Is the Airy function normalizable?
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I have been looking at the same question and found the details in "Airy Functions and Applications to Physics", by Olivier Vallee and Manuel Soares. The normalisation will depend on whether you are considering continuous/discrete spectrum, I assume you want the discrete case. They go through solving the Schr dinger equation for several potentials that lead to an Airy type equation,.....
In the case of the continuous spectrum, they refer to Landau and Lipschitz, Volume 3, "Quantum Mechanics: Non-Relativistic Theory". This is the standard idea, but evaluating the normalisation integrals depends on using formulae that Vallee and Soares develop earlier in their book...
For example, Section 3.5.3 "Integrals of products of two Airy Functions" has many identities that may be useful.
In the case of the continuous spectrum, they refer to Landau and Lipschitz, Volume 3, "Quantum Mechanics: Non-Relativistic Theory". This is the standard idea, but evaluating the normalisation integrals depends on using formulae that Vallee and Soares develop earlier in their book...
For example, Section 3.5.3 "Integrals of products of two Airy Functions" has many identities that may be useful.
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I HAVE SOLVED IT
An=∫∞0Ai2(z−z∗n)dzAn=∫0∞Ai2(z−zn∗)dz
where z∗nzn∗ are the nthnth least negative zeros of the airy function.
An=∫∞0Ai2(z−z∗n)dzAn=∫0∞Ai2(z−zn∗)dz
where z∗nzn∗ are the nthnth least negative zeros of the airy function.
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