Sturm-Liouville and Rayleigh Quotient Problem

[FAS1998]

Homework Statement

I’ve included an image of the problem below. This isn’t technically a homework problem. It’s a practice test question that I have the solutions for. I’m stuck at part 2. I have to show that eigenvalues are positive with the Rayleigh quotient.

Relevant Equations

d/dx(p(x)d*phi(x)/dx) + q(x)*phi(x) + lamda*sigma(x)*phi(x)

By comparing the given equation to the equation for the Sturm-Liouville form, I see that p(x) must equal 1, q(x) must equal 0 and sigma(x) must equal 1.

After this, I have no idea what I should be doing for part 2.

 

[FAS1998]

I see now that by plugging in values of p(x), q(x), and σ(x) into the equation, and then applying boundary conditions, it can be reduced to (∫(dΦ(x)/dx)^2)/(∫Φ(x)^2), with both integrals bounded from 0 to L.

I also understand that Φ(x) cannot be identically 0 because it's an eigenfunction.

I still don't understand how we know that both the top and bottom function are not less than 0.

 

[pasmith]

I see now that by plugging in values of p(x), q(x), and σ(x) into the equation, and then applying boundary conditions, it can be reduced to (∫(dΦ(x)/dx)^2)/(∫Φ(x)^2), with both integrals bounded from 0 to L.

I also understand that Φ(x) cannot be identically 0 because it's an eigenfunction.

I still don't understand how we know that both the top and bottom function are not less than 0.

This follows from the following straightforward proposition:

Let f : [a,b] \to \mathbb{R} be continuous, non-negative and not identically zero. Then \int_a^b f(x)\,dx > 0.
Proof: As f is non-negative and not identically zero, there exists x_0 \in [a,b] such that f(x_0) &gt; 0. By continuity of f, there exists an interval [c,d] \subseteq [a,b] of strictly positive width containing x_0 such that if x \in [c,d] then f(x) \geq \frac12 f(x_0). But then <br /> \int_a^b f(x)\,dx \geq \int_c^d f(x)\,dx \geq \int_c^d \tfrac12 f(x_0)\,dx = \tfrac12 (d - c)f(x_0) &gt; 0 as required.