  The quantum Ising chain with the interaction decaying as a power law
$1/r^{1+\sigma}$ of the distance between spins $r$ was investigated
numerically. A particular attention was paid to the low-energy spectrum,
namely, the single-magnon and two-magnon-bound-state masses, $m_{1,2}$,
respectively, in the ordered phase. It is anticipated that for each $\sigma$,
the scaled bound-state mass $m_2/m_1$ should take a universal constant
(critical amplitude ratio) in the vicinity of the critical point. In this
paper, we calculated the amplitude ratio $m_2/m_1$ with the exact
diagonalization method, which yields the spectral information such as $m_{1,2}$
directly. As a result, we found that the scaled mass $m_2/m_1$ exhibits a
non-monotonic dependence on $\sigma$; that is, the bound state is stabilized by
an intermediate value of $\sigma$. Such a feature is accordant with a recent
observation based on the non-perturbative-renormalization-group method.
