Sang Pyo Kim, Jewan Kim, Kwang Sup Soh
The Moody-Shapere-Wilczek's adiabatic effective Hamiltonian and Lagrangian method is developed further into the matrix effective Hamiltonian (MEH) and Lagrangian (MEL) approach to a parameter-dependent quantum system. The matrix operator approach formulated in the product integral (PI) provides not only a method to find wave function efficiently in the MEH approach but also higher order corrections to the effective action systematically in the MEL approach, a la the Magnus expansion and the Kubo's cumulant expansion. A coupled quantum system of a light particle of harmonic oscillator is worked out, and as a by-product a new kind of gauge potential (Berry's connection) is found even for nondegenerate case (real eigenfunctions). Moreover, in the PI formulation the holonomy of the induced gauge potential is related to the Schlesinger's exact formula for the gauge field tensor. A superadiabatic expansion is also constructed and a generalized Dykhne formula, depending on the contour integrals of homotopy class of complex degenerate points, is rephrased in the PI formulation.
View original:
http://arxiv.org/abs/1212.2680
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