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Matthias Lesch

Publications and source records attributed to Matthias Lesch.

46 records · Page 3Linked to original sources
arXiv1999

On boundary value problems for Dirac type operators. I. Regularity and self-adjointness

In a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index theorems in subsequent publications. Along with a number of new results, we extend and simplify the proofs of many known theorems. Our point of departure is the simple structure which is displayed by Dirac type operators near the boundary. Thus our proofs are given in an abstract functional analytic setting, generalizing considerably the framework of compact manifolds with boundary. The results of this paper have been announced in math.DG/9902100

math.FA↗
arXiv1999

The inverse spectral problem for first order systems on the half line

On the half line $[0,\infty)$ we study first order differential operators of the form $B 1/i d/(dx) + Q(x)$, where $B:=\mat{B_1}{0}{0}{-B_2}$, $B_1,B_2\in M(n,\C)$ are self--adjoint positive definite matrices and $Q:\R_+\to M(2n,\C)$, $\R_+:=[0,\infty)$, is a continuous self-adjoint off-diagonal matrix function. We determine the self-adjoint boundary conditions for these operators. We prove that for each such boundary value problem there exists a unique matrix spectral function $σ$ and a generalized Fourier transform which diagonalizes the corresponding operator in $L^2_σ(R, C)$. We give necessary and sufficient conditions for a matrix function $σ$ to be the spectral measure of a matrix potential $Q$. Moreover we present a procedure based on a Gelfand-Levitan type equation for the determination of $Q$ from $σ$. Our results generalize earlier results of M. Gasymov and B. Levitan. We apply our results to show the existence of $2n\times 2n$ Dirac systems with purely absolute continuous, purely singular continuous and purely discrete spectrum of multiplicity $p$, where $1\le p \le n$ is arbitrary.

math.SP↗
arXiv1999

Spectral theory of boundary value problems for Dirac type operators

The purpose of this note is to describe a unified approach to the fundamental results in the spectral theory of boundary value problems, restricted to the case of Dirac type operators. Even though many facts are known and well presented in the literature (cf. the monograph of Booss-Wojciechowski), we simplify and extend or sharpen most results by using systematically the simple structure which Dirac type operators display near the boundary. Thus our approach is basically functional analytic, and consequently we achieve results which apply to more general situations than compact manifolds with boundary. The details of the material presented here will be published elsewhere.

math.DG↗
arXiv1999

On index formulas for manifolds with metric horns

In this paper we discuss the index problem for geometric differential operators (Spin-Dirac operator, Gauß-Bonnet operator, Signature operator) on manifolds with metric horns. On singular manifolds these operators in general do not have unique closed extensions. But there always exist two extremal extensions $D_{min}$ and $D_{max}$. We describe the quotient ${\cal D}(D_{max}) / {\cal D}(D_{min})$ explicitely in geometric resp. topologic terms of the base manifolds of the metric horns. We derive index formulas for the Spin-Dirac and Gauß-Bonnet operator. For the Signature operator we present a partial result. The first version of this paper was completed August 1995 at the University of Augsburg.

dg-ga↗
arXiv1999

Determinants of regular singular Sturm-Liouville operators

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known that the $ζ$-function of this operator $ζ_L(s)=\sum_{λ\in\spec(L)\setminus\{0\}} λ^{-s}$ has a meromorphic continuation to the whole complex plane with 0 being a regular point. Then, according to Ray and Singer the $ζ$-regularized determinant of $L$ is defined by $\detz(L):=\exp(-ζ_L'(0)).$ In this paper we are going to express this determinant in terms of the solutions of the homogeneous differential equation $Ly=0$ generalizing earlier work of S. Levit and U. Smilansky, T. Dreyfus and H. Dym, and D. Burghelea, L. Friedlander and T. Kappeler. More precisely we prove the formula $\detz(L)=\frac{πW(ψ,ϕ)} {2^{ν_0+ν_1} Γ(ν_0+1)Γ(ν_1+1)}.$ Here $ϕ, ψ$ is a certain fundamental system of solutions for the homogeneous equation $Ly=0$, $W(ϕ, ψ)$ denotes their Wronski determinant, and $ν_0, ν_1$ are numbers related to the characteristic roots of the regular singular points $0, 1$.

math.DG↗
arXiv1999

On the eta-invariant of certain nonlocal boundary value problems

Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.

dg-ga↗
arXiv1999

On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibly with higher order poles. Our algebra of operators has a bigrading given by the order and the highest log-power occuring in the symbol expansion. We construct "higher" noncommutative residue functionals on the subspaces given by the log-grading. However, in contrast to the classical case we prove that the whole algebra does not admit any nontrivial traces. Finally we show that the analogue of the Kontsevich-Vishik trace also exists on our algebra. Our method also provides an alternative approach to the Kontsevich-Vishik trace.

dg-ga↗
arXiv1999

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\R^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

math.FA↗
arXiv1996

Differential operators of Fuchs type, conical singularities, and asymptotic methods

This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more general, stratified spaces. This book provides a self--contained treatment of the analysis of the heat equation and index theory for these operators. Major topics are short--time asymptotics, $η$-- and $ζ$--functions, (relative) index theorems. Another chapter is devoted to the discussion of deficiency indices and Dirac Schrödinger operators.

dg-ga↗