  Let $\Gamma$ be a discrete group acting properly discontinuously and
isometrically on the three-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$,
and $\square$ the Laplacian which is a second-order hyperbolic differential
operator. We study linear independence of a family of generalized Poincar&#233;
series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236,
arXiv:1209.4075], which are defined by the $\Gamma$-average of certain
eigenfunctions on $\mathrm{AdS}^{3}$. We prove that the multiplicities of
$L^{2}$-eigenvalues of the hyperbolic Laplacian $\square$ on
$\Gamma\backslash\mathrm{AdS}^{3}$ are unbounded when $\Gamma$ is finitely
generated. Moreover, we prove that the multiplicities of stable
$L^{2}$-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.
