A Chebychev propagator for inhomogeneous Schrödinger equations

Area of Science:

• Quantum mechanics
• Computational chemistry
• Applied mathematics

Background:

• Time-dependent Schrödinger equations are crucial in quantum dynamics.
• These equations appear in optimal control theory and reactive scattering.
• Efficient numerical methods are needed for these complex problems.

Purpose of the Study:

• To present a novel propagation scheme for time-dependent inhomogeneous Schrödinger equations.
• To apply this scheme to problems in optimal control theory.
• To analyze the numerical efficiency and convergence of the proposed method.

Main Methods:

• Derivation of a formal solution using polynomial expansion of the inhomogeneous term.
• Approximation of the solution using Chebyshev polynomials.
• Demonstration of different variants of the inhomogeneous propagator.

Main Results:

• The proposed scheme was applied to two optimal control theory examples.
• Convergence behavior and numerical efficiency were analyzed.
• The method provides an effective approach for solving the targeted equations.

Conclusions:

• The developed propagation scheme offers an efficient and accurate method for time-dependent inhomogeneous Schrödinger equations.
• The findings are relevant for advancing computational approaches in optimal control and quantum dynamics.
• The study highlights the utility of Chebyshev polynomial approximations in this context.