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Sturm-Liouville Eigenvalue problem

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data
    We are to show that for 0<β<1, eigenvalues are strictly positive and for β>1, we have to determine how many negative eigenvalues there are.

    u''+λ2u=0, u(0)=0, βu(π)-u'(π) = 0

    2. Relevant equations

    I've already shown that the eigenvalues are determined by tan(λπ)=λ/β (was told this is correct).

    Some how just from this, we are supposed to determine what I listed in part 1. I tried using the Rayliegh quotient, but that is no good as you can't definitely say what β should and shouldn't be to make it positive in this case.

    3. The attempt at a solution

    We know that L is self-adjoint so our eigenvalues, λ2, are real. Therefore λ is either strictly real or strictly imaginary. I was told that if you plug in λ=i*ω, leading to negative eigenvalues, this would force β>1. And then if you assume λ is real, this would force β<1. I do not see why this is true at all. If you assume λ is imaginary, you wind up with

    tanh(ωπ)=ω/β,

    and this has solutions for β<1, which indicates that you still have negative eigenvalues for β<1, which I was told should not be possible.

    Can anyone explain what I was supposed to be seeing when I put a complex or real number in, or have an alternate way of explaining it?

    EDIT: Professor finally admitted there was an error. We should actually be using the interval [0,1], not [0,π]. tan(λ)=λ/β, which is easy to show now what beta you need.
     
    Last edited: Oct 9, 2012
  2. jcsd
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