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Continous spcetrum

  1. Aug 26, 2007 #1
    given a self-adjoint operator

    [tex] \mathcal L [y(x)]=-\lambda_{n} y(x) [/tex]

    where the index 'n' can be any positive real number (continous spectrum) then my question is if

    [tex] \int_{c}^{d}\int_{a}^{b}dn y_{n}(x)y_{m} (x)w(x)dx = 1 [/tex]

    deduced from the fact that for continous n and m then the scalar product

    [tex] <y_{n} |y_{m} > =\delta (n-m) [/tex] (Dirac delta --> continous Kronecker delta --> discrete case )

    am i right ?? ... for the problem we have a continous set of eigenvalues [tex] \lambda _{n}=h(n) [/tex] where n >0 is any real and positive number
  2. jcsd
  3. Aug 26, 2007 #2
    I believe you should use the inner product [itex]<L[y_n],y_m>[/itex] and use the fact that L is self-adjoint, but im not sure. Why don't you see the discrete proof for orthogonal functions and try to extend it?
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