nLab Airy function

Idea

An Airy function is a special function satisfying the ordinary differential equation

$$y''(x) - x \cdot y(x) = 0$$

and with the Airy integral representation

$$Ai(x) = \frac{1}{\pi}\int_0^\infty cos\left(\frac{t^3}{3}+tx\right) dt = \frac{1}{2\pi}\int_{-\infty}^\infty exp\left(\frac{i t^3}{3}+i t x\right) dt$$

Properties

This function appears often in the study of oscillating (path) integrals, e.g. in semiclassical approximation to quantum mechanics and in the geometric approximation to wave mechanics/optics. Its asymptotics is important in the study of the singular behaviour of light in the vicinity of caustics?.

The asymptotic expansions for the Airy function have sharp changes at certain lines, observed by G. G. Stokes, and present often in the stationary phase method (cf. semiclassical approximation). This is called the Stokes phenomenon and is a special case of the wall crossing.

Airy function appears in the subject of integrable models, related to Painleve transcendents and also in the study of Hermitean random matrices (work of Tracy and Widom). Airy function has also a remarkable role in the Kontsevich‘s solution to the Witten conjecture.

References

• wikipedia

• G. B. Airy, On the intensity of light in the neighbourhood of a caustic, Trans. Camb. Phil. Soc., 6 (1838), 379-403.

• Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23, euclid

• C. A. Tracy, H. Widom, Level-spacing distributions and the Airy kernel, Physics Letters B 305 (1-2): 115–118 (1993) hep-th/9210074, doi; Level-spacing distributions and the Airy kernel, Commun. in Math. Physics 159 (1): 151–174 (1994) euclid doi, MR1257246; On orthogonal and symplectic matrix ensembles, Commun. in Math. Phys. 177 (3): 727–754 (1996) doi, MR1385083

See also sec. 7.2 in

• Alain Connes, Caterina Consani, The universal thickening of the field of real numbers, arxiv/1202.4377

For generalizations see the references

• R. N. Fernandez, V. S. Varadarajan, Matrix Airy functions for compact Lie groups, Internat. J. Math. 20 (2009), no. 8, 945–977, doi, MR2554728
• R. N. Fernandez, V. S. Varadarajan, D. Weisbart, Airy functions over local fields, Lett. Math. Phys. 88 (2009), no. 1-3, 187–206, MR2010d:11138, doi
• Marco Bertola, Boris Dubrovin, Di Yang, Simple Lie algebras and topological ODEs, arxiv/1508.03750

category: analysis