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 $f(x_0) > 0$. 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 $$\int_a^b f(x)\,dx \geq \int_c^d f(x)\,dx \geq \int_c^d \tfrac12 f(x_0)\,dx = \tfrac12 (d - c)f(x_0) > 0$$ as required.