### Abstract

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 ℝ^{N} and asymptotically flat manifolds. It is shown that the absolute or relative Euler characteristic of an obstacle in 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.

Original language | English |
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Pages (from-to) | 475-513 |

Number of pages | 39 |

Journal | Communications in Mathematical Physics |

Volume | 114 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 1988 |

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### Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*114*(3), 475-513. https://doi.org/10.1007/BF01242140

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*Communications in Mathematical Physics*, vol. 114, no. 3, pp. 475-513. https://doi.org/10.1007/BF01242140

**Relative index theorems and supersymmetric scattering theory.** / Borisov, N. V.; Müller, W.; Schrader, R.

Research output

TY - JOUR

T1 - Relative index theorems and supersymmetric scattering theory

AU - Borisov, N. V.

AU - Müller, W.

AU - Schrader, R.

PY - 1988/9/1

Y1 - 1988/9/1

N2 - 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 ℝN and asymptotically flat manifolds. It is shown that the absolute or relative Euler characteristic of an obstacle in ℝN 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 L2-Euler characteristic.

AB - 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 ℝN and asymptotically flat manifolds. It is shown that the absolute or relative Euler characteristic of an obstacle in ℝN 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 L2-Euler characteristic.

UR - http://www.scopus.com/inward/record.url?scp=0001417932&partnerID=8YFLogxK

U2 - 10.1007/BF01242140

DO - 10.1007/BF01242140

M3 - Article

AN - SCOPUS:0001417932

VL - 114

SP - 475

EP - 513

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -