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We provide a detailed study of the properties of a few interacting spin $1/2$fermions trapped in a one-dimensional harmonic oscillator potential. Theinteraction is assumed to be well represented by a contact delta potential.Numerical results obtained by means of exact diagonalization techniques arecombined with analytical expressions for both the non-interacting and stronglyinteracting regime. The $N=2$ case is used to benchmark our numericaltechniques with the known exact solution of the problem. After a detaileddescription of the numerical methods, in a tutorial-like manner, we present thestatic properties of the system for $N=2, 3, 4$ and 5 particles, e.g.low-energy spectrum, one-body density matrix, ground-state densities. Then, weconsider dynamical properties of the system exploring first the excitation ofthe breathing mode, using the dynamical structure function and correspondingsum-rules, and then a sudden quench of the interaction strength.

Static and dynamic properties of a few spin $1/2$ interacting fermions trapped in an harmonic potential