Global folds between Banach spaces as perturbations

TL;DR

This paper develops a geometric scheme to construct global folds between Banach spaces using perturbations of Fredholm operators, encompassing many known examples and introducing new ones with diverse applications.

Contribution

It introduces a unified geometric framework for global folds in Banach spaces via perturbations of Fredholm operators, extending classical results to broader contexts.

Findings

The scheme captures most known examples of global folds.

Concrete examples include operators like Laplacian, Schrödinger, and fractional Laplacian.

The approach relies on spectral properties and positivity conditions of eigenfunctions.

Abstract

Global folds between Banach spaces are obtained from a simple geometric construction: a Fredholm operator $T$ of index zero with one dimensional kernel is perturbed by a compatible nonlinear term $P$. The scheme encapsulates most of the known examples and suggests new ones. Concrete examples rely on the positivity of an eigenfunction. For the standard Nemitskii case $P(u) = f(u)$ (but $P$ might be nonlocal, non-variational), $T$ might be the Laplacian with different boundary conditions, as in the Ambrosetti-Prodi theorem, or the Schr\"{o}dinger operators associated with the quantum harmonic oscillator or the Hydrogen atom, a spectral fractional Laplacian, a (nonsymmetric) Markov operator. For self-adjoint operators, we use results on the nondegeneracy of the ground state. On Banach spaces, a similar role is played by a recent extension by Zhang of the Krein-Rutman theorem.