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The spectral flow and the Maslov index. (English) Zbl 0859.58025
In this article are studied operators of the form $D_A={d\over dt}- A(t),$ where for each $$t$$, $$A(t)$$ is an unbounded, selfadjoint operator on a Hilbert space. It is proved that the operator $$D_A$$ is Fredholm and its index is given by the spectral flow of the operator family $$\{A(t)\}_{t\in\mathbb{R}}$$. The spectral flow is characterized axiomatically, and it is proved that the Fredholm index satisfies these axioms. Also there are considered properties of the Maslov index. Using the spectral flow and Maslov index, the authors prove the Morse index theorem. This result is applied to the Cauchy-Riemann operator on the infinite cylinder with general nonlocal boundary conditions.

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 47A53 (Semi-) Fredholm operators; index theories 58J32 Boundary value problems on manifolds
##### Keywords:
Fredholm index; spectral flow; Maslov index; Morse index theorem
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