DOCSLIB.ORG

Math212a1419 the Discrete and the Essential Spectrum, Compact and Relatively Compact Operators, the Schr¨Odingeroperator

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math212a1419 the Discrete and the Essential Spectrum, Compact and Relatively Compact Operators, the Schr¨Odingeroperator Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Shlomo Sternberg November 20, 2014 Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The main results of today's lecture are about the Schr¨odinger operator H = H0 + V . They are: If V is bounded and V 0 as x then ! ! 1 σess(H) = σess(H0): So this is a generalization of the results of the lecture on the square well. I will review some definitions and facts about the discrete and the essential spectrum. If V as x then σess(H) is empty. ! 1 ! 1 Both of these results will require a jazzed up version of Weyl's theorem on the stability of the essential spectrum as worked out in lecture on the square well. This in turn will require some results on compact operators that we did not do in Lecture 3. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. 1 The discrete and the essential spectrum. 2 Finite rank operators. A normal form for norm limits of finite rank operators. 3 Compact operators. 4 Hilbert Schmidt operators Hilbert Schmidt integral operators. Hilbert Schmidt operators in general. 5 Weyl's theorem on the essential spectrum. Applications to Schr¨odingeroperators. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. The complement in σd (H) in σ(H) is called the essential spectrum of H and is denoted by σess(H) or simply by σess when H is fixed in the discussion. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The discrete spectrum and the essential spectrum. Let H be a self-adjoint operator on a Hilbert space H and let σ = σ(H) R denote is spectrum. ⊂ Recall that the discrete spectrum of H is defined to be those eigenvalues λ of H which are of finite multiplicity and are also isolated points of the spectrum. This latter condition says that there is some > 0 such that the intersection of the interval (λ , λ + ) with σ consists of the single point λ . The discrete − f g spectrum of H will be denoted by σd (H) or simply by σd when H is fixed in the discussion. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The discrete spectrum and the essential spectrum. Let H be a self-adjoint operator on a Hilbert space H and let σ = σ(H) R denote is spectrum. ⊂ Recall that the discrete spectrum of H is defined to be those eigenvalues λ of H which are of finite multiplicity and are also isolated points of the spectrum. This latter condition says that there is some > 0 such that the intersection of the interval (λ , λ + ) with σ consists of the single point λ . The discrete − f g spectrum of H will be denoted by σd (H) or simply by σd when H is fixed in the discussion. The complement in σd (H) in σ(H) is called the essential spectrum of H and is denoted by σess(H) or simply by σess when H is fixed in the discussion. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Characterizing the discrete spectrum. If λ σd (H) then for sufficiently small > 0 the spectral 2 projection P = P((λ , λ + )) has the property that it − commutes with H and the restriction of H to the image of P has only λ in its spectrum and hence P(H) is finite dimensional, since the multiplicity of λ is finite by assumption. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. The converse. Conversely, suppose that λ σ(H) and that P(λ , λ + ) is 2 − finite dimensional. This means that in a multiplication spectral representation of H, the subset S(λ−,λ+) of S = σ N × consisting of all (s; n) λ < s < λ + has the property that f j − g L2(S(λ−,λ+); dµ) is finite dimensional. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. L2(S(λ−,λ+); dµ) is finite dimensional. If we write [ S = (λ , λ + ) n (λ−,λ+) − × f g n2N then since M^ L2(S(λ−,λ+)) = L2((λ , λ + ); n ) n − f g we conclude that all but finitely many of the summands on the right are zero, which implies that for all but finitely many n we have µ ((λ , λ + ) n ) = 0: − × f g Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. For all but finitely many n we have µ ((λ , λ + ) n ) = 0: − × f g For each of the finite non-zero summands, we can apply the case N = 1 of the following lemma: Lemma N N Let ν be a measure on R such that L2(R ; ν) is finite dimensional. Then ν is supported on a finite set in the sense that there is some finite set of m distinct points x1;:::; xm each of positive measure and such that the complement of the union of these points has ν measure zero. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Proof. N Partition R into cubes whose vertices have all coordinates of the form t=2r for a vector t with integer coordinates and with integer r and so that this is a disjoint union. The corresponding decomposition of the L2 spaces shows that only finitely many of these cubes have positive measure, and as we increase r the cubes with positive measure are nested downward, and can not increase N in number beyond n = dim L2(R ; ν). Hence they converge in measure to at most n distinct points each of positive ν measure and the complement of their union has measure zero. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. We conclude from this lemma that there are at most finitely many points (sr ; k) with sr (λ , λ + ) which have finite measure in 2 − the multiplicative spectral representation of H, each giving rise to an eigenvector of H with eigenvalue sr , and the complement of these points has measure zero. This shows that λ σd (H). We 2 have proved Proposition. λ σ(H) belongs to σd (H) if and only if there is some > 0 such 2 that P((λ , λ + ))(H) is finite dimensional. − Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Characterizing the essential spectrum This is simply the contrapositive of the preceding proposition: Proposition. λ σ(H) belongs to σess(H) if and only if for every > 0 the space 2 P((λ , λ + ))(H) − is infinite dimensional. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. If 1; : : : ; n is an orthonormal basis of Ran(K) and is any element of H then n X K = (K ; j ) j : j=1 ∗ If we set φj := K j we can write this last equation as n X K = ( ; φj ) j : j=1 Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Finite rank operators. An operator K L(H) is said to be of finite rank if its range, 2 Ran(K), is finite dimensional, in which case the dimension of its range is called the rank of K. Shlomo Sternberg Math212a1419 The discrete and the essential spectrum, Compact and relatively compact operators, The Schr¨odingeroperator. Outline The discrete and the essential spectrum. Finite rank operators. Compact operators. Hilbert Schmidt operators Weyl's theorem on the essential spectrum. Finite rank operators. An operator K L(H) is said to be of finite rank if its range, 2 Ran(K), is finite dimensional, in which case the dimension of its range is called the rank of K.
Load more

Recommended publications
  • Discrete-Continuous and Classical-Quantum
    Under consideration for publication in Math. Struct. in Comp. Science Discrete-continuous and classical-quantum T. Paul C.N.R.S., D.M.A., Ecole Normale Sup´erieure Received 21 November 2006 Abstract A discussion concerning the opposition between discretness and continuum in quantum mechanics is presented. In particular this duality is shown to be present not only in the early days of the theory, but remains actual, featuring different aspects of discretization. In particular discreteness of quantum mechanics is a key-stone for quantum information and quantum computation. A conclusion involving a concept of completeness linking dis- creteness and continuum is proposed. 1. Discrete-continuous in the old quantum theory Discretness is obviously a fundamental aspect of Quantum Mechanics. When you send white light, that is a continuous spectrum of colors, on a mono-atomic gas, you get back precise line spectra and only Quantum Mechanics can explain this phenomenon. In fact the birth of quantum physics involves more discretization than discretness : in the famous Max Planck’s paper of 1900, and even more explicitly in the 1905 paper by Einstein about the photo-electric effect, what is really done is what a computer does: an integral is replaced by a discrete sum. And the discretization length is given by the Planck’s constant. This idea will be later on applied by Niels Bohr during the years 1910’s to the atomic model. Already one has to notice that during that time another form of discretization had appeared : the atomic model. It is astonishing to notice how long it took to accept the atomic hypothesis (Perrin 1905).
    [Show full text]
  • 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 =+∞.
    [Show full text]
  • A 2-Ghz Discrete-Spectrum Waveband-Division Microscopic Imaging System
    Optics Communications 338 (2015) 22–26 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate/optcom A 2-GHz discrete-spectrum waveband-division microscopic imaging system Fangjian Xing, Hongwei Chen n, Cheng Lei, Minghua Chen, Sigang Yang, Shizhong Xie Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronic Engineering, Tsinghua University, Beijing 100084, P.R. China article info abstract Article history: Limited by dispersion-induced pulse overlap, the frame rate of serial time-encoded amplified microscopy Received 26 June 2014 is confined to the megahertz range. Replacing the ultra-short mode-locked pulse laser by a multi-wa- Received in revised form velength source, based on waveband-division technique, a serial time stretch microscopic imaging sys- 1 September 2014 tem with a line scan rate of in the gigahertz range is proposed and experimentally demonstrated. In this Accepted 4 September 2014 study, we present a surface scanning imaging system with a record line scan rate of 2 GHz and 15 pixels. Available online 22 October 2014 Using a rectangular spectrum and a sufficiently large wavelength spacing for waveband-division, the Keywords: resulting 2D image is achieved with good quality. Such a superfast imaging system increases the single- Real-time imaging shot temporal resolution towards the sub-nanosecond regime. Ultrafast & 2014 Elsevier B.V. All rights reserved. Discrete spectrum Wavelength-to-space-to-time mapping GHz imaging 1. Introduction elements with a broadband mode-locked laser to achieve ultrafast single pixel imaging. An important feature of STEAM is the spec- Most of the current research in optical microscopic imaging trum of the broadband laser source.
    [Show full text]
  • 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].
    [Show full text]
  • Improving Frequency Resolution of Discrete Spectra
    AGH University of Science and Technology, Kraków, Poland Faculty of Electrical Engineering, Automatics, Computer Science and Electronics Marek GĄSIOR Improving Frequency Resolution of Discrete Spectra A doctoral dissertation done under guidance of Professor Mariusz ZIÓŁKO Kraków 2006 - II - Marek Gąsior, Improving frequency resolution of discrete spectra Akademia Górniczo-Hutnicza w Krakowie Wydział Elektrotechniki, Automatyki, Informatyki i Elektroniki Marek GĄSIOR Poprawa rozdzielczości częstotliwościowej widm dyskretnych Rozprawa doktorska wykonana pod kierunkiem prof. dra hab. inż. Mariusza ZIÓŁKO Kraków 2006 Marek Gąsior, Improving frequency resolution of discrete spectra - III - - IV - Marek Gąsior, Improving frequency resolution of discrete spectra To my Parents Moim Rodzicom Acknowledgements I would like to thank all those who influenced this work. They were in particular the Promotor, Professor Mariusz Ziółko, who also triggered my interest in digital signal processing already during mastership studies; my CERN colleagues, especially Jeroen Belleman and José Luis González, from whose experience I profited so much. I am also indebted to Jean Pierre Potier, Uli Raich and Rhodri Jones for supporting these studies. Separately I would like to thank my wife Agata for her patience, understanding and sacrifice without which this work would never have been done, as well as my parents, Krystyna and Jan, for my education. To them I dedicate this doctoral dissertation. Podziękowania Pragnę podziękować wszystkim, którzy mieli wpływ na kształt niniejszej pracy. W szczególności byli to Promotor, Pan profesor Mariusz Ziółko, który także zaszczepił u mnie zainteresowanie cyfrowym przetwarzaniem sygnałów jeszcze w czasie studiów magisterskich; moi koledzy z CERNu, przede wszystkim Jeroen Belleman i José Luis González, z których doświadczenia skorzystałem tak wiele.
    [Show full text]
  • Ultra-Weak Time Operators of Schrodinger Operators
    Title Ultra-Weak Time Operators of Schrodinger Operators Author(s) Arai, Asao; Hiroshima, Fumio Annales Henri Poincaré, 18(9), 2995-3033 Citation https://doi.org/10.1007/s00023-017-0586-x Issue Date 2017-09 Doc URL http://hdl.handle.net/2115/71420 Rights The final publication is available at link.springer.com Type article (author version) File Information arai-hiroshima-rev.pdf Instructions for use Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Ultra-Weak Time Operators of SchrÄodinger Operators Asao Arai and Fumio Hiroshima Abstract. In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra- weak time operator are established. As an application of the theorems, it is shown that SchrÄodingeroperators HV with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions f : R ! R such that f(HV ) has an ultra-weak time operator is found. Keywords. Time operators, Weyl relations, CCR, SchrÄodingeroperators. 1. Introduction The present paper concerns a time operator in quantum theory which is de- ¯ned, in a ¯rst stage of cognition, as a symmetric operator canonically con- jugate to a Hamiltonian if it exists. The uncertainty relation which is derived from the canonical commutation relation of a time operator and a Hamil- tonian may be interpreted as a mathematically rigorous form of time-energy uncertainty relation. Moreover time operators may play important roles in quantum phenomena [Miy01, Ara05, Ara08b, AM08b , MME08, MME09].
    [Show full text]
  • Wave Mechanics (PDF)
    WAVE MECHANICS B. Zwiebach September 13, 2013 Contents 1 The Schr¨odinger equation 1 2 Stationary Solutions 4 3 Properties of energy eigenstates in one dimension 10 4 The nature of the spectrum 12 5 Variational Principle 18 6 Position and momentum 22 1 The Schr¨odinger equation In classical mechanics the motion of a particle is usually described using the time-dependent position ix(t) as the dynamical variable. In wave mechanics the dynamical variable is a wave- function. This wavefunction depends on position and on time and it is a complex number – it belongs to the complex numbers C (we denote the real numbers by R). When all three dimensions of space are relevant we write the wavefunction as Ψ(ix, t) C . (1.1) ∈ When only one spatial dimension is relevant we write it as Ψ(x, t) C. The wavefunction ∈ satisfies the Schr¨odinger equation. For one-dimensional space we write ∂Ψ 2 ∂2 i (x, t) = + V (x, t) Ψ(x, t) . (1.2) ∂t −2m ∂x2 This is the equation for a (non-relativistic) particle of mass m moving along the x axis while acted by the potential V (x, t) R. It is clear from this equation that the wavefunction must ∈ be complex: if it were real, the right-hand side of (1.2) would be real while the left-hand side would be imaginary, due to the explicit factor of i. Let us make two important remarks: 1 1. The Schr¨odinger equation is a first order differential equation in time. This means that if we prescribe the wavefunction Ψ(x, t0) for all of space at an arbitrary initial time t0, the wavefunction is determined for all times.
    [Show full text]
  • Spectrum (Functional Analysis) - Wikipedia, the Free Encyclopedia
    Spectrum (functional analysis) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Spectrum_(functional_analysis) Spectrum (functional analysis) From Wikipedia, the free encyclopedia In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand 0 is in the spectrum because the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to densely-defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)−1:X→D.
    [Show full text]
  • Spectral Stability of the Critical Front in the Extended Fisher-KPP Equation
    Spectral stability of the critical front in the extended Fisher-KPP equation Montie Avery∗ Louis Garénaux† September 4, 2020 Abstract We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp t−3/2 decay rate, as previously obtained in the classical Fisher-KPP equation. The main challenges are to regularize the singular perturbation in the extended Fisher-KPP equation and to track eigenvalues near the essential spectrum, and we overcome these difficulties with functional analytic methods. 1 Introduction 1.1 Background and main results We consider the extended Fisher-KPP equation 2 ut = δ uxxxx + uxx + f(u), (1.1) − for small values of the parameter δ. Notice that the sign of δ R will play no role in what follows. Here we ∈ assume f is of Fisher-KPP type: f(0) = f(1) = 0, f 0(0) > 0, f 0(1) < 0, and f 00(u) < 0 for all u (0, 1); see ∈ Section 1.2 for comments on this last assumption. Equation (1.1) may be derived as an amplitude equation near certain co-dimension 2 bifurcations in reaction-diffusion systems [27], and for δ = 0, it reduces to the classical Fisher-KPP equation, which is a fundamental model for front propagation into unstable states.
    [Show full text]
  • A Primary Introduction to Spectral Theory
    A primary introduction to spectral theory Jianwei Yang∗ Assistant Professor at School of Mathematics and Statistics, Beijing Institute of Technology Chaire internationale au LAGA de l’Université Paris 13 En contriburant au projet scientifique du Labex MME-DII Novembre 2019 ∗http://math.bit.edu.cn/szdw/azcpl/tbfyjy/yjw/index.htm i Contents Introduction iii 1 Hilbert spaces and the spectrum of operators 1 1.1 Hilbert space and its subspaces . 1 1.1.1 Hilbert spaces . 1 1.1.2 Subspaces . 5 1.2 Operators on Hilbert spaces . 7 1.3 Spectrum . 11 2 Self-adjointness 13 2.1 Spectrum of self-adjoint operators . 13 2.2 Fundamental criteria for self-adjointness . 15 3 The essential spectrum and Compact operators 17 3.1 Weyl’s criterion on the essential spectrum . 17 3.2 Compact operators . 20 4 Stability of the essential spectrum: Weyl’s theorem 24 4.1 Relatively compact operators . 24 4.2 Weyl’s Theorem . 24 4.3 An application . 26 Bibliography 28 ii Introduction This course will be at very elementary level with prerequisite only on limited knowledge of integration theory and linear algebra. We will start with the theory of Hilbert space and operators on it, focusing our attention on self-adjoint operators and compact operators. After introducing the general notion of spectrum of an operator and its classification, we move on to certain properties of these operators in connection with their spectral behavior, including a simple version of the Fredholm alternative and Weyl’s theorem, which is about the invariance of the essential spectrum of an operator under relatively compact perturbation.
    [Show full text]
  • On the Discrete Spectrum of Exterior Elliptic Problems by Rajan Puri A
    On the discrete spectrum of exterior elliptic problems by Rajan Puri A dissertation submitted to the faculty of The University of North Carolina at Charlotte in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Charlotte 2019 Approved by: Dr. Boris Vainberg Dr. Stanislav Molchanov Dr. Yuri Godin Dr. Hwan C. Lin ii c 2019 Rajan Puri ALL RIGHTS RESERVED iii ABSTRACT RAJAN PURI. On the discrete spectrum of exterior elliptic problems. (Under the direction of Dr. BORIS VAINBERG) In this dissertation, we present three new results in the exterior elliptic problems with the variable coefficients that describe the process in inhomogeneous media in the presence of obstacles. These results concern perturbations of the operator H0 = −div((a(x)r) in an exterior domain with a Dirichlet, Neumann, or FKW boundary condition. We study the critical value βcr of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. −1 2 ˚2 Our main technical tool of the study is the resolvent operator (H0 − λ) : L ! H near point λ = 0: The dependence of βcr on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated. iv ACKNOWLEDGEMENTS It is hard to even begin to acknowledge all those who influenced me personally and professionally starting from the very beginning of my education. First and foremost I want to thank my advisor, Professor Dr.
    [Show full text]
  • Schr¨Odinger Operators with Purely Discrete Spectrum 1
    Methods of Functional Analysis and Topology Vol. 15 (2009), no. 1, pp. 61–66 SCHROD¨ INGER OPERATORS WITH PURELY DISCRETE SPECTRUM BARRY SIMON DedicatedtoA.Ya.Povzner Abstract. We prove that −∆+V has purely discrete spectrum if V ≥ 0 and, for all M, |{x | V (x) <M}| < ∞ and various extensions. 1. Introduction Our main goal in this note is to explore one aspect of the study of Schrod¨ inger operators (1.1) H = −∆+V 1 Rν which we will suppose have V ’s which are nonnegative and in Lloc( ), in which case (see, e.g., Simon [15]) H can be defined as a form sum. We are interested here in criteria under which H has purely discrete spectrum, that is, σess(H)isempty. Thisiswellknownto be equivalent to proving (H +1)−1 or e−sH for any (and so all) s>0 is compact (see [9, Thm. XIII.16]). One of the most celebrated elementary results on Schr¨odinger operators is that this is true if (1.2) lim V (x)=∞. |x|→∞ But (1.2) is not necessary. Simple examples where (1.2) fails but H still has compact resolvent were noted first by Rellich [10]—one of the most celebrated examples is in ν =2,x =(x1,x2), and 2 2 (1.3) V (x1,x2)=x1x2 where (1.2) fails in a neighborhood of the axes. For proof of this and discussions of eigenvalue asymptotics, see [11, 16, 17, 20, 21]. There are known necessary and sufficient conditions on V for discrete spectrum in terms of capacities of certain sets (see, e.g., Maz’ya [6]), but the criteria are not always so easy to check.
    [Show full text]