The regularized free fall II -- Homology computation via heat flow

TL;DR

This paper investigates a non-local heat flow derived from a regularized free fall functional, constructing a Morse complex and revealing a rich solution structure with implications for Schrödinger equations.

Contribution

It introduces a Morse complex for the heat flow of a non-local functional, showing the homology is simple despite the complex solution space.

Findings

Constructed a Morse complex with one generator per degree

Found the Morse homology is surprisingly simple with just one generator

Revealed a wealth of solutions to the heat flow equation

Abstract

In [BOV20] Barutello, Ortega, and Verzini introduced a non-local functional which regularizes the free fall. This functional has a critical point at infinity and therefore does not satisfy the Palais-Smale condition. In this article we study the $L^2$ gradient flow which gives rise to a non-local heat flow. We construct a rich cascade Morse complex which has one generator in each degree $k\ge 1$. Calculation reveals a rather poor Morse homology having just one generator. In particular, there must be a wealth of solutions of the heat flow equation. These can be interpreted as solutions of the Schr\"odinger equation after a Wick rotation.