  There is a matrix model corresponding to a scalar field theory called
Grosse-Wulkenhaar model, which is renormalizable by adding a harmonic
oscillator potential to scalar $\Phi^{4}$ theory on Moyal spaces. There are
more unknowns in $\Phi^{4}$ matrix model than in $\Phi^{3}$ matrix model, for
example, in terms of integrability. We then construct a one-matrix model
($\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model) with multiple potentials, which is
a combination of a $3$-point interaction and a $4$-point interaction, where the
$3$-point interaction of $\Phi^{3}$ is multiplied by some positive definite
diagonal matrix $M$. This model is solvable due to the effect of this $M$. In
particular, the connected $\displaystyle\sum_{i=1}^{B}N_{i}$-point function
$G_{|a_{N_{1}}^{1}\cdots a_{N_{1}}^{1}|\cdots|a_{1}^{B}\cdots a_{N_{B}}^{B}|}$
of $\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model is studied in detail. This
$\displaystyle\sum_{i=1}^{B}N_{i}$-point function can be interpreted
geometrically and corresponds to the sum over all Feynman diagrams (ribbon
graphs) drawn on Riemann surfaces with $B$ boundaries (punctures). Each
$|a_{1}^{i}\cdots a_{N_{i}}^{i}|$ represents $N_{i}$ external lines coming from
the $i$-th boundary (puncture) in each Feynman diagram. First, we construct
Feynman rules for $\Phi^{3}$-$\Phi^{4}$ Hybrid-Matrix-Model and calculate
perturbative expansions of some multipoint functions in ordinary methods.
Second, we calculate the path integral of the partition function
$\mathcal{Z}[J]$ and use the result to compute exact solutions for $1$-point
function $G_{|a|}$ with $1$-boundary, $2$-point function $G_{|ab|}$ with
$1$-boundary, $2$-point function $G_{|a|b|}$ with $2$-boundaries, and $n$-point
function $G_{|a^{1}|a^{2}|\cdots|a^{n}|}$ with $n$-boundaries. They include
contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams
or higher genus surfaces.
