# Study of various few-body systems using Gaussian expansion method (GEM)

## Abstract

We review our calculation method, Gaussian expansion method (GEM), to solve accurately the Schrödinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to *S*-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body (3- to 5-body) systems, such as 1) few-nucleon systems, 2) few-body structure of hypernuclei, 3) clustering structure of light nuclei and unstable nuclei, 4) exotic atoms/molecules, 5) cold atoms, 6) nuclear astrophysics and 7) structure of exotic hadrons. Showing examples in our published papers, we explain i) high accuracy of GEM calculations and its reason, ii) wide applicability of GEM to various few-body systems, iii) successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in *geometric progression* work very well both for shortrange and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes (oscillations). These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.

## Keywords

few-body problems Gaussian expansion method Gaussian ranges in geometric progression## Notes

### Acknowledgements

It is our great pleasure to submit this invited review paper to the international symposium in honor of Professor Akito Arima to the celebration of his 88th birthday. We are very grateful for his continuous encouragement on our work. We would like to thank Professor Y. Kino for valuable discussions on GEM and its applications. The writing of this review was partially supported by the Japan Society for the Promotion of Science under grants 16H03995 and 16H02180 and by the RIKEN Interdisciplinary Theoretical Science Research Group project.

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