The $N$-body harmonic oscillator system and its generalized system, the Calogero model, are known as quantum integrable systems, i.e. their Schr\"odinger equations are solvable and eigenstates of the Hamiltonians can be constructed. This talk is concerned with the new correspondence of the quantum solvable systems with matrix models discovered last year. These matrix models are given as the Grosse-Wulkenhaar models, known as renormalizable scalar $\Phi^4$-theories on Moyal spaces, which are non-commutative spaces. The Moyal space has Fock representation, so field theories can be expressed by using matrix representation. A scalar $\Phi^4$-theory on the Moyal space corresponds to a Hermitian $\Phi^4$-matrix model or a real symmetric $\Phi^4$-matrix model. In particular, $\Phi^4$-matrix model corresponding to the Grosse-Wulkenhaar models have kinetic terms $\mathrm{Tr}( E \Phi^2)$, where $E$ is a positive diagonal matrix without degenerate eigenvalues. We show that their partition functions of these Hermitian matrix model and real symmetric matrix model correspond to zero-energy solutions of a Schr\"odinger equation with $N$-body harmonic oscillator Hamiltonian and Calogero-Moser Hamiltonian, respectively.
This presentation is based on joint research with Harald Grosse, and with Harald Grosse, Naoyuki Kanomata ,Raimar Wulkenhaar.