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Orthogonality condition for Airy functions

  1. Jun 24, 2014 #1
    Hi, all. I'll be brief. Can Airy functions [those who solve the differential equation y''-xy=0] be considered orthogonal over some interval? If so, what is their orthogonality condition? Given that the Airy functions have a representation in terms of Bessel functions, I would be inclined to think that the answer is yes; however, is there a compact form for their orthogonality condition? I've looked everywhere without success. Thanks.
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  3. Jun 25, 2014 #2


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    I think you want to look at [tex]
    y'' = \lambda x y,[/tex] whose solutions are Airy functions of [itex]r = \lambda^{1/3}x[/itex], and the eigenfunctions will be orthogonal with respect to [tex]
    \int_a^b x y_1(x) y_2(x)\,dx.
    [/tex] where [itex]\lambda^{1/3}a[/itex] and [itex]\lambda^{1/3}b[/itex] should be zeroes of Airy functions or their derivatives. Unhelpfully [itex]x = 0[/itex] is not a such a zero.
  4. Jun 26, 2014 #3
    Thanks a lot. I'll look it up. I'm working in the problem of a particle trapped in quantum well with infinite walls at x=0 and x=H>0 within which we have gravity, i.e., V = mgy, but I need the orthogonality relation of the wavefunction for a further calculation.
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