Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting multiplicities). By a result of Colin de Verdière, given any finite sequence $a_1,\cdots,a_k$, there exists a Riemannian manifold and a Laplace operator on it such that the first k eigenvalues are exactly, $a_1,\cdots,a_k$.

My question is about what's happening at infinity. More precisely, since usually the spectrum of a Laplace operator is a quadratic polynomial, is there a Laplace operator (on a closed manifold) such that there is no $n_0$ such that {$(\lambda)_{n\geq n_0}$} is of the form {$P(n) : n\geq 0$} where $P$ is a quadratic polynomial ?