Maslov-Type Index Theory for Symplectic Paths and Spectral Flow (I)

Zhu, Chaofeng; LONG, YIMING

doi:10.1142/s0252959999000485

Cited by 65 publications

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“…We call I (B 1 , B 2 ) the relative Morse index, following [Fei 1995;Zhu and Long 1999;Long and Zhu 2000;Long 2002]. …”

Section: Maslov Type Index Theory For Linear Hamiltonian Systems Satimentioning

confidence: 99%

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

Dong¹

2005

Pacific J. Math.

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“…We call I (B 1 , B 2 ) the relative Morse index, following [Fei 1995;Zhu and Long 1999;Long and Zhu 2000;Long 2002]. …”

Section: Maslov Type Index Theory For Linear Hamiltonian Systems Satimentioning

confidence: 99%

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

Dong¹

2005

Pacific J. Math.

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“…The spectral flow has the following properties (cf. [23] and Lemma 2.6 and Proposition 2.2 in [33]). Proposition 2.3.…”

Section: 2mentioning

confidence: 86%

“…It counts the number of spectral lines of A s coming from the negative side of ℓ to the non-negative side of ℓ. For the details, see [33]. The spectral flow has the following properties (cf.…”

Section: 2mentioning

confidence: 99%

General spectral flow formula for fixed maximal domain

Booss-Bavnbek

Zhu

2005

Open Mathematics

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Abstract. We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.1. Statement of the problem and main result 1.1. Statement of the problem. Roughly speaking, the spectral flow counts the net number of eigenvalues changing from the negative real half axis to the non-negative one. The definition goes back to a famous paper by M. Atiyah, V. Patodi, and I. Singer [3], and was made rigorous by J. Phillips [23] and Y. Long [33] in various non-self-adjoint cases, and by B. BoossBavnbek, M. Lesch, and J. Phillips [7] in the unbounded self-adjoint case. We shall give a rigorous definition of spectral flow, most suitable for our purpose, below in Subsection 2.1 together with a review of its basic properties. For a definition of spectral flow admitting zero in the continuous spectrum, we refer to A. Carey and J. Phillips [13].In various branches of mathematics one is interested in the calculation of the spectral flow of a continuous family of closed densely defined (not necessarily bounded) self-adjoint Fredholm operators in a fixed Hilbert space. We consider the following typical problem of this kind. Sobolev space over the boundary. (Note that in this paper the symbols x and y do not denote points of the underlying manifolds M or Σ, but points in Hilbert spaces, sections of vector bundles, etc., following the conventions of functional analysis and dynamical systems.) Assume that each P s defines a self-adjoint elliptic boundary condition for A s , i.e., A s,Ps is a self-adjoint Fredholm operator for each s ∈ [0, 1].Then the spectral flow sf{A s,Ps ; s ∈ [0, 1]} or, shortly, sf{A s,Ps } is well defined. As a spectral invariant it is essentially a quantum variable which one may not always be able to determine directly by eigenvalue calculations. As an alternative, one is looking for a classical method of calculating the spectral flow. There are two different approaches. One setting expresses the spectral flow (of a loop of Dirac operators on a closed manifold) as an integral over a 1-form induced by the heat kernel (for a review see [13]). The other setting is reduction to the boundary, i.e., one expresses the spectral flow (of a path of self-adjoint boundary value problems on a compac...

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“…In this section, we show that the R-valued variation formula for 77-invariants (which has been used in (3.28)) admits an extension to a C-valued variation formula valid also for the non-self-adjoint operators discussed in the present paper. First, the concept of spectral flow can be extended to non-self-adjoint operators, and this has been done in 18 in a general context.…”

Section: More On Rj-invariants Spectral Flow and The Phase Of The Rementioning

confidence: 99%

“…Finally, using the extension (to the case of non-self-adjoint operators) given in 18 of the concept of spectral flow, 3 we propose a refinement in C of the above variation formula for ^-invariants.…”

Section: Introductionmentioning

confidence: 99%

Η-Invariant AND FLAT VECTOR BUNDLES II

Ma¹,

Zhang²

2006

Nankai Tracts in Mathematics

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We first apply the method and results in the previous paper to give a new proof of a result (holds in C/Z) of Gilkey on the variation of ninvariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain 77-invariant appearing in recent papers of Braverman-Kappeler on what they call refined analytic torsion, and propose an alternate formulation of their definition of the refined analytic torsion. A refinement in C of the above variation formula is also proposed.

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Maslov-Type Index Theory for Symplectic Paths and Spectral Flow (I)

Cited by 65 publications

References 0 publications

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

General spectral flow formula for fixed maximal domain

Η-Invariant AND FLAT VECTOR BUNDLES II

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