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Borel summable semiclassical expansions in 1D quantum mechanics areconsidered. These are the Borel summable expansions of fundamental solutionsand of quantities constructed with their help. An expansion, calledtopological,is constructed for the corresponding Borel functions. Its mainproperty is to order the singularity structure of the Borel plane in ahierarchical way by an increasing complexity of this structure starting fromthe analytic one. This allows us to study the Borel plane singularity structurein a systematic way. Examples of such structures are considered for linear,harmonic and anharmonic potentials. Together with the best approximationprovided by the semiclassical series the exponentially small contributioncompleting the approximation are considered. A natural method of constructingsuch an exponential asymptotics relied on the Borel plane singularitystructures provided by the topological expansion is developed. The method isused to form the semiclassical series including exponential contributions forthe energy levels of the anharmonic oscillator.Comment: 46 pages, 22 EPS figure

Topological expansion and exponential asymptotics in 1D quantum mechanics