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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)
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- ellipticity (3)
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- Atiyah-Bott obstruction (2)
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- Mellin transform (2)
- Zaremba problem (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic complexes (2)
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- 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)
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- regularizers (2)
- spectral flow (2)
- surgery (2)
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- weighted edge spaces (2)
- APS problem (1)
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- Euler operator (1)
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- Hardy‘s inequality (1)
- Hodge theory (1)
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- 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)
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- cohomology (1)
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- continuity in Sobolev spaces with double weights (1)
- corner Sobolev spaces with double weights (1)
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- edge Sobolev spaces (1)
- edge algebra (1)
- 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)
- manifold with edge (1)
- manifolds with cusps (1)
- meromorphic family (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)
- pseudodifferential subspaces (1)
- relative cohomology (1)
- relative index formulas (1)
- residue (1)
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- spectral boundary value problems (1)
- spectral resolution (1)
- star product (1)
- 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)
We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.
We give a survey on the calculus of (pseudo-differential) boundary value problems with the transmision property at the boundary, and ellipticity in the Shapiro-Lopatinskij sense. Apart from the original results of the work of Boutet de Monvel we present an approach based on the ideas of the edge calculus. In a final section we introduce symbols with the anti-transmission property.
On a method for solution of the ordinary differential equations connected with Huygens' equations
(2010)
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.