Eigenvalue asymptotics for the one-particle density matrix

TL;DR

This paper establishes the asymptotic behavior of eigenvalues of the one-particle density matrix in quantum systems, revealing a specific decay rate as the eigenvalue index grows large.

Contribution

It provides a rigorous proof of the eigenvalue asymptotics for the one-particle density matrix in atomic and molecular bound states.

Findings

Eigenvalues decay as (Ak)^{-8/3} for large k

The asymptotic formula applies to the density matrix of bound states

Enhances understanding of spectral properties in quantum chemistry

Abstract

The one-particle density matrix $\gamma(x, y)$ for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula $\lambda_k \sim (Ak)^{-8/3}$, $A \ge 0$, as $k\to\infty$, for the eigenvalues $\lambda_k$ of the self-adjoint operator $\boldsymbol\Gamma\ge 0$ with kernel $\gamma(x, y)$.