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Relative index theorems and supersymmetric scattering theory. (English) Zbl 0663.58032

We discuss supersymmetric scattering theory and employ Krein’s theory of spectral shift functions to investigate supersymmetric scattering systems. This is the basis for the derivation of relative index theorems on some classes of open manifolds. As an example we discuss the de Rham complex for obstacles in \({\mathbb{R}}^ N\) and asymptotically flat manifolds. It is shown that the absolute or relative Euler characteristic of an obstacle in \({\mathbb{R}}^ N\) may be obtained from scattering data for the Laplace operator on forms with absolute or relative boundary conditions respectively. In the case of asymptotically flat manifolds we obtain the Chern-Gauss-Bonnet theorem for the \(L^ 2\)-Euler characteristic.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58Z05 Applications of global analysis to the sciences
81T60 Supersymmetric field theories in quantum mechanics
81U99 Quantum scattering theory
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