Error analysis of splitting methods for the time dependent Schrodinger equation

A typical procedure to integrate numerically the time dependent Schr"o-din-ger equation involves two stages. In the first one carries out a space discretization of the continuous problem. This results in the linear system of differential equations $i du/dt = H u$, where $H$ is a real symmetric matrix, whose solution with initial value $u(0) = u_0 \in \mathbb{C}^N$ is given by $u(t) = \e^{-i t H} u_0$. Usually, this exponential matrix is expensive to evaluate, so that time stepping methods to construct approximations to $u$ from time $t_n$ to $t_{n+1}$ are considered in the second phase of the procedure. Among them, schemes involving multiplications of the matrix $H$ with vectors, such as Lanczos and Chebyshev methods, are particularly efficient. In this work we consider a particular class of splitting methods which also involves only products $Hu$. We carry out an error analysis of these integrators and propose a strategy which allows us to construct different splitting symplectic methods of different order (even of order zero) possessing a large stability interval that can be adapted to different space regularity conditions and different accuracy ranges of the spatial discretization. The validity of the procedure and the performance of the resulting schemes are illustrated on several numerical examples.

💡 Research Summary

The paper addresses the numerical solution of the time‑dependent Schrödinger equation after spatial discretization, focusing on time‑stepping schemes that require only matrix‑vector products with the Hamiltonian matrix H. Starting from a Fourier‑collocation (spectral) discretization on a periodic interval, the continuous problem is reduced to a large linear ODE system i du/dt = H u, where H is a real symmetric (Hermitian) matrix and the exact solution is u(t) = exp(−i t H) u₀. Direct computation of the matrix exponential is prohibitive for large N, so the authors investigate splitting methods that exploit the symplectic structure of the problem.

By rewriting the complex unitary evolution as a real symplectic rotation O(t H) =