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 HNHN in Fock space. We prove that there exists a choice of the Fock space frequency such that HNHN can be written as a sum of a non-interacting Hamiltonian H0,NH0,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 H0,NH0,N, and show that the spectral gap remains finite at N=∞N=∞ at least up to the second order. We also apply the method to the U(N)U(N)-invariant anharmonic oscillator, and demonstrate that our bounds agree with the exact result of Brezin et al.