Topological expansion and exponential asymptotics in 1D quantum mechanics

Borel summable semiclassical expansions in 1D quantum mechanics are conside red. These are the Borel summable expansions of fundamental solutions and o f quantities constructed with their help. An expansion, called topological, is constructed for the corresponding Borel functions. This allows us to st udy the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic and anharmonic potentia ls. Together with the best approximation provided by the semiclassical seri es the exponentially small contributions completing the approximation are c onsidered. A natural method of constructing such exponential asymptotics ba sed on the Borer plane singularity structures provided by the topological e xpansion is developed. The method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator.