A Jensen inequality for partial traces and applications to partially semiclassical limits

TL;DR

This paper introduces a new matrix inequality for convex functions of Hermitian matrices on bipartite spaces and applies it to extend results on eigenvalue asymptotics of Schrödinger operators in semiclassical limits.

Contribution

It presents a Jensen-type inequality for partial traces and uses it to generalize eigenvalue asymptotics in semiclassical analysis.

Findings

Proved a matrix inequality for convex functions of Hermitian matrices.

Extended eigenvalue asymptotics results to cases with infinite Weyl expressions.

Applied the inequality to partially semiclassical limits in quantum mechanics.

Abstract

We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application we reprove and extend some theorems about eigenvalue asymptotics of Schr\"odinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.