Algorithm for reduction of boundary-value problems in multistep adiabatic approximation

TL;DR

This paper introduces a step-by-step averaging algorithm based on the multi-step Kantorovich method to reduce and solve multidimensional boundary-value problems in quantum physics, demonstrated on a helium atom.

Contribution

It presents a novel symbolic-numerical algorithm for multi-step adiabatic reduction applicable to multidimensional boundary-value problems, extending the Kantorovich method.

Findings

Successfully calculated ground and excited states of helium atom.

Demonstrated effectiveness of the algorithm in complex quantum systems.

Enhanced computational efficiency for multidimensional problems.

Abstract

The adiabatic approximation is well-known method for effective study of few-body systems in molecular, atomic and nuclear physics, using the idea of separation of "fast" and "slow" variables. The generalization of the standard adiabatic ansatz for the case of multi-channel wave function when all variables treated dynamically is presented. For this reason we are introducing the step-by-step averaging methods in order to eliminate consequently from faster to slower variables. We present a symbolic-numerical algorithm for reduction of multistep adiabatic equations, corresponding to the MultiStep Generalization of Kantorovich Method, for solving multidimensional boundary-value problems by finite element method. An application of the algorithm to calculation of the ground and first exited states of a Helium atom is given.