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The different facets of the R-matrix method are presented pedagogically in a general framework. Two variants have been developed over the years: (i) The 'calculable' R-matrix method is a calculational tool to derive scattering properties from the Schrödinger equation in a large variety of physical problems. It was developed rather independently in atomic and nuclear physics with too little mutual influence. (ii) The 'phenomenological' R-matrix method is a technique to parametrize various types of cross sections. It was mainly (or uniquely) used in nuclear physics. Both directions are explained by starting from the simple problem of scattering by a potential. They are illustrated by simple examples in nuclear and atomic physics. In addition to elastic scattering, the R-matrix formalism is applied to inelastic and radiative-capture reactions. We also present more recent and more ambitious applications of the theory in nuclear physics. This article was invited by P-H Heenen. 3.5. On the basis and boundary parameter choices 12 3.6. Resonances 13 3.7. Bound states 14 3.8. Capture cross sections 15 3.9. Propagation methods 15 3.10. Extension to multichannel collisions 16 4. Applications of the calculable R matrix 17 4.1. Conditions of the calculations 17 4.2. Basis functions 18 4.3. Application to a narrow resonance: 12 C + p 18 4.4. Application to a broad resonance: α + α 20 4.5. Application to a non-resonant system: α + 3 He 20 4.6. Application to a deep potential 21 4.7. Application to a non-local potential: e − -H scattering in the static-exchange approximation 21 4.8. Discussion of resonances 23 4.9. Application to bound states 23 4.10. Application to a multichannel problem: α + d 23 4.11. Application to propagation methods 24 4.12. Application to capture reactions: 12 C(p,γ ) 13 N 25 5. The phenomenological R matrix 25 5.1. Introduction 25 5.2. Single-pole approximation of elastic scattering 27 5.3. Multiresonance elastic scattering 28 5.4. Phenomenological parametrization of multichannel collisions 28 5.5. Application to the 12 C + p elastic scattering 29 5.6. Application to the 18 Ne(p,p ) 18 Ne(2 + ) inelastic scattering 30 5.7. Radiative-capture reactions 31 6. Recent applications of the R-matrix method 32 6.1. Introduction 32 6.2. Microscopic cluster models 32 6.3. The continuum discretized coupled channel (CDCC) method 36 6.4. Three-body continuum states 38 7. Conclusion 40 Acknowledgments 40 Appendix A. Collision matrix and K matrix 41 Appendix B. Proof of relation (3.27) 41 Appendix C. Matrix elements for various basis functions 41 References 42

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

Three-body continuum states are investigated with the hyperspherical method on a Lagrange mesh. The $R$-matrix theory is used to treat the asymptotic behaviour of scattering wave functions. The formalism is developed for neutral as well as for charged systems. We point out some specificities of continuum states in the hyperspherical method. The collision matrix can be determined with a good accuracy by using propagation techniques. The method is applied to the $^6$He (=$\alpha$+n+n) and $^6$Be (=$\alpha$+p+p) systems, as well as to $^{14}$Be (=$^{12}$Be+n+n). For $^6$He, we essentially recover results of the literature. Application to $^{14}$Be suggests the existence of an excited $2^+$ state below threshold. The calculated B(E2) value should make this state observable with Coulomb excitation experiments.Comment: 17 pages, 8 figures. Accepted for publication in Nucl.Phys.