On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential

In this paper, we study the convergence behaviour of high-order exponential operator splitting methods for the time integration of linear Schrödinger equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{i}\,\partial_{\,t}\psi(x,t)=-\,\tfrac{1}{2}\,\Delta\,\psi(x,t)+V(x)\,\psi(x,t)\,,\quad x\in\mathbb{R}^{d}\,,\ t\geq 0\,,$$\end{document} involving unbounded potentials; in particular, our analysis applies to potentials V defined by polynomials. We deduce a global error estimate which implies that any time-splitting method retains its classical convergence order for linear Schrödinger equations, provided that the exact solution of the considered problem fulfills suitable regularity requirements. Numerical examples illustrate the theoretical result.