**IMAGE CREDIT: NASA/JPL: This shows Jupiter’s Great Red Spot; a storm that has been occurring for over 300 years now. Quite recently, however, observations show that the Spot appears to be shrinking in size. **

About a week ago, I was looking through my notebooks and came across an unfinished problem posed by one of my professors. Unfortunately, I was not able to solve the problem during the semester. However, I thought it might be something interesting to consider. I did a quick search and found that the problem he gave us was to prove the Sturm-Liouville Theorem.

The main brute-force method to analytically solving a given partial differential equation is the separation of variables. This method is heavily used by physicists and in doing so transforms the initial boundary value problem (IBVP) into a Sturm-Liouville problem in which we have an ordinary differential equation and linear homogeneous boundary conditions.

A quick statement of the problem:

Consider the following Sturm-Liouville Problem:

In the context of the problem, let . About the , I am aware that the original differential equation had a single . However, I am not sure of the implications to the problem by removing one of the , so I am including the problem as stated by my professor. So, Eq.(1.1) becomes

So the problem is to prove that if there exists two unique non-trivial eigenfunctions, such that and that for , then on the interval

The problem also has the properties

- An infinite sequence of eigenvalues: .
- For each eigenvalue there exists a unique, nontrivial (i.e. non-zero) solution , not including multiples of said solution.

Much of the background information (e.g. the properties and general form of the problem) was obtained from the following sources:

https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory#Application_to_PDEs

Farlow, S.J., *Partial Differential Equations for Scientists and Engineers*. Dover. 1993.

** “Proof”*. ** Recall from above the Eqs. (1.4), (1.2), and (1.3). Let the following equations be indexed for :

Similarly, let the following equations be indexed for :

Consider Eqs.(i) and (iv.), let us multiply Eq.(i) by and the same with Eq.(iv.), however we multiply by . Taking the difference yields

Let us now integrate Eq.(2) over the closed interval to give

Let us focus on the first integral on the left-hand side. We can separate each term to give

We can integrate by parts for the first term

and doing the same for the second term we arrive at

Finding the difference, we see that the remaining integrals in (5) and (6) cancel leaving the terms in the braces. If we evaluate those terms over the interval we get the following

Let us suppose now that we equate Eqs.(ii) and (v.) and Eqs.(iii) and (vi) and multiply equations (ii) and (v) by and , respectively, and multiply equations (iii) and (vi) by and . For Eqs.(ii) and (v) it follows that

From which we observe that if we assume that then

The first term in the sum above vanishes which leads us to conclude that

By similar logic we find that for the second boundary condition

Therefore, returning (all the way back) to Eq.(7), we arrive at

where we assume that the eigenvalues satisfy , for . As a check, suppose that , in which case the integral (Eq.(10)) becomes

Furthermore, if , then the entire equation vanishes. Thus, this “proof”is complete. Q.E.D.

* Once again, I hesitate to call this a proof since it is not what a mathematician would call rigorous.