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- elliptic operators (6)
- manifolds with singularities (6)
- Fredholm property (5)
- boundary value problems (5)
- index (5)
- pseudodifferential operators (5)
- Boundary value problems (4)
- relative index (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- K-theory (3)
- conical singularities (3)
- conormal symbol (3)
- differential operators (3)
- ellipticity (3)
- index theory (3)
- Atiyah-Bott obstruction (2)
- Atiyah-Patodi-Singer theory (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- Zaremba problem (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic complexes (2)
- elliptic families (2)
- elliptic family (2)
- elliptic operator (2)
- homotopy classification (2)
- index formulas (2)
- manifold with singularities (2)
- manifolds with conical singularities (2)
- manifolds with edges (2)
- monodromy matrix (2)
- pseudo-differential boundary value problems (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- surgery (2)
- symmetry conditions (2)
- weighted edge spaces (2)
- APS problem (1)
- Atiyah-Singer theorem (1)
- Boundary-contact problems (1)
- C0−semigroup (1)
- Calderón projections (1)
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- Cauchy Riemann operator (1)
- Chern character (1)
- Corner boundary value problems (1)
- Crack theory (1)
- Edge-degenerate operators (1)
- Elliptic operators in domains with edges (1)
- Euler operator (1)
- G-index (1)
- G-trace (1)
- Green and Mellin edge operators (1)
- Hardy‘s inequality (1)
- Hodge theory (1)
- Korn’s weighted inequality (1)
- Lefschetz number (1)
- Meromorphic operator functions (1)
- Operators on manifolds with conical singularities (1)
- Operators on manifolds with edge (1)
- Operators on manifolds with edge and conical exit to infinity (1)
- Operators on manifolds with second order singularities (1)
- Pseudo-differential operators (1)
- Pseudodifferential operators (1)
- Sobolev spaces with double weights on singular cones (1)
- Surface potentials with asymptotics (1)
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- asymptotics of solutions (1)
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- cohomology (1)
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- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- elliptic operators in subspaces (1)
- elliptic operators on non-compact manifolds (1)
- elliptic problem (1)
- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with respect to interior and edge symbols (1)
- eta-invariant (1)
- exponential stability (1)
- exterior tensor product (1)
- index formula (1)
- index of elliptic operator (1)
- index of elliptic operators in subspaces (1)
- integral formulas (1)
- interfaces with conical singularities (1)
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- norm estimates with respect to a parameter (1)
- operator algebras on manifolds with singularities (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- order reduction (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- principal symbolic hierarchies (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudo-differential operators (1)
- pseudo-differentialboundary value problems (1)
- pseudodifferential boundary value problems (1)
- pseudodifferential operator (1)
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- relative cohomology (1)
- relative index formulas (1)
- residue (1)
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- spectral resolution (1)
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- symmetry group (1)
- symplectic (canonical) transformations (1)
- uniform compact attractor (1)
- vibration (1)
- weighted spaces (1)
- weighted spaces with asymptotics (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (115)
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
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.
The homotopy classification and the index of boundary value problems for general elliptic operators
(1999)
We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.
Boundary value problems for pseudodifferential operators (with or without the transmission property) are characterised as a substructure of the edge pseudodifferential calculus with constant discrete asymptotics. The boundary in this case is the edge and the inner normal the model cone of local wedges. Elliptic boundary value problems for non-integer powers of the Laplace symbol belong to the examples as well as problems for the identity in the interior with a prescribed number of trace and potential conditions. Transmission operators are characterised as smoothing Mellin and Green operators with meromorphic symbols.
Ellipticity of operators on manifolds with conical singularities or parabolicity on space-time cylinders are known to be linked to parameter-dependent operators (conormal symbols) on a corresponding base manifold. We introduce the conormal symbolic structure for the case of corner manifolds, where the base itself is a manifold with edges and boundary. The specific nature of parameter-dependence requires a systematic approach in terms of meromorphic functions with values in edge-boundary value problems. We develop here a corresponding calculus, and we construct inverses of elliptic elements.
Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.
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.
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.
The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to shortrange correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.