DOCSLIB.ORG

Realisations of Elliptic Operators on Compact Manifolds with Boundary 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Realisations of Elliptic Operators on Compact Manifolds with Boundary 3 REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY LASHI BANDARA, MAGNUS GOFFENG, AND HEMANTH SARATCHANDRAN Abstract. This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of B¨ar-Ballmann to first order elliptic opera- tors. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calder´on projectors which, in the first order case, is equivalent to results of B¨ar-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven. Contents 1. Introduction 1 2. Elliptic differential operators and boundary conditions 9 3. Calder´on projectors and the Douglis-Nirenberg calculus 21 4. The Cauchy data space of an elliptic differential operator 33 5. Characterisations of regularity via graphical decompositions 39 6. Higher order boundary regularity 50 7. Differential operators with positive principal symbols and their Weyl laws 55 8. Rigidity results for the index of elliptic differential operators 59 9. Symbol computations with Calder´on projectors in the first order case 63 Appendix A. Lemmas on dimensions and subspaces 70 Appendix B. B¨ar-Ballmann’s approach from an abstract perspective 72 References 83 1. Introduction Elliptic boundary value problems have a long history emerging from classical problems in physics and engineering. Their mathematical description dates back to classical works such as [32, 34, arXiv:2104.01919v2 [math.AP] 22 Jul 2021 36, 37, 50, 57, 61, 69]. Elliptic operators – their index theory and spectral theory – is broadly used in mathematics connecting seemingly disjoint areas of mathematics, for instance through noncommutative geometry and geometric analysis. The spectral theory of elliptic boundary value problems has primarily been focused on Laplace type operators (for an overview see [52]) or Dirac type operators (see for instance [15, 18, 19, 44]), but has also been studied in larger generality (e.g. in [29, 36, 38, 40, 42]). The index theory of elliptic boundary value problems is also a well studied area. For instance, the index theory of Dirac operators with the spectral APS boundary condition was computed by Atiyah-Patodi-Singer [4], and has since been well studied, see for instance [15,18,41,43,59]. The index theory for boundary value problems in the Boutet de Monvel calculus was computed by [16] relating to ideas of Agranovich-Dynin. This formula covers the case of local elliptic boundary conditions. Boutet de Monvel’s results have been extensively studied, see Date: July 23, 2021. 2010 Mathematics Subject Classification. 35J58, 35J56, 58J05, 58J32. Key words and phrases. Elliptic differential operator, Fredholm boundary conditions, boundary regularity, Calder´on projector. 1 2 LASHIBANDARA, MAGNUS GOFFENG, ANDHEMANTHSARATCHANDRAN [26, 27, 58, 60], and were generalised to an extended Boutet de Monvel calculus by Schulze-Seiler [63]. In this paper we take a new approach to higher order elliptic operators on manifolds with boundary. The approach is based on work of B¨ar-Ballmann [5]. The paper [5] consists of a coherent overview of first order boundary value problems (admitting a self-adjoint adapted boundary operator) based on an analytic description of the Cauchy data space as well as a novel graphical decomposition of regular realisations of first order operators. The results were later extended to general first order problems by the first listed author and B¨ar in [6]. Although this approach was first fully utilised and made explicit in [5], the ideas have been around since the ’60s and was implicitly contained in a paper by Seeley [65, top of page 782]. They were visible already in Agranovich-Dynin’s classical works on index theory for elliptic boundary value problems of general order. Similar approaches predating B¨ar-Ballmann’s have been studied in the abstract in [13], been applied to Dirac-Schr¨odinger operators in [8] as well as having been used in the study of Maxwell’s equations on Lipschitz domains [20, 48]. Related ideas can also be seen in for instance [4, 19, 34, 37, 41, 43]. Since its appearance, the coherence of B¨ar-Ballmann’s work [5] has paved the way for numerous applications [7, 9, 17, 21, 66]. The aim of this paper is to present a perspective similar to [5], providing an overview of the theory for general order elliptic boundary value problems from the vantage point of the Cauchy data space. As such, the main novelty lies in the framework and the methods. Indeed, several of the results in this paper are, as stand alone results, at best mild generalisations of known results and are unlikely to surprise experts in the field of boundary value problems. A key feature in several of the results is that they can treat all (or at least the regular extensions with the occasional restriction to the pseudo-local case) closed extensions of an elliptic differential operator on an equal footing with the special classes of boundary conditions studied classically. We have attempted to the best of our abilities to indicate where the results we are generalising can be found in the abundance of literature on boundary value problems. The approach we take lies close in spirit to the ideas of noncommutative geometry [22] whose methods have proven their worth in index theory, relating to recent work on boundaries in non- commutative geometry [28]. It takes a more abstract perspective than the semiclassical methods ordinarily deployed to study boundary value problems, e.g. in [32, 37, 50, 57, 61, 69], and lies closer to methods for order one elliptic operators seen in [5, 6, 13, 15, 18, 41, 43] rather than the abstract methods of, for instance, [10,23,34,36]. The level of abstraction makes the method feasible for further development of index theory and applications to a larger class of problems than elliptic differential operators on manifolds with boundary. We anticipate this in- cludes elliptic operators on singular manifolds, hypoelliptic operators (cf. the subelliptic boundary conditions in [25]) or geometric operators in Lorentzian geometry (cf. [7]). These facts raises opti- mism for demystifying the general index formulas appearing in the (extended) Boutet de Monvel calculus relying on abstract homotopies [16,60], non-explicit inverses to isomorphisms in K-theory [58] or Chern characters of operator valued symbols [26, 27]. The method of B¨ar-Ballmann for first order operators can be summarised as describing boundary value problems by means of the Cauchy data space, i.e. the range of the corresponding trace map onto a mixed Sobolev space on the boundary. For higher order operators we apply the same ideas to the full trace map (taking into account all boundary traces up to the order of the operator) into a mixed Sobolev space on the boundary. The range of the full trace map – the Cauchy data space – characterises the realisation defining the boundary value problem when topologising it so that the full trace map is a quotient. The analytic details involved in this procedure produces precise information about regularity and Fredholm properties of the realisations in terms of properties of the boundary condition relative to the range of the full trace mapping on the maximal domain of the elliptic operator. The general theme in this framework is to reduce properties of realisations, and their proofs, to neat functional analysis arguments and Fredholm theory. REALISATIONS OF ELLIPTIC OPERATORS ON COMPACT MANIFOLDS WITH BOUNDARY 3 The description of the possible realisations of an elliptic operator as subspaces of a Hilbert space of functions on the boundary relies on a precise description of the Cauchy data space. Abstractly, the Cauchy data space is isomorphic to the quotient of the maximal domain by the minimal domain. By classical results of Lions-Magenes [56, 57] the Cauchy data space is a concrete subspace of a mixed order Sobolev space on the boundary. For an elliptic first order differential operator this is described in detail in [6] using an adapted boundary operator. To describe the Cauchy data space of a general order elliptic differential operator, we employ the Calder´on projection of Seeley [65]. Although our setup is based on the work of B¨ar-Ballmann, similar ideas date back much further as discussed above. We emphasise that this paper, despite allowing for elliptic differential operators of general order, by no means supersedes [5, 6]. A significant novelty distinguishing [6] from this paper is that [6] links analysis on the Cauchy data space to the H∞-functional calculus, which in [6] forms the key to obtaining the estimates required to topologise the Cauchy data space. In contrast, in this paper we use Seeley’s work on Calder´on projectors to analyse the Cauchy data space. 1.1. Main results and overview of paper. Let M be a compact manifold with boundary Σ := ∂M. We fix a smooth measure µ on M. In our convention, Σ M and the interior of M ⊂ is denoted by M˚ = M Σ. We also choose a smooth interior pointing vectorfield T~ transversal to \ Σ= ∂M. The smooth measure on Σ induced by µ and T~ will be denoted by ν. In this paper, we are mainly concerned with elliptic differential operators D : C∞(M; E) C∞(M; F ) acting between hermitian vector bundles (E,hE ) M and (F,hF ) M. The hermit-→ ian metric will be implicit in the notations. We let m denote→ the order of D.
Load more

Recommended publications
  • Arxiv:1908.09677V4 [Math.AG] 13 Jul 2021 6 Hr Ro Fterm12.1 Theorem of Proof Third a 16
    AN ANALYTIC VERSION OF THE LANGLANDS CORRESPONDENCE FOR COMPLEX CURVES PAVEL ETINGOF, EDWARD FRENKEL, AND DAVID KAZHDAN In memory of Boris Dubrovin Abstract. The Langlands correspondence for complex curves is traditionally for- mulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamilto- nians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to formulate a function-theoretic correspondence. We conjecture the existence of a canonical self-adjoint extension of the symmetric part of this al- gebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G = GL1 and in the simplest non-abelian case. Contents 1. Introduction 2 Part I 8 2. Differential operators on line bundles 8 3. Differential operators on BunG 11 4. The spectrum and opers 15 5. The abelian case 19 6. Bundles with parabolic structures 26 1 7. The case of SL2 and P with marked points 28 arXiv:1908.09677v4 [math.AG] 13 Jul 2021 8. Proofs of two results 30 Part II 33 9. Darboux operators 34 10. EigenfunctionsandmonodromyforDarbouxoperators 38 11. Essentially self-adjoint algebras of unbounded operators 43 12. The main theorem 49 13. Generalized Sobolev and Schwartz spaces attached to the operator L. 49 14. Proof of Theorem 12.1 60 15.
    [Show full text]
  • Arxiv:2107.01242V1 [Math.OA] 2 Jul 2021 Euneo Ievle ( Eigenvalues of Sequence a Eetdacrigt T Agbac Utpiiy Y[ by Multiplicity
    CONNES’ INTEGRATION AND WEYL’S LAWS RAPHAEL¨ PONGE Abstract. This paper deal with some questions regarding the notion of integral in the frame- work of Connes’s noncommutative geometry. First, we present a purely spectral theoretic con- struction of Connes’ integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue’s integral. This answers another question of Alain Connes. We further clarify the relationship of Connes’ integration with Weyl’s laws for compact operators and Birman-Solomyak’s perturbation theory. We also give a ”soft proof” of Birman-Solomyak’s Weyl’s law for negative order pseudodifferential operators on closed mani- fold. This Weyl’s law yields a stronger form of Connes’ trace theorem. Finally, we explain the relationship between Connes’ integral and semiclassical Weyl’s law for Schr¨odinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis. 1. Introduction The quantized calculus of Connes [18] aims at translating the main tools of the classical infin- itesimal calculus into the operator theoretic language of quantum mechanics. As an Ansatz the integral in this setup should be a positive trace on the weak trace class L1,∞ (see Section 2). Natural choices are given by the traces Trω of Dixmier [23] (see also [18, 34] and Section 2). These traces are associated with extended limits. Following Connes [18] we say that an operator A L is measurable when the value of Tr (A) is independent of the extended limit.
    [Show full text]
  • WEYL's LAW on RIEMANNIAN MANIFOLDS Contents 1. the Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifol
    WEYL'S LAW ON RIEMANNIAN MANIFOLDS SETH MUSSER Abstract. We motivate Weyl's law by demonstrating the relevance of the distribution of eigenvalues of the Laplacian to the ultraviolet catastrophe of physics. We then introduce Riemannian geometry, define the Laplacian on a general Riemannian manifold, and find geometric analogs of various concepts in Rn, along the way using S2 as a clarifying example. Finally, we use analogy with Rn and the results we have built up to prove Weyl's law, making sure at each step to demonstrate the physical significance of any major ideas. Contents 1. The Role of Weyl's Law in the Ultraviolet Catastrophe 1 2. Riemannian Manifolds 3 3. Weyl's Law 11 4. Acknowledgements 20 References 20 1. The Role of Weyl's Law in the Ultraviolet Catastrophe In the year 1900 Lord Rayleigh used the equipartition theorem of thermodynamics to de- duce the famous Rayleigh-Jeans law of radiation. Rayleigh began with an idealized physical concept of the blackbody; a body which is a perfect absorber of electromagnetic radiation and which radiates all the energy it absorbs independent of spatial direction. We sketch his proof to find the amount of radiation energy emitted by the blackbody at a given frequency. Take a blackbody in the shape of a cube for definiteness, and assume it is made of con- ducting material and is at a temperature T . We will denote the cube by D = [0;L]3 ⊆ R3. Inside this cube we have electromagnetic radiation that obeys Maxwell's equations bouncing around. The equipartition theorem of thermodynamics says that every wave which has con- stant spatial structure with oscillating amplitude, i.e.
    [Show full text]
  • Scalar Curvature for Noncommutative Four-Tori
    SCALAR CURVATURE FOR NONCOMMUTATIVE FOUR-TORI FARZAD FATHIZADEH AND MASOUD KHALKHALI Abstract. In this paper we study the curved geometry of noncommutative 4 4-tori Tθ. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which corre- 4 spond to the volume and scalar curvature of Tθ. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe 4 the scalar curvature of Tθ. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action. Contents 1. Introduction 1 2. Noncommutative Tori 3 2.1. Noncommutative real tori. 3 2.2. Noncommutative complex tori. 4 3. Laplacian and its Heat Kernel 5 4 3.1. Perturbed Laplacian on Tθ. 5 4 3.2. Connes' pseudodifferential calculus for Tθ. 7 3.3. Small time asymptotic expansion for Trace(e−t4' ). 8 4. Weyl's Law and Connes' Trace Theorem 10 4.1. Asymptotic distribution of the eigenvalues of 4'. 10 4 4.2. A noncommutative residue for Tθ. 12 4 4.3. Connes' trace theorem for Tθ. 15 5. Scalar Curvature and Einstein-Hilbert Action 16 5.1. Scalar curvature for 4. 17 arXiv:1301.6135v1 [math.QA] 25 Jan 2013 Tθ 4 5.2.
    [Show full text]
  • Location and Weyl Formula for the Eigenvalues of Some Non Self-Adjoint Operators
    Location and Weyl formula for the eigenvalues of some non self-adjoint operators Vesselin Petkov Abstract We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator G of the contraction semigroup etG, t ≥ 0, related to the wave equation in an unbounded domain Ω with dissipative boundary conditions on ∂Ω. Also one examines the interior transmission eigenvalues (ITE) in a bounded domain K obtaining a Weyl formula with remainder for the counting function N(r) of complex (ITE). The analysis is based on a semi-classical approach. 1 Introduction ∞ Let P(x,Dx) be a second order differential operator with C (K) real-valued coef- d ficients in a bounded domain K ⊂ R , d ≥ 2, with C∞ boundary ∂K. Consider a boundary problem ( P(x,D )u = f in K, x (1.1) B(x,Dx)u = g on ∂K, where B(x,Dx) is a differential operator with order less or equal to 1 and the princi- 2 pal symbol P(x,ξ) of P(x,Dx) satisfies p(x,ξ) ≥ c0|ξ| , c0 > 0. Assume that there exists 0 < ϕ < π such that the problem ( (P(x,D ) − z)u = f in K, x (1.2) B(x,Dx)u = g on ∂K. is parameter-elliptic for every z ∈ Γψ = {z : argz = ψ}, 0 < |ψ| ≤ ϕ. Then following a classical result of Agranovich-Vishik [1] we can find a closed operator A with Vesselin Petkov Institut de Mathematiques´ de Bordeaux, 351, Cours de la Liberation,´ 33405 Talence, France , e- mail: [email protected] 1 2 Vesselin Petkov domain D(A) ⊂ H2(K) related to the problem (1.1).
    [Show full text]
  • An Introduction to Pseudo-Differential Operators
    An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality).
    [Show full text]
  • What Is Noncommutative Geometry ? How a Geometry Can Be Commutative and Why Mine Is Not
    What is Noncommutative Geometry ? How a geometry can be commutative and why mine is not Alessandro Rubin Junior Math Days 2019/20 SISSA - Scuola Internazionale Superiore di Studi Avanzati This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. 1/41 Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... 1/41 Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic.
    [Show full text]
  • Notes on the Quantum Harmonic Oscillator Brief Discussion of Coherent States, Weyl's Law and the Mehler Kernel
    Notes on the Quantum Harmonic Oscillator Brief Discussion of Coherent States, Weyl's Law and the Mehler Kernel Brendon Phillips 250817875 April 22, 2015 1 The Harmonic Oscillators Recall that any object oscillating in one dimension about the point 0 (say, a mass-spring system) obeys Hooke's law Fx = −ksx, where x is the compression of the spring, and ks is the spring constant of the system. By Newton's second law of motion, we then have the equation of the Classical Harmonic Oscillator @2x m = −k x (1) @t2 s 1 Now, Fx is conservative, so we can recover the potential energy function as follows : @V 1 −F = =) V (x) = k x2 (2) x @x 2 s The solution to (1) is given by r ! k x(t) = A cos s t + (3) m The frequency of oscillation is then r 1 1 k f = = s (4) T 2π m Let the oscillating object be a particle. Now, classical mechanics only provides a good ap- proximation of the system as long as E >> hf, whereE is the total energy of the particle, f is the classical frequency, and h is the Planck constant. Assume that this is not the case; we now see the system as a Quantum Mechanical Os- cillator. As a consequence of the Heisenberg Uncertainty Principle ∆^x∆^p ≥ } (5) 2 1We assume without loss of generality that the constant of integration is 0. 1 (wherex ^ represents the position operator,p ^ represents the momentum operator, and } is the reduced Planck constant), we lose our grasp of the particle's trajectory, since we now switch our focus from the position of the particle to its wavefunction (x; t).
    [Show full text]
  • 100 Years of Weyl's
    100 years of Weyl's law Victor Ivrii February 28, 2017 Abstract We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymp- totics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical mi- crolocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schr¨odingerand Dirac operators with the strong magnetic field and, finally, applica- tions to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field. Contents1 arXiv:1608.03963v2 [math.SP] 27 Feb 2017 1 Introduction2 1.1 A bit of history . .2 1.2 Method of the hyperbolic operator . .6 2 Local semiclassical spectral asymptotics7 2.1 Asymptotics inside the domain . .7 2.2 Boundary value problems . 23 1 1. Introduction 2 3 Global asymptotics 31 3.1 Weyl asymptotics . 32 3.2 Non-Weyl asymptotics . 41 4 Non-smooth theory 50 5 Magnetic Schr¨odingeroperator 53 5.1 Introduction . 53 5.2 Standard theory . 54 5.3 2D case, degenerating magnetic field . 58 5.4 2D case, near the boundary . 60 5.5 Pointwise asymptotics and short loops . 64 5.6 Magnetic Dirac operators . 66 6 Magnetic Schr¨odingeroperator. II 67 6.1 Higher dimensions . 67 6.2 Non-smooth theory . 70 6.3 Global asymptotics . 71 7 Applications to multiparticle quantum theory 75 7.1 Problem set-up . 75 7.2 Reduction to one-particle problem .
    [Show full text]
  • QMATH14: Mathematical Results in Quamtum Physics – Abstracts
    QMath14: Mathematical Results in Quantum Physics Aarhus University, 12–16 August 2019 Programme and abstracts – including posters Programme . .next page Abstracts . page 18 Posters . page 63 http://conferences.au.dk/qmath14/ Last update: 2019-08-13 09:53:41 Foto: Anders Trærum, AU Communication Programme Monday, 12 August 09:00–12:00 Mini course A quantum information travel guide for mathematicians Lecturer: David Pérez García Location: Aud E (1533.103) 12:00–14:00 Lunch Location: MATH Canteen (1536, ground floor) 14:00–15:00 Plenary talk Zeno and the bomb Michael Wolf Chair: Pavel Exner Location: Aud E (1533.103) 15:00–15:30 Coffee break 15:30–16:30 Plenary talk An update on Many-Body Localization Wojciech De Roeck Chair: Pavel Exner Location: Aud E (1533.103) 16:30–17:00 Break 17:00– Welcome reception 1 Tuesday, 13 August 09:30–10:30 Plenary talk Quantized quantum transport in interacting systems Sven Bachmann Chair: Simone Warzel Location: Aud E (1533.103) 10:30–11:00 Break 11:00–12:00 Plenary talk Universal Singularities of Random Matrices Torben Krüger Chair: Simone Warzel Location: Aud E (1533.103) 12:00–14:00 Lunch Location: MATH Canteen (1536, ground floor) 14:00–15:30 Parallel sessions – Main speakers Spectral Theory in Aud F (1534.125) see page 8 Quantum Information in Aud G1 (1532.116) see page 9 Many-Body Systems in Aud G2 (1532.122) see page 10 Random Systems in Koll G3 (1532.218) see page 11 Condensed Matter in Koll G4 (1532.222) see page 12 15:30–16:00 Coffee break 2 16:00–17:55 Parallel sessions – Contributed speakers Spectral
    [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]
  • Weyl Law on Asymptotically Euclidean Manifolds
    Ann. Henri Poincar´e 22 (2021), 447–486 c 2020 The Author(s) 1424-0637/21/020447-40 published online December 23, 2020 Annales Henri Poincar´e https://doi.org/10.1007/s00023-020-00995-1 Weyl Law on Asymptotically Euclidean Manifolds Sandro Coriasco and Moritz Doll Abstract. We study the asymptotic behaviour of the eigenvalue count- ing function for self-adjoint elliptic linear operators defined through clas- sical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geomet- ric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the bound- ary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator Q =(1+|x|2)(1 − Δ)on Rd. Mathematics Subject Classification. Primary 58J50; Secondary 58J40, 35P20. 1. Introduction Let (X, g)bead-dimensional asymptotically Euclidean manifold. More explic- itly, X belongs to a class of compact manifolds with boundary, whose interior is equipped with a Riemannian metric g which assumes a specific form close to the boundary ∂X (see Definition 29 in Section A.1 of “Appendix”). The elements of such class are also known as scattering manifolds, asymptotically conic manifolds,ormanifolds with large conic ends. A typical example is the unit ball Bd, equipped with a scattering metric. On X, we consider a self-adjoint positive operator P , elliptic in the SG- calculus of order (m, n) with m, n ∈ (0, ∞).1 By the compact embedding of 1We refer to Sect.
    [Show full text]