Search

In this chapter we study what are without doubt the most important functions in all of analysis and topology, the continuous functions.

This chapter is devoted entirely to the theory and application of infinite products, and as a consolation prize, we also talk about partial fractions.

In this chapter we present a small sample of some of the most beautiful formulas in the world.

One of the goals of this text is to get you proving mathematical statements in real analysis. Set theory provides a safe environment in which to learn about math statements, “if ... then,” “if and only if,” etc., and to learn the logic behind proofs. Since this is an introductory book on analysis, our treatment of sets is “very naive,” in the sense that we actually don’t define sets rigorously, only informally; we are mostly interested in how “they work,” not really what they are.

This chapter is on the study of numbers. Of course, we all have a working understanding of the real numbers, and we use many aspects of these numbers in everyday life: tallying up tuition and fees, figuring out how much we have left on our food cards, etc. We have accepted from our childhood all the properties of numbers that we use every day.

We dabbled a little into the theory of continued fractions (that is, fractions that continue on and on and on ...) way back on the exercises of Section 3.4. In this chapter we concentrate on this fascinating subject.

We report on a particular case of the paper [7], joint with Raphael Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n− 1, n− 3, n− 5, . . .. Version francaise abregee Soit D la classe des operateurs differentiels elliptiques symmetriques d’ordre 1 sur une variete Riemannienne fermee M , agissant sur les sections d’un fibre vectoriel Hermitien E. Il est connu que pour un operateur D ∈ D , le spectre de D en tant qu’operateur non-borne dans L(M,E) est discret. Les fonctions eta et zeta associees a D sont defines par les series

In this paper we show how semiclassical pseudodifferential operators together with Getzler’s rescaling can be used to give a systematic proof of the Atiyah-Singer index theorem using the resolvent or the heat kernel.

We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends., 25 pages

We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace–Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.

In this paper, a complete description of the zeta functions and corresponding zeta determinants for Dirac and Laplace-type operators over finite cylinders using the contour integration method, for example described in [K. Kirsten, Spectral Functions in Mathematics and Physics, Chapman & Hall/CRC Press, Boca Raton, 2001] is given. Different boundary conditions, local and non-local ones, are considered. The method is shown to be very powerful in that it is easily adapted to each situation and in that answers are very elegantly obtained.

In this paper, we solve the gluing problem for the ζ -determinant of a Dirac Laplacian. To do so, we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems, the analysis of the resolvent of the Dirac operator, and the introduction of an auxiliary model problem. Moreover, as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis.

The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac-type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and ‘b-admissible’ partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderon projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulae, and solve the Bojarski conjecture.

We relate zeta determinants of Dirac operators with generalized APS boundary conditions for compact manifolds with boundary and parallel b-zeta determinants of perturbed Dirac operators on the corresponding complete manifolds with cylindrical end. We also derive, without invertibility conditions, corresponding relative formulae for the (b-) zeta determinants.

In this paper, we derive a formula for the ratio of the ζ-determinants of the Laplacian with Neumann and Dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the ζ-determinant of the Dirichlet to Neumann map.

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.

On compact manifolds with corners of arbitrary codimension, we characterize the ‘multi-cylindrical end’ (or b-type) pseudodifferential operators that are Fredholm on weighted Sobolev spaces and we compute their indices. The index formula contains the usual interior term manufactured from the local symbols of the operator and also contains boundary correction terms corresponding to eta-type invariants of the induced operators on the boundary faces.

(2005). Dirichlet and Fresnel Integrals via Iterated Integration. Mathematics Magazine: Vol. 78, No. 1, pp. 63-67.

In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. 1. The gluing problem for the spectral invariants Since their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invariants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that [2, p. 89] “no precise pasting formulas are obtained but only adiabatic ones.” In [23, 24], we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in [25, 24]. We begin with describing the gluing problem for compact manifolds. Let D be a Dirac type operator acting on C∞(M,S) where M is a closed compact Riemannian manifold of arbitrary dimension and S is a Clifford bundle over M . Let Y be an embedded hypersurface in M and let M = M− ∪ M+ be decomposition of M into manifolds with boundary such that ∂M− = ∂M+ = Y . We assume that all geometric structures are of product type over a tubular neighborhood N = [−1, 1]×Y of Y where the Dirac operator takes the product form D = G(∂u+DY ), where G is a unitary operator on S0 := S|Y and DY is a Dirac type operator over Y satisfying G = −Id and DY G = −GDY . Recall that the eta function of D and the zeta function of D2 are defined through the heat operator e−tD2 via (1.1) ηD(s) = 1 Γ( s+1 2 ) (∫ 1

In this paper, we explicitly compute the ζ-determinant of a Dirac Laplacian with Atiyah–Patodi–Singer (APS) boundary conditions over a finite cylinder. Using this exact result, we illustrate the gluing and comparison formulae for the ζ-determinants of Dirac Laplacians proved by Loya and Park.

We study mapping properties of the heat operator e tA of an m-th order elliptic b-differential operator in appropriately defined spaces of whole and fractional (Holder) derivatives. An application is made to short time existence of solutions to certain semilinear parabolic equations.

We derive asymptotic properties for the heat kernel of elliptic cone (or Fuchs type) differential operators on compact manifolds with boundary. Applications include asymptotic formulas for the heat trace, counting function, spectral function, and zeta function of cone operators.

We construct the complex powersAz for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatAz exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofAz, both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(Az).

Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 + of the trace Tr Pe -tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)2 terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals.

We show that the resolvent kernel of an elliptic b -pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is a polyhomogeneous, or classical, function on a certain manifold with corners. The singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity. Explicit descriptions of the expansions, including logarithmic terms, are given. In particular, the asymptotics of the resolvent restricted to the diagonal follows as a corollary. Applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given.

The resolvent (A – λ)−1 of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is shown to lie in a calculus of operators, tempered in the parameter λ in a special way. We show that the Laplace and Mellin transforms, with respect to λ, of these tempered operators can be defined and that they have Schwartz kernels which can be described geometrically. As a corollary, we obtain the structures of the kernels of the heat operator and complex powers of b-pseudodifferential operators, as the heat operator and complex powers are the Laplace and Mellin transforms, respectively, of the resolvent. The heat operator and complex powers are then used to generalize the index formula of Atiyah, Patodi, and Singer for Dirac operators on manifolds with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners.

In this paper we study the asymptotic behavior (in the sense of meromorphic functions) of the zeta function of a Laplace-type operator on a closed manifold when the underlying manifold is stretched in the direction normal to a dividing hypersurface, separating the manifold into two manifolds with infinite cylindrical ends. We also study the related problem on a manifold with boundary as the manifold is stretched in the direction normal to its boundary, forming a manifold with an infinite cylindrical end. Such singular deformations fall under the category of "analytic surgery", developed originally by Hassell, Mazzeo and Melrose \cite{mazz95-5-14,hass95-3-115,hass98-6-255} in the context of eta invariants and determinants., Comment: 33 pages, 12 figures

Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, selfadjoint first-order differential operators, and selfadjoint elliptic pseudodifferential operators. As a result, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a new proof of a well known result of Branson-Gilkey, which was obtained by means of Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of the third author's paper [Po1]. Corrections to that statement are given in Appendix B., Final version. To appear in International Journal of Mathematics. 20 pages

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire., Comment: 22 pages, 2 figures

It is shown how the pre-exponential factor of the Feynman propagator for a large class of potentials can be computed using contour integrals. This is of direct relevance in the context of tunnelling processes in quantum theories. The prerequisites for this analysis are accessible to advanced undergraduate students and involve only introductory courses in ordinary differential equations and complex variables., 15 pages, to appear in American Journal of Physics

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For {\it arbitrary} self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with \cite{BKD}, a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained., Comment: 27 pages, 2 figures; to appear in Manuscripta Mathematica

In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator $-d^2/dr^2 - 1/(4r^2)$ over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for $k\in \N$ we find terms like $(\log t)^{-k}$ in the small-$t$ asymptotic expansion of the heat kernel. Furthermore, the zeta function has $s=0$ as a logarithmic branch point., 28 pages, 4 figures, LaTeX

In this paper we combine elements of the b-calculus and elliptic boundary problems to solve the decomposition problem for the (regularized) �-determinant of the Laplacian on a manifold with cylin- drical end into the �-determinants of the Laplacians with Dirichlet con- ditions on the manifold with boundary and on the half infinite cylinder. We also compute all the contributions to this formula explicitly. We investigate the 'Mayer-Vietoris' or 'cut and paste' decomposition for- mula of the �-determinant for a Laplacian on a manifold with cylindrical end into the �-determinants of the Laplacians with Dirichlet conditions on the manifold with boundary and on the half infinite cylinder. We also introduce a new method to attack such surgery problems by comparing the problem to a corresponding model problem. This approach works for compact manifolds as well as manifolds with cylindrical ends and will be used to solve related decomposition problems for the spectral invariants of Dirac type operators in (12), (13). We remark that the noncompactness of the underlying manifold introduces many new facets and obstacles not found in the compact case, as we will explain later. We begin with a brief account of zeta determinants. The �-determinant of a Laplace type operator was pioneered in the seminal paper (21) by Ray and Singer. They were seeking an analytic version of the so- called Reidemeister torsion, a combinatorial-topological invariant introduced by Reidemeister (22) and Franz (6). They conjectured that their analytic in- variant was the same as the Reidemeister torsion. Later, this conjecture was proved independently by Cheeger (4) and Muller (17). The �-determinants have also been of great use in quantum field theory where they are being used to develop rigorous models for Feynman path integral techniques (10). Because of their use in differential topology and quantum field theory, much work has been done on understanding the nature of�-determinants, especially their behavior under 'cutting and pasting' of manifolds. This was initiated

In this expository article, we survey index theory of Dirac operators using the Gauss-Bonnet formula as the catalyst to discuss index formulas on manifolds with and without boundary. Considered in detail are the Atiyah-Singer and Atiyah-Patodi-Singer index theorems, their heat kernel proofs, and their generalizations to manifolds with corners of codimension two via the method of `attaching cylindrical ends’.

The goal of this paper is to present, from the $b$-calculus perspective, a program for analyzing generalized resolvent families and heat kernels of pseudodifferential operators on conic manifolds. To carry this out, a new class of parameter-dependent cone pseudodifferential operators is developed and studied.

In this article we analyze the spectral zeta function, the heat kernel, and the resolvent of the operator −d2/dr2+κ/r2+r2 over the interval (0,∞) for κ≥−1/4. Depending on the self-adjoint extension chosen, nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed. In particular, for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s=−k, k∊N0, do occur. This implies that the small-t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk/(ln t)l+1 for k,l∊N0. The corresponding statements for the resolvent are also shown. Furthermore, we evaluate the zeta determinant of the operator for all values of κ and any self-adjoint extension.

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions of these $\z$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities., Comment: Paper presented at the 8th Workshop on Quantum Field Theory under the Influence of External Conditions (Leipzig, Germany, 16-21 September, 2007)

In this note, we explicitly compute the functional determinant of a Dirac Laplacian with nonlocal pseudodifferential boundary conditions over a finite cylinder in terms of the ζ-function of the Dirac operator on the cross section and the pseudodifferential operators defining the boundary conditions. In particular, this result reduces to our previous formula [J. Phys. AJPHAC5 37, 7381 (2004)] for the special case of generalized Atiyah–Patodi–Singer conditions. To prove our main result, we use the gluing and comparison formulas established by the present authors in Refs 14 and 15.