TY  - GEN
A1  - Mickelsson, Jouko
A1  - Paycha, Sylvie
T1  - The logarithmic residue density of a generalized Laplacian
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of
classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a
twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 649 
KW  - residue
KW  - index
KW  - Dirac operators
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413680
SN  - 1866-8372
IS  - 649
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Euler solutions of pseudodifferential equations
N2  - We consider a homogeneous pseudodifferential equation on a cylinder C = IR x X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent on the variable t element IR, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points since C can be thought of as a 'stretched' manifold with conical points at t = -infinite and t = infinite.
T3  - Preprint - (1998) 09 
KW  - pseudodifferential operator
KW  - meromorphic family
KW  - residue
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25211
ER  -