Proof of the BMV Conjecture

TL;DR

This paper proves the BMV conjecture, demonstrating that the trace exponential function of Hermitian matrices is the Laplace transform of a positive measure, with a semi-explicit measure representation.

Contribution

It provides a proof of the BMV conjecture and offers a semi-explicit form for the associated positive measure.

Findings

Confirmed the BMV conjecture for Hermitian matrices.

Established the trace exponential as a Laplace transform of a positive measure.

Derived a semi-explicit representation for the measure.

Abstract

We prove the BMV (Bessis, Moussa, Villani, 1975) conjecture, which states that the function t -> Tr exp(A-tB), t \geq 0, is the Laplace transform of a positive measure on [0,\infty) if A and B are n x n Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.