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

A. Voros
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Abstract

An exact WKB treatment of 1-d homogeneous Schrödinger 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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A. Voros. 2003-08-26. Airy function (exact WKB results for potentials of odd degree). https://doi.org/10.1088/0305-4470%2F32%2F7%2F020

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