DOCSLIB.ORG

Rigged Hilbert Spaces and the Properties of Self-Adjoint Operators in Them']

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rigged Hilbert Spaces and the Properties of Self-Adjoint Operators in Them'] Appendix A Rigged Hilbert Spaces and the Properties of Self-Adjoint Operators in them'] A Hilbert space in quantum mechanics, as well as in analysis, arises from completing the space (/J of 'sufficiently good' (for example, smooth and decrea­ sing in infinity) functions with respect to a norm defined by a scalar product. The RHS theory assumes that (/J is the nuclear'] space in which the convergence is defined by a countable system of norms (the rtj)-convergence). If the scalar product (qJ , 1/1) is introduced in (/J continuous relative to the rtj)-convergence, then (/J will not be complete relative to the new convergence in the norm IIqJlI = = J(qJ , qJ) (the r£-convergence). However, (/J can be completed with respect to this new r£-convergence to the Hilbert space £ . The space £ * ofall antilinear functionals on £ is isomorphic to £ itself. The functionals from £ * are also continuous linear functionals on the space (/J c £ . The space (/J* of all linear functionals on (/J turns out to be broader and to include the Hilbert space £. The set of the three spaces embedded densely into each other (A.t) is called the rigged Hilbert space (or the Gel'fand triad). Henceforth in the Appendix we shall use the following notation: small letters g, h, ... denote the vectors from the Hilbert space £ ; the Greek letters qJ, 1/1 , ... denote the vectors from (/J; the capital letters F, R, ... denote the vectors from (/J* . The notation for the scalar product in the sense of RHS (A1) should be understood as follows: (h Ig) is the conventional scalar product in £; h, g E £ ; (qJ IF) = (F I q» is the value of the functional FE (/J* on the vector qJ E (/J (the horizontal bar indicates the complex conjugation). Generally, the quantity (F IG) (where F, G E (/J* , but neither F no G belong to (/J) is not defined. In particular, the norm of the Gamow state in the RHS formalism is not defined. Let a linear operator A be given in £ with the definition domain D(A) :::> (/J such that the operator A*, adjoint of A, is defined in the domain D(A*) :::> (/J . 1) Since Appendix A pertains mainly to Chapter 4, the references here are those from that chapter. 2) For the definition see [22]. APPENDIX A 345 Here, A and A* do not map outside the space cP (i.e. cP is invariant with respect to A and A*): ACP c cP , A*CP c cP . It should be reminded that the operator A* adjoint of A in the Hilbert space is called such an operator that (A*h Ig) = (hi Ag) (A2) at arbitrary 9 E D(A) and hE D(A*) . If A = A* provided D(A) = D(A*), the operator is called self-adjoint. Throughout the space cP* the extension Aofthe operator A is defined by the relation (AF I rp) = (F I A*rp) or (rp IAF) = (A*rp IF) (A3) which must be satisfied for all rp E D(A*) and F E cP* . The extension A+ of the adjoint operator A* may be defined analogously: (A+F I rp) = (F IArp), rp E D(A), FE CP*. (A4) For these extensions it is possible to formulate the generalized problems for the eigenvalues: AR(A) = AR(A) , (A5a) A +L(Ii) = iiL(Ii) (A5b) where the complex number A is called the right eigenvalue of the operator A corresponding to the right generalized eigenvector (GEV) R(A) E cP* ; Ii and L(Ii) are the left eigenvalue and the left GEV of the operator A, respectively. Using the Dirac notation, we can write equations (A5) in a more common manner, whence the meaning of the terms 'right' and 'left' becomes clear: A IR(A) = A IR(A) , (A6a) (L(Ii)1 A = Ii (L(Ii)1 . (A6b) However, it should be remembered that the meaning of these equations must conform to the definitions (A3) and (A4): (A*rp I R(A) = A(rp IR(A), rp E D(A*) ; (A7a) (L(Ii) I Arp) = Ii (L(Ii) Irp), rp E D(A) . (A7b) 346 APPENDIX A IfR(A.) is the right GEV of the operator A belonging to the eigenvalue A., then it proves to be the left GEV ofthe operator A* corresponding to the eigenvalue A. Indeed, we have or i.e. For the self-adjoint operator A = A*, the extensions A and A+ coincide with each other, so any GEV is simultaneously the right vector with the eigenva­ lue A. and the left vector with the eigenvalue A . For the real eigenvalues, in particular, the right and left GEV of the self-adjoint operator are the same. Nevertheless, we retain the differencein the notation (R and L) bearing in mind the continuation of GEV to the complex plane. Furthermore, the right GEV will usually appear in the position of a ket, and the left GEV in the position ofa bra. Generally, the self-adjoint operators in the Hilbert space, e.g. such as the Hamiltonians of physical systems, have not only a discrete, but also a continu­ ous, spectrum; only the eigenvectors of the discrete spectrum belong to Jt1, however. Ifwe use the RHS, we may also treat the (generalized)eigenvectors of the continuous spectrum belonging to (/)*. Moreover, a self-adjoint operator in the RHS has a complete GEV system. The following (Gel'fand-Maurin) nuclear spectral theorem holds [18, 23]: for any self-adjoint operator A in the complex separable Hilbert space Jt1 with the definition domain D(A) there exists a nucle­ ar space (/) E D(A) densely and continuously imbedded in Jt1 which is invariant with respect to A such that the system of generalized eigenvectors of the operator A corresponding to the real eigenvalues from the spectrum a(A) is complete, i.e. for all rp, If/ E cP we get (rp 11f/) = f dp(A.) ~ (rp IRi(A.) (Li(A.) 11f/) . (A8) utA) In fact, this is the conventional spectral expansion of unity expressed in terms of RHS. The sum in (A8) allows for the multiplicity of the spectrum (for example, in the case of a one-particle Hamiltonian, i is a set of quantum numbers of the angular momentum I proper and of its projection m) and the Stieltjes integral taken over the spectrum of the operator A. The physical Hamiltonians have usually a discrete and absolute continuous spectrum APPENDIX A 347 A.. Now, having introduced the Lebesgue measure dPaAA.) = h(A. )dA. , we can write (for simplicity we omit the multiplicity indices (9' I '1') = L ( 9' l.fj) (jj I '1') + j + f(9' I R(A.) (L(A.) I '1') dA. (A9) If where .fj E Jf are the proper eigenvectors ofthe discrete spectrum; R(A.) = L(A.) are the GEV of the continuous spectrum. Going over from (A8) to (A9), we change their normalization in such a way that the weight function h(A.) disappe­ ars. Itshould be remembered again that R(A.) in (A9)coincides with L(A.) because A. E A. is real. Apart from the GEV entering the expansion (A9) and corresponding to the real eigenvalue A. E u(A), there exist other GEV with eigenvalues located both inside and outside the spectrum. In fact, the GEV of the continuous spectrum entering (A9) form an analytic GEV family, i.e. a vector-valued function with the value in (/J*, analytic in the region Q (which includes A.) such that at every A. E Q R(A.) is a GEV and A. is the corresponding eigenvalue. The analyticity of R(A.) means that the function (9' I R(A.) is analytic at any 9' E (/J • Let us examine an isolated singular point A.oof the function R(A.) . From the conventional Laurent expansion of the function (9' IR(A.) the expansion for the GEV arises: (AIO) n= -00 whose coefficients are ~ C = f R(A.) dA. (All) 1 n 21ti (A. - A. t + c o where C is a simple closed and positively orientated curve inside the analyticity region Q of the function R(A.). The integral (All) defines the vector C; E (/J* • The properties of the Laurent series coefficients C; in the expansion of GEV R(A.) of the operator A are similar to the properties of the same Laurent series coefficients in the expansion of the resolvent near its singular points, thereby resulting in (AI2) This follows immediately from (All) and from the weak continuity ofA in (/J*. Thus, if Cn _ 1 = 0, then C; is GEV of the operator A corresponding to A.o. In particular, if A.o is a first-order pole, the first non-zero coefficient is C-1 and 348 APPENDIX A C_2 = o. Therefore, C_1 is a GEV corresponding to the eigenvalue AO' Thus, the first-order-pole residue of the GEV R(A) belonging to continuous spectrum and continued to the complex plane Ais also the GEV of the operator A. This residue in the resonance pole is a Gamow state. Consider the simple example of the Schrodinger equation for spinless partic­ les with central local potential V(r) which behaves 'properly' at the origin and at infinity: V(r) '" (O(r- 3/2+8 ), e > 0; V(r) '" '" (O(r- 3-"), b > O. Any r~O r~oo 2 function 'II(r) E L (1Ii) can be expanded in spherical functions: (A13) and the scalar product is defined as 00 ('III 17J) = r fdr Vilm(r)l7Jlm(r) = r('IIlm I I7Jlm)' lm lm o The functions 'IIlm(r) belong to the Hilbert space L2(R) which is denoted as .Tflm.
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]
  • 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]
  • 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]
  • 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]
  • An Approach to Nonstandard Quantum Mechanics
    An approach to nonstandard quantum mechanics A. Raab St.-Ilgener-Str. 2, 69190 Walldorf Federal Republic of Germany E-mail: [email protected] July 24, 2018 Abstract We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite- dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formula- tion of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to non-relativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scatter- arXiv:math-ph/0612082v1 27 Dec 2006 ing theory, we derive rigorously explicit formulas for the Møller wave operators and the S-Matrix. 1 Introduction Quantum mechanics is conventionally formulated in a complex Hilbert space, . The possible states of a quantum system are associated with unit vectors inH , and the observables are associated with self-adjoint linear operators H on . A central role plays the Hamiltonian of the quantum system. The eigenvaluesH of the Hamiltonian are commonly interpreted as the energies of 1 bound states of the system. Moreover, the values of the continuous part of the Hamiltonians spectrum are interpreted as the energies of ”continuum” or scattering states.
    [Show full text]
  • Unbounded Linear Operators Jan Derezinski
    Unbounded linear operators Jan Derezi´nski Department of Mathematical Methods in Physics Faculty of Physics University of Warsaw Ho_za74, 00-682, Warszawa, Poland Lecture notes, version of June 2013 Contents 3 1 Unbounded operators on Banach spaces 13 1.1 Relations . 13 1.2 Linear partial operators . 17 1.3 Closed operators . 18 1.4 Bounded operators . 20 1.5 Closable operators . 21 1.6 Essential domains . 23 1.7 Perturbations of closed operators . 24 1.8 Invertible unbounded operators . 28 1.9 Spectrum of unbounded operators . 32 1.10 Functional calculus . 36 1.11 Spectral idempotents . 41 1.12 Examples of unbounded operators . 43 1.13 Pseudoresolvents . 45 2 One-parameter semigroups on Banach spaces 47 2.1 (M; β)-type semigroups . 47 2.2 Generator of a semigroup . 49 2.3 One-parameter groups . 54 2.4 Norm continuous semigroups . 55 2.5 Essential domains of generators . 56 2.6 Operators of (M; β)-type ............................ 58 2.7 The Hille-Philips-Yosida theorem . 60 2.8 Semigroups of contractions and their generators . 65 3 Unbounded operators on Hilbert spaces 67 3.1 Graph scalar product . 67 3.2 The adjoint of an operator . 68 3.3 Inverse of the adjoint operator . 72 3.4 Numerical range and maximal operators . 74 3.5 Dissipative operators . 78 3.6 Hermitian operators . 80 3.7 Self-adjoint operators . 82 3.8 Spectral theorem . 85 3.9 Essentially self-adjoint operators . 89 3.10 Rigged Hilbert space . 90 3.11 Polar decomposition . 94 3.12 Scale of Hilbert spaces I . 98 3.13 Scale of Hilbert spaces II .
    [Show full text]
  • Spectral Theory - Wikipedia, the Free Encyclopedia Spectral Theory from Wikipedia, the Free Encyclopedia
    6/2/12 Spectral theory - Wikipedia, the free encyclopedia Spectral theory From Wikipedia, the free encyclopedia In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.[1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations.[2] The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.[3] Contents 1 Mathematical background 2 Physical background 3 A definition of spectrum 4 What is spectral theory, roughly speaking? 5 Resolution of the identity 6 Resolvent operator 7 Operator equations 8 Notes 9 General references 10 See also 11 External links Mathematical background The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. There have been three main ways to formulate spectral theory, all of which retain their usefulness. After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics; particularly at the hands of von Neumann.[4] The further theory built on this to include Banach algebras, which can be given abstractly.
    [Show full text]
  • A Gelfand Triple Approach to Wigner and Husimi Representations
    A GELFAND TRIPLE APPROACH TO WIGNER AND HUSIMI REPRESENTATIONS R.BEUKEMA, W.M. DE MUYNCK, AND J. DE GRAAF Abstract. The notion of Gelfand triples is applied to interpret mathe- matically a family of phase-space representations of quantum mechanics interpolating between the Wigner and Husimi representations. Gelfand triples of operators on Hilbert space, and Gelfand triples of functions on phase-space are introduced in order to get isomorphic correspondences between operators and their phase-space representations. The phase- space Gelfand triples are characterized by means of growth conditions on the analytic continuation of the functions. We give integral expressions for the sesquilinear forms belonging to the phase-space Gelfand triples. This provides mathematically rigorous phase-space analogues for quan- tum mechanical expectation values of bounded operators. 1. Introduction It is shown in [9] and in [1] that quantum mechanical expectation values can be expressed as phase-space averages. However, little care has been taken to provide a mathematically rigorous formulation. For the special case of the Wigner distribution several mathematically rigorous formulations of the idea to express expectation values in terms of ‘averages’ of functions on phase-space have been carried out using appropri- ate theories of generalized functions. For example [12] and more recently [14]. The generalized functions in [12] are introduced by means of a spe- cial choice for a one-parameter semi-group of ‘smoothing’ operators. In [27] the approach taken in [12] is generalized and used to give a mathematically rigorous version of the Dirac formalism. The concept of Dirac basis presented in [27] puts the concept of (continu- ous) bases of (generalized) eigenvectors of (unbounded) self-adjoint operators into an elegant functional analytic framework.
    [Show full text]