Generalized semiclassical-phase-index formulas via sequential stationary phase

Sequential stationary phase, i.e., the replacement of a multidimensional stationary phase evaluation by an ordered set of lower-dimensional stationary phase integrations, is applied to uniformly asymptotic path integral forms for the semiclassical propagator. The results are useful formulas for computing trajectory indices (generalized Maslov indices) in any quantum representation for general types of time-dependent Hamiltonians. Index connecting relations that relate the indices for the different representations are also obtained. We also demonstrate a general canonical structure for the semiclassical phase indices that arises naturally through the application of sequential stationary phase.