Exact factorization ansatz of the ground state many-body wave function: A case study of quartic potential problem

Abstract

We propose an explicit factorized formula for the ground state many-body wave function of a model quartic potential problem in two and three dimensions, using our recently developed method for solving Schrödinger equations. Our novel formula has a clear real-space structure: it is exactly factorized into three parts, including a non-integer pre-exponential power factor, dominant decaying terms on the exponent, and a modulator function, which is of minor importance. This result is a generalization of our previously obtained exact formula for the 1-body wave function. As in the 1-body case, here we show that for many-body ground wave functions that cannot achieve variable separation, our method is still far more efficient than the conventional basis expansion method both in representing the wave function and in calculating the energy. The exact factorization ansatz proposed in the present case study shall provide important insight into the general structure of many-body wave functions.