Which finite sequences $0=\lambda_0<\lambda_1<\dots<\lambda_k$ can be obtained as the first $k+1$ eigenvalues of the Laplacian operator $\Delta_g$ on the circle $S^1$? Of course the metric $g$ is allowed to be arbitrary.
Thoughts: In this case, the metric is specified by a positive function $S^1\to\mathbb{R}$, and by expanding this as a Fourier series, the Laplace equation $\Delta f=\lambda f$ becomes an infinite-dimensional matrix equation for the fourier coefficients of $f$, however, I am not sure how useful this viewpoint is.
By way of motivation, it is a theorem of de Verdiere that any such sequence can be realized on any closed manifold of dimension $\ge 3$ by appropriate choice of metric .
In the case of $\mathbb S^1$ (a one dimensional compact manifold), they are all iosmetric to $\mathbb R \setminus L\mathbb Z$ by some $L>0$ (via arc length parametrization). Thus the eigevalues are always of the form
$$ 0, \frac{4\pi^2}{L^2}, \frac{4\pi^2}{L^2}, \frac{16\pi^2}{L^2}, \frac{16\pi^2}{L^2} , \cdots, \frac{(2k\pi)^2}{L^2}, \cdots. $$
Thus quite a lot of sequence $\lambda_0 <\lambda_1\le \lambda_2 \le \cdots \lambda_{k}$ cannot be realized as eigenvalues of $(\mathbb S^1, g)$.