  Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is
called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists
$xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the
action of $G$ on $G/H$ by (left) multiplication is the rational number
$\rho(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where
$\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The
\emph{intersection spectrum} of the group $G$ is then defined to be the set
  $$
  \sigma(G) := \left\{ \rho(G,H) : H\leq G \right\}.
  $$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.},
42(1):111-128, 2015] that if $\sigma(G) = \{1\}$, then $G$ is necessarily
solvable. The natural question that arises is, therefore, which rational
numbers larger than $1$ belong to $\sigma(G)$, whenever $G$ is non-solvable. In
this paper, we study the intersection spectrum of the linear group
$\operatorname{PSL}_2(q)$. It is shown that $2 \in
\sigma\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3
\pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that
$\rho(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing
$\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes
\mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.
