Recursive generation of the semiclassical expansion in arbitrary dimension

We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semiclassical expansion of the quantum action for a quantum mechanical potential in arbitrary dimensions. In the method, we use the spectral information emerges from the singularities of the propagator on the complex $t$ plane, which are handled by the $iϵ$ prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in nonrelativistic quantum mechanics.