DOCSLIB.ORG

Quantum Mechanics in Rigged Hilbert Space Language

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Mechanics in Rigged Hilbert Space Language Quantum Mechanics in Rigged Hilbert Space Language by Rafael de la Madrid Modino DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Physics DEPARTAMENTO DE FISICA TEORICA FACULTAD DE CIENCIAS. UNIVERSIDAD DE VALLADOLID May 2001 c 2002 Rafael de la Madrid Modino Last updated: March 28, 2002 To my parents, to my siblings, and to those who always believed in me If you are lucky enough to have lived in Austin as a young man, then wherever you go for the rest of your life, it stays with you, for Austin is a movable feast Acknowledgments To write a dissertation is an experience that is at once challenging and rewarding. Although one has total control over his work, this dissertation would not have been possible without the help of many people. Here they are: My deepest appreciation to my supervisor Professor Manuel Gadella for his support, for the insight into the RHS that I gained from him, and for letting me do my own thing. For the past three years, I have been fortunate to work at the University of Texas at Austin. It has been the fulfillment of all my dreams. I have been in Austin because of one single reason: Professor Arno Bohm. Thanks from the bottom of my heart. Love and affection to my family for the enormous sacrifice they have made and for their extraordinary sense of world wonder. Many friends have helped me keep my mental balance. Especially Al for opening a whole new world to me, Arturo for putting me up so many times, Toro for being always so sweet, and Casey for her love. Special thanks to Ram´on Trecet for his radio show “Dialogos 3,” that is definitely one of the greatest influences on my life. Also thanks to the the rest of the radio shows in Radio 3. So sorry I can’t listen to you that often any more. My gratitude to my coworkers for the exchange of ideas; especially to Manish and Nathan for their help with the English, to Luismi and Jose Manuel for their help with the application forms, to Mariano del Olmo for his advise on dissertation style and to the students of Phy 381N and Steve Newberry for proofreading Chapters 2 and 3 and for making invaluable suggestions. Thanks also to Prof. Galindo for invaluable suggestions, which prompted this revised version of the dissertation. Finally, thanks so much to the city of Austin for being a movable feast. This dissertation was typeset with LATEX1 by the author. 1Latex is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEXProgram. Contents 1 Introduction 1 1.1 A Brief History of the Rigged Hilbert Space . 3 1.2 HarmonicOscillator................................ 5 1.3 A Rigged Hilbert Space of the Square Barrier Potential . .... 9 1.4 ScatteringofftheSquareBarrierPotential . .... 12 1.5 The Gamow Vectors of the Square Barrier Potential Resonances....... 15 1.6 TimeReversal................................... 19 1.7 Synopsis...................................... 20 2 Mathematical Framework of Quantum Mechanics 23 2.1 LinearSpaces ................................... 25 2.1.1 Introduction................................ 25 2.1.2 LinearSpacesandScalarProduct . 25 2.1.3 LinearOperators ............................. 28 2.1.4 Antilinear Functionals . 32 2.2 TopologicalSpaces ................................ 34 2.2.1 Introduction................................ 34 2.2.2 OpenSetsandNeighborhoods. 34 2.2.3 SeparationAxioms ............................ 37 2.2.4 ContinuityandHomeomorphicSpaces . 38 2.3 LinearTopologicalSpaces ............................ 39 2.3.1 Introduction................................ 39 2.3.2 CauchySequences............................. 41 2.3.3 Normed, Scalar Product and Metric Spaces . 43 2.3.4 Continuous Linear Operators and Continuous Antilinear Functionals 45 2.4 CountablyHilbertSpaces ............................ 49 2.4.1 Introduction................................ 49 2.4.2 Dual Space of a Countably Hilbert Space . 53 2.4.3 Countably Hilbert Spaces in Quantum Mechanics . 54 2.5 LinearOperatorsonHilbertSpaces . 56 2.5.1 Introduction................................ 56 2.5.2 BoundedOperatorsonaHilbertSpace . 57 2.5.3 UnboundedOperatorsonaHilbertSpace. 62 xi xii Contents 2.6 Nuclear Rigged Hilbert Spaces . 65 2.6.1 Introduction................................ 65 2.6.2 Nuclear Rigged Hilbert Spaces . 66 3 The Rigged Hilbert Space of the Harmonic Oscillator 69 3.1 Introduction.................................... 71 3.2 AlgebraicOperations ............................... 72 3.3 ConstructionoftheTopologies. 77 3.3.1 Introduction................................ 77 3.3.2 HilbertSpaceTopology.......................... 78 3.3.3 NuclearTopology............................. 83 3.3.4 Physical Interpretation of Ψ, Φ and ................. 85 H 3.3.5 ExtensionoftheAlgebraofOperators . 85 3.4 The RHS of the Harmonic Oscillator . 89 3.4.1 TheConjugateSpace........................... 89 3.4.2 Construction of the Rigged Hilbert Space . 91 3.4.3 Continuous Linear Operators on the Rigged Hilbert Space . 93 3.5 Basis Systems, Eigenvector Decomposition and the Gelfand-Maurin Theorem 94 3.5.1 Basis Systems and Eigenvector Decomposition—a Heuristic Introduc- tion .................................... 94 3.5.2 Gelfand-MaurinTheorem . 104 3.6 Gelfand-Maurin Theorem Applied to the Harmonic Oscillator . 109 3.6.1 Spectral Theorem Applied to the Energy Operator . 109 3.6.2 Spectral Theorem Applied to the Position and Momentum Operators 110 3.6.3 Realizations of the RHS of the Harmonic Oscillator by Spaces of Func- tions....................................117 3.6.4 Summary ................................. 125 3.7 A Remark Concerning Generalizations . 128 3.7.1 Realization of the Abstract RHS by Spaces of Functions . 128 3.7.2 General Statement of the Gelfand-Maurin Theorem . 133 3.7.3 Generalization of the Algebra of Operators . 134 3.7.4 Appendix: Continuity of the Algebra of the Harmonic Oscillator . 135 4 A Rigged Hilbert Space of the Square Barrier Potential 137 4.1 Introduction.................................... 139 4.2 Sturm-Liouville Theory Applied to the Square Barrier Potential . ..... 141 4.2.1 Schr¨odinger Equation in the Position Representation . 141 4.2.2 Self-AdjointExtension . 142 4.2.3 ResolventandGreenFunctions . 143 4.2.4 Diagonalization of H and Eigenfunction Expansion . 146 4.2.5 TheNeedoftheRHS........................... 152 4.2.6 Construction of the Rigged Hilbert Space . 154 4.2.7 DiracBasisVectorExpansion . 155 Contents xiii 4.2.8 EnergyRepresentationoftheRHS . 156 4.2.9 Meaning of the δ-normalization of the Eigenfunctions . 157 4.3 ConclusiontoChapter4 ............................. 159 4.4 AppendicestoChapter4............................. 160 4.4.1 Appendix 1: Self-Adjoint Extension . 160 4.4.2 Appendix 2: Resolvent and Green Function . 161 4.4.3 Appendix 3: Diagonalization and Eigenfunction Expansion . 163 4.4.4 Appendix4: ConstructionoftheRHS . 165 4.4.5 Appendix 5: Dirac Basis Vector Expansion . 167 4.4.6 Appendix6: EnergyRepresentationoftheRHS . 168 5 Scattering off the Square Barrier Potential 171 5.1 Introduction.................................... 173 5.2 Lippmann-SchwingerEquation. 174 5.2.1 Lippmann-SchwingerKets . 174 5.2.2 Radial Representation of the Lippmann-Schwinger Equation . 175 5.2.3 Solution of the Radial Lippmann-Schwinger Equation . 177 5.2.4 Direct Integral Decomposition Associated to the In-States . ..... 178 5.2.5 Direct Integral Decomposition Associated to the Observables ..... 183 5.3 Construction of the Lippmann-Schwinger Kets and Dirac Basis Vector Ex- pansion ......................................185 5.4 S-matrixandMøllerOperators . 188 5.5 AppendicestoChapter5............................. 190 5.5.1 Appendix 7: Free Hamiltonian . 190 5.5.2 Appendix8: SpacesofHardyFunctions . 198 6 The Gamow Vectors of the Square Barrier Potential Resonances 203 6.1 Introduction.................................... 205 6.2 S-matrixResonances ............................... 206 6.3 TheGamowVectors ............................... 208 6.3.1 Lippmann-Schwinger Equation of the Gamow Vectors . 208 6.3.2 The Gamow Vectors in Position Representation . 209 6.3.3 TheGamowVectorsinEnergyRepresentation . 212 6.4 ComplexBasisVectorExpansion . 216 6.5 Semigroup Time Evolution of the Gamow Vectors . 217 6.6 Time Asymmetry of the Purely Outgoing Boundary Condition . 219 6.6.1 Outgoing Boundary Condition in Phase . 219 6.6.2 Outgoing Boundary Condition in Probability Density . 220 6.7 ExponentialDecayLawoftheGamowVectors . 221 6.8 ConclusiontoChapter6 ............................. 223 6.9 Appendix9:Figures ............................... 224 xiv Contents 7 The Time Reversal Operator in the Rigged Hilbert Space 227 7.1 Introduction.................................... 229 7.2 The Standard Time Reversal Operator (ǫT = ǫI =1) ............. 230 7.3 The Time Reversal Doubling (ǫ = ǫ = 1)..................234 T I − 7.4 Appendix10:TimeReversal. .. .. 238 8 Conclusions 245 Chapter 1 Introduction In this chapter, we sketch the contents of this dissertation. These contents will be mostly concerned with the properties of Dirac kets, Lippmann-Schwinger kets, and Gamow vectors. Jim looked at the trash, and then looked at me, and back at the trash again. He had got the dream fixed so strong in his head that he couldn’t seem to shake it loose and get the facts back into place again, right away. But when he did get the things straightened around, he looked at me steady, without ever smiling, and says: “What do dey stan’ for? I’s gwyne to tell you. When I got all wore out wid work, en
Load more

Recommended publications
  • Quantum Errors and Disturbances: Response to Busch, Lahti and Werner
    entropy Article Quantum Errors and Disturbances: Response to Busch, Lahti and Werner David Marcus Appleby Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia; [email protected]; Tel.: +44-734-210-5857 Academic Editors: Gregg Jaeger and Andrei Khrennikov Received: 27 February 2016; Accepted: 28 April 2016; Published: 6 May 2016 Abstract: Busch, Lahti and Werner (BLW) have recently criticized the operator approach to the description of quantum errors and disturbances. Their criticisms are justified to the extent that the physical meaning of the operator definitions has not hitherto been adequately explained. We rectify that omission. We then examine BLW’s criticisms in the light of our analysis. We argue that, although the BLW approach favour (based on the Wasserstein two-deviation) has its uses, there are important physical situations where an operator approach is preferable. We also discuss the reason why the error-disturbance relation is still giving rise to controversies almost a century after Heisenberg first stated his microscope argument. We argue that the source of the difficulties is the problem of interpretation, which is not so wholly disconnected from experimental practicalities as is sometimes supposed. Keywords: error disturbance principle; uncertainty principle; quantum measurement; Heisenberg PACS: 03.65.Ta 1. Introduction The error-disturbance principle remains highly controversial almost a century after Heisenberg wrote the paper [1], which originally suggested it. It is remarkable that this should be so, since the disagreements concern what is arguably the most fundamental concept of all, not only in physics, but in empirical science generally: namely, the concept of measurement accuracy.
    [Show full text]
  • The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
    The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric.
    [Show full text]
  • A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II: Supmech and Quantum Systems
    A Stepwise Planned Approach to the Solution of Hilbert’s Sixth Problem. II : Supmech and Quantum Systems Tulsi Dass Indian Statistical Institute, Delhi Centre, 7, SJS Sansanwal Marg, New Delhi, 110016, India. E-mail: [email protected]; [email protected] Abstract: Supmech, which is noncommutative Hamiltonian mechanics (NHM) (developed in paper I) with two extra ingredients : positive ob- servable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the ‘CC condition’ [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phe- nomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative su- perselection rules. Traditional features of quantum mechanics of finite parti- cle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr¨odinger wave functions with traditional Born interpretation appear as natural objects for the descrip- tion of their pure states and the Schr¨odinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given. arXiv:1002.2061v4 [math-ph] 18 Dec 2010 1 I. Introduction This is the second of a series of papers aimed at obtaining a solution of Hilbert’s sixth problem in the framework of a noncommutative geome- try (NCG) based ‘all-embracing’ scheme of mechanics.
    [Show full text]
  • Riesz-Like Bases in Rigged Hilbert Spaces, in Preparation [14] Bonet, J., Fern´Andez, C., Galbis, A
    RIESZ-LIKE BASES IN RIGGED HILBERT SPACES GIORGIA BELLOMONTE AND CAMILLO TRAPANI Abstract. The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂H⊂D×[t×]. A Riesz- like basis, in particular, is obtained by considering a sequence {ξn}⊂D which is mapped by a one-to-one continuous operator T : D[t] → H[k · k] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces. 1. Introduction Riesz bases (i.e., sequences of elements ξn of a Hilbert space which are trans- formed into orthonormal bases by some bounded{ } operator withH bounded inverse) often appear as eigenvectors of nonself-adjoint operators. The simplest situation is the following one. Let H be a self-adjoint operator with discrete spectrum defined on a subset D(H) of the Hilbert space . Assume, to be more definite, that each H eigenvalue λn is simple. Then the corresponding eigenvectors en constitute an orthonormal basis of . If X is another operator similar to H,{ i.e.,} there exists a bounded operator T withH bounded inverse T −1 which intertwines X and H, in the sense that T : D(H) D(X) and XT ξ = T Hξ, for every ξ D(H), then, as it is → ∈ easily seen, the vectors ϕn with ϕn = Ten are eigenvectors of X and constitute a Riesz basis for .
    [Show full text]
  • Spectral Theory
    SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensional case. Namely, we first show that an eigenvector exists, and then we show that there is an orthonormal basis of eigenvectors by an inductive argument (in the form of an application of Zorn's lemma). We first recall a few facts about self-adjoint operators. Proposition 1.1. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. (1) If W ⊂ V is a T -invariant subspace, then W ? is T -invariant. (2) For all v 2 V , hT v; vi 2 R. (3) Let Vλ = fv 2 V j T v = λvg. Then Vλ ? Vµ whenever λ 6= µ. We will need one further technical fact that characterizes the norm of a self-adjoint operator. Lemma 1.2. Let V be a Hilbert space, and T : V ! V a bounded, self-adjoint operator. Then kT k = supfjhT v; vij j kvk = 1g. Proof. Let α = supfjhT v; vij j kvk = 1g. Evidently, α ≤ kT k. We need to show the other direction. Given v 2 V with T v 6= 0, setting w0 = T v=kT vk gives jhT v; w0ij = kT vk.
    [Show full text]
  • Quantum Mechanical Observables Under a Symplectic Transformation
    Quantum Mechanical Observables under a Symplectic Transformation of Coordinates Jakub K´aninsk´y1 1Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics. E-mail address: [email protected] March 17, 2021 We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quan- tized version of a finite-dimensional system with configuration space iso- morphic to Rq. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper. 1 Introduction From a mathematical perspective we can view quantum mechanics as a science of finite- dimensional quantum systems, i.e. systems whose classical pre-image has configuration space of finite dimension. The single most important representative from this family is a quantized version of the classical system whose configuration space is isomorphic to Rq with some q N, which corresponds e.g. to the system of finitely many coupled ∈ harmonic oscillators. Classically, the state of such system is given by a point in the arXiv:2007.10858v2 [quant-ph] 16 Mar 2021 phase space, which is a vector space of dimension 2q equipped with the symplectic form. One would typically prefer to work in the abstract setting, without the need to choose any particular set of coordinates: in principle this is possible. In practice though, one will often end up choosing a symplectic basis in the phase space aligned with the given symplectic form and resort to the coordinate description.
    [Show full text]
  • Operators in Rigged Hilbert Spaces: Some Spectral Properties
    OPERATORS IN RIGGED HILBERT SPACES: SOME SPECTRAL PROPERTIES GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI Abstract. A notion of resolvent set for an operator acting in a rigged Hilbert space D⊂H⊂D× is proposed. This set depends on a family of intermediate locally convex spaces living between D and D×, called interspaces. Some properties of the resolvent set and of the correspond- ing multivalued resolvent function are derived and some examples are discussed. 1. Introduction Spaces of linear maps acting on a rigged Hilbert space (RHS, for short) × D⊂H⊂D have often been considered in the literature both from a pure mathematical point of view [22, 23, 28, 31] and for their applications to quantum theo- ries (generalized eigenvalues, resonances of Schr¨odinger operators, quantum fields...) [8, 13, 12, 15, 14, 16, 25]. The spaces of test functions and the dis- tributions over them constitute relevant examples of rigged Hilbert spaces and operators acting on them are a fundamental tool in several problems in Analysis (differential operators with singular coefficients, Fourier trans- arXiv:1309.4038v1 [math.FA] 16 Sep 2013 forms) and also provide the basic background for the study of the problem of the multiplication of distributions by the duality method [24, 27, 34]. Before going forth, we fix some notations and basic definitions. Let be a dense linear subspace of Hilbert space and t a locally convex D H topology on , finer than the topology induced by the Hilbert norm. Then D the space of all continuous conjugate linear functionals on [t], i.e., D× D the conjugate dual of [t], is a linear vector space and contains , in the D H sense that can be identified with a subspace of .
    [Show full text]
  • Mathematical Work of Franciszek Hugon Szafraniec and Its Impacts
    Tusi Advances in Operator Theory (2020) 5:1297–1313 Mathematical Research https://doi.org/10.1007/s43036-020-00089-z(0123456789().,-volV)(0123456789().,-volV) Group ORIGINAL PAPER Mathematical work of Franciszek Hugon Szafraniec and its impacts 1 2 3 Rau´ l E. Curto • Jean-Pierre Gazeau • Andrzej Horzela • 4 5,6 7 Mohammad Sal Moslehian • Mihai Putinar • Konrad Schmu¨ dgen • 8 9 Henk de Snoo • Jan Stochel Received: 15 May 2020 / Accepted: 19 May 2020 / Published online: 8 June 2020 Ó The Author(s) 2020 Abstract In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and influence. Keywords Szafraniec Á Mathematical work Á Biography Mathematics Subject Classification 01A60 Á 01A61 Á 46-03 Á 47-03 1 Biography Professor Franciszek Hugon Szafraniec’s mathematical career began in 1957 when he left his homeland Upper Silesia for Krako´w to enter the Jagiellonian University. At that time he was 17 years old and, surprisingly, mathematics was his last-minute choice. However random this decision may have been, it was a fortunate one: he succeeded in achieving all the academic degrees up to the scientific title of professor in 1980. It turned out his choice to join the university shaped the Krako´w mathematical community. Communicated by Qingxiang Xu. & Jan Stochel [email protected] Extended author information available on the last page of the article 1298 R. E. Curto et al. Professor Franciszek H. Szafraniec Krako´w beyond Warsaw and Lwo´w belonged to the famous Polish School of Mathematics in the prewar period.
    [Show full text]
  • Generalized Eigenfunction Expansions
    Universitat¨ Karlsruhe (TH) Fakult¨atf¨urMathematik Institut f¨urAnalysis Arbeitsgruppe Funktionalanalysis Generalized Eigenfunction Expansions Diploma Thesis of Emin Karayel supervised by Prof. Dr. Lutz Weis August 2009 Erkl¨arung Hiermit erkl¨areich, dass ich diese Diplomarbeit selbst¨andigverfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Emin Karayel, August 2009 Anschrift Emin Karayel Werner-Siemens-Str. 21 75173 Pforzheim Matr.-Nr.: 1061189 E-Mail: [email protected] 2 Contents 1 Introduction 5 2 Eigenfunction expansions on Hilbert spaces 9 2.1 Spectral Measures . 9 2.2 Phillips Theorem . 11 2.3 Rigged Hilbert Spaces . 12 2.4 Basic Eigenfunction Expansion . 13 2.5 Extending the Operator . 15 2.6 Orthogonalizing Generalized Eigenvectors . 16 2.7 Summary . 21 3 Hilbert-Schmidt-Riggings 23 3.1 Unconditional Hilbert-Schmidt Riggings . 23 3.1.1 Hilbert-Schmidt-Integral Operators . 24 3.2 Weighted Spaces and Sobolev Spaces . 25 3.3 Conditional Hilbert-Schmidt Riggings . 26 3.4 Eigenfunction Expansion for the Laplacian . 27 3.4.1 Optimality . 27 3.5 Nuclear Riggings . 28 4 Operator Relations 31 4.1 Vector valued multiplication operators . 31 4.2 Preliminaries . 33 4.3 Interpolation spaces . 37 4.4 Eigenfunction expansion . 38 4.5 Semigroups . 41 4.6 Expansion for Homogeneous Elliptic Selfadjoint Differential Oper- ators . 42 4.7 Conclusions . 44 5 Appendix 47 5.1 Acknowledgements . 47 5.2 Notation . 47 5.3 Conventions . 49 3 1 Introduction Eigenvectors and eigenvalues are indispensable tools to investigate (at least com- pact) normal linear operators. They provide a diagonalization of the operator i.e.
    [Show full text]
  • A Nonstandard Proof of the Spectral Theorem for Unbounded Self-Adjoint
    A NONSTANDARD PROOF OF THE SPECTRAL THEOREM FOR UNBOUNDED SELF-ADJOINT OPERATORS ISAAC GOLDBRING Abstract. We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab. 1. Introduction Throughout this note, all Hilbert spaces will be over the complex numbers and (following the convention in physics) inner products are conjugate-linear in the first coordinate. The goal of this note is to provide a nonstandard proof of the Spectral theorem for unbounded self-adjoint operators: Theorem 1.1. Suppose that A is an unbounded self-adjoint operator on the Hilbert space H. Then there is a projection-valued measure P : Borel(R) B(H) such that A = id dP. → ZR Here, Borel(R) denotes the σ-algebra of Borel subsets of R and id denotes the identity function on R. All of the terms appearing in the previous theorem will arXiv:2104.01949v1 [math.FA] 5 Apr 2021 be defined precisely in the next section. We refer to the expression A = R id dP as a spectral resolution of A. Thus, the theorem says that every unboundedR self-adjoint operator admits a spectral resolution. In fact, this spectral resolu- tion is unique (that is, the projection-valued measure yielding the resolution is unique), although we will not address the uniqueness issue here. The Spectral theorem for unbounded operators is especially important in quantum mechan- ics, for it provides one with the probability distributions for measuring observ- ables with continuous spectra such as position and momentum.
    [Show full text]
  • Partial Inner Product Spaces, Metric Operators and Generalized Hermiticity
    Partial inner product spaces, metric operators and generalized hermiticity Jean-Pierre Antoine a1 and Camillo Trapani b2 a Institut de Recherche en Math´ematique et Physique Universit´ecatholique de Louvain B-1348 Louvain-la-Neuve, Belgium b Dipartimento di Matematica e Informatica, Universit`adi Palermo I-90123, Palermo, Italy Abstract Motivated by the recent developments of pseudo-hermitian quantum mechanics, we ana- lyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (pip-space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular pip-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism. PACS numbers: 03.65.-w, 03.65.Ca, 02.30.Sa, 02.30.Tb 1 Introduction Pseudo-hermitian quantum mechanics (QM) is a recent, unconventional, approach to QM, based on the use of non-hermitian Hamiltonians, whose hermitian character can be restored by chang- ing the ambient Hilbert space, via a so-called metric operator. Such Hamiltonians are not self- adjoint (this is the proper mathematical term, rather than the physicists’ hermitian), but have a real spectrum, usually discrete. Instead they are in general PT -symmetric, that is, invariant under the joint action of space reflection (P) and complex conjugation (T ).
    [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]