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

Abstract

We review an exact analytical resolution method for general one-dimensional (1D) quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving spectral (usual) vs "classical" (newer) zeta-regularisations in parallel. The central results are a set of Bohr--Sommerfeld-like but exact quantisation conditions, directly drawn from Wronskian identities, and appearing to extend Bethe-Ansatz formulae of integrable systems. Such exact quantisation conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position.

17 pages, 2 figures. V2: minor amendments throughout, with one typo corrected in eq.(19)

References in corpus (3)

• The ODE/IM Correspondence
• From exact-WKB toward singular quantum perturbation theory II
• The general 1D Schrödinger equation as an exactly solvable problem

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