Is it possible to create a Laplacian matrix of the form:
$$\left(\matrix{ & a&0&0&0&0\\a&&b&0&0&0\\0&b&&c&0&0\\0&0&c&&d&0\\0&0&0&d&&e\\0&0&0&0&e&}\right)$$
such that we can specify a metric $g$ in a 1D spatial coordinate system using the definition given under "N-dimensions" https://en.wikipedia.org/wiki/Laplace_operator
$$\nabla^2=\frac{1}{\det g}\frac{\partial}{\partial\xi^i}\left(\sqrt{\det{g}}g^{ij}\frac{\partial}{\partial\xi^j}\right)$$
Presumably we would just have a $1\times 1$ metric $g$? So we would have something of the form
$$\nabla^2=\frac{1}{g(x)}\frac{\partial}{\partial\xi}\left(g^{3/2}(x)\frac{\partial}{\partial\xi}\right)$$ but I'm not sure whether it is possible to get it into a symmetric form which can be made effectively tridiagonal with the addition of a diagonal V matrix