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# Program of the lectures 2015

**A) Introduction (12 hours)**

Since the course covers a wide range of topics, it will be necessary to unify the different backgrounds of the students and to introduce the notation. Moreover an outline of the main ingredients appearing in the construction of modern nucleon-nucleon (NN) interaction and three-nucleon forces will be given. Specific topics are:

- the NN interaction, from phenomenological potentials to chiral perturbation theory;
- the two-nucleon problem: bound states;
- the two-nucleon problem: scattering states;
- some properties of scattering states: low energy behavior and integral relations.

*Exercises will include:*

- brief introduction to programming languages and the use of computing resources;
- writing a code to solve the two-nucleon system using partial wave decomposition;
- calculation of observables and comparison to the experimental data.

**B) The Faddeev and Faddeev-Yakubovsky Equations (12 hours)**

Here the Faddeev and Faddeev-Yakobovsky equations will be discussed in configuration as well as in momentum space. The different numerical techniques used to solve the equations will be illustrated.The discrete and the continuum spectrum will be analyzed. The connection between the solution of the equations and the description of three- and four-nucleon reactions will be discussed. Important aspects of this chapter will be the discussion of the three-nucleon force and the treatment of the Coulomb interaction. Complex scaling will be introduced as a method to compute the breakup amplitude in configuration space.

*Exercises will include:*

- derivation of the Faddeev and Faddeev-Yakobovsky equations in the case of separable and s-wave interactions;
- applications to the calculation of the Efimov spectrum;
- use of the Faddeev and Faddeev-Yakobovsky equations to describe simple reactions as neutron-deuteron and neutron-triton scattering processes.

**C) Methods Based on Basis Expansions (12 hours)**

The construction of a general basis for fermions will be addressed. The specific case of the hyperspherical harmonics (HH) basis will be discussed in detail, however the relation with the harmonic oscillator (HO) will be given. The particular case of the adiabatic expansion will be illustrated. The direct applications of large basis to a bound state problem will be discussed in the context of the Rayleigh-Ritz variational principle. Application to scattering states using the Kohn variational principle will be discussed as well. The description of few-nucleon reactions using expansion basis will be illustrated. In this context the Resonating Group Method will be introduced.

*Exercises will include:*

- convergence studies of the HH/HO basis using different types of potentials;
- numerical implementation of large matrix algebra.

**D) Few-nucleon Reactions with External Probes (12 hours)**

The study of few-nucleon systems using external probes will be discussed. The main ingredients for the calculation of electro-weak reactions will be introduced. Bound state type techniques to calculate reactions (Lorentz Integral Transform, Complex Scaling and other methods) will be explained and put in practice in physical cases.

*Exercises will include:*

- Calculations of the dipole response function to the deuteron, 3He and 4He for simplified potentials with different techniques;
- Use of different routines for inversion of integral transforms.