  We study the spectrum of rank $2$ torsion free sheaves on $\mathbb{P}^3$ with
aim of producing examples of distinct irreducible components of the moduli
space with the same spetrcum answering the question presented by Rao for the
case of torsion free sheaves. In order to do so, we provide a full description
of the spectrum of the sheaves in the moduli space of semistable rank $2$
torsion free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2,c_3)$
equals to $(-1,2,0)$ and $(0,3,0)$.
