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Boundary problem

  1. Feb 18, 2010 #1
    given a boundary Hermitian eigenvalue problem

    [tex]L= -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y [/tex]

    with y=y(x) , in one dimension, can we always find two operators

    [tex] D_{1} = \frac{d}{dx}+f(x) [/tex] and [tex] D_{2} = -\frac{d}{dx}+U(x) [/tex]

    so [tex]L= D_{1} D_{2} [/tex], with [tex]Adj( D_{1}) = D_{2} [/tex] [tex]Adj( D_{2}) = D_{1} [/tex] ???

    Also the eigenfunctions of L are of the form [tex] \Psi (x)= \phi_{2} (x) \phi _{1}(x) [/tex]

    and the eigenvalues of L are of the form [tex] \lambda _{n} = s.s^{*} [/tex]
     
  2. jcsd
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