Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and Schr\H{o}dinger equations

TL;DR

This paper introduces new separable coordinate systems in four-dimensional flat spaces using maximal Abelian subgroups, and explicitly solves the Schrödinger equation in these coordinates, enhancing understanding of variable separation in quantum mechanics.

Contribution

It develops a systematic method for generating separable coordinates via subgroup chains and provides explicit solutions to the Schrödinger equation in these systems.

Findings

New coordinate systems with maximal ignorable variables are constructed.

Explicit solutions of the Schrödinger equation are obtained in these coordinates.

Graphical representations of subgroup chains are provided.

Abstract

Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally, the explicit solutions of the Schr\H{o}dinger equation in the separable coordinate systems are computed.