TY - INPR
A1 - Nazaikinskii, Vladimir
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris
T1 - Surgery and the relative index theorem for families of elliptic operators
N2 - We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.
T3 - Preprint - (2002) 11
KW - elliptic operators
KW - index theory
KW - surgery
KW - relative index
KW - boundary value problems
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26300
ER -
TY - INPR
A1 - Harutjunjan, G.
A1 - Schulze, Bert-Wolfgang
T1 - Reduction of orders in boundary value problems without the transmission property
N2 - Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary.
T3 - Preprint - (2002) 03
KW - Boundary value problems
KW - elliptic operators
KW - order reduction
KW - Volterra symbols
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26220
ER -
TY - INPR
A1 - Nazaikinskii, Vladimir
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris
T1 - Localization problem in index theory of elliptic operators
N2 - This is a survey of recent results concerning the general index locality principle, associated surgery, and their applications to elliptic operators on smooth manifolds and manifolds with singularities as well as boundary value problems. The full version of the paper is submitted for publication in Russian Mathematical Surveys.
T3 - Preprint - (2001) 34
KW - elliptic operators
KW - index theory
KW - surgery
KW - relative index
KW - manifold with singularities
Y1 - 2001
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26175
ER -
TY - INPR
A1 - Schulze, Bert-Wolfgang
A1 - Nazaikinskii, Vladimir E.
A1 - Sternin, Boris
T1 - On the homotopy classification of elliptic operators on manifolds with singularities
N2 - We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol.
T3 - Preprint - (1999) 21
KW - elliptic operators
KW - homotopy classification
KW - manifold with singularities
KW - Atiyah-Singer theorem
Y1 - 1999
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25574
ER -
TY - INPR
A1 - Fedosov, Boris
A1 - Schulze, Bert-Wolfgang
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - The index of elliptic operators on manifolds with conical points
N2 - For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
T3 - Preprint - (1997) 24
KW - manifolds with singularities
KW - pseudodifferential operators
KW - elliptic operators
KW - index
Y1 - 1997
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25096
ER -
TY - INPR
A1 - Schulze, Bert-Wolfgang
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - The Riemann-Roch theorem for manifolds with conical singularities
N2 - The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points.
T3 - Preprint - (1997) 18
KW - manifolds with singularities
KW - elliptic operators
KW - divisors
Y1 - 1997
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25051
ER -