DOCSLIB.ORG

Functional Analysis and Quantum Mechanics: an Introduction for Physicists Kedar S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Functional Analysis and Quantum Mechanics: an Introduction for Physicists Kedar S Fortschritte der Physik Fortschr. Phys. 63, No. 9–10, 644–658 (2015) / DOI 10.1002/prop.201500023 Progress Review Paper of Physics Functional analysis and quantum mechanics: an introduction for physicists Kedar S. Ranade∗ Received 4 May 2015, revised 4 May 2015, accepted 8 June 2015 Published online 20 August 2015 It is the intention of this tutorial to introduce basic We give an introduction to certain topics from functional anal- concepts of functional analysis to physicists and to point ysis which are relevant for physics in general and in particular out their influence on quantum mechanics. We aim at for quantum mechanics. Starting from some examples, we dis- mathematical rigour in terminology, but we will leave out cuss the theory of Hilbert spaces, spectral theory of unbounded proofs. Our approach is not necessarily consistent with a operators, distributions and their applications and present mathematics textbook building up from bottom to top by proving every theorem from some basic axioms. Rather, some facts from operator algebras. We do not give proofs, but we strive at an intuitive understanding from concepts present examples and analogies from physics which should be and analogies known either from basic linear algebra or useful to get a feeling for the topics considered. used in basic quantum mechanics. Prerequisites for reading this tutorial are a knowledge of textbook quantum mechanics (such as wave mechan- ics and Dirac notation) including some basic quantum 1 Introduction optics (harmonic oscillator, ladder operators, coherent states etc.), but excluding quantum field theory; from It is well-known that physics and mathematics are mathematics, knowledge of analysis and linear algebra is closely interconnected sciences. New results in physics required.1 lead to new branches of mathematics, and new concepts from mathematics are used to describe physical situa- tions and to predict new physical results. In the study 2 Hilbert space theory of physics quantum mechanics plays a fundamental role for almost all fields of modern physics: from quan- The formalism of quantum mechanics uses the concept tum optics, quantum information to solid-state physics, of a Hilbert space. In this section we will elaborate on the high-energy physics, particle physics etc. A similar role theory and give a full classification of Hilbert spaces, i. e. is enjoyed in mathematics by functional analysis in the we list (in some sense) all possible Hilbert spaces. The theory of differential equations, numerical methods and mathematics covered here can be found in several text- elsewhere. books on functional analysis, e. g. [1–3]. While every physicist knows quantum mechanics and every mathematician knows functional analysis, they of- ten do not know the connection between these two fields—both of which were founded at the beginning of the 20th century. While every student of physics at- 1 This manuscript is partially based on a series of lectures enti- tends lectures on higher mathematics, such as analysis of tled Mathematische Aspekte der Quantenmechanik presented one or several variables, linear algebra, differential equa- (in german) at the Institut fur¨ Quantenphysik, Universit¨at Ulm, tions and complex analysis, it is not so common that from April to August 2014. he gets to know functional analysis. Though it is possi- Institut fur¨ Quantenphysik, Universit¨at Ulm, and Center for Inte- ble to understand the basics of quantum mechanics with grated Quantum Science and Technology (IQST), Albert-Einstein- pure linear algebra, a deeper understanding is gained Allee 11, D-89081 Ulm, Deutschland, Germany ∗ by knowing at least the very structure of functional Corresponding author: E-mail: [email protected], analysis. Phone: +49/731/50-22783 644 Wiley Online Library C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Fortschritte der Physik Fortschr. Phys. 63, No. 9–10 (2015) Progress Review Paper of Physics 2.1 Introductory examples function on [0; ∞) with a possibly complex eigenvalue α, provided that Im α>0. How is this possible? We start this tutorial by presenting four examples, which show that the na¨ıve use of linear algebra fails in certain situations.2 2.1.4 Eigenvalues of hermitian operators II In the previous example, one may argue that the radius 2.1.1 Commutators and traces cannot be negative and some part of the eigenfunction is cut off. Now we give a more dramatic example. First It is a simple exercise in linear algebra to show that we consider an abstract hermitian operator A with eigen- there holds Tr AB = Tr BA for any two matrices A and B vector v and eigenvalue λ: Av = λv. Hermiticity of A is (prove it by coordinate representation), which may be defined by Ax|y=x|Ay for all x, y ∈ H, and textbook rewritten as a trace of a commutator: Tr [A, B] = 0. In quantum mechanics (cf. e. g. Schwabl [10]) tells us that quantum mechanics, one requires, by analogy to the eigenvalues of hermitian operators are real by the follow- Poisson bracket of classical mechanics, [xˆ, pˆ] = i1H for ing simple argument: we calculate position and momentum operators xˆ and pˆ, respectively (known as canonical quantisation). Taking the trace on ∗ λ v|v=λv|v=Av|v=v|Av=v|λv=λv|v, (1) both sides of this equation yields 0 = i dim H.3 So does quantum mechanics really exist? so v|v = 0 yields λ = λ∗, and this implies λ ∈ R. Now take a spinless pointlike one-dimensional par- 2.1.2 Distributions ticle in position space, which is described by a square- integrable function on R. Consider the hermitian opera- ˆ = 3 + 3 = = ∂ Consider a particle in a box (square-well potential) with tor A xˆ pˆ pˆxˆ with the usual xˆ x and pˆ i ∂x and infinitely high walls: V (√x) = 0, if |x| ≤ a and V (x) =∞ take the function (plotted in the diagram) = 15 2 − 2 otherwise. Let (x) 4a5/2 (a x ) be the wavefunc- 1 −3/2 − 1 tion in the interior part which shall vanish outside f (x) = √ |x| e 4x2 . (2) 2 of the box. If we want to calculate the variance of the energy Hˆ =Hˆ 2−Hˆ2, we need the expectation Setting f (0) := 0, the function f is everywhere defined ˆ 2 ˆ 2 = 4 ∂4 = ˆ 2= R 4 value of H .FromH 4m2 ∂x4 0, we have H on and “smooth”. Further—unlike plane waves, a ∗(x)Hˆ 2(x)dx = 0. Since Hˆ2 is strictly positive, x=−a delta functions and the like—it is square-integrable 2 +∞ − − 1 the variance Hˆ is negative, which obviously is impossi- = 3 2x2 = and normalised: x∈R f (x) dx x=0 x e dx − 1 +∞ − 1 ble. What is wrong here? 2x2 = = 4x2 e x=0 1. For the derivative of g(x): e we find = − 1 −2 · = 1 −3 > g (x) ( 4 x ) g(x) 2 x g(x), thus for x 0, there 2.1.3 Eigenvalues of hermitian operators I holds −3 − / 3 − / − / x Consider a radially-symmetric potential V (r )inthree xˆ3pxˆ 3 2g(x) = x3 − x 5 2 + x 3 2 g(x) and i 2 2 space-dimensions. Using the ansatz (r ) = R(r)Y (ϑ, ϕ) lm (3) in spherical coordinates we get the spherical harmon- u(r) ics Y and by substituting R(r) = we get a radial − / d / lm r pˆxˆ3x 3 2g(x) = x3 2g(x) Schrodinger¨ equation for u(r) by adding a centrifugal i dx 2 + potential l(l 1) to V (r). We thus have a radial position −3 2mr 3 1/2 3/2 x = ∂ = x + x g(x). (4) operator rˆ and a corresponding momentum pˆr i ∂r , i 2 2 which fulfil the canonical commutator relation. Now, pˆr i α is hermitian, but ψ(r) = e r is a normalisable eigen- As f is even, and each power of xˆ and pˆ changes sym- ˆ = metries once, there holds Af i f for the function as a whole. Altogether, Aˆ is a hermitian operator with an 2 For these and other related examples see the articles by Gieres eigenfunction f ,butitseigenvalueisnotarealnumber. [4] and Bonneau et al. [5], which are very much recommended Where is the error? to the reader. 3 For a real commutator (without the imaginary prefactor i), 4 It can be arbitrarily often differentiated. It should not bother us one can use the ladder operators in a harmonic oscillator with that as a complex function there is an essential singularity in [aˆ, aˆ†] = 1 in a similar fashion. the origin, since we only consider real functions here. C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Wiley Online Library 645 Fortschritte der Physik K. S. Ranade: Functional analysis and quantum mechanics—an introduction Progress of Physics sum is, i. e. in the course of calculation we secretly leave 0.7 the Hilbert space for some time, which is not allowed. 0.6 Review Paper 0.5 2.3 Hilbert spaces 0.4 We start with the definition of a Hilbert space. We will use 0.3 Dirac’s bra-ket notation when we find it appropriate. 0.2 Definition 1 (Hilbert space). A pre-Hilbert space is a vec- tor space over some field K equipped with a scalar product 0.1 (or inner product), i. e. a function ·|·: V × V → Kwith the following three properties: 2 4 1. sesquilinearity: x|λv + μw=λx|v+μx|w and Figure 1 A wavefunction with imaginary eigenvalue. λv + μw|x=λ∗v|x+μ∗w|x, 2. anti-symmetry: w|v=v|w∗ and 2.2 Linear algebra and functional analysis 3.
Load more

Recommended publications
  • Studies in the Geometry of Quantum Measurements
    Irina Dumitru Studies in the Geometry of Quantum Measurements Studies in the Geometry of Quantum Measurements Studies in Irina Dumitru Irina Dumitru is a PhD student at the department of Physics at Stockholm University. She has carried out research in the field of quantum information, focusing on the geometry of Hilbert spaces. ISBN 978-91-7911-218-9 Department of Physics Doctoral Thesis in Theoretical Physics at Stockholm University, Sweden 2020 Studies in the Geometry of Quantum Measurements Irina Dumitru Academic dissertation for the Degree of Doctor of Philosophy in Theoretical Physics at Stockholm University to be publicly defended on Thursday 10 September 2020 at 13.00 in sal C5:1007, AlbaNova universitetscentrum, Roslagstullsbacken 21, and digitally via video conference (Zoom). Public link will be made available at www.fysik.su.se in connection with the nailing of the thesis. Abstract Quantum information studies quantum systems from the perspective of information theory: how much information can be stored in them, how much the information can be compressed, how it can be transmitted. Symmetric informationally- Complete POVMs are measurements that are well-suited for reading out the information in a system; they can be used to reconstruct the state of a quantum system without ambiguity and with minimum redundancy. It is not known whether such measurements can be constructed for systems of any finite dimension. Here, dimension refers to the dimension of the Hilbert space where the state of the system belongs. This thesis introduces the notion of alignment, a relation between a symmetric informationally-complete POVM in dimension d and one in dimension d(d-2), thus contributing towards the search for these measurements.
    [Show full text]
  • 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]
  • Asymptotic Spectral Measures: Between Quantum Theory and E
    Asymptotic Spectral Measures: Between Quantum Theory and E-theory Jody Trout Department of Mathematics Dartmouth College Hanover, NH 03755 Email: [email protected] Abstract— We review the relationship between positive of classical observables to the C∗-algebra of quantum operator-valued measures (POVMs) in quantum measurement observables. See the papers [12]–[14] and theB books [15], C∗ E theory and asymptotic morphisms in the -algebra -theory of [16] for more on the connections between operator algebra Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout [1], is integrally related K-theory, E-theory, and quantization. to positive asymptotic morphisms on locally compact spaces In [1], Martinez and Trout showed that there is a fundamen- via an asymptotic Riesz Representation Theorem. Examples tal quantum-E-theory relationship by introducing the concept and applications to quantum physics, including quantum noise of an asymptotic spectral measure (ASM or asymptotic PVM) models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed. A~ ~ :Σ ( ) { } ∈(0,1] →B H I. INTRODUCTION associated to a measurable space (X, Σ). (See Definition 4.1.) In the von Neumann Hilbert space model [2] of quantum Roughly, this is a continuous family of POV-measures which mechanics, quantum observables are modeled as self-adjoint are “asymptotically” projective (or quasiprojective) as ~ 0: → operators on the Hilbert space of states of the quantum system. ′ ′ A~(∆ ∆ ) A~(∆)A~(∆ ) 0 as ~ 0 The Spectral Theorem relates this theoretical view of a quan- ∩ − → → tum observable to the more operational one of a projection- for certain measurable sets ∆, ∆′ Σ.
    [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]
  • Quantum Minimax Theorem
    Quantum Minimax Theorem Fuyuhiko TANAKA July 18, 2018 Abstract Recently, many fundamental and important results in statistical decision the- ory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples. In the present paper, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by Le Cam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics. 1 Introduction Quantum statistical inference is the inference on a quantum system from relatively small amount of measurement data. It covers also precise analysis of statistical error [34], ex- ploration of optimal measurements to extract information [27], development of efficient arXiv:1410.3639v1 [quant-ph] 14 Oct 2014 numerical computation [10]. With the rapid development of experimental techniques, there has been much work on quantum statistical inference [31], which is now applied to quantum tomography [1, 9], validation of entanglement [17], and quantum bench- marks [18, 28]. In particular, many fundamental and important results in statistical deci- sion theory [38] have been extended to the quantum system. Theoretical framework was originally established by Holevo [20, 21, 22].
    [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]
  • Categorical Characterizations of Operator-Valued Measures
    Categorical characterizations of operator-valued measures Frank Roumen Inst. for Mathematics, Astrophysics and Particle Physics (IMAPP) Radboud University Nijmegen [email protected] The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras. 1 Introduction The logic governing quantum measurements differs from classical logic, and it is still unknown which mathematical structure is the best description of quantum logic. The first attempt for such a logic was discussed in the famous paper [2], in which Birkhoff and von Neumann propose to use the orthomod- ular lattice of projections on a Hilbert space. However, this approach has been criticized for its lack of generality, see for instance [22] for an overview of experiments that do not fit in the Birkhoff-von Neumann scheme. The operational approach to quantum physics generalizes the approach based on pro- jective measurements. In this approach, all measurements should be formulated in terms of the outcome statistics of experiments. Thus the logical and probabilistic aspects of quantum mechanics are combined into a unified description. The basic concept of operational quantum mechanics is an effect on a Hilbert space, which is a positive operator lying below the identity.
    [Show full text]