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

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