Isogeometric Analysis of Bound States of a Quantum Three-Body Problem in 1D

TL;DR

This paper explores the application of isogeometric analysis (IGA) to solve the quantum three-body problem in one dimension, demonstrating its potential to accurately compute bound states despite challenges like boundary conditions and high degrees of freedom.

Contribution

The paper introduces the use of IGA with B-spline basis functions for quantum three-body problems, providing a new numerical approach to handle complex many-body quantum states.

Findings

IGA effectively computes bound state energies and wavefunctions.

Numerical experiments show promising results for three-body quantum problems.

IGA offers a viable alternative to traditional finite element methods for quantum systems.

Abstract

In this paper, we initiate the study of isogeometric analysis (IGA) of a quantum three-body problem that has been well-known to be difficult to solve. In the IGA setting, we represent the wavefunctions by linear combinations of B-spline basis functions and solve the problem as a matrix eigenvalue problem. The eigenvalue gives the eigenstate energy while the eigenvector gives the coefficients of the B-splines that lead to the eigenstate. The major difficulty of isogeometric or other finite-element-method-based analyses lies in the lack of boundary conditions and a large number of degrees of freedom for accuracy. For a typical many-body problem with attractive interaction, there are bound and scattering states where bound states have negative eigenvalues. We focus on bound states and start with the analysis for a two-body problem. We demonstrate through various numerical experiments that IGA provides a promising technique to solve the three-body problem.