Operator Separation Of Variables For Adiabatic Problems In Quantum And Wave Mechanics

TL;DR

This paper introduces a general operator-based scheme for adiabatic approximation in quantum and wave mechanics, unifying and extending existing methods to derive effective reduced equations for complex states.

Contribution

It presents a novel, operator-valued symbol approach that generalizes classical adiabatic methods like Born-Oppenheimer and Maslov, enabling broader applicability.

Findings

Unified framework for adiabatic approximations

Derivation of effective reduced equations

Handles degeneracy in quantum states

Abstract

We study linear problems of mathematical physics in which the adiabatic approximation is used in the wide sense. Using the idea that all these problems can be treated as problems with operator-valued symbol, we propose a general regular scheme of adiabatic approximation based on operator methods. This scheme is a generalization of the Born-Oppenheimer and Maslov methods, the Peierls substitution, etc. The approach proposed in this paper allows one to obtain "effective" reduced equations for a wide class of states inside terms (i.e., inside modes, subregions of dimensional quantization, etc.) with the possible degeneration taken into account.