# Confusion regarding examples of Sturm-Liouville operators

1. Jul 29, 2015

### spaghetti3451

I have reading a paper where the functional determinants of differential operators are discussed: http://arxiv.org/abs/0711.1178

In the beginning of Section 3, the paper explains that functional determinants of operators of the form $-\frac{d^{2}}{dx^{2}} + V(x)$ can be found using a so-called Gel'fand-Yaglom theorem. The paper goes on to state the theorem and give some examples. That's all good and well.

Now, at the end of Section 3, the author mentions that the Gel'fand-Yaglom theorem is, in fact, applicable for all Sturm-Liouville problems. Now, what I took from this statement was that the Gel'fand-Yaglom theorem is applicable for all second-order differential operators. However, the author then goes on to use $-\frac{d^{2}}{dr^{2}}+\frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^{2}}+V(r)$ as an example of a Sturm-Liouville operator, the determinant of which he calculates.

Now, I don't see how the operator $-\frac{d^{2}}{dr^{2}}+\frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^{2}}+V(r)$ is any different from the operator $-\frac{d^{2}}{dx^{2}} + V(x)$, as both do not have a first-order derivative term, and so the second differential operator is not really a fitting example of a general Sturm-Liouville operator, is it?

2. Aug 3, 2015