  We investigate the explicit expression for the principal eigenvalue
$\lambda_{1}^{X}(D)$ for a large class of compound Poisson processes $X$ on a
bounded open set $D$ by examining its spectral heat content. When the jump
density of the compound Poisson process is radially symmetric and strictly
decreasing, we demonstrate that balls are the unique minimizers for
$\lambda_{1}^{X}(D)$ among all sets with equal Lebesgue measure. Furthermore,
we show that this uniqueness fails if the jump density is not strictly
decreasing.
