Integrability of \( \Phi ^4\) matrix model as N-body harmonic oscillator system

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