Note on a differentiation formula, with application to the two-dimensional Schrödinger equation.

Area of Science:

• Numerical analysis
• Quantum mechanics
• Computational physics

Background:

• Standard numerical methods for the two-dimensional Schrödinger equation exhibit slow convergence.
• Discretization of function derivatives is crucial for numerical solutions.

Purpose of the Study:

• To present a generalized divided difference method for derivative discretization.
• To apply this method to the eigenvalue problem of the 2D Schrödinger equation.
• To demonstrate improved accuracy and efficiency compared to standard techniques.

Main Methods:

• Generalization of divided differences by modifying the dependent variable.
• Application of the modified divided differences to derive discretization formulas.
• Numerical solution of the two-dimensional Schrödinger equation using the proposed method.

Main Results:

• The generalized divided difference method provides accurate results for the 2D Schrödinger equation.
• This approach converges faster than standard numerical methods for this problem.
• The method is conceptually simple and easy to implement.

Conclusions:

• The proposed method offers an effective and accurate approach for solving the 2D Schrödinger eigenvalue problem.
• Generalized divided differences present a viable alternative for derivative discretization in numerical methods.
• The simplicity and accuracy suggest potential utility in other scientific and engineering applications.