Integrability of $\Phi^4$ Matrix Model as $N$-body Harmonic Oscillator System

Abstract

We study a Hermitian matrix model with a kinetic term given by $ Tr (H \Phi^2 )$, where $H$ is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential $\Phi^3$ replaced by $\Phi^4$. We show that its partition function solves an integrable Schr\"odinger-type equation for a non-interacting $N$-body Harmonic oscillator system.