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Elmar Schrohe

Publications and source records attributed to Elmar Schrohe.

47 records · Page 3Linked to original sources
arXiv2008

Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid $\cT^-X$ generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C*_r(\cT^-X) of C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.

math.FA↗
arXiv2007

Uniqueness of the Kontsevich-Vishik Trace

Let M be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on M, whose (complex) order is not an integer greater than or equal to -dim M, is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the L^2-operator trace on trace class operators. Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique.

math.FA↗
arXiv2006

Boutet de Monvel's Calculus and Groupoids I

Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra $Ψ^0(G)$ of pseudodifferential operators on some Lie groupoid $G$? If it could, the kernel ${\mathcal G}$ of the principal symbol homomorphism would be isomorphic to the groupoid {$C^*$-algebra} $C^*(G)$. While the answer to the above question remains open, we exhibit in this paper a groupoid $G$ such that $C^*(G)$ possesses an ideal ${\mathcal I}$ isomorphic to ${\mathcal G}$. %ES, the kernel of the principal symbol homomorphism on Boutet de Monvel's algebra. In fact, we prove first that ${\mathcal G}\simeqΨ\otimes{\mathcal K}$ with the $C^*$-algebra $Ψ$ generated by the zero order pseudodifferential operators on the boundary and the algebra $\mathcal K$ of compact operators. As both $Ψ\otimes \mathcal K$ and $\mathcal I$ are extensions of $C(S^*Y)\otimes {\mathcal{K}}$ by ${\mathcal{K}}$ ($S^*Y$ is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

math.KT↗
arXiv2005

A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

math.KT↗
arXiv2004

Traces and Quasi-traces on the Boutet de Monvel Algebra

We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class zero), and an auxiliary operator B, formed of the Dirichlet realization of a strongly elliptic second-order differential operator and an elliptic operator on the boundary, we consider the coefficient C_0(A,B) of (-λ)^{-N} in the asymptotic expansion of the resolvent trace Tr(A(B-λ)^{-N}) (with N large) in powers and log-powers of λas λtends to infinity in a suitable sector of the complex plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series for the meromorphic extension of the generalized zeta function ζ(A,B,s)= Tr(AB^{-s}) at s=0, when B is invertible. We show that C_0(A,B) is in general a quasi-trace, in the sense that it vanishes on commutators [A,A'] modulo local terms, and has a specific value independent of B modulo local terms; and we single out particular cases where the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A. Our main tool is a precise analysis of the asymptotic expansion of the resolvent trace, based on pseudodifferential calculations involving rational functions (in particular Laguerre functions) of the normal variable.

math.AP↗
arXiv2003

Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties

Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on 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 spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.

math-ph↗
arXiv2002

The Resolvent of Closed Extensions of Cone Differential Operators

We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-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| tends to infinity. 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 straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem \dot{u}-Δu=f, u(0)=0.

math.AP↗
arXiv2001

Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

For a pseudodifferential boundary operator A of integer order νand class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(AB^{-s}) where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+ν-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero equals the noncommutative residue of A, as defined by Fedosov, Golse, Leichtnam, and Schrohe by a different method. To achieve this, we establish a full asymptotic expansion of Trace(A(B-λ)^{-k}) in powers of λ^{-j/2} and log-powers λ^{-j/2} log λ, where the noncommutative residue equals the coefficient of the highest log-power. There is a related expansion for Trace(A exp(-tB)).

math.AP↗