Mapping of Two-Dimensional Schr\"odinger Equation under the Point Transformation

TL;DR

This paper investigates point transformations of the two-dimensional Schrödinger equation that lead to models with position-dependent mass, enabling the construction of new solvable quantum systems with diverse effective mass functions.

Contribution

It introduces a general framework for point transformations resulting in Schrödinger equations with effective mass, expanding the class of solvable models beyond conventional methods.

Findings

Derived a wide class of models with different effective mass functions.

Constructed solvable partner models from a known solvable system.

Provided examples of models not solvable by standard separation of variables.

Abstract

For the two-dimensional Schr\"odinger equation, the general form of the point transformations such that the result can be interpreted as a Schr\"odinger equation with effective (i.e. position dependent) mass is studied. A wide class of such models with different forms of mass function is obtained in this way. Starting from the solvable two-dimensional model, the variety of solvable partner models with effective mass can be built. Several illustrating examples not amenable to the conventional separation of variables are given.