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The Moody-Shapere-Wilczek's adiabatic effective Hamiltonian and Lagrangianmethod is developed further into the matrix effective Hamiltonian (MEH) andLagrangian (MEL) approach to a parameter-dependent quantum system. Thematrix-operator approach formulated in the product integral (PI) provides notonly a method to find the wave function efficiently in the MEH approach butalso higher order corrections to the effective action systematically in the MELapproach, a la the Magnus expansion and the Kubo cumulant expansion. A coupledquantum system of a light particle of a harmonic oscillator is worked out, andas a by-product, a new kind of gauge potential (Berry's connection) is foundeven for nondegenerate cases (real eigenfunctions). Moreover, in the PIformulation the holonomy of the induced gauge potential is related toSchlesinger's exact formula for the gauge field tensor. A superadiabaticexpansion is also constructed, and a generalized Dykhne formula, depending onthe contour integrals of the homotopy class of complex degenerate points, isrephrased in the PI formulation.

Matrix Operator Approach to the Quantum Evolution Operator and the Geometric Phase