Orthogonality condition for Airy functions

[skujesco2014]

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.

 

[pasmith]

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.

I think you want to look at $$y'' = \lambda x y,$$ whose solutions are Airy functions of $r = \lambda^{1/3}x$, and the eigenfunctions will be orthogonal with respect to $$\int_a^b x y_1(x) y_2(x)\,dx.$$ where $\lambda^{1/3}a$ and $\lambda^{1/3}b$ should be zeroes of Airy functions or their derivatives. Unhelpfully $x = 0$ is not a such a zero.

 

[skujesco2014]

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.