Eigenvalue asymptotics for the one-particle kinetic energy density operator

TL;DR

This paper establishes the asymptotic behavior of eigenvalues of the one-particle kinetic energy density operator in quantum systems, providing insight into its spectral properties as eigenvalue index grows large.

Contribution

It proves an asymptotic formula for the eigenvalues of the kinetic energy density operator, advancing understanding of its spectral characteristics in quantum mechanics.

Findings

Eigenvalues $\lambda_k$ satisfy $\lambda_k o (Bk)^{-2}$ as $k o

Provides a precise asymptotic relation for the spectral distribution of the operator

Enhances theoretical understanding of quantum kinetic energy operators

Abstract

The kinetic energy of a multi-particle system is described by the one-particle kinetic energy density matrix $\tau(x, y)$. Alongside the one-particle density matrix $\gamma(x, y)$, it is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula $\lambda_k \sim (Bk)^{-2}$, $B \ge 0$, as $k\to\infty$, for the eigenvalues $\lambda_k$ of the self-adjoint operator $\boldsymbol{\sf T}\ge 0$ with kernel $\tau(x, y)$.