Real symmetric \( \Phi ^4\)-matrix model as Calogero–Moser model

We study a real symmetric \(\Phi ^4\) ${\mathrm{\Phi }}^4$ -matrix model whose kinetic term is given by \(\textrm{Tr}( E \Phi ^2)\) $\text{Tr}(E{\mathrm{\Phi }}^2)$ , 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 Schrödinger type equation with Calogero–Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.