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Higher spectral flow. (English) Zbl 0932.37062

A notion of higher spectral flow is introduced. It is defined for a continuous one parameter family of families of Dirac-type operators parametrized by a compact space. The authors show that this higher spectral flow can be computed analytically. Moreover, restricting to a one parameter family of self-adjoint elliptic pseudodifferential operators with the standard Atiyah-Patodi-Singer spectral projections at the endpoints, the higher spectral flow coincides with the usual spectral flow. The authors also show that this higher version of spectral flow satisfies the basic properties of spectral flows. Applications to study the index theory for families of manifolds with boundary are also given.

MSC:

37K99 Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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