Accelerating the solution of the quantum-mechanical three-body problem

Abstract

A fundamental difficulty of describing few-body systems is that the Schrödinger equation has no analytic solution for more than two interacting particles. Hence, we must resort to numerical methods to obtain a description of the system. The Schrödinger equation is discretised using the pseudo-spectral method. As a result, the Schrödinger equation is transformed into an eigenvalue problem. The resulting eigenvalue problem contains a large sparse linear system. For this reason, the Jacobi Davidson QR (JDQR) eigenvalue solver has been chosen. It excels at finding eigenpairs of large sparse matrices. Within JDQR a system of equations must be solved using an iterative solver such as Generalized minimal residual method (GMRES). Therefore, with increasing grid sizes solving the system becomes more troublesome for iterative methods. This problem is exacerbated in the 2D case, where the size of the matrices grows much faster than in the 1D case. In the 2D case the system matrix has a size of 𝑛^4 × 𝑛^4 instead of 𝑛^2 × 𝑛^2 in the one-dimensional case. As a result, a direct method for approximating this system is imperative, giving rise to the need for preconditioners. The purpose of the preconditioner is to approximate the inverse of the system matrix, thereby significantly reducing the computational time required to solve the system. In contrast to the 1D scenario[14], there is no direct method to precondition the system matrix in two dimensions. In this report we propose a preconditioner for the quantum-mechanical three-body problem in two spatial dimensions. The proposed preconditioner, called the ’2D inexact preconditioner’ reduces the amount of time needed for convergence and the number of iterations. The computation time was decreased by 86% on the largest grid size we have considered. Additionally, we attempt to expand upon the preconditioner proposed for the 1D case by [14]. Their approach neglects the potential in the eigenvalue problem. The idea is to incorporate the potential for an improved convergence. We derive two algorithms that attempt to achieve this. Lastly, the bosonic ground states of the three-body system are plotted and compared to those already present in literature