TY  - INPR
A1  - Schrohe, Elmar
A1  - Seiler, Jörg
T1  - The resolvent of closed extensions of cone differential operators
N2  - We study an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent (λ - A )-¹ exists in a sector of the complex plane and decays like 1/|λ| as |λ| -> ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with striaght conical degeneracy and establish maximal regularity for the Cauchy problem u - Δu = f, u(0) = 0.
T3  - Preprint - (2002) 19 
Y1  - 2002
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26378
ER  - 
TY  - INPR
A1  - Melo, S. T.
A1  - Nest, R.
A1  - Schrohe, Elmar
T1  - C*-structure and K-theory of Boutet de Monvel's algebra
N2  - We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get Ki(A,k) congruent Ki(C(X))⊕Ksub(1-i)(Csub(0)(T*X)),i = 0,1, with k denoting the compact ideal, and T*X denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, Ksub(0)(A) congruent Z up(2g+m) and Ksub(1)(A) congruent Z up(2g+m-1), where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ k ⊂ G ⊂ A, with A/G commutative and G/k isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values in compact operators on L²(R+).
T3  - Preprint - (2001) 33 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26166
ER  - 
TY  - INPR
A1  - Coriasco, Sandro
A1  - Schrohe, Elmar
A1  - Seiler, Jörg
T1  - Bounded imaginary powers of differential operators on manifolds with conical singularities
N2  - We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds.
T3  - Preprint - (2001) 12 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25962
ER  - 
TY  - INPR
A1  - Junker, Wolfgang
A1  - Schrohe, Elmar
T1  - Adiabatic vacuum states on general spacetime manifolds : definition, construction, and physical properties
N2  - Adiabatic vacuum states are a well-known class of physical states for linear quantum fields n Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetme manifolds (factoriality, quasiequivalence, local definteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
T3  - Preprint - (2001) 27 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26100
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
T1  - A short introduction to Boutet de Monvel's calculus
N2  - This paper provides an introduction to Boutet de Monvel's calculus on the half-space IRn (positiv) in the framework of a pseudodifferential calculus with operator-valued symbols.
T3  - Preprint - (2000) 03 
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25696
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
A1  - Schulze, Bert-Wolfgang
T1  - Edge-degenerate boundary value problems on cones
N2  - We consider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their push-forwards as the cylinder is blown up to a cone near infinity. We show that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredholm theory of boundary value problems on manifolds with edges.
T3  - Preprint - (1999) 06 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25436
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
A1  - Seiler, Jörg
T1  - Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces
N2  - Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces, it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B).
T3  - Preprint - (1999) 28 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25621
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
T1  - Noncommutative residues, Dixmier's Trace, and heat trace expansions on manifolds with boundary
N2  - For manifolds with boundary, we define an extension of Wodzicki's noncommutative residue to boundary value problems in Boutet de Monvel's calculus. We show that this residue can be recovered with the help of heat kernel expansions and explore its relation to Dixmier's trace.
T3  - Preprint - (1999) 13 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25486
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
A1  - Walze, Markus
A1  - Warzecha, Jan-Martin
T1  - Construction de Triplets Spectraux à Partir de Modules de Fredholm
N2  - Soit (A, H, F) un module de Fredholm p-sommable, où l'algèbre A = CT est engendrée par un groupe discret Gamma d'éléments unitaires de L(H) qui est de croissance polynomiale r. On construit alors un triplet spectral (A, H, D) sommabilité q pour tout q > p + r + 1 avec F = signD. Dans le cas où (A, H, F) est (p, infini)-sommable on obtient la (q, infini)-sommabilité de (A, H, D)pour tout q > p + r + 1.
N2  - Let (A, H, F) be a p-summable Fredholm module where the algebra A = CT is generated by a discrete group of unitaries in L(H) which is of polynomial growth r. Then we construct a spectral triple (A, H, D) with F = signD which is q-summable for each q > p + r + 1. In case (A, H, F) is (p, infinite)-summable we obtain (q, infinite)-summability of (A, H, D) for each q > p + r + 1.
T3  - Preprint - (1998) 12 
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25247
ER  - 
TY  - INPR
A1  - Hieber, Matthias
A1  - Schrohe, Elmar
T1  - Lρ spectral independence of elliptic operators via commutator estimates
N2  - Let {Tsub(p) : q1 ≤ p ≤ q2} be a family of consistent Csub(0) semigroups on Lφ(Ω) with q1, q2 ∈ [1, ∞)and Ω ⊆ IRn open. We show that certain commutator conditions on Tφ and on the resolvent of its generator Aφ ensure the φ independence of the spectrum of Aφ for φ ∈ [q1, q2]. Applications include the case of Petrovskij correct systems with Hölder continuous coeffcients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coeffcients.
T3  - Preprint - (1997) 17 
KW  - Lφ spectrum
KW  - spectral independence
KW  - elliptic systems
Y1  - 1997
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25047
ER  -