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From multi-instantons to exact results. (English) Zbl 1073.81043

Summary: Conjectures about the exact semi-classical expansion of eigenvalues of hamiltonians corresponding to potentials with degenerate minima, are recalled. They were initially motivated by semi-classical calculations of quantum partition functions using a path integral representation and have later been proven to a large extent, using the theory of resurgent functions. They take the form of generalized Bohr-Sommerfeld quantization formulae. We explain here their relation with the corresponding WKB expansion of the Schrödinger equation. We show how these conjectures naturally emerge from an evaluation of multi-instanton contributions in the path integral formulation of quantum mechanics.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M37 Resurgence phenomena (MSC2000)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
81S40 Path integrals in quantum mechanics
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