DOCSLIB.ORG

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Distributions of Self-Adjoint Extensions of Symmetric Operators Journal of Stochastic Analysis Volume 2 Number 2 Article 6 June 2021 On Distributions of Self-Adjoint Extensions of Symmetric Operators Franco Fagnola Politecnico di Milano, Milan, 20133, Italy, [email protected] Zheng Li Politecnico di Milano, Milan, 20133, Italy, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/josa Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Fagnola, Franco and Li, Zheng (2021) "On Distributions of Self-Adjoint Extensions of Symmetric Operators," Journal of Stochastic Analysis: Vol. 2 : No. 2 , Article 6. DOI: 10.31390/josa.2.2.06 Available at: https://digitalcommons.lsu.edu/josa/vol2/iss2/6 Journal of Stochastic Analysis Vol. 2, No. 2 (2021) Article 6 (14 pages) digitalcommons.lsu.edu/josa DOI: 10.31390/josa.2.2.06 ON DISTRIBUTIONS OF SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS FRANCO FAGNOLA AND ZHENG LI* Abstract. In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribu- tion with respect to a given state. In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamil- tonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these prob- ability distributions on boundary conditions. 1. Introduction In Quantum Probability a self-adjoint operator defines a real random variable. Precisely, any self-adjoint⟨ operator⟩ A on Hilbert space H determines a probabil- ity measure by B 7! u; P A(B)u , where u 2 H is a unit vector and P A is the projection-valued measure associated with A. The problem of studying the prob- ability distribution of A with respect to some state u 2 H naturally arises. In particular, the problem of finding the vacuum state distribution of field operators in interacting Fock spaces has recently attracted a lot of interest (see [2, 3, 5] and the references therein). When the self-adjoint operator is an extension of a symmetric operator, it is interesting to understand the dependence of the distribution on the special self- adjoint extension chosen. This problem is highly non-trivial because different self- adjoint extensions define non-commuting random variables and, moreover, they do not have a common essential domain. In addition, it may not be possible to determine the distribution computing moments, as usually done for field operators in interacting Fock spaces (see [1, 5]), because either moments do not exist or the distribution is not determined by moments. In this paper, we consider momentum (imaginary unit i times the first deriva- tive) and Hamiltonian (minus second derivative) operators defined on compactly supported smooth functions on (0; 1) or (0; +1), and self-adjoint extensions of their closures in L2(0; 1) or L2(0; +1). The paper is organized as follows. In Section 2 we compute explicitly the distributions of self-adjoint extensions of momentum on L2(0; 1) finding a family of discrete distributions (Theorem 2.1) without moments, except in the special Received 2021-3-7; Accepted 2021-5-25; Communicated by the editors. 2010 Mathematics Subject Classification. Primary 81P16; Secondary 81Q80. Key words and phrases. Quantum probability, distribution, self-adjoint extension. * Corresponding author. 1 2 FRANCO FAGNOLA AND ZHENG LI case where one gets the delta δ0 at 0 with respect to the pure state determined by the constant function 1 equal to 1. This result can be achieved either by a spectral theoretic or a probabilistic method. In Section 3 we consider self-adjoint extensions of minus second derivative on smooth compactly supported functions on (0; +1). These are determined by boundary conditions u0(0) + ru(0) = 0 (r 2 R) or u(0) = 0. In this case the spec- trum is the half-line [0; +1) for r ≤ 0 and {−r2g [ [0; +1) for r > 0 (Proposition 3.1). We can compute explicitly the distribution of self-adjoint extensions with respect to some reference states related with Hermite polynomials in the special cases where the boundary condition is u0(0) = 0 or u(0) = 0. Indeed, we find the explicit formula for the Laplace transform of the distribution and, after in- version, we get a family of Gamma distributions (Theorem 3.2). Apparently, this computation does not give an explicit (invertible) Laplace transform for r =6 0. Finally, in Section 4 we consider self-adjoint extensions of minus second deriv- ative on smooth compactly supported functions on (0; 1). In this case boundary conditions become more complicated therefore we consider only some special cases. We compute the Laplace transform of the distribution, which can be easily tackled by separation of variables, and find a collection of discrete distributions (Theorem 4.1) with respect to the reference state 1 after the inversion. Throughout the paper we write L2(I) for some interval I ⊂ R to denote L2(I; C), i.e. we consider the complex-valued functions. 2. The Momentum Operator on L2(0; 1) The momentum operator on L2(0; 1) can be defined as a self-adjoint extension of the first-order differential operator A on L2(0; 1), i.e. { } Au = iu0; D(A) = u 2 H1(0; 1) j u(0) = u(1) = 0 (2.1) Obviously, A is densely-defined, closed and symmetric. As shown in Reed and Simon ([8] Ch.X, Ex.1, p.141), the self-adjoint extensions of A, denoted by Aθ (0 ≤ θ < 2π), have domain { } 1 −iθ D(Aθ) = u 2 H (0; 1) j u(1) = e u(0) : Thus, if we consider as reference state 1 2 L2(0; 1), where 1 represents the constant function equal to 1, the distribution of Aθ is given by the following 2 Theorem 2.1. The self-adjoint extension Aθ of the operator A on L (0; 1) defined by (2.1), with respect to the state 1, has the distribution δ0 when θ = 0 and X 2(1 − cos θ) · δ (2.2) (θ + 2kπ)2 θ+2kπ k2Z when 0 < θ < 2π. Proof. We first study the point spectrum of the self-adjoint operator Aθ. To this end we solve the eigenvalue problem 0 Aθu = iu = λu, u 2 D(Aθ); λ 2 C; SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS 3 finding solutions −iλx uλ(x) = e ; λ = θ + 2kπ; k 2 Z: Then, we can calculate the probability mass of Aθ at points λ = θ + 2kπ, in the state 1. For θ =6 0 we find − 2 − 2 sin θ i(1 cos θ) 2(1 cos θ) h1;Aθ (fλg) 1i = jhuλ; 1ij = + = (2.3) θ + 2kπ θ + 2kπ (θ + 2kπ)2 Noting that, for all θ the sequence of functions (uθ+2kπ)k2Z forms a complete 2 orthogonal system in L (0; 1), and so we found the spectral resolution of Aθ, this proves (2.2). As a byproduct we find the summation formula X 2(1 − cos θ) = 1 (θ + 2kπ)2 k2Z for all θ =6 0. When θ = 0, we have ( 1; k = 0 h1;A0 (f2kπg) 1i = (2.4) 0; k 2 Z n f0g namely, A0 has the distribution δ0. One may also notice that, if we let θ ! 0 in (2.3), it gives ( 2(1 − cos θ) 1; k = 0 lim = θ!0 (θ + 2kπ)2 0; k 2 Z n f0g in other words, as a random variable, Aθ converges in distribution to A0 as θ ! 0. Remark 2.2. One can get the same conclusion by a probabilistic method first com- puting explicitly the characteristic function( ) and then the density by L´evyinversion itAθ formula. Indeed the unitary group e t2R solves the transport equation @u(t; x) @u(t; x) = iA u(t; x) = − ; 8t 2 R; 8x 2 (0; 1) @t θ @x with initial value u(0; x) = u(x) and boundary condition u(t; 1) = e−iθu(t; 0). If we assume limt!0+ u(t; 1) = u(1), then the solution will be determined by ( u(t; x) = u(x − t) (2.5) u(t; 1) = e−iθu(t; 0) Solving the differential equation we find itAθ ([t]+1)iθ [t]iθ e 1(x) = e 1[0;t−[t]](x) + e 1]t−[t];1](x) where [t] denotes the integer part of the real number t and 1A the indicator function of a Borel set A. Similarly, for t < 0, itAθ [t]iθ ([t]−1)iθ e 1(x) = e 1[0;t+1−[t]](x) + e 1]t+1−[t];1](x) 4 FRANCO FAGNOLA AND ZHENG LI itA Then, the Fourier transform ϕ(t) = h1; e θ 1i of the distribution of Aθ with respect to the state 1 becomes ( e([t]+1)iθ(t − [t]) + e[t]iθ(1 − (t − [t])); t > 0 ϕ(t) = (2.6) e([t]−1)iθ(−(t − [t])) + e[t]iθ(1 + (t − [t])); t < 0 In order to find the discrete part of the distribution we apply the L´evyinversion formula computing Z Z n nX−1 k+1 [ ] e−ityϕ(t)dt = e−ity ek+1θ(t − k) + eikθ(1 − t + k) dt − n k=−n k − [ ] nX1 1 + iy − eiy 1 − iy − e−iy = ei(k+1)(θ−y) + eik(θ−y) y2 y2 k=−n for n ≥ 1. Recalling the summation formulae ( nX−1 ik(θ−y) 2n if y = θ + 2hπ e = ein(θ−y)−e−in(θ−y) − if y =6 θ + 2hπ k=−n ei(θ y)−1 for y = θ we find Z [ ] 1 n 1 1 + iy − eiy 1 − iy − e−iy lim e−ityϕ(t)dt = lim · 2n · + !1 !1 2 2 n 2n −n n 2n y y 2(1 − cos y) 2(1 − cos θ) = = : y2 (θ + 2hπ)2 The above limit is 0 for y =6 θ and we recover the result of Theorem 2.1.
Load more

Recommended publications
  • Chapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
    Chapter 12. Ordinary Differential Equation Boundary Value (BV) Problems In this chapter we will learn how to solve ODE boundary value problem. BV ODE is usually given with x being the independent space variable. y p(x) y q(x) y f (x) a x b (1a) and the boundary conditions (BC) are given at both end of the domain e.g. y(a) = and y(b) = . They are generally fixed boundary conditions or Dirichlet Boundary Condition but can also be subject to other types of BC e.g. Neumann BC or Robin BC. 14.1 LINEAR FINITE DIFFERENCE (FD) METHOD Finite difference method converts an ODE problem from calculus problem into algebraic problem. In FD, y and y are expressed as the difference between adjacent y values, for example, y(x h) y(x h) y(x) (1) 2h which are derived from the Taylor series expansion, y(x+h) = y(x) + h y(x) + (h2/2) y(x) + … (2) y(x‐h) = y(x) ‐ h y(x) + (h2/2) y(x) + … (3) Kosasih 2012 Chapter 12 ODE Boundary Value Problems 1 If (2) is added to (3) and neglecting the higher order term (O (h3)), we will get y(x h) 2y(x) y(x h) y(x) (4) h 2 The difference Eqs. (1) and (4) can be implemented in [x1 = a, xn = b] (see Figure) if few finite points n are defined and dividing domain [a,b] into n‐1 intervals of h which is defined xn x1 h xi1 xi (5) n -1 [x1,xn] domain divided into n‐1 intervals.
    [Show full text]
  • Fokas's Unified Transform Method for Linear Systems
    QUARTERLY OF APPLIED MATHEMATICS VOLUME LXXVI, NUMBER 3 SEPTEMBER 2018, PAGES 463–488 http://dx.doi.org/10.1090/qam/1484 Article electronically published on September 28, 2017 FOKAS’S UNIFIED TRANSFORM METHOD FOR LINEAR SYSTEMS By BERNARD DECONINCK (Department of Applied Mathematics, University of Washington, Campus Box 353925, Seattle, WA, 98195 ), QI GUO (Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095-1555 ), ELI SHLIZERMAN (Department of Applied Mathematics & Department of Electrical Engineering, University of Washington, Campus Box 353925, Seattle, WA, 98195 ), and VISHAL VASAN (International Centre for Theoretical Sciences, Bengaluru, India) Abstract. We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial dif- ferential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, as they arise symmetrically in the global relations. This allows the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. Wave equations are treated separately in an appendix. 1. Introduction. The Unified Transform Method (UTM) or Method of Fokas pre- sents a new approach to the solution of boundary value problems (BVPs) for integrable nonlinear equations [6,8], or even more successfully for linear, constant coefficients partial differential equations (PDEs) [5, 8]. In this latter context, the method has resulted in new results for one-dimensional problems involving more than two spatial derivatives [7,8], for elliptic problems [3,8], and for interface problems [4,11], to name but a few.
    [Show full text]
  • Reaction-Diffusion Equations with Applications
    Reaction-Diffusion equations with applications Christina Kuttler Sommersemester 2011 Contents 1 Reaction & Diffusion 3 1.1 Introduction....................................... ..... 3 1.2 Examples for typical reactions . ...... 3 1.2.1 Population dynamics . 3 1.2.2 Predator-prey . 4 1.2.3 Competition .................................... ... 4 1.2.4 Symbiosis ....................................... 4 1.2.5 Chemical reactions . 5 1.3 Diffusionequation .................................... .... 9 1.3.1 Fick’s law / Conservation of mass . ....... 9 1.3.2 (Weak) Maximum principle . 10 1.3.3 Uniqueness . 11 1.3.4 Fundamental solution of the Diffusion equation . 12 1.3.5 Including a source . 14 1.3.6 Random motion and the Diffusion equation . 15 1.3.7 “Time”ofDiffusion ................................. 17 2 Mathematical basics / Overview 19 2.1 Some necessary basics from ODE theory . 19 2.2 Boundary conditions . 21 2.3 Separation of Variables . ...... 22 2.4 Similaritysolutions................................ ........ 24 2.5 Blowup........................................... 26 3 Comparison principles 29 4 Existence and uniqueness 31 4.1 Problemformulation ................................. ...... 31 4.2 Abstract linear problems . 33 4.3 Abstract semilinear problems . 34 4.4 Existence and uniqueness for reaction-diffusion problems . 37 4.5 Globalsolutions.................................... ...... 39 5 Stationary solutions / Stability 40 5.1 Basics ........................................... 40 5.2 Example: The spruce budworm model . 41 6 Reaction-diffusion systems 50 6.1 What can be expected from Reaction-diffusion systems? . 50 6.2 Conservativesystems................................ ....... 50 6.2.1 Lotka-Volterrasystem . ..... 50 6.3 Some more basics for Reaction-diffusion systems . ..... 52 7 Waves 57 7.1 Planewaves........................................ 57 7.1.1 Introduction ..................................... 57 7.1.2 Different types of waves . 64 7.1.3 Radially symmetric wave .
    [Show full text]
  • On Resonances in Transversally Vibrating Strings Induced by an External Force and a Time- Dependent Coefficient in a Robin Boundary Condition
    Delft University of Technology On resonances in transversally vibrating strings induced by an external force and a time- dependent coefficient in a Robin boundary condition Wang, Jing; van Horssen, Wim T.; Wang, Jun-Min DOI 10.1016/j.jsv.2021.116356 Publication date 2021 Document Version Final published version Published in Journal of Sound and Vibration Citation (APA) Wang, J., van Horssen, W. T., & Wang, J-M. (2021). On resonances in transversally vibrating strings induced by an external force and a time-dependent coefficient in a Robin boundary condition. Journal of Sound and Vibration, 512, 1-16. [116356]. https://doi.org/10.1016/j.jsv.2021.116356 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10. Journal of Sound and Vibration 512 (2021) 116356 Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi On resonances in transversally vibrating strings induced by an external force and a time-dependent coefficient in a Robin boundary condition Jing Wang a,b,<, Wim T.
    [Show full text]
  • Inverse Sturm-Liouville Problems Using Multiple Spectra Mihaela Cristina Drignei Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2008 Inverse Sturm-Liouville problems using multiple spectra Mihaela Cristina Drignei Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Drignei, Mihaela Cristina, "Inverse Sturm-Liouville problems using multiple spectra" (2008). Retrospective Theses and Dissertations. 15694. https://lib.dr.iastate.edu/rtd/15694 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Inverse Sturm-Liouville problems using multiple spectra by Mihaela Cristina Drignei A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Applied Mathematics Program of Study Committee: Paul E. Sacks, Major Professor Fritz Keinert Justin Peters James Wilson Stephen Willson Iowa State University Ames, Iowa 2008 Copyright c Mihaela Cristina Drignei, 2008. All rights reserved. 3316216 Copyright 2008 by Drignei, Mihaela Cristina All rights reserved 3316216 2008 ii TABLE OF CONTENTS LIST OF FIGURES . v ABSTRACT . vii CHAPTER 1. INTRODUCTION . 1 1.1 A short history of direct and inverse Sturm-Liouville problems . 1 1.2 The work of Pivovarchik 1999 . 4 1.3 The contributions of this thesis . 5 1.4 A motivation for the inverse three spectra problem . 6 1.5 Inverse problems on graphs .
    [Show full text]
  • Optimization of the Shape of Regions Supporting Boundary Conditions Charles Dapogny, Nicolas Lebbe, Edouard Oudet
    Optimization of the shape of regions supporting boundary conditions Charles Dapogny, Nicolas Lebbe, Edouard Oudet To cite this version: Charles Dapogny, Nicolas Lebbe, Edouard Oudet. Optimization of the shape of regions sup- porting boundary conditions. Numerische Mathematik, Springer Verlag, 2020, 146, pp.51-104. 10.1007/s00211-020-01140-0. hal-02064477 HAL Id: hal-02064477 https://hal.archives-ouvertes.fr/hal-02064477 Submitted on 11 Mar 2019 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. OPTIMIZATION OF THE SHAPE OF REGIONS SUPPORTING BOUNDARY CONDITIONS C. DAPOGNY1, N. LEBBE1,2, E. OUDET1 1 Univ. Grenoble Alpes, CNRS, Grenoble INP1, LJK, 38000 Grenoble, France 2 Univ. Grenoble Alpes, CEA, LETI, F38000 Grenoble, France Abstract. This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a `physical' partial differential equation. At first, we analyze a model situation involving the solution uΩ to a Laplace equation in a domain Ω; the boundary @Ω is divided into three parts ΓD, Γ and ΓN , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative J0(Ω)(θ) of a general objective function J(Ω) of the domain is calculated in the framework of Hadamard's method when the considered deformations θ are allowed to modify the geometry of ΓD, Γ and ΓN (i.e.
    [Show full text]
  • A New Derivation of Robin Boundary Conditions Through Homogenization of a Stochastically Switching Boundary∗
    SIAM J. APPLIED DYNAMICAL SYSTEMS c 2015 Society for Industrial and Applied Mathematics Vol. 14, No. 4, pp. 1845–1867 A New Derivation of Robin Boundary Conditions through Homogenization of a Stochastically Switching Boundary∗ † ‡ Sean D. Lawley and James P. Keener Abstract. We give a new derivation of Robin boundary conditions and interface jump conditions for the diffusion equation in one dimension. To derive a Robin boundary condition, we consider the diffusion equation with a boundary condition that randomly switches between a Dirichlet and a Neumann condition. We prove that, in the limit of infinitely fast switching rate with the proportion of time spent in the Dirichlet state, denoted by ρ, approaching zero, the mean of the solution satisfies a Robin condition, with conductivity parameter determined by the rate at which ρ approaches zero. We carry out a similar procedure to derive an interface jump condition by considering the diffusion equation with a no flux condition in the interior of the domain that is randomly imposed/removed. Our results also provide the effective deterministic boundary condition for a randomly switching boundary. Key words. Robin boundary conditions, interface jump conditions, stochastic hybrid systems, piecewise deter- ministic Markov process, random PDEs AMS subject classifications. 35R60, 35A99, 37H99, 60J99, 92C40 DOI. 10.1137/15M1015182 1. Introduction. The Robin boundary condition (also known as third type, impedance, radiation, convective, partially absorbing, or reactive condition) for a partial differential equa- tion (PDE) enjoys a long history in science and engineering applications. While the Dirichlet boundary condition specifies the value of the solution and the Neumann boundary condition specifies the value of the derivative of the solution, the Robin boundary condition specifies a relationship between the solution and its derivative.
    [Show full text]
  • Lecture Notes
    Lecture notes May 2, 2017 Preface These are lecture notes for MATH 247: Partial differential equations with the sole purpose of providing reading material for topics covered in the lectures of the class. Several examples used here are reproduced from Partial Differential Equations, an Introduction: Walter Strauss. Notation Default notation unless mentioned. • Independent variables: x; y; z (Spatial variables), t (time variable) • Functions: u; f; g; h; φ, @u @u • ux = @x , uy = @y 1 Lec 1 1.1 What is a PDE? Definition 1.1. PDE Consider an unknonw function u(x; y; : : :) of several independent variables (x; y; : : :). A partial dif- ferential equation is a relation/identity/equation which relates the independent variables, the unknown function u and partial derivatives of the function u. Definition 1.2. Order of a PDE The order of a PDE is the highest derivative that appears in the equation. The most general first order partial differential equation of two independent variables can be expressed as F (x; y; u; ux; uy) = 0 : (1) Similarly, the most general second order differential equation can be expressed as F (x; y; u; ux; uy; uxx; uyy; uxy) = 0 : (2) Example 1.1. Here are a few of examples of first order partial differential equations: ut + ux = 0 ; Transport equation: (3) ut + uux = 0 ; Burger’s equation – model problem for shock waves (4) utux = sin (t + x) : (5) Example 1.2. Here are a few examples of second order differential equations uxx + uyy + uzz = 0 Laplace’s equation – Governing equations for electrostatics, potential flows, . : (6) ut = uxx + uyy ; Heat equation/Diffusion equation : (7) utt = uxx ; Vibrating string.
    [Show full text]
  • Boundary Conditions for Macroscale Waves in an Elastic System with Microscale Heterogeneity
    Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity Chen Chen∗ A. J. Robertsy J. E. Bunderz October 16, 2018 Abstract Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially resulting in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problems. Contents 1 Introduction 2 2 A spring-mass system with microscale structure 3 3 Macroscale modelling in the interior 6 3.1 Fourier transform establishes the basis . .6 3.2 There exists a slow manifold model . .8 arXiv:1603.06686v1 [math.DS] 22 Mar 2016 3.2.1 There exists eigenvalues of zero with multiplicity one .8 3.2.2 All nonzero eigenvalues are real and positive . .8 3.2.3 All positive eigenvalues are separated from zero . .9 ∗School of Mathematical Sciences, University of Adelaide, South Australia 5005, Aus- tralia. mailto:[email protected] ySchool of Mathematical Sciences, University of Adelaide, South Australia 5005, Aus- tralia. mailto:[email protected] zSchool of Mathematical Sciences, University of Adelaide, South Australia 5005, Aus- tralia.
    [Show full text]
  • Robin Boundary Conditions for the Hodge Laplacian and Maxwell's
    Robin Boundary Conditions for the Hodge Laplacian and Maxwell's Equations A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Bo Yang IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Advisor: Prof. Douglas N. Arnold September, 2019 c Bo Yang 2019 ALL RIGHTS RESERVED Acknowledgements I am most grateful to my advisor Prof. Douglas N. Arnold. His enthusiasm toward mathematics, his superb lectures, his patience have inspired me greatly throughout my graduate study. Without his encouragement and support, it would be impossible for me to overcome many hard times and complete this work. I am also thankful to my family for their unconditional support for years. i Dedication To my family. ii Abstract In this thesis we study two problems. First, we generalize the Robin boundary condition for the scalar Possoin equation to the vector case and derive two kinds of general Robin boundary value problems. We propose finite elements for these problems, and adapt the finite element exterior calculus (FEEC) framework to analyze the methods. Second, we study the time-harmonic Maxwell's equations with impedance boundary condition. We work with the function space H (curl) consisting of L2 vector fields whose curl are square integrable in the domain and whose tangential traces are square integrable on the boundary. We will show convergence of our numerical solution using H (curl)- conforming finite element methods. iii Contents Acknowledgements i Dedication ii Abstract iii List of Tables vii 1 Introduction 1 1.1 Poisson problem and Hodge Laplacian . .2 1.2 FEEC theory .
    [Show full text]