Searcharxiv⌕ Search

arXiv subjects

Nils Waterstraat

Publications and source records attributed to Nils Waterstraat.

At least 19 recordsLinked to original sources
arXiv2026

An Index Theorem in Relative K-Theory for First-Order Systems

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and Portaluri. The main novelties of our approach firstly concern the analytical setting, where we lift the common assumption that the equations are asymptotically hyperbolic. Secondly, we consider general compact parameter spaces instead of a single parameter, which results in a remarkably simple index formula in relative $K$-theory.

math.DS↗
arXiv2025

On the Uniqueness of the $G$-Equivariant Spectral Flow

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators. Lesch as well as Pejsachowicz, Fitzpatrick and Ciriza independently showed that it is uniquely characterised by its elementary properties. The authors recently introduced a $G$-equivariant spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the action of a compact Lie group $G$. The purpose of this paper is to show that the $G$-equivariant spectral flow is uniquely characterised by the same elementary properties when appropriately restated. As an application, we introduce an alternative definition of the $G$-equivariant spectral flow via a $G$-equivariant Maslov index.

math.FA↗
arXiv2024

Local and Global Bifurcation for Periodic Solutions of Hamiltonian Systems via Comparison Theory for the Spectral Flow

We obtain local and global bifurcation for periodic solutions of Hamiltonian systems by using a new way to apply a comparison principle of the spectral flow that was originally introduced by Pejsachowicz in a joint work with the third author. A particular novelty is the study of global bifurcation, which to the best of our knowledge has not been done via the spectral flow.

math.DS↗
arXiv2024

The relative cup-length in local Morse cohomology

Local Morse cohomology associates cohomology groups to isolating neighborhoods of gradient flows of Morse functions on (generally non-compact) Riemannian manifolds $M$. We show that local Morse cohomology is a module over the cohomology of the isolating neighborhood, which allows us to define a cup-length relative to the cohomology of the isolating neighborhood that gives a lower bound on the number of critical points of functions on $M$ that are not necessarily Morse. Finally, we illustrate by an example that this lower bound can indeed be stronger than the lower bound given by the absolute cup-length.

math.GT↗
arXiv2023

The Equivariant Spectral Flow and Bifurcation for Functionals with Symmetries -- Part I

We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a recent construction of an equivariant spectral flow by the authors, we obtain a bifurcation theorem that generalises well-established results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and Recht. Finally, we discuss some elementary examples of strongly indefinite systems of PDEs and Hamiltonian systems where the mentioned classical approaches fail but an invariance under an orthogonal action of $\mathbb{Z}_2$ makes our methods applicable and yields the existence of bifurcation.

math.FA↗
arXiv2021

The Equivariant Spectral Flow and Bifurcation of Periodic Solutions of Hamiltonian Systems

We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This $G$-equivariant spectral flow shares all common properties of the integer valued classical spectral flow, and it can be non-trivial even if the classical spectral flow vanishes. Our main theorem uses the $G$-equivariant spectral flow to study bifurcation of periodic solutions for autonomous Hamiltonian systems with symmetries.

math.FA↗
arXiv2020

The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems

The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and Jänich in the sixties, and it is readily seen that on complex Hilbert spaces this so called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that for every real Hilbert space and every compact topological space $X$ there is a family of selfadjoint Fredholm operators parametrised by $X\times S^1$ which has a non-trivial index bundle. Further, we use this observation and a family index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.

math.FA↗
arXiv2020

Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is the existence of an exponential dichotomy. In this paper we generalize all our previous results to the latter setting, which requires substantial modifications of our arguments. In addition, we generalize previous results on continuity and differentiability of Nemitski operators for discrete dynamical systems to obtain even better bifurcation results.

math.FA↗
arXiv2019

On a Comparison Principle and the Uniqueness of Spectral Flow

The spectral flow is a well-known quantity in spectral theory that measures the variation of spectra about $0$ along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly, we consider homotopy invariance properties of the spectral flow and establish a simple formula which comprises its classical homotopy invariance and yields a comparison theorem for the spectral flow under compact perturbations. We apply our result to the existence of non-trivial solutions of boundary value problems of Hamiltonian systems. Secondly, the spectral flow was axiomatically characterised by Lesch, and by Ciriza, Fitzpatrick and Pejsachowicz under the assumption that the endpoints of the paths of selfadjoint Fredholm operators are invertible. We propose a different approach to the uniqueness of spectral flow which lifts this additional assumption. As application of the latter result, we discuss the relation between the spectral flow and the Maslov index in symplectic Hilbert spaces.

math.FA↗
arXiv2019

Parity flow as ${\mathbb Z}_2$-valued spectral flow

This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray-Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the ${\mathbb Z}_2$-valued spectral flow of an associated path of chiral skew-adjoints. Furthermore the related notion of ${\mathbb Z}_2$-index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non-trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non-linear partial differential equation.

math-ph↗
arXiv2018

On the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems

We show the existence of homoclinic type solutions of second order Hamiltonian systems with a potential satisfying a relaxed superquadratic growth condition and a forcing term that is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger time-periods. We prove that the latter systems admit periodic solutions of mountain-pass type, and obtain homoclinic type solutions of the original system from them by passing to the limit (in the topology of almost uniform convergence) when the periods go to infinity.

math.DS↗
arXiv2018

The Maslov Index and the Spectral Flow - revisited

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of selfadjoint first-order operators. We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller's theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.

math.DS↗
arXiv2018

On the Fredholm Lagrangian Grassmannian, Spectral Flow and ODEs in Hilbert Spaces

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main theorem relates the spectra of infinite dimensional Hamiltonian systems under homoclinic boundary conditions to intersections of their stable and unstable spaces. Our proof has some interest in its own. Firstly, we extend a celebrated theorem by Cappell, Lee and Miller about the classical Maslov index in $\mathbb{R}^{2n}$ to symplectic Hilbert spaces. Secondly, we generalise the classical index bundle for families of Fredholm operators of Atiyah and Jänich to unbounded operators for applying it to Hamiltonian systems under varying boundary conditions. Finally, we substantially make use of striking results by Abbondandolo and Majer to study Fredholm properties of infinite dimensional Hamiltonian systems.

math.DS↗
arXiv2017

The $E$-cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on $T^{2n}$

We give a new proof of the strong Arnold conjecture for $1$-periodic solutions of Hamiltonian systems on tori, that was first shown by C. Conley and E. Zehnder in 1983. Our proof uses other methods and is shorter than the previous one. We first show that the $E$-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals. Then an existence result for the $E$-cohomological Conley index, which applies to the setting of the Arnold conjecture, paves the way to a new proof of it on tori.

math.DS↗
arXiv2017

Not every conjugate point of a semi-Riemannian geodesic is a bifurcation point

We revisit an example of a semi-Riemannian geodesic that was discussed by Musso, Pejsachowicz and Portaluri in 2007 to show that not every conjugate point is a bifurcation point. We point out a mistake in their argument, showing that on this geodesic actually every conjugate point is a bifurcation point. Finally, we provide an improved example which yields that the claim in our title is nevertheless true.

math.DG↗
arXiv2017

Bifurcation of critical points along gap-continuous families of subspaces

We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations.

math.FA↗
arXiv2017

On the existence of homoclinic type solutions of inhomogenous Lagrangian systems

We study the existence of homoclinic type solutions for second order Lagrangian systems of the type $\ddot{q}(t)-q(t)+a(t)\nabla G(q(t))=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$, $a\colon\mathbb{R}\to\mathbb{R}$ is a continuous positive bounded function, $G\colon\mathbb{R}^n\to\mathbb{R}$ is a $C^1$-smooth potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and $f\colon\mathbb{R}\to\mathbb{R}^n$ is a continuous bounded square integrable forcing term. A homoclinic type solution is obtained as limit of $2k$-periodic solutions of an approximative sequence of second order differential equations.

math.DS↗