  We provide a detailed study of the properties of a few interacting spin $1/2$
fermions trapped in a one-dimensional harmonic oscillator potential. The
interaction is assumed to be well represented by a contact delta potential.
Numerical results obtained by means of exact diagonalization techniques are
combined with analytical expressions for both the non-interacting and strongly
interacting regime. The $N=2$ case is used to benchmark our numerical
techniques with the known exact solution of the problem. After a detailed
description of the numerical methods, in a tutorial-like manner, we present the
static 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, we
consider dynamical properties of the system exploring first the excitation of
the breathing mode, using the dynamical structure function and corresponding
sum-rules, and then a sudden quench of the interaction strength.

Quantum Gases;Quantum Physics