An Algebraic Approach to Solving Evolution Problems in Some Nonlinear Quantum Models

TL;DR

This paper introduces a Lie-algebraic method for solving evolution equations in certain nonlinear quantum models, utilizing polynomially deformed Lie algebras and reducing problems to solvable $su(2)$ cases.

Contribution

The paper presents a novel algebraic technique that simplifies solving nonlinear quantum evolution problems by expanding operators and reducing to known solvable models.

Findings

Effective expansion of evolution operators using $su_{pd}(2)$ algebra

Reduction of complex problems to solvable $su(2)$ Hamiltonians

Applicable to nonlinear quantum physics models

Abstract

A new general Lie-algebraic approach is proposed to solving evolution tasks in some nonlinear problems of quantum physics with polynomially deformed Lie algebras $su_{pd}(2)$ as their dynamic symmetry algebras. The method makes use of an expansion of the evolution operators by power series in the $su_{pd}(2)$ shift operators and a (recursive) reduction of finding coefficient functions to solving auxiliary exactly solvable $su(2)$ problems with quadratic Hamiltonians. PACS numbers: 03.70; 02.20; 42.50