinetic-energy densities for $N$ non-interacting fermions confined in a local potential. Using Gutzwiller's semi-classical Green function, we describe the oscillating parts of the densities in terms of closed non-periodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions, and a ``local virial theorem'' valid also for arbitrary non-integrable potentials. We give simple analytical formulae for the density oscillations in a one-dimensional potential.