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

Eric A. Carlen, Rupert L. Frank, Simon Larson

TL;DR

The paper proves a bipartite Jensen inequality for convex functions applied to partial traces and leverages it to derive eigenvalue asymptotics in partially semiclassical regimes. The authors develop a density-matrix version of Jensen’s inequality and demonstrate its use in obtaining sharp heat-kernel estimates via coherent states, Golden–Thompson bounds, and Tauberian arguments. They establish two main results: (i) a standard semiclassical limit for Schrödinger operators with homogeneous potentials (possibly infinite Weyl terms) and (ii) a partially semiclassical limit where variables separate into semiclassical and quantum components, yielding an integral over a reduced phase space of effective operators. The approach unifies and streamlines earlier works by Berezin, Lieb, Solomyak, Simon, and others, providing simpler proofs and a transparent inequality-based framework for partially semiclassical spectral asymptotics.

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 Schrödinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.

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

TL;DR

The paper proves a bipartite Jensen inequality for convex functions applied to partial traces and leverages it to derive eigenvalue asymptotics in partially semiclassical regimes. The authors develop a density-matrix version of Jensen’s inequality and demonstrate its use in obtaining sharp heat-kernel estimates via coherent states, Golden–Thompson bounds, and Tauberian arguments. They establish two main results: (i) a standard semiclassical limit for Schrödinger operators with homogeneous potentials (possibly infinite Weyl terms) and (ii) a partially semiclassical limit where variables separate into semiclassical and quantum components, yielding an integral over a reduced phase space of effective operators. The approach unifies and streamlines earlier works by Berezin, Lieb, Solomyak, Simon, and others, providing simpler proofs and a transparent inequality-based framework for partially semiclassical spectral asymptotics.

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 Schrödinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs.

Paper Structure

This paper contains 6 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction and main results
  2. A Jensen inequality partial traces
  3. Partially semiclassical limits
  4. Proof of Theorem \ref{['mainjensen']}
  5. Semiclassical limit: Proof of Theorem \ref{['warmupheat']}
  6. Partially semiclassical limit: Proof of Theorem \ref{['mainheat']}

Key Result

Theorem 1

Let $\mathcal{H}_1$, $\mathcal{H}_2$ be finite dimensional Hilbert spaces, let $H$ be a Hermitian matrix in $\mathcal{H}_1\otimes\mathcal{H}_2$ and let $f$ be a convex function on the convex hull of the spectrum of $H$. Then for any density matrix $\rho$ on $\mathcal{H}_1$

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Lemma 8
  • proof