A Schrödinger-type algorithm for solving the Schrödinger equations via Phragmén–Lindelöf inequalities

In this article, we consider the numerical method for solving the Schrödinger equations via Phragmén–Lindelöf inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on the Phragmén–Lindelöf inequalities, the underlying system of inequalities is reformulated as a system of smooth equations, and a Schrödinger-type method is proposed to solve it iteratively so that a solution of the system of the Schrödinger equations is found. By means of the Schrödinger type inequalities, the algorithm is proved to be well defined and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm is effective.