Airy function (exact WKB results for potentials of odd degree)
A. Voros
Abstract
An exact WKB treatment of 1-d homogeneous Schrödinger operators (with the confining potentials qN, 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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