Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Waterstraat, Nils

doi:10.1112/plms/pdv028

Cited by 20 publications

(43 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is worth noticing that Proposition 3.7 is the generalization of the main result recently proved in by author in [Wat15].…”

Section: Let Us Define the Operatorssupporting

confidence: 52%

“…The invariant stable and unstable subspaces defined above are Lagrangian subspaces (cf., for instance, [CH07,Wat15] and references therein) and the following convergence result holds:…”

Section: Index Theory For Unbounded Orbitsmentioning

confidence: 99%

“…One can prove that there exists ε 0 > 0 sufficiently small such that sf(A t ; t ∈ [a, b]) = sf(A t + εI; t ∈ [a, b]) (where I denotes the identity operator on V ) for ε ∈ [0, ε 0 ]; furthermore for almost every such ε the path t → A t + εI has regular crossings. We refer the interested reader to [CH07,HS09,Wat15] and references therein.…”

Section: The Spectral Flow For Paths Of Closed Selfadjoint Fredholm Omentioning

confidence: 99%

“…(We refer the interested reader to [HS09,HS10,HLS14,BJP14,BJP16] and references therein). Except these results in which an index theory for homoclinic motions was developed, we only mention the paper [CH07], where the authors assigned a geometrical index to any unbounded motion of a Hamiltonian system, and [Wat15] where a suitable spectral flow formula for a one-parameter family of Hamiltonian systems under homoclinic boundary conditions was proved.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Index theory for heteroclinic orbits of Hamiltonian systems

Порталури

2017

Calc. Var.

View full text Add to dashboard Cite

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits.Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrised by bounded intervals.

show abstract

“…It is worth noticing that Proposition 3.7 is the generalization of the main result recently proved in by author in [Wat15].…”

Section: Let Us Define the Operatorssupporting

confidence: 52%

“…The invariant stable and unstable subspaces defined above are Lagrangian subspaces (cf., for instance, [CH07,Wat15] and references therein) and the following convergence result holds:…”

Section: Index Theory For Unbounded Orbitsmentioning

confidence: 99%

Section: The Spectral Flow For Paths Of Closed Selfadjoint Fredholm Omentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Index theory for heteroclinic orbits of Hamiltonian systems

Порталури

2017

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…This is Proposition 6.1 in [12]. Note that in this setting the path A = {A λ } λ∈I has the constant domain D(A λ ) = {u ∈ H 1 (I, R 2n ) : u(0), u(1) ∈ {0} × R n }, which allows to compute its spectral flow by crossing forms (see [18] and [22]) and yields the different proof of (27) given in [12].…”

Section: A Spectral Flow Formula For Hamiltonian Systemsmentioning

confidence: 85%

The Maslov index and the spectral flow—revisited

Izydorek¹,

Janczewska²,

Waterstraat³

2019

Fixed Point Theory Appl

Self Cite

View full text Add to dashboard Cite

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.

show abstract

On a comparison principle and the uniqueness of spectral flow

Starostka

Waterstraat

2022

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

Cited by 20 publications

References 37 publications

Index theory for heteroclinic orbits of Hamiltonian systems

Index theory for heteroclinic orbits of Hamiltonian systems

The Maslov index and the spectral flow—revisited

On a comparison principle and the uniqueness of spectral flow

Contact Info

Product

Resources

About