Numerical approximation of an initial–boundary value problem for a variable-coefficient hyperbolic Schrödinger equation

We study an initial–boundary value problem for a three-dimensional variable-coefficient hyperbolic Schrödinger equation whose spatial operator has an indefinite structure. Under suitable regularity and positivity assumptions on the variable coefficient, a uniqueness result is first established for the continuous problem. The main contribution is the development and analysis of a Crank–Nicolson finite difference scheme combining trapezoidal time averaging with second-order central spatial differences. The method leads to a sparse complex linear system at each time step, with a time-independent coefficient matrix that allows a single factorization to be reused. The local truncation error is of second order in both time and space. A frozen-coefficient von Neumann analysis gives an amplification factor of unit modulus. For the variable-coefficient Dirichlet problem, a discrete weighted-energy argument establishes conservation of the weighted norm for the homogeneous perturbation equation and yields stability and second-order convergence without a time-step restriction. Numerical experiments based on two manufactured solutions examine spatial and temporal convergence separately. The errors decrease under refinement and generally support the predicted behavior, although the more oscillatory second example exhibits somewhat lower spatial convergence rates. Monte Carlo simulations with additive complex Gaussian noise further assess the sensitivity of the computed solutions to perturbations in the input data. These results provide a numerical framework for the considered variable-coefficient hyperbolic Schrödinger problem.