Geodesic
Eta-invariant of elliptic parameter-dependent boundary-value problems
Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 599-620

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In this paper, we study the eta-invariant of elliptic parameter-dependent boundary value problems and its main properties. Using Melrose's approach, we define the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundary value problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary value problems and is based on the reduction of parameter-dependent boundary value problems to boundary value problems with no parameter.
Mots-clés : eta-invariant
Keywords: elliptic parameter-dependent boundary value problem, pseudodifferential boundary value problem, Boutet de Monvel operator, regularized trace. 
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K. N. Zhuikov; A. Yu. Savin. Eta-invariant of elliptic parameter-dependent boundary-value problems. Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 599-620. http://geodesic.mathdoc.fr/item/CMFD_2023_69_4_a3/