Regarding the asymptotic behavior of the spectrum of the Laplacian (or, as the OP puts it, the behavior at infinity), the most basic result is **Weyl's asymptotic formula** (see [Chavel's book][1], p.172): let $(M,g)$ be a compact manifold with $\dim M=n$ and $0=\lambda_0<\lambda_1\leq \lambda_2\leq\dots$ be the eigenvalues of the Laplacian, each distinct eigenvalue repeated according to its multiplicity. Denote by $N(\lambda)=\sum_{\lambda_j\leq\lambda} 1$ the number of eigenvalues (counted with multiplicity) that are $\leq\lambda$. Then

$$N(\lambda)\sim vol(M,g)\frac{vol(B^n)}{(2\pi)^n}\lambda^{n/2}, \quad \mbox{as} \quad\lambda\to+\infty,$$

where $vol(B^n)=\frac{\pi^{n/2}}{\Gamma(n/2+1)}$ is the volume of the unit ball of $\mathbb R^n$. In particular,

$$(\lambda_k)^{n/2}\sim\frac{(2\pi)^n}{vol(B^n)}\frac{k}{vol(M,g)}, \quad \mbox{as}\quad k\to+\infty.$$

> Thus, the asymptotic behavior of the eigenvalues **cannot be prescribed** - it has to satisfy the above.


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Also, as far as I understand, Colin de Verdière's result is stronger than stated. Namely, given **any** compact connected manifold M, with $\dim M\geq 3$, and any finite sequence $0\leq a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0\leq a_1\leq\dots\leq a_k$.


  [1]: http://www.amazon.com/Eigenvalues-Riemannian-Geometry-Applied-Mathematics/dp/0121706400