Real symmetric ${\mathrm{\Phi }}^4$-matrix model as Calogero-Moser model

We study a real symmetric ${\mathrm{\Phi }}^4$-matrix model whose kinetic term is given by $\mathrm{T}\mathrm{r}(E{\mathrm{\Phi }}^2)$, where $E$ is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schödinger type equation with Calogero-Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.