QMATH14: Mathematical Results in Quamtum Physics – Abstracts
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![Arxiv:1908.09677V4 [Math.AG] 13 Jul 2021 6 Hr Ro Fterm12.1 Theorem of Proof Third a 16](https://docslib.b-cdn.net/cover/1460/arxiv-1908-09677v4-math-ag-13-jul-2021-6-hr-ro-fterm12-1-theorem-of-proof-third-a-16-101460.webp)
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. -
Spectral Analysis of Quasiperiodic Schrödinger Operators
UNIVERSITY OF CALIFORNIA, IRVINE Spectral analysis of quasiperiodic Schr¨odingeroperators DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics by Shiwen Zhang Dissertation Committee: Professor Svetlana Jitomirskaya, Chair Professor Anton Gorodetski Professor Abel Klein 2016 c 2016 Shiwen Zhang Dedication This thesis is dedicated to my beloved parents. For their endless love, support and encouragement. ii Table of Contents Acknowledgements v Curriculum Vitae vi Abstract of the Dissertation x Introduction 1 0.1 Discrete Schr¨odingeroperators . 1 0.2 Motivation and Background . 1 1 Quantitative continuity of singular continuous spectral measures and arithmetic criteria. 5 1.1 Introduction . 5 1.1.1 Main application . 9 1.1.2 Spectral singularity, continuity and proof of Theorem 1.1.4 . 10 1.1.3 Relation with other dimensions; Corollaries for the AMO, Stur- mian potentials, and Transport exponents. 14 1.1.4 Preliminaries . 18 1.2 Spectral Continuity . 22 1.2.1 Proof of Theorem 1.1.6 . 22 1.2.2 Proof of Theorem 1.2.1 . 26 1.2.3 The hyperbolic case: Proof of Lemma 1.2.4 . 27 1.2.4 Energies with Trace close to 2: Proof of Lemma 1.2.5 . 33 1.2.5 Proof of Lemmas 1.2.2 and 1.2.3 . 39 1.3 Spectral Singularity . 40 1.3.1 Power law estimates and proof of Theorem 1.1.5 . 40 1.3.2 Proof of the density lemmas . 45 1.4 Sturmian Hamiltonian . 47 2 Mixed spectral types for the one frequency discrete quasi-periodic Schr¨odingeroperator 50 2.1 Introduction . -
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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. -
Spectral Gap and Asymptotics for a Family of Cocycles of Perron-Frobenius Operators
Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators by Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics c Joseph Anthony Horan, 2020 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. We acknowledge with respect the Lekwungen peoples on whose traditional territory the university stands, and the Songhees, Esquimalt, and WSANE´ C´ peoples whose ¯ historical relationships with the land continue to this day. Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators by Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013 Supervisory Committee Dr. Christopher Bose, Co-Supervisor (Department of Mathematics and Statistics) Dr. Anthony Quas, Co-Supervisor (Department of Mathematics and Statistics) Dr. Sue Whitesides, Outside Member (Department of Computer Science) ii ABSTRACT At its core, a dynamical system is a set of things and rules for how they change. In the study of dynamical systems, we often ask questions about long-term or average phe- nomena: whether or not there is an equilibrium for the system, and if so, how quickly the system approaches that equilibrium. These questions are more challenging in the non-autonomous (or random) setting, where the rules change over time. The main goal of this dissertation is to develop new tools with which to study random dynamical systems, and demonstrate their application in a non-trivial context. -
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. -
Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Integr. Equ. Oper. Theory 69 (2011), 151–170 DOI 10.1007/s00020-010-1856-x Published online January 11, 2011 c The Author(s) This article is published Integral Equations with open access at Springerlink.com 2010 and Operator Theory Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples Anne Boutet de Monvel, Jan Janas and Serguei Naboko Abstract. We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum fills any finite number of bounded intervals is considered. Their point spectrum accumulates to +∞ and −∞. The asymptotics of large eigenvalues is also found. Mathematics Subject Classification (2010). Primary 47B36; Secondary 47A10, 47B25. Keywords. Jacobi matrix, essential spectrum, gaps, asymptotics. 1. Introduction In this paper we look for examples of unbounded Jacobi matrices with sev- 2 2 eral gaps in the essential spectrum. Let 0 = 0(N) be the space of sequences ∞ {fk}1 with a finite number of nonzero coordinates. For given real sequences ∞ ∞ 2 {ak}1 and {bk}1 the Jacobi operator J 0 acts in 0 by the formula (J 0 f)k = ak−1fk−1 + bkfk + akfk+1 (1.1) where k =1, 2, ... and a0 = f0 =0.Theak’s and bk’s are called the weights and diagonal terms, respectively. ∞ In what follows we deal with positive sequences {ak}1 . By Carleman’s 2 2 criterion [1] one can extend J 0 to a self-adjoint operator J in = (N) provided that 1 =+∞. -
Nonlinear Spectral Calculus and Super-Expanders
NONLINEAR SPECTRAL CALCULUS AND SUPER-EXPANDERS MANOR MENDEL AND ASSAF NAOR Abstract. Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Ces`aroaver- ages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space. Contents 1. Introduction 2 1.1. Coarse non-embeddability 3 1.2. Absolute spectral gaps 5 1.3. A combinatorial approach to the existence of super-expanders 5 1.3.1. The iterative strategy 6 1.3.2. The need for a calculus for nonlinear spectral gaps 7 1.3.3. Metric Markov cotype and spectral calculus 8 1.3.4. The base graph 10 1.3.5. Sub-multiplicativity theorems for graph products 11 2. Preliminary results on nonlinear spectral gaps 12 2.1. The trivial bound for general graphs 13 2.2. γ versus γ+ 14 2.3. Edge completion 18 3. Metric Markov cotype implies nonlinear spectral calculus 19 4. An iterative construction of super-expanders 20 5. The heat semigroup on the tail space 26 5.1. Warmup: the tail space of scalar valued functions 28 5.2. Proof of Theorem 5.1 31 5.3. Proof of Theorem 5.2 34 5.4. Inverting the Laplacian on the vector-valued tail space 37 6. Nonlinear spectral gaps in uniformly convex normed spaces 39 6.1. -
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. -
The Absolutely Continuous Spectrum of the Almost Mathieu Operator
THE ABSOLUTELY CONTINUOUS SPECTRUM OF THE ALMOST MATHIEU OPERATOR ARTUR AVILA Abstract. We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schr¨odinger operator problems for the twenty-first century. 1. Introduction This work is concerned with the almost Mathieu operator H = Hλ,α,θ defined on ℓ2(Z) (1.1) (Hu)n = un+1 + un−1 +2λ cos(2π[θ + nα])un where λ = 0 is the coupling, α R Q is the frequency and θ R is the phase. This is the6 most studied quasiperiodic∈ \ Schr¨odinger operator, arisin∈ g naturally as a physical model (see [L3] for a recent historical account and for the physics back- ground). We are interested in the decomposition of the spectral measures in atomic (corre- sponding to point spectrum), singular continuous and absolutely continuous parts. Our main result is the following. Main Theorem. The spectral measures of the almost Mathieu operator are abso- lutely continuous if and only if λ < 1. | | 1.1. Background. Singularity of the spectral measures for λ 1 had been pre- viously established (it follows from [LS], [L1], [AK]). Thus| | the ≥ Main Theorem reduces to showing absolute continuity of the spectral measures for λ < 1, which is Problem 6 of Barry Simon’s list [S3]. | | We recall the history of this problem (following [J]). Aubry-Andr´econjectured the following dependence on λ of the nature of the spectral measures: arXiv:0810.2965v1 [math.DS] 16 Oct 2008 (1) (Supercritical regime) For λ > 1, spectral measures are pure point, (2) (Subcritical regime) For λ| <| 1, spectral measures are absolutely continu- ous. -
The Essential Spectrum of Schr¨Odinger, Jacobi, And
THE ESSENTIAL SPECTRUM OF SCHRODINGER,¨ JACOBI, AND CMV OPERATORS YORAM LAST1;3 AND BARRY SIMON2;3 Abstract. We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein’s results on orthogonal polynomials with finite essential spectra. 1. Introduction One of the most simple but also most powerful ideas in spectral the- ory is Weyl’s theorem, of which a typical application is (in this intro- duction, in order to avoid technicalities, we take potentials bounded): Theorem 1.1. If V; W are bounded functions on Rº and limjxj!1[V (x) ¡ W (x)] = 0, then σess(¡∆ + V ) = σess(¡∆ + W ) (1.1) Our goal in this paper is to find a generalization of this result that allows “slippage” near infinity. Typical of our results are the following: Theorem 1.2. Let V be a bounded periodic function on (¡1; 1) d2 2 and HV the operator ¡ dx2 + V (x) on L (R). For x > 0, define p d2 2 W (x) = V (x + x) and let HW be ¡ dx2 + W (x) on L (0; 1) with some selfadjoint boundary conditions at zero. Then σess(HW ) = σ(HV ) (1.2) Date: March 7, 2005. 1 Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel. E- mail: [email protected]. -
Spectral Gap and Definability
SPECTRAL GAP AND DEFINABILITY ISAAC GOLDBRING ABSTRACT. We relate the notions of spectral gap for unitary representations and subfactors with definability of certain important sets in the corresponding structures. We give several applications of this relationship. CONTENTS 1. Introduction 2 1.1. A crash course in continuous logic 3 2. Definability in continuous logic 5 2.1. Generalities on formulae 5 2.2. Definability relative to a theory 7 2.3. Definability in a structure 9 3. Spectral gap for unitary group representations 11 3.1. Generalities on unitary group representations 11 3.2. Introducing spectral gap 12 3.3. Spectral gap and definability 13 3.4. Spectral gap and ergodic theory 15 4. Basic von Neumann algebra theory 17 4.1. Preliminaries 17 4.2. Tracial von Neumann algebras as metric structures 19 4.3. Property Gamma and the McDuff property 20 5. Spectral gap subalgebras 21 5.1. Introducing spectral gap for subalgebras 21 arXiv:1805.02752v1 [math.LO] 7 May 2018 5.2. Spectral gap and definability 22 5.3. Relative bicommutants and e.c. II1 factors 23 5.4. Open questions 24 6. Continuum many theories of II1 factors 25 6.1. The history and the main theorem 25 6.2. The base case 26 6.3. A digression on good unitaries and generalized McDuff factors 27 6.4. The inductive step 28 References 29 Goldbring’s work was partially supported by NSF CAREER grant DMS-1349399. 1 2 ISAAC GOLDBRING 1. INTRODUCTION The notion of definable set is one of (if not the) most important concepts in clas- sical model theory. -
Spectral Perturbation Bounds for Selfadjoint Operators I∗
Spectral perturbation bounds for selfadjoint operators I∗ KreˇsimirVeseli´c† Abstract We give general spectral and eigenvalue perturbation bounds for a selfadjoint op- erator perturbed in the sense of the pseudo-Friedrichs extension. We also give sev- eral generalisations of the aforementioned extension. The spectral bounds for finite eigenvalues are obtained by using analyticity and monotonicity properties (rather than variational principles) and they are general enough to include eigenvalues in gaps of the essential spectrum. 1 Introduction The main purpose of this paper is to derive spectral and eigenvalue bounds for selfadjoint operators. If a selfadjoint operator H in a Hilbert space H is perturbed into T = H + A (1) with, say, a bounded A then the well-known spectral spectral inclusion holds σ(T ) ⊆ {λ : dist(λ, σ(H)) ≤ kAk} . (2) Here σ denotes the spectrum of a linear operator. (Whenever not otherwise stated we shall follow the notation and the terminology of [3].) If H, A, T are finite Hermitian matrices then (1) implies |µk − λk| ≤ kAk, (3) where µk, λk are the non-increasingly ordered eigenvalues of T,H, respectively. (Here and henceforth we count the eigenvalues together with their multiplicities.) Whereas (2) may be called an upper semicontinuity bound the estimate (3) contains an existence statement: each of the intervals [λk − kAk, λk + kAk] contains ’its own’ µk. Colloquially, bounds like (2) may be called ’one-sided’ and those like (3) ’two-sided’. As it is well-known (3) can be refined to another two-sided bound min σ(A) ≤ µk − λk ≤ max σ(A). (4) In [9] the following ’relative’ two-sided bound was derived |µk − λk| ≤ b|λk|, (5) ∗This work was partly done during the author’s stay at the University of Split, Faculty of Electrotechnical Engineering, Mechanical Engineering and Naval Archtecture while supported by the National Foundation for Science, Higher Education and Technological Development of the Republic of Croatia.