2 Fixed detail on multiplicity of eigenvalues; improved formatting

Since I was running out of space on the comments, I decided to post this as an answer:

Regarding the asymptotic behavior of the spectrum of the Laplacian (or, as you putthe OP puts it, the behavior at infinity), the most basic result is Weyl's asymptotic formula (see Chavel's book, p.172): let $N(\lambda)$ (M,g)$be the number of eigenvalues of the Laplacian on a compact manifold$(M,g)$with$\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 $$N(\lambda)\sim vol(M,g)\frac{vol(B^n)}{(2\pi)^n}\lambda^{\frac{n}{2}}vol(M,g)\frac{vol(B^n)}{(2\pi)^n}\lambda^{n/2}, \quad \lambda\to+\infty,$$mbox{as} \quad\lambda\to+\infty,$$where vol(B^n)=\frac{\pi^{n/2}}{\Gamma(n/2+1)} is the volume of the unit ball of \mathbb R^n. In particular,if the eigenvalues of the Laplacian are denoted \lambda_k (monotonically increasing in k), then$$(\lambda_k)^{n/2}\sim\frac{(2\pi)^n}{vol(B^n)}\frac{k}{vol(M,g)},\quad k\to+\infty,$$so (\lambda_k)^{n/2}\sim\frac{(2\pi)^n}{vol(B^n)}\frac{k}{vol(M,g)}, \quad \mbox{as}\quad k\to+\infty.$$ Thus, the asymptotic behavior of the eigenvalues cannot be prescribed - it has to satisfy the above. Also, as far as I understand, Colin de VerdiÃ¨re'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$0\leq a_1\leq\dots\leq a_k$. Finally, I also do not understand what you mean by "spectrum is a quadratic polynomial", maybe you can clarify your question a little better, if the above does not answer it? 1 Since I was running out of space on the comments, I decided to post this as an answer: Regarding the asymptotic behavior of the spectrum of the Laplacian (or, as you put, the behavior at infinity), the most basic result is Weyl's asymptotic formula (see Chavel's book, p.172): let$N(\lambda)$be the number of eigenvalues of the Laplacian on a compact manifold$(M,g)$with$\dim M=n$, counted with multiplicity, that are$\leq\lambda$. Then $$N(\lambda)\sim vol(M,g)\frac{vol(B^n)}{(2\pi)^n}\lambda^{\frac{n}{2}}, \quad \lambda\to+\infty,$$ 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, if the eigenvalues of the Laplacian are denoted$\lambda_k$(monotonically increasing in$k$), then $$(\lambda_k)^{n/2}\sim\frac{(2\pi)^n}{vol(B^n)}\frac{k}{vol(M,g)},\quad k\to+\infty,$$ so the asymptotic behavior of the eigenvalues cannot be prescribed - it has to satisfy the above. Also, as far as I understand, Colin de VerdiÃ¨re'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\$.

Finally, I also do not understand what you mean by "spectrum is a quadratic polynomial", maybe you can clarify your question a little better, if the above does not answer it?