Zeta-regularization for exact-WKB resolution of a general 1D Schrödinger equation

We review an exact analytical resolution method for general one-dimensional quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving 'spectral' (usual) versus 'classical' (newer) zeta-regularizations in parallel. The central results are a set of Bohr–Sommerfeld-like but exact quantization conditions, directly drawn from Wronskian identities, and appearing to extend the Bethe-ansatz formulae of integrable systems. Such exact quantization conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. (review)