Maslov index of Hamiltonian systems

Iteration formulae of the Maslov-type index theory in weak symplectic Hilbert spaces

In this paper, we prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then we give a direct proof of iteration formulae for the Maslov-type indices of symplectic paths.

局部 Maslov 指标的公式及其应用

A formula of local Maslov index and applications

本文在有限维情形下，无需辛约化，将局部Maslov指标显式表示为Maslov指标，并计算三角形式下Lagrange子空间对的Maslov指标，进而得到辛路径的Maslov型指标，应用于计算分裂数、迭代理论对三角框架的依赖性及模2指标。

The Maslov and Morse indices for Schrödinger operators on $\mathbb{R}$

Assuming a symmetric potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schrödinger operators on $\mathbb{R}$. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index.

The Nonlinear Maslov index and the Calabi homomorphism

In this paper we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard 2n-1 dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard n-1 complex projective space. We prove that the value of the pull back of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, restricted to elements supported in sufficiently small open subsets of M, equals the Calabi invariant of these elements up to a multiplication by a…

The Maslov index and spectral counts for linear Hamiltonian systems on $\mathbb{R}$

Working with a general class of linear Hamiltonian systems specified on $\mathbb{R}$, we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $[λ_1, λ_2)$ and $(-\infty, λ_2)$. We verify that our framework can be implemented for Sturm-Liouville systems, fourth-order potential systems, and a family of systems nonlinear in the spectral parameter. The analysis is primarily motivated by applications to the analysis of spectral stability for nonlinear waves, and aspects of such analyses are emphasized.

Counting spectrum via the Maslov index for one dimensional $θ-$periodic Schrödinger operators

We study the spectrum of the Schrödinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $θ-$periodic boundary conditions. For two such operators, corresponding to different values of $θ$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $θ$ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a …

Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems

We obtain geometric insight into the stability of traveling pulses for reaction-diffusion equations with skew-gradient structure. For such systems, a Maslov index of the traveling wave can be defined and related to the eigenvalue equation for the linearization $L$ about the wave. We prove two main results about this index. First, for general skew-gradient systems, it is shown that the Maslov index gives a lower bound on the number of real, unstable eigenvalues of $L$. Second, we show how the Maslov index gives an exact count of all unstable eigenvalues for fast traveling waves in a FitzHugh-Na…

Maslov 指数、退化交叉与 Swift-Hohenberg 方程脉冲解的稳定性

The Maslov index, degenerate crossings and the stability of pulse solutions to the Swift-Hohenberg equation

本文通过扩展 Robbin-Salamon 方法计算 Maslov 指数，提出了一种确定标量 Swift-Hohenberg 方程对称脉冲解谱稳定性的方法，并验证该指数与线性化方程的不稳定特征值数量一致。

Bifurcation to instability through the lens of the Maslov index

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems. In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest…

Gutzwiller's Semiclassical Trace Formula and Maslov-Type Index Theory for Symplectic Paths

Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…

Generalized Maslov indices for non-Hamiltonian systems

We extend the definition of the Maslov index to a broad class of non-Hamiltonian dynamical systems. To do this, we introduce a family of topological spaces--which we call Maslov-Arnold spaces--that share key topological features with the Lagrangian Grassmannian, and hence admit a similar index theory. This family contains the Lagrangian Grassmannian, and much more. We construct a family of examples, called hyperplane Maslov-Arnold spaces, that are dense in the Grassmannian, and hence are much larger than the Lagrangian Grassmannian (which is a submanifold of positive codimension). The resultin…

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.

A Maslov Map for Coisotropic Submanifolds, Leaf-wise Fixed Points and Presymplectic Non-Embeddings

Let $(M,ω)$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $Σ$ a compact oriented (real) surface. I define a natural Maslov index for each continuous map $u:Σ\to M$ that sends every connected component of $\partialΣ$ to some isotropic leaf of $N$. This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of $N$ to compensate for the loss of boundary data in the case codimension $N<\dim M/2$. The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplect…

On the Maslov class rigidity for coisotropic submanifolds

We define the Maslov index of a loop tangent to the characteristic foliation of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the group of linear symplectic transformations, incorporating the "rotation" of the tangent space of the leaf -- this is the standard Lagrangian counterpart -- and the holonomy of the characteristic foliation. Furthermore, we show that, with this definition, the Maslov class rigidity extends to the class of the so-called stable coisotropic submanifolds including Lagrangian tori and stable hypersurfaces.

Computing the Maslov index for large systems

We address the problem of computing the Maslov index for large linear symplectic systems on the real line. The Maslov index measures the signed intersections (with a given reference plane) of a path of Lagrangian planes. The natural chart parameterization for the Grassmannian of Lagrangian planes is the space of real symmetric matrices. Linear system evolution induces a Riccati evolution in the chart. For large order systems this is a practical approach as the computational complexity is quadratic in the order. The Riccati solutions, however, also exhibit singularites (which are traversed by c…

哈密顿谱流、Maslov 指标与非线性薛定谔方程中驻波的稳定性

Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schrödinger equation

本文用Maslov指标分析线性哈密顿算子谱，通过非正则交叉处特征值曲线的Jordan链凹性公式与同伦方法计算指标，给出正实特征值下界，并据此建立紧区间上非线性薛定谔方程驻波在不同边界条件下的新稳定性判据。

The Morse and Maslov indices for matrix Hill's equations

For Hill's equations with matrix valued periodic potential, we discuss relations between the Morse index, counting the number of unstable eigenvalues, and the Maslov index, counting the number of signed intersections of a path in the space of Lagrangian planes with a fixed plane. We adapt to the one dimensional periodic setting the strategy of a recent paper by J. Deng and C. Jones relating the Morse and Maslov indices for multidimensional elliptic eigenvalue problems.

Instability of pulses in gradient reaction-diffusion systems: A symplectic approach

In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evo…

A Stability Index for Traveling Waves in Activator-Inhibitor Systems

We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Masl…

On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System

The FitzHugh-Nagumo equation, which was derived as a simplification of the Hodgkin-Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system in which two agents diffuse at an equal rate. Using geometric singular perturbation theory, we prove the existence and stability of fast traveling pulses. The stability proof makes use of the Maslov index--an invariant of symplectic geometry--to count unstable eigenvalues for the linearization about the wave. The calculation of the Maslov index is carried o…

马斯洛夫$S^1$丛与马斯洛夫数据

Maslov $S^{1}$ Bundles and Maslov Data

本文在辛流形上定义马斯洛夫$S^1$丛，分析其与紧李群辛作用的关联，提出非平凡丛情形下的马斯洛夫数据概念，并给出其在可积哈密顿系统等场景的三类应用。

The Morse index of Chaperon's generating families

This is an expository paper devoted to the Morse index of Chaperon's generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott's iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate the precise dependence of the Morse index from the choice of the generating family associated to a given Hamiltonian diffeomorphism, which will allow to see the Morse index as a Maslov index for the linea…

Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\bf R}^{2n}$

In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface $Σ\subset{\bf R}^{2n}$, there exist at least $n$ non-hyperbolic closed characteristics with even Maslov-type indices on $Σ$ when $n$ is even. When $n$ is odd, there exist at least $n$ closed characteristics with odd Maslov-type indices on $Σ$ and at least $(n-1)$ of them are non-hyperbolic. Here we call a compact star-shaped hypersurface $Σ\subset {\bf R}^{2n}$ {\rm index perfect} if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed charac…

Brake subharmonic solutions of first order Hamiltonian systems

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the $L$-Maslov type index theory to study the properties of brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that when the positive integers $j$ and $k$ satisfies the certain conditions, there exists a $jT$-periodic nonconstant brake solution $z_{j}$ such that $z_{j}$ and $z_{kj}$ are distinct.

$\mathbf{R}^{6}$ 中非退化星形超曲面上的三条闭特征

Three closed characteristics on non-degenerate star-shaped hypersurfaces in $\mathbf{R}^{6}$

该研究证明在 $\mathbf{R}^{6}$ 的非退化 $C^3$ 紧星形超曲面上，只要缺少 Maslov 型指标 0 或 -1 的本原闭特征，就至少存在三条本原闭特征。

Subharmonic Solutions and Minimal Periodic Solutions of First-order Hamiltonian Systems with Anisotropic Growth

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

非哈密顿系统的广义指标理论

A generalized index theory for non-Hamiltonian system

本文针对非哈密顿耗散系统，引入基于布劳威尔度的新拓扑不变量“度指标”作为马斯洛夫指标的替代，并通过抽象迹公式证明其与莫尔斯指标的等式，最后应用于一维反应扩散系统。

Loops of Infinite Order and Toric Foliations

In 2005 Dullin et al. proved that the non-zero vector of Maslov indices is an eigenvector with eigenvalue 1 of the monodromy matrices of an integrable Hamiltonian system. We take a close look at the geometry behind this result and extend it to a more general context. We construct a bundle morphism defined on the lattice bundle of an (general) integrable system, which can be seen as a generalization of the vector of Maslov indices. The non-triviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue 1 of the monodromy matrices, and gives rise to a corank 1 tor…

Hill-type formula for Hamiltonian system with Lagrangian boundary conditions

In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the brake symmetry periodic orbits in $n$-body problem naturally. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined …

非紧量子图的索引理论

Index theory for non-compact quantum graphs

本文发展了非紧量子图变分问题的索引理论，提出谱流公式和莫塞塔数定理，将特征值变化与边界数据的Maslov指数关联，并应用于非线性波方程。

Collision index and stability of elliptic relative equilibria in planar n-body problem

It is well known that a planar central configuration of the $n$-body problem gives rise to solutions where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homographic motion. When the eccentricity $e$ of the Keplerian orbit belongs in $[0,1)$, following Meyer and Schmidt, we call such solutions elliptic relative equilibria (shortly, ERE). In order to study the linear stability of ERE in the near-collision case, namely when $1-e$ is small enough, we introduce the collision index for planar central configurations. The collision index is a Maslov-ty…