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

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.