Journal of Dynamics and Differential Equations

, Volume 28, Issue 3–4, pp 1031–1038 | Cite as

The Airy Function is a Fredholm Determinant



Let G be the Green’s function for the Airy operator
$$\begin{aligned} L\varphi := -\varphi ''+ x \varphi , \quad 0< x < \infty , \quad \varphi (0)=0. \end{aligned}$$
We show that the integral operator defined by G is Hilbert–Schmidt and that the 2-modified Fredholm determinant
$$\begin{aligned} {\mathrm {det}}_2(1+zG) = \frac{{\mathrm {Ai}}(z)}{{\mathrm {Ai}}(0)} , \quad z \in {\mathbb {C}}. \end{aligned}$$


Airy function Fredholm determinant Hilbert–Schmidt operators 

Mathematics Subject Classification

MSC 47G10 MSC 33C10 



Supported by NSF Grant 1411278.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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