Confusion regarding examples of Sturm-Liouville operators

[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?

 

[jvicens]

Thank you for bringing up this interesting paper and for your questions regarding the Gel'fand-Yaglom theorem. I would like to clarify and expand upon the information presented in the paper.

Firstly, the Gel'fand-Yaglom theorem is indeed applicable for all Sturm-Liouville problems, which includes second-order differential operators of the form $-\frac{d^{2}}{dx^{2}} + V(x)$. However, it is also applicable for more general Sturm-Liouville operators, such as the one mentioned in the paper, $-\frac{d^{2}}{dr^{2}}+\frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^{2}}+V(r)$.

To understand the difference between these two operators, it is important to note that the Sturm-Liouville problem is a general framework for solving second-order differential equations of the form $Lu(x) = \lambda w(x)u(x)$, where $L$ is a linear differential operator, $w(x)$ is a weight function, and $\lambda$ is a constant. This includes the operator $-\frac{d^{2}}{dx^{2}} + V(x)$, but it also allows for more complex operators, such as the one mentioned in the paper.

In the case of the operator $-\frac{d^{2}}{dr^{2}}+\frac{(l+\frac{d-3}{2})(l+\frac{d-1}{2})}{r^{2}}+V(r)$, the weight function $w(r)$ is not equal to 1, as it is in the case of $-\frac{d^{2}}{dx^{2}} + V(x)$. This means that the Sturm-Liouville problem for this operator is more general and requires the use of the Gel'fand-Yaglom theorem for its solution.

I hope this explanation helps clarify the use of the Gel'fand-Yaglom theorem in the paper and the difference between the two operators mentioned. Please let me know if you have any further questions or concerns. Thank you.