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Inspired by an example of Gueritaud-Kassel [Geom. Topol. 2017], we constructa family of infinitely generated discontinuous groups $\Gamma$ for the3-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$. These groups are notnecessarily sharp (a kind of "strong" properly discontinuous conditionintroduced by Kassel and Kobayashi [Adv. Math. 2016]), and we give itscriterion. Moreover, we find upper and lower bounds of the counting$N_{\Gamma}(R)$ of a $\Gamma$-orbit contained in a pseudo-ball $B(R)$ as theradius $R$ tends to infinity. We then find a non-sharp discontinuous group$\Gamma$ for which there exist infinitely many $L^2$-eigenvalues of theLaplacian on the noncompact anti-de Sitter manifold$\Gamma\backslash\mathrm{AdS}^{3}$, by applying the method established byKassel-Kobayashi. We also prove that for any increasing function $f$, thereexists a discontinuous group $\Gamma$ for $\mathrm{AdS}^{3}$ such that thecounting $N_{\Gamma}(R)$ of a $\Gamma$-orbit is larger than $f(R)$ forsufficiently large $R$.

Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space