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Sturm-Liouville and Rayleigh Quotient Problem

  • Thread starter FAS1998
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Problem 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.
 

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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

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

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