Hamiltonian partial differential equations and symplectic scale manifolds

TL;DR

This paper develops a framework for Hamiltonian PDEs on symplectic scale manifolds, introducing strong sc-smoothness and demonstrating invariance under symplectomorphisms, with applications to the Schrödinger equation.

Contribution

It introduces the concept of strong sc-smoothness for Hamiltonian vector fields on symplectic scale manifolds, extending the theory to PDEs.

Findings

Strong sc-smoothness formalizes smoothness for Hamiltonian functions.

The theory is invariant under sc-smooth symplectomorphisms.

Application to the Schrödinger equation verifies the framework.

Abstract

This paper defines symplectic scale manifolds based on Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields and flows on these by narrowing down sc-smoothness to what we denote by strong sc-smoothness, a concept which effectively formalizes the desired smoothness properties for Hamiltonian functions. We show the concept to be invariant under sc-smooth symplectomorphisms, whence it is compatible with Hofer's scale manifolds. We develop and verify the theory at the hand of the free Schr\"odinger equation.