Airy function (exact WKB results for potentials of odd degree)

Details Airy function (exact WKB results for potentials of odd degree) Airy function (exact WKB results for potentials of odd degree)

1998-11-02

arxiv.org/abs/math-ph/9811001v2

J. Phys. A32 (1999) 1301-1311

Physics

Mathematical Physics

tex txt.tex, 2 files; v2: corrections as indicated in footnotes

Scientific paper

10.1088/0305-4470/32/7/020

An exact WKB treatment of 1-d homogeneous Schr\"odinger operators (with the confining potentials $q^N$, $N$ even) is extended to odd degrees $N$. The resulting formalism is first illustrated theoretically and numerically upon the spectrum of the cubic oscillator (potential $|q|^3$). Concerning the linear potential (N=1), the theory exhibits a duality in which the Airy functions Ai, Ai' become paired with the spectral determinants of the quartic oscillator (N=4). Classic identities for the Airy function, as well as some less familiar ones, appear in this new perspective as special cases in a general setting.

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