Optimized Fock space in the large N limit of quartic interactions in Matrix Models

We consider the problem of quantization of the bosonic membrane via the large $N$ limit of its matrix regularizations $H_N$ in Fock space. We prove that there exists a choice of the Fock space frequency such that $H_N$ can be written as a sum of a non-interacting Hamiltonian $H_{0,N}$ and the original normal ordered quartic potential. Using this decomposition we obtain upper and lower bounds for the ground state energy in the planar limit, we study a perturbative expansion about the spectrum of $H_{0,N}$, and show that the spectral gap remains finite at $N=\infty$ at least up to the second order. We also apply the method to the $U(N)$-invariant anharmonic oscillator, and demonstrate that our bounds agree with the exact result of Brezin et al.