Institut für Mathematik
Refine
Year of publication
Document Type
- Preprint (45)
- Monograph/Edited Volume (17)
- Article (1)
Language
- English (63)
Keywords
- K-theory (7)
- elliptic operators (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
- Fredholm property (5)
- index theory (5)
- boundary value problems (4)
- elliptic operator (4)
- index (4)
- manifold with singularities (4)
- surgery (4)
- conical singularities (3)
- conormal symbol (3)
- eta invariant (3)
- index of elliptic operators in subspaces (3)
- Atiyah-Bott condition (2)
- Atiyah-Bott obstruction (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- boundary value problem (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic families (2)
- elliptic family (2)
- ellipticity (2)
- eta-invariant (2)
- homotopy classification (2)
- index formulas (2)
- linking coefficients (2)
- manifolds with conical singularities (2)
- modn-index (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- symmetry conditions (2)
- (co)boundary operator (1)
- APS problem (1)
- Atiyah-Singer theorem (1)
- Calderón projections (1)
- Cauchy Riemann operator (1)
- Chern character (1)
- Dirac operators (1)
- Euler operator (1)
- Green operator (1)
- Pontrjagin duality (1)
- Riemann-Roch theorem (1)
- Sobolev problem (1)
- analytic index (1)
- contact transformations (1)
- covering (1)
- dimension functional (1)
- edge symbol (1)
- elliptic morphism (1)
- elliptic operators in subspaces (1)
- elliptic problem (1)
- exterior tensor product (1)
- finiteness theorem (1)
- index formula (1)
- index of elliptic operator (1)
- manifold with edge (1)
- manifolds with edges (1)
- mod k index (1)
- modulo n index (1)
- nonhomogeneous boundary value problems (1)
- nonlocal problem (1)
- parameter-dependent ellipticity (1)
- parity condition (1)
- parity conditions (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- relative η-invariant (1)
- spectral boundary value problems (1)
- spectral resolution (1)
- symplectic (canonical) transformations (1)
- η-invariant (1)
Institute
- Institut für Mathematik (63) (remove)
An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established.
In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms.
The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.
For elliptic operators on manifolds with boundary, we define spectral boundary value problems, which generalize the Atiyah-Patodi-Singer problem to the case of nonhomogeneous boundary conditions, operators of arbitrary order, and nonself-adjoint conormal symbols. The Fredholm property is proved and equivalence with certain elliptic equations on manifolds with conical singularities is established.
We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.
The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.