# Geometric Numerical Integration and Schrödinger Equations

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*Erwan Faou*

A publication of the European Mathematical Society

The goal of geometric numerical integration is the simulation
of evolution equations possessing geometric properties over long
periods of time. Of particular importance are Hamiltonian partial
differential equations typically arising in application fields such as
quantum mechanics or wave propagation phenomena. They exhibit many
important dynamical features such as energy preservation and
conservation of adiabatic invariants over long periods of time. In
this setting, a natural question is how and to which extent the
reproduction of such long-time qualitative behavior can be ensured by
numerical schemes.

Starting from numerical examples, these notes provide a detailed
analysis of the Schrödinger equation in a simple setting
(periodic boundary conditions, polynomial nonlinearities) approximated
by symplectic splitting methods. Analysis of stability and instability
phenomena induced by space and time discretization are given, and
rigorous mathematical explanations are provided for them.

The book grew out of a graduate-level course and is of interest to
researchers and students seeking an introduction to the subject matter.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in geometric numerical integration, symplectic integrators, backward error analysis, and Schrödinger equations.