TY  - GEN
A1  - Ginoux, Nicolas
T1  - Dirac operators on Lagrangian submanifolds
N2  - We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.
KW  - Dirac operators
KW  - Global Analysis
KW  - Spectral Geometry
KW  - Spin Geometry
KW  - Lagrangian submanifolds
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-5627
ER  - 
TY  - JOUR
A1  - Mickelsson, Jouko
A1  - Paycha, Sylvie
T1  - The logarithmic residue density of a generalized Laplacian
JF  - Journal of the Australian Mathematical Society
N2  - We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form.  We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension.  These correspond to special cases of a moregeneral formula by Scott and Zagier.  In our approach, which is of perturbative nature, we use either aCampbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.
KW  - residue
KW  - index
KW  - Dirac operators
Y1  - 2011
U6  - https://doi.org/10.1017/S144678871100108X
SN  - 0263-6115
SN  - 1446-8107
VL  - 90
IS  - 1
SP  - 53
EP  - 80
PB  - Cambridge Univ. Press
CY  - Cambridge
ER  - 
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  - Savin, Anton
A1  - Sternin, Boris
T1  - Eta invariant and parity conditions
N2  - We give a formula for the η-invariant of odd order operators on even-dimensional manifolds, and for even order operators on odd-dimensional manifolds. Geometric second order operators are found with nontrivial η-invariants. This solves a problem posed by P. Gilkey.
T3  - Preprint - (2000) 21 
KW  - eta invariant
KW  - parity conditions
KW  - K-theory
KW  - linking coefficients
KW  - Dirac operators
KW  - spectral flow
KW  - elliptic operators
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25869
ER  -