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FewBodyToolkit.jl

A Julia package for solving quantum systems of 2 or 3 particles with general potentials in 1D, 2D, and 3D.

Features

The FewBodyToolkit.jl package currently provides three modules with the following features:

GEM2B - Two-body solver (1D,2D,3D)

  • Supports 1D, 2D, and 3D geometries
  • Symmetric interaction potentials
  • Real-ranged Gaussian basis functions
  • Optional complex-ranged Gaussian basis functions for highly excited states
  • Complex scaling method (CSM) for resonances
  • Basis optimization routine
  • Routine for scaling the interaction to yield a desired eigenenergy (inverse problem)
  • Coupled-channel problems with additional derivative terms (up to 4th order)
  • Output of the wave function

GEM3B1D - Three-body solver (1D)

  • Supports 1D three-body problems
  • Symmetric two-body pair-interactions, no three-body forces
  • Expansion in up to three Fadeev components (rearrangement channels)
  • Automatic (anti-)symmetrization for identical bosons or fermions
  • Product basis in Jacobi coordinates
  • Complex scaling method (CSM) for resonances
  • No separate output of the wave function

ISGL - Three-body solver (3D)

  • Supports 3D three-body problems
  • Central two-body pair-interactions, no three-body forces
  • Employs infinitesimally-shifted Gaussian basis functions to handle arbitrary high intrinsic angular momenta (computationally expensive for high values)
  • Expansion in up to three Fadeev components (rearrangement channels)
  • Automatic (anti-) symmetrization for identical bosons or fermions
  • Product basis in Jacobi coordinates
  • Complex scaling method (CSM) for resonances
  • On-the-fly calculation of central observables and mean-square radii
  • No separate output of the wave function

Installation

using Pkg
Pkg.add(url="https://github.com/lhapp27/FewBodyToolkit.jl")

Quick Start

using FewBodyToolkit

# 1) Build physical and numerical parameters for a 2-body problem: 3D Coulomb problem
phys = make_phys_params2B(vint_arr=[r->-1/r], dim=3)
num  = make_num_params2B(gem_params=(nmax=10, r1=0.3, rnmax=20.0))

# 2) Solve generalized eigenvalue problem:
E = GEM_solve(phys,num)

# 3) Print 4 lowest eigenenergies
println("E_1 = ", E[1]) # should be close to -0.50000
println("E_2 = ", E[2]) # should be close to -0.12500
println("E_3 = ", E[3]) # should be close to -0.05555
println("E_4 = ", E[4]) # should be close to -0.03125

Modules

  • GEM2B — two-body systems in 1D, 2D, and 3D
  • GEM3B1D — three-body systems in 1D
  • ISGL — three-body systems in 3D

Examples

Check out the Examples for examples of each module.

API Reference

See the full API documentation for a list of all exported types and functions.

GEM2B Module

The GEM2B module provides tools for solving two-body problems using the Gaussian Expansion Method (GEM). The following functions and types are exported:

Parameter Setup

  • make_phys_params2B: Create a named tuple with the physical parameters of the two-body system.
  • make_num_params2B: Set up numerical parameters such as the number of basis functions.
  • PreallocStruct2B: Preallocates and stores relevant arrays for two-body calculations as well as the outputs (energies, coefficient-eigenvector)

Solvers and Optimization

  • GEM_solve: Solves the two-body Schrödinger equation for given physical and numerical parameters
  • v0GEMOptim: Optimizes the interaction strength to match a target eigenenergy.
  • GEM_Optim_2B: Optimizes Gaussian basis widths for improved convergence.

Wave Function Utilities

  • wavefun_arr: Returns the full wave function as an array sampled on a grid.
  • wavefun_point: Evaluates the wave function at a given point.