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First European Congress of Mathematics pp 511–545Cite as

Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow

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Part of the book series: Progress in Mathematics ((PM,volume 3))

Abstract

Atiyah, Patodi and Singer [3] observed that the Fredholm index of the operator

$$ {D_A}\, = \,\frac{d}{{dt}}\, + \,A\left( t \right) $$

with invertible limits \( {A^ \pm }\, = \,\mathop {\lim }\limits_{t \to \pm \infty } \,A\left( t \right) \) is given by the spectral flow of the self-adjoint operator family A(t) (the number of eigenvalues crossing 0 counted with signs). Such operators appear in infinite dimensional analogues of Morse theory as the linearisation of the gradient flow equation. The Fredholm index is the dimension of the space of gradient flow lines connecting two critical points. It can be thought of as the relative Morse index in cases where the absolute Morse index (the number of negative eigenvalues of the Hessian) is infinite.

This research has been partially supported by the SERC.

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Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, Great Britain

    Stamatis Dostoglou & Dietmar A. Salamon

Authors
  1. Stamatis Dostoglou
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  2. Dietmar A. Salamon
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Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex 05, France

    Anthony Joseph  & Rudolf Rentschler  & 

  2. The Weizmann Institute of Science, 76100, Rehovot, Israel

    Anthony Joseph

  3. Mathématiques Bâtiment 425, Université de Paris-Sud, 91405, Orsay Cedex, France

    Fulbert Mignot

  4. Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex, France

    François Murat

  5. Laboratoire de Statistique Médicale, Université Paris V, 45, rue des Saints Pères, 75270, Paris Cedex, France

    Bernard Prum

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© 1994 Birkhäuser Verlag

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Dostoglou, S., Salamon, D.A. (1994). Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow. In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics . Progress in Mathematics, vol 3. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9110-3_17

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  • DOI: https://doi.org/10.1007/978-3-0348-9110-3_17

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9911-6

  • Online ISBN: 978-3-0348-9110-3

  • eBook Packages: Springer Book Archive

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