DOCSLIB.ORG

Spectral Bounds for Dirac Operators on Open Manifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Bounds for Dirac Operators on Open Manifolds SPECTRAL BOUNDS FOR DIRAC OPERATORS ON OPEN MANIFOLDS CHRISTIAN BAR¨ ABSTRACT. We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the au- thor’s estimate on surfaces. 1. INTRODUCTION This paper is concerned with spectral bounds for Dirac operators on noncompact Rie- mannian manifolds. We work with generalized Dirac operators in the sense of Gromov and Lawson as explained in Section 2. Examples are the classical Dirac operatorp acting on spinors, the operator D = d + d acting on forms, and the operator D = 2(¶¯ + ¶¯ ∗) on a Kahler¨ manifold. In Section 3 we show that Friedrich’s lower bound for the eigenvalues of the classical Dirac operator on a compact spin manifold of positive scalar curvature extends to a lower bound for the fundamental tone of the square of any Dirac operator even if the manifold M is incomplete. If M is complete this implies a spectral gap for the Dirac operator itself. We also discuss the equality case. If the spectral bound is an eigen- value of the Dirac operator, then the corresponding eigensection satisfies the Killing spinor equation, an overdetermined elliptic equation of first order. In the case of the classical Dirac operator this implies that M is compact and Einstein. As another spe- cial case we give a simple proof of Lichnerowicz’ classical lower bound for the first positive eigenvalue of the Laplace operator on a compact manifold with positive Ricci curvature. In Section 4 we study the essential spectrum. We show that if the curvature endo- morphism of the Dirac operator is bounded from below at infinity, then a Friedrich- arXiv:0810.5598v1 [math.DG] 31 Oct 2008 Lichnerowicz type estimate holds for the essential spectrum. In particular, this yields a sufficient criterion for a Dirac operator on a complete manifold to be a Fredholm operator. If the curvature endomorphism tends to infinity at infinity, then the spectrum of the square of the Dirac operator is discrete just as in the case of a compact manifold. In the last section we study the classical Dirac operator on surfaces. We show that if M is a connected surface of genus 0 with finite area and if the spin structure is bounding Date: October 29, 2018. 2000 Mathematics Subject Classification. 53C27. Key words and phrases. Dirac operators, point spectrum, continuous spectrum, discrete spectrum, essential spectrum, Killing spinor, Friedrich inequality, Lichnerowicz inequality. 1 2 CHRISTIAN BAR¨ at infinity, then 4p l (D2) ≥ : ∗ area(M) Here M need not be complete and may have infinitely many ends. If one drops the assumption on the spin structure, then the estimate fails. The estimate is proved by reducing it to the case M = S2 which was established by the author almost two decades ago. 2. GENERALIZED DIRAC OPERATORS Let M denote a connected Riemannian manifold. There will be no completeness or compactness assumption on M unless we explicitly say so. We will work with gen- eralized Dirac operators in the sense of Gromov and Lawson [6]. Let E ! M be a Riemannian or Hermitian vector bundle over M equipped with a metric connection ∇E . ¥ For j;y 2 Cc (M;E), the space of smooth, compactly supported sections of E, the L2-product and norms are defined, Z (j;y) = hj;yidV; kjk2 = (j;j): M Here dV denotes the volume element induced by the Riemannian metric of M. The ¥ Hilbert space obtained by completing Cc (M;E) with respect to this scalar product is denoted by L2(M;E). Pointwise scalar products will be denoted by h·;·i and pointwise norms by j · j. We assume that tangent vectors act by Clifford multiplication on E, i. e. there is a vector bundle homomorphism TM ⊗ E ! E; X ⊗ j 7! X · j; satisfying • the Clifford relations X ·Y · j +Y · X · j + 2hX;Yij = 0 for all X;Y 2 TpM, j 2 Ep, p 2 M. • skew symmetry hX · j;yi = −hj;X · yi for all X 2 TpM, j;y 2 Ep, p 2 M. • the product rule E E ∇X (Y · j) = (∇XY) · j +Y · ∇X j for all differentiable vector fields X and Y and for all differentiable sections j in E. Here ∇ is the Levi-Civita connection of M. The Dirac operator is now defined by n D := e · ∇E j ∑ i ei j i=1 SPECTRAL BOUNDS FOR DIRAC OPERATORS ON OPEN MANIFOLDS 3 where e1;:::;en denotes any orthonormal tangent basis, n = dim(M). This definition is independent of the choice of basis. Example 2.1. Let M carry a spin structure. Then one can define the spinor bundle E = SM which has all the required structure. The corresponding generalized Dirac operator is the classical Dirac operator, sometimes also called the Atiyah-Singer operator. See e. g. [11] for the definitions. Example 2.2. Let M be an n-dimensional Riemannian manifold and E = Ln p ∗ p=0 L T M. Using the wedge product and its adjoint one can define a Clifford multi- plication such that the corresponding generalized Dirac operator is given by D = d +d. Here d is the exterior differential and d is its adjoint. See e. g. [16, Ch. 3] for details. Example 2.3. Let M be a Kahler¨ manifold of complex dimension m. Then E = Lm L0;qT ∗M can be equipped with a Clifford multiplication such that the corre- q=0 p sponding generealized Dirac operator is given by D = 2(¶¯ + ¶¯ ∗), compare [16, Ch. 3]. Any generalized Dirac operator is a formally self-adjoint elliptic differential operator of first order. By [6, Thm. 1.17] the Dirac operator is essentially self-adjoint on the ¥ domain Cc (M;E), smooth sections of E with compact support, in the Hilbert space L2(M;E) provided that M is complete. In this case, when we speak about the spectrum of the Dirac operator we will always mean the spectrum of its self-adjoint closure. The spectrum s(D) decomposes into two disjoint parts, the point spectrum sp(D) consisting of all eigenvalues (with square-integrable eigensections), and the continu- ous spectrum sc(D). By elliptic regularity theory eigensections are smooth. If M is compact, then sc(D) is empty and sp(D) is discrete and all eigenvalues have finite multiplicity, compare Corollary 4.4. 2 ¥ Since the operator D is symmetric and nonnegative on the domain Cc (M;E) it has a canonical self-adjoint extension even if M is not complete. This is the Friedrichs 1 extension, the unique self-adjoint extension whose domain is contained in HD(M;E), ¥ the closure of Cc (M;E) with respect to the norm induced by the scalar product (j;y) 1 = (j;y) + (Dj;Dy): HD If M is complete, then this is the only self-adjoint extension of D2 and it coincides with the square of the self-adjoint extension of D. If M is the interior of a compact manifold M with boundary such that the metric extends, then taking the Friedrichs extension corresponds to imposing Dirichlet boundary conditions. When we speak about the spectrum of D2 we always mean the spectrum of the Friedrichs extension. Again, the spectrum decomposes disjointly into the point spectrum and the continuous spectrum, 2 2 2 s(D ) = sp(D ) t sc(D ): 2 2 Since D is nonnegative its spectrum is contained in [0;¥). We call l∗(D ) := mins(D2) the Dirac fundamental tone. One has (D2j;j) (Dj;Dj) (Dj;Dj) l (D2) = inf = inf = inf : ∗ ¥ ¥ j2Cc (M;E) (j;j) j2Cc (M;E) (j;j) j2H1 (M;E) (j;j) 6= 6= D j 0 j 0 j6=0 See [17, Sec. 4.2] for the functional analytic background. 4 CHRISTIAN BAR¨ For a differentiable section j and a differentiable function f : M ! C we have (1) D( f j) = grad f · j + f Dj by the very definition of D. The curvature endomorphism of D is defined by n 1 E K := ∑ ei · e j · R (ei;e j) 2 i; j=1 where RE is the curvature tensor of ∇E . The endomorphism field K is symmetric and we have the Bochner-Lichnerowicz formula [6, Prop. 2.5] (2) D2 = (∇E )∗∇E + K : 1 If D is the classical Dirac operator acting on spinors, then K = 4 scal. If D = d + d, then the restriction of K to 1-forms is Ricci curvature. 3. A FRIEDRICH INEQUALITY Now we show that D has a spectral gap provided K is uniformly positive, a result which is due to Friedrich in the compact case [5, Thm. A]. Since we cannot work with eigensections here, the proof needs modification. Theorem 3.1. Let M be a (possibly incomplete) n-dimensional Riemannian mani- fold. Let D be a generalized Dirac operator on M whose curvature endomorphism is bounded from below by a positive constant k 2 R, i. e. K ≥ k > 0 in the sense of symmetric endomorphisms. Then nk l (D2) ≥ : ∗ n − 1 ¥ Proof. Let j 2 Cc (M;E). By (2) we have (3) kDjk2 = (D2j;j) = (((∇E )∗∇E + K )j;j) ≥ k∇E jk2 + kkjk2: By the Cauchy-Schwarz inequality1 we have n n jD j = j e · ∇E j ≤ j∇E j j ∑ i ei j ∑ ei j i=1 i=1 s s n n p ≤ 12 · j∇E j2 = nj∇E j ∑ ∑ ei j j i=1 i=1 hence 1 j∇E jj2 ≥ jDjj2: n Plugging this into (3) we get 1 kDjk2 ≥ kDjk2 + kkjk2 n 1This trick was shown to me by O.
Load more

Recommended publications
  • Geometry of Noncommutative Three-Dimensional Flat Manifolds
    Philosophiae Doctor Dissertation GEOMETRY OF NONCOMMUTATIVE THREE-DIMENSIONAL FLAT MANIFOLDS Piotr Olczykowski Krakow´ 2012 Dla moich Rodzic´ow Contents Introduction ix 1 Preliminaries 1 1.1 C∗−algebras . .1 1.2 Gelfand-Naimark-Seagal Theorem . .3 1.2.1 Commutative Case . .4 1.2.2 Noncommutative Case . .4 1.3 C∗-dynamical systems . .7 1.3.1 Fixed Point Algebras of C(M)..............8 1.4 K−theory in a Nutshell . .8 1.5 Fredholm Modules in a Nutshell . 10 1.5.1 Pairing between K−theory and K−homology . 12 1.5.2 Unbounded Fredholm Modules . 12 2 Spectral Triples 15 2.1 Spin Structures . 15 2.1.1 Clifford Algebras . 15 2.1.2 SO(n) and Spin(n) Groups . 16 2.1.3 Representation of the Clifford Algebra . 17 2.1.4 Spin Structures and Bundles . 18 2.2 Classical Dirac Operator . 20 2.3 Real Spectral Triple { Definition . 23 2.3.1 Axioms . 24 2.3.2 Commutative Real Spectral Triples . 27 3 Noncommutative Spin Structures 29 3.1 Noncommutative Spin Structure . 29 3.2 Equivariant Spectral Triples - Definition . 31 3.3 Noncommutati Tori . 32 3.3.1 Algebra . 33 3.3.2 Representation . 34 v vi CONTENTS T3 3.3.3 Equivariant real spectral triples over A( Θ)...... 35 3.4 Quotient Spaces . 36 3.4.1 Reducible spectral triples . 36 Z 3.4.2 Spectral Triples over A(T1) N .............. 38 3.4.3 Summary . 40 4 Noncommutative Bieberbach Manifolds 41 4.1 Classical Bieberbach Manifolds . 42 4.2 Three-dimensional Bieberbach Manifolds . 44 4.2.1 Spin structures over Bieberbach manifolds .
    [Show full text]
  • Dirac Spectra, Summation Formulae, and the Spectral Action
    Dirac Spectra, Summation Formulae, and the Spectral Action Thesis by Kevin Teh In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Submitted May 2013) i Acknowledgements I wish to thank my parents, who have given me their unwavering support for longer than I can remember. I would also like to thank my advisor, Matilde Marcolli, for her encourage- ment and many helpful suggestions. ii Abstract Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A; H; D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation. iii Contents Acknowledgements i Abstract ii 1 Introduction 1 2 Quaternionic Space, Poincar´eHomology Sphere, and Flat Tori 5 2.1 Introduction . 5 2.2 The quaternionic cosmology and the spectral action . 6 2.2.1 The Dirac spectra for SU(2)=Q8.................... 6 2.2.2 Trivial spin structure: nonperturbative spectral action . 7 2.2.3 Nontrivial spin structures: nonperturbative spectral action . 9 2.3 Poincar´ehomology sphere . 10 2.3.1 Generating functions for spectral multiplicities . 10 2.3.2 The Dirac spectrum of the Poincar´esphere .
    [Show full text]
  • Spectra of Orbifolds with Cyclic Fundamental Groups
    SPECTRA OF ORBIFOLDS WITH CYCLIC FUNDAMENTAL GROUPS EMILIO A. LAURET Abstract. We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamen- tal group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra by using generating functions. We also include many isospectral examples. 1. Introduction Let M be a connected compact Riemannian manifold, and let Γ1 and Γ2 be cyclic subgroups of Iso(M). In this paper we will consider the question: Under what conditions the (good) orbifolds Γ1\M and Γ2\M are isospectral? Here, isospectral means that the Laplace-Beltrami operators on Γ1\M and on Γ2\M have the same spectrum. Such operators are given by the Laplace-Beltrami operator on M acting on Γj-invariant smooth functions on M. Clearly, the condition on Γ1 and Γ2 of being conjugate in Iso(M) is sufficient since in this case Γ1\M and Γ2\M are isometric. However, it is well known that this condition is not necessary due to examples of non-isometric isospectral lens spaces constructed by Ikeda [Ik80a]. Lens spaces are spherical space forms with cyclic fundamental groups, that is, they are the only compact manifolds with constant sectional curvature and cyclic fundamental group. In the recent paper [LMR15], R. Miatello, J.P. Rossetti and the author show an isospectral characterization among lens spaces in terms of isospectrality of their associated congruence lattices with respect to the one-norm k·k1.
    [Show full text]
  • Calculation of Supercritical Dirac Resonances in Heavy-Ion Collisions
    CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS EDWARD ACKAD A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY arXiv:0809.4256v2 [physics.atom-ph] 26 Sep 2008 GRADUATE PROGRAM IN DEPARTMENT OF PHYSICS AND ASTRONOMY YORK UNIVERSITY TORONTO, ONTARIO 2021 CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS by Edward Ackad a dissertation submitted to the Faculty of Graduate Stud- ies of York University in partial fulfilment of the require- ments for the degree of DOCTOR OF PHILOSOPHY c 2021 Permission has been granted to: a) YORK UNIVER- SITY LIBRARIES to lend or sell copies of this disserta- tion in paper, microform or electronic formats, and b) LI- BRARY AND ARCHIVES CANADA to reproduce, lend, distribute, or sell copies of this dissertation anywhere in the world in microform, paper or electronic formats and to authorise or procure the reproduction, loan, distribu- tion or sale of copies of this dissertation anywhere in the world in microform, paper or electronic formats. The author reserves other publication rights, and nei- ther the dissertation nor extensive extracts for it may be printed or otherwise reproduced without the author’s written permission. CALCULATION OF SUPERCRITICAL DIRAC RESONANCES IN HEAVY-ION COLLISIONS by Edward Ackad By virtue of submitting this document electronically, the author certifies that this is a true electronic equivalent of the copy of the dissertation approved by York University for the award of the degree. No alteration of the content has occurred and if there are any minor variations in formatting, they are as a result of the coversion to Adobe Acrobat format (or similar software application).
    [Show full text]
  • Horizontal Dirac Operators in CR Geometry
    Horizontal Dirac Operators in CR Geometry DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universitat¨ zu Berlin von Dipl.-Math. Christoph Martin Stadtm¨uller Pr¨asidentin der Humboldt-Universit¨atzu Berlin Prof. Dr.-Ing. Dr. Sabine Kunst Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke Gutachter 1. Prof. Dr. Helga Baum 2. Prof. Dr. Uwe Semmelmann 3. Prof. Dr. Bernd Ammann Tag der m¨undlichen Pr¨ufung:14. Juli 2017 Contents Introduction v 1 Contact and CR manifolds and their adapted connections 1 1.1 Contact manifolds . 1 1.2 CR manifolds . 13 1.3 Differential forms on metric contact manifolds . 19 1.3.1 The forms over the contact distribution . 21 1.3.2 The other parts . 23 1.3.3 Application: K¨ahlerform and Nijenhuis tensor . 24 1.4 Adapted connections . 27 1.4.1 Definition and basic properties . 28 1.4.2 The torsion tensor of an adapted connection . 29 1.4.3 The Tanaka-Webster connection and CR connections . 34 1.4.4 Adapted connections with skew-symmetric torsion . 36 1.5 The curvature tensors of an adapted connection . 37 2 Spin geometry on metric contact and CR manifolds 43 2.1 Crash course spin geometry . 43 2.1.1 Clifford algebras and the spin group . 43 2.1.2 Spin manifolds . 47 2.2 Connections and Dirac operators . 49 2.3 The spinor bundle over metric contact manifolds . 54 3 Horizontal Dirac operators 59 3.1 Definition and basic properties .
    [Show full text]
  • Finite Spectral Triples for the Fuzzy Torus Arxiv:1908.06796V1 [Math.QA
    Finite spectral triples for the fuzzy torus John W. Barrett, James Gaunt School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD, UK E-mail [email protected] [email protected] August 19th, 2019 Abstract Finite real spectral triples are defined to characterise the non- commutative geometry of a fuzzy torus. The geometries are the non- commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac opera- tors, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus. 1 Introduction arXiv:1908.06796v1 [math.QA] 19 Aug 2019 A Riemannian spin geometry can be expressed in algebraic terms using the algebra of smooth functions on the manifold and the Dirac operator on the spinor bundle. This point of view allows for a significant generalisation of geometry by removing the restriction that the algebra is commutative, ad- justing the axioms in as natural a way as possible. The mathematical struc- ture that results is called a real spectral triple [1] and encompasses both commutative and non-commutative geometries. 1 Among the non-commutative geometries, it is of significant interest to examine examples that are analogues of Riemannian spin manifolds, or, more specifically, approximations of Riemannian spin manifolds. This pa- per contributes to this by defining and studying the properties of some non- commutative analogues of the flat torus called fuzzy tori.
    [Show full text]
  • Spectral Action in Noncommutative Geometry Springerbriefs in Mathematical Physics
    SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 27 Michał Eckstein · Bruno Iochum Spectral Action in Noncommutative Geometry SpringerBriefs in Mathematical Physics Volume 27 Series editors Nathanaël Berestycki, Cambridge, UK Mihalis Dafermos, Princeton, USA Tohru Eguchi, Tokyo, Japan Atsuo Kuniba, Tokyo, Japan Matilde Marcolli, Pasadena, USA Bruno Nachtergaele, Davis, USA SpringerBriefs are characterized in general by their size (50–125 pages) and fast production time (2–3 months compared to 6 months for a monograph). Briefs are available in print but are intended as a primarily electronic publication to be included in Springer’s e-book package. Typical works might include: • An extended survey of a field • A link between new research papers published in journal articles • A presentation of core concepts that doctoral students must understand in order to make independent contributions • Lecture notes making a specialist topic accessible for non-specialist readers. SpringerBriefs in Mathematical Physics showcase, in a compact format, topics of current relevance in the field of mathematical physics. Published titles will encompass all areas of theoretical and mathematical physics. This series is intended for mathematicians, physicists, and other scientists, as well as doctoral students in related areas. More information about this series at http://www.springer.com/series/11953 Michał Eckstein • Bruno Iochum Spectral Action in Noncommutative Geometry 123 Michał Eckstein Bruno Iochum Copernicus Center for Interdisciplinary Aix-Marseille Univ, Université
    [Show full text]
  • The Dirac Spectrum on Manifolds with Gradient Conformal Vector Fields Andrei Moroianu, Sergiu Moroianu
    The Dirac spectrum on manifolds with gradient conformal vector fields Andrei Moroianu, Sergiu Moroianu To cite this version: Andrei Moroianu, Sergiu Moroianu. The Dirac spectrum on manifolds with gradient conformal vector fields. Journal of Functional Analysis, Elsevier, 2007, 253 (1), pp.207-219. 10.1016/j.jfa.2007.04.013. hal-00125939v2 HAL Id: hal-00125939 https://hal.archives-ouvertes.fr/hal-00125939v2 Submitted on 7 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE DIRAC SPECTRUM ON MANIFOLDS WITH GRADIENT CONFORMAL VECTOR FIELDS ANDREI MOROIANU AND SERGIU MOROIANU Abstract. We show that the Dirac operator on a spin manifold does not admit L2 eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing. 1. Introduction The Dirac operator on a closed spin manifold is essentially self-adjoint as an unbounded operator in L2, and has purely discrete spectrum. Its eigenvalues grow at a certain speed determined by the volume of the manifold and its dimension.
    [Show full text]
  • Spectral Asymmetry of the Massless Dirac Operator on a 3-Torus
    Spectral asymmetry of the massless Dirac operator on a 3-torus Article Accepted Version Downes, R. J., Levitin, M. and Vassiliev, D. (2013) Spectral asymmetry of the massless Dirac operator on a 3-torus. Journal of Mathematical Physics, 54 (11). 111503. ISSN 0022- 2488 doi: https://doi.org/10.1063/1.4828858 Available at http://centaur.reading.ac.uk/34885/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . To link to this article DOI: http://dx.doi.org/10.1063/1.4828858 Publisher: American Institute of Physics All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online Spectral asymmetry of the massless Dirac operator on a 3-torus Robert J. Downes,1, a) Michael Levitin,2, b) and Dmitri Vassiliev1, c) 1)Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 2)Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explic- itly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric.
    [Show full text]
  • Spectral Geometry Bruno Iochum
    Spectral Geometry Bruno Iochum To cite this version: Bruno Iochum. Spectral Geometry. A. Cardonna, C. Neira-Jemenez, H. Ocampo, S. Paycha and A. Reyes-Lega. Spectral Geometry, Aug 2011, Villa de Leyva, Colombia. World Scientific, 2014, Geometric, Algebraic and Topological Methods for Quantum Field Theory, 978-981-4460-04-0. hal- 00947123 HAL Id: hal-00947123 https://hal.archives-ouvertes.fr/hal-00947123 Submitted on 14 Feb 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spectral Geometry Bruno Iochum Aix-Marseille Université, CNRS UMR 7332, CPT, 13288 Marseille France Abstract The goal of these lectures is to present some fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea here is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein–Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one.
    [Show full text]
  • Spectral Asymptotics for Coupled Dirac Operators Nikhil Savale
    Spectral Asymptotics for Coupled Dirac Operators by Nikhil Savale S.B., Massachusetts Institute of Technology (2007) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2012 c Massachusetts Institute of Technology 2012. All rights reserved. Author.............................................................. Department of Mathematics August 9, 2012 Certified by. Tomasz Mrowka Singer Professor of Mathematics Thesis Supervisor Accepted by . Paul Seidel Chairman, Department Committee on Graduate Students 2 Spectral Asymptotics for Coupled Dirac Operators by Nikhil Savale Submitted to the Department of Mathematics on August 9, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an n n+1 n dimensional manifold is of order r 2 followed by a remainder of O(r 2 ). We perform n−1 computations of spectral flow on the sphere which show that O(r 2 ) is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show n+3 that its odd functional trace exhibits cancellations in its first 2 terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained. Thesis Supervisor: Tomasz Mrowka Title: Singer Professor of Mathematics 3 4 Acknowledgments I am extremely grateful to my advisor Tom Mrowka for everything that I have learnt from him.
    [Show full text]
  • Emmanuel Lorin De La Grandmaison Research Proposal RESEARCH
    Emmanuel Lorin de la Grandmaison Research Proposal RESEARCH PROPOSAL This research proposal is devoted to the development and analysis of mathematical and nu- merical methods, as well as computer implementation in quantum relativistic physics. With the advent of strong laser physics, accurate and efficient computational methods become cru- cial to not only simulate but also to understand fundamental quantum physics phenomena. This research program is a contribution from the mathematical point of view to theoretical and experimental research in Attosecond Science, strong field physics, Quantum ElectroDy- namics (QED), and nonperturbative nonlinear optics, research fields in which Canada is an international leader. Mathematically, the program is mainly applied-analysis oriented, more specifically towards modern tools in applied spectral theory, and the theory of Partial Differ- ential Equations (PDEs) and their numerical approximation. HQPs will be highly involved throughout the program, as detailed in the HQP-training form. The references can be found in the file List of References. Long-term objectives. • Derivation and analysis of mathematical models in quantum non-relativistic and rela- tivistic physics, laser-physics and nonperturbative nonlinear optics. • Derivation of generic, accurate and efficient computational methods, as well as their mathematical analysis for relativistic and non-relativistic nonlinear quantum physics. • Development of parallel simulation codes implementing these numerical models, with specific targeted areas of application: laser-filamentation, attosecond science, QED (production of particles-antiparticles, graphene). I plan to make these codes accessible to the Canadian and international applied mathematics and physics communities. • Transition towards quantum simulation: development of quantum algorithms for forth- coming quantum computers. These long-term objectives (15 years) could, in particular, be reached thanks to shorter-term objectives (5 years) described below, which constitute essential components of my research program.
    [Show full text]