Searcharxiv⌕ Search

arXiv subjects

Matthias Lesch

Publications and source records attributed to Matthias Lesch.

At least 37 records · Page 2Linked to original sources
arXiv2011

Spectral flow and the unbounded Kasparov product

We present a fairly general construction of unbounded representatives for the interior Kasparov product. As a main tool we develop a theory of C^1-connections on operator * modules; we do not require any smoothness assumptions; our sigma-unitality assumptions are minimal. Furthermore, we use work of Kucerovsky and our recent Local Global Principle for regular operators in Hilbert C*-modules. As an application we show that the Spectral Flow Theorem and more generally the index theory of Dirac-Schrödinger operators can be nicely explained in terms of the interior Kasparov product.

math.OA↗
arXiv2011

Perturbation of Sectorial Projections of Elliptic Pseudo-differential Operators

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderon projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley's original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.

math.SP↗
arXiv2011

Connes-Chern character for manifolds with boundary and eta cochains

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.

math.OA↗
arXiv2011

Regular singular Sturm-Liouville operators and their zeta-determinants

We consider Sturm-Liouville operators on the line segment [0, 1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski- determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.

math.SP↗
arXiv2010

Pseudodifferential Operators and Regularized Traces

This paper is based on talks delivered in summer 2008 at the Conference on Motives, QFT and Pseudodifferential Operators in Boston, and at the Trimester programme Geometry and Physics, Hausdorff Institute for Mathematics in Bonn The paper surveys the pseudodifferential calculus with parameter, resolvent expansions, Dixmier trace, noncommutative residue, Kontsevich-Vishik canonical trace as well some aspects of regularized integrals on spaces of symbols.

math.OA↗
arXiv2009

The Calderon Projection: New Definition and Applications

We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze various ways of writing A|U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data.

math.DG↗
arXiv2009

The invertible double of elliptic operators

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderon projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case.

math.DG↗
arXiv2007

Quantum gravity: unification of principles and interactions, and promises of spectral geometry

Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme.

hep-th↗
arXiv2006

Relative pairing in cyclic cohomology and divisor flows

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to all the essential features of the divisor flows, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

math.KT↗
arXiv2006

Selected Aspects of the Mathematical Work of Krzysztof P. Wojciechowski

I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of elliptic boundary problems. The theme of the 90s is the eta invariant. The paper with Douglas was fundamental for establishing spectral invariants for manifolds with boundary and for the investigation of the behavior of spectral invariants under analytic surgery. This was so influential that many different proofs of the gluing formula for the eta-invariant were published. Finally turning to the new millennium we will look at the zeta--determinant. Compared to eta this is a much more rigid spectral invariant which is technically challenging. This paper was presented at the workshop "Krzysztof Wojciechowski - 50 Years" held in May 2005 near Roskilde/Denmark.

math.SP↗
arXiv2004

The uniqueness of the spectral flow on spaces of unbounded self--adjoint Fredholm operators

We discuss several natural metrics on spaces of unbounded self--adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of non-closed paths which is path additive and stable under homotopies with endpoints varying in the space of invertible self-adjoint operators. Furthermore we show that for certain Riesz continuous paths of self--adjoint Fredholm operators the spectral flow can be expressed in terms of the index of the pair of positive spectral projections at the endpoints. Finally we review the Cordes--Labrousse theorem on the stability of the Fredholm index with respect to the graph metric in a modern language and we generalize it to the Clifford index and to the equivariant index.

math.FA↗
arXiv2004

Unbounded Fredholm Operators and Spectral Flow

We study the gap (= "projection norm" = "graph distance") topology of the space of (not necessarily bounded) self--adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

math.FA↗
arXiv2003

On the rho invariant for manifolds with boundary

This article is a follow up of the previous article of the authors on the analytic surgery of eta- and rho-invariants. We investigate in detail the (Atiyah-Patodi-Singer)-rho-invariant for manifolds with boundary. First we generalize the cut-and-paste formula to arbitrary boundary conditions. A priori the rho-invariant is an invariant of the Riemannian structure and a representation of the fundamental group. We show, however, that the dependence on the metric is only very mild: it is independent of the metric in the interior and the dependence on the metric on the boundary is only up to its pseudo--isotopy class. Furthermore, we show that this cannot be improved: we give explicit examples and a theoretical argument that different metrics on the boundary in general give rise to different rho-invariants. Theoretically, this follows from an interpretation of the exponentiated rho-invariant as a covariantly constant section of a determinant bundle over a certain moduli space of flat connections and Riemannian metrics on the boundary. Finally we extend to manifolds with boundary the results of Farber-Levine-Weinberger concerning the homotopy invariance of the rho-invariant and spectral flow of the odd signature operator.

math.DG↗
arXiv2003

Calderon projector for the Hessian of the perturbed Chern-Simons function on a 3-manifold with boundary

The existence and continuity for the Calderon projector of the perturbed odd signature operator on a 3-manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3-sphere with the difference in local intersection numbers of the character varieties coming from a Heegard decomposition.

math.GT↗
arXiv2002

The eta-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderon projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott-Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Bruning-Lesch to the mod Z gluing formula. Our presentation includes careful constructions of the Maslov index and triple index in a symplectic Hilbert space. As a byproduct we give intuitively appealing proofs of two theorems of Nicolaescu on the spectral flow of Dirac operators. As an application of our methods, we carry out a detailed analysis of the eta-invariant of the odd signature operator coupled to a flat connection using adiabatic methods. This is used to extend the definition of the Atiyah-Patodi-Singer rho-invariant to manifolds with boundary. We derive a ``non-additivity'' formula for the Atiyah-Patodi-Singer rho-invariant and relate it to Wall's non-additivity formula for the signature of even-dimensional manifolds.

math.DG↗
arXiv2000

Essential self-adjointness of symmetric linear relations associated to first order systems

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. This part is based on joint work with Mark Malamud. Details will be published elsewhere. In the second part we consider a complete Riemannian manifold, $M$, and a first order differential operator, $D:\cinfz{E}\to \cinfz{F}$, acting between sections of the hermitian vector bundles $E,F$. Moreover, let $V:\cinf{E}\to L^{\infty}_{\loc}(E)$ be a self-adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator $H=D^tD+V$ to be essentially self-adjoint. This generalizes recent work of I. Oleinik \cite{Ole:ESA,Ole:CCQ,Ole:ESAG}, M. Shubin \cite{Shu:CQC,Shu:ESA}, and M. Braverman \cite{Bra:SAS}.

math.SP↗
arXiv2000

On the number of square integrable solutions and self-adjointness of symmetric first order systems of differential equations

The main purpose of this paper is to investigate the formal deficiency indices ${\cal N}_{\pm}(I)$ of a symmetric first order system $$ Jf'+Bf=λ{\cal H} f $$ on an interval $I$, where $I=\mathbb{R}$ or $I=\mathbb{R}_\pm.$ Here $J,B,{cal H}$ are $n\times n$ matrix valued functions and the Hamiltonian ${\cal H}\ge 0$ may be singular even everywhere. We obtain two results for such a system to have minimal numbers ${\cal N}_\pm(\mathbb{R})=0$ (resp. ${\cal N}_\pm(\mathbb{R}_\pm)=n$) and a criterion for their maximality ${\cal N}_{\pm}(\mathbb{R}_+)=2n.$ Some conditions for a canonical system to have intermediate numbers ${\cal N}_\pm(\mathbb{R}_+)$ are presented, too. We also obtain a generalization of the well-known Titchmarsh-Sears theorem for second order Sturm-Liouville type equations. This contains results due to Lidskii and Krein as special cases. It is important to note that in general the above system does not give rise to an operator but rather to a symmetric linear relation in a Hilbert space. These relations are investigated in detail. As a byproduct we obtain very short proofs of (generalizations of) the main results of a paper by Kogan and Rofe-Beketov (Proc. Roy. Soc. Edinb. 74 (1974/75))as well as a criterion for the quasi-regularity of canonical systems. This covers the Kac-Krein theorem and some results from the quoted paper of Kogan and Rofe-Beketov.

math.FA↗