Consider a 1D Riemannian manifold defined on the unit segment $[0,1]$ and equipped with an unknown metric $g=g(x)$ which is positive for all $x$.
In this case the Laplace-Beltrami operator $\Delta f= \frac{1}{\sqrt{|G|}} \sum_{i} \frac{\partial}{\partial x_i}(\sqrt{|G|} \sum_{j} g^{ij} \frac{\partial f}{\partial x_j})$
(where $G$ is the metric tensor, $g^{ij}$ is its inverse and $f=f(x)$)
reduces to $\Delta f= g^{-1/2} \frac{d}{dx}(g^{-1/2} \frac{df}{dx})=\frac{1}{g}\frac{d^2f}{dx^2}-\frac{1}{2g^2}\frac{dg}{dx}\frac{df}{dx}$
Which boundary conditions on $f$ and $g$ do I need to solve the eigenproblem $\Delta f+\lambda f= \frac{1}{g}\frac{d^2f}{dx^2}-\frac{1}{2g^2}\frac{dg}{dx}\frac{df}{dx}+\lambda f= 0$ in this case?