Relationship between Φ⁴-matrix model and N-body harmonic oscillator or Calogero-Moser model

Abstract We study some Hermitian Φ 4 -matrix model and some real symmetric Φ 4 -matrix model whose kinetic terms are given by Tr( E Φ 2 ), where E is a positive diagonal matrix without degenerate eigenvalues. We show that the partition functions of these matrix models correspond to zero-energy solutions of a Schödinger type equation with N -body harmonic oscillator Hamiltonian and Calogero-Moser Hamiltonian, respectively. The first half of this paper is primarily a review of previous work of us. The discussion of the properties of zero-energy solutions and the discussion of systems of differential equations satisfied by partition functions derived from the Virasoro algebra in the latter half of this paper contain novel material.