DOCSLIB.ORG

Spectral Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectral Theory Spectral Theory Kim Klinger-Logan November 25, 2016 Abstract: The following is a collection of notes (one of many) that compiled in prepa- ration for my Oral Exam which related to spectral theory for automorphic forms. None of the information is new and it is rephrased from Rudin's Functional Analysis, Evans' Theory of PDE, Paul Garrett's online notes and various other sources. Background on Spectra To begin, here is some basic terminology related to spectral theory. For a continuous linear operator T 2 EndX, the λ-eigenspace of T is Xλ := fx 2 X j T x = λxg: −1 If Xλ 6= 0 then λ is an eigenvalue. The resolvent of an operator T is (T − λ) . The resolvent set ρ(T ) := fλ 2 C j λ is a regular value of T g: The value λ is said to be a regular value if (T − λ)−1 exists, is a bounded linear operator and is defined on a dense subset of the range of T . The spectrum of is the collection of λ such that there is no continuous linear resolvent (inverse). In oter works σ(T ) = C − ρ(T ). spectrum σ(T ) := fλ j T − λ does not have a continuous linear inverseg discrete spectrum σdisc(T ) := fλ j T − λ is not injective i.e. not invertibleg continuous spectrum σcont(T ) := fλ j T −λ is not surjective but does have a dense imageg residual spectrum σres(T ) := fλ 2 σ(T ) − (σdisc(T ) [ σcont(T ))g Bounded Operators Usually we want to talk about bounded operators on Hilbert spaces. Recall that a Hilbert space is a vector space with an inner product space that is complete with respect to the distance function induced by the metric. We can sometimes deduce nice properties for operators on Banach spaces (a complete normed metric space) when the operator satisfies more specific conditions but most of the time we will stick with Hilbert spaces. 1 A linear map T : X ! Y is bounded if for all > 0 there is a δ > 0 such that if jxjX < δ then jT xjY < . For a linear map of Hilbert spaces, the following are equivalent: (i) continuous (ii) continuous at 0 (iii) bounded. ∗ ∗ The adjoint of a map T : X ! Y is the map T : Y ! X so that hT x; yiY = hx; T yiX : Any continuous linear map T have a unique adjoint T ∗ and T ∗∗ = T . For a self-adjoint maps, all eigenvalues are real. If T 2 EndX and T ∗T = TT ∗ then T is normal. For a normal operator there is no residual spectrum{it is all discrete or continuous. Any normal operator has σres(T ) = ;. A normal operator is unitary if T ∗T = TT ∗ = I. Compact Operators All compact operators are bounded operators. Thus everything that has previously been said still applies. There are many ways to define a compact operator T : X ! Y : • maps the unit ball in X to a precompact (has compact closure) set in Y • maps bounded subsequences in X to sequences in Y with convergent subsequences. Let fTng be a sequence of compact operators on a normed linear space X. Suppose that Tn ! T (in the space of bounded operators). Then T is also compact. For compact operators we do not need to restrict the domain to Hilbert spaces in order to have some spectral theorem. We have a spectral theorem for compact operators on Banach spaces. One method of proof involves the following two Lemmas: Reisz's Lemma: Let X be a Banach space and Y ⊂ X a proper closed subspace. For all > 0 there is an x 2 X with jjxjj = 1 and 1 ≥ d(x; Y ) ≥ 1 − (We can think of this as the analogue to orthogonality in Hilbert spaces but for Banach spaces where there is no notion of orthogonality.) proof: Let x1 2= Y and set R := infy2Y jy − x1j. Note that R > 0. For > 0 let y1 2 Y so that jx1 − y1j < R + . x − y Set x := 1 1 and note that jjxjj = 1 and y1 − x1 x1 − y1 y1 x1 infy2Y jy − x1j R inf jy − xj = inf y − = inf y + − = = y2Y y2Y y1 − x1 y2Y y1 − x1 y1 − x1 jy1 − x1j R + We can make this quotient arbitrarily close to 1 by taking smaller . 2 We can think of this as the analogue to orthogonality in Hilbert spaces but for Banach spaces where there is no notion of orthogonality. Lemma 2: If T is compact then Im(T − I) is closed. We can also use the Fredholm Alternative to prove the first part of the Spectral Theorem for Compact Operators. Fredholm Alternative: Let T : X ! X be a compact operator on Banach spaces and λ 6= 0, either (A) T − λ is a bijection or (B) Im(T − λ) is closed and dim (X=Im(T − λ)) = dim (ker(T − λ)) : We can also use Reisz' Lemma to prove the Fredholm Alternative. Spectral Theorem for Compact Operators on Banach Spaces: Let T be a compact on an infinite-dimensional Banach space then (1) the nonzero spectrum is discrete (for both T and T ∗) (2) λi ! 0 as i ! 1 (and this is the only accumulation point) (3) the spectrum is countable (4) the number of eigenvalues outside a disk jλj ≤ r is finite for r ≥ 0 proof: (1) This follows nicely from the Fredholm Alternative: Suppose that λ 6= 0 is in σ(T ) but is not an eigenvalue for T . Then T − λ is injective. Thus dim (ker(T − λ)) = 0 and by the Fredholm Alternative, dim (X=Im(T − λ)) = 0. Thus T − λ is surjective and so λ must be in the resolvent. We can also prove this result using Reisz' Lemma: The idea of the proof is by contradiction. Also WLOG, assume λ = 1. Assume λ 2 σ(T ) is not an eigenvalue. Thus T − λ = T − I is injective but not surjective. Since T is compact, by Lemma 2, Im(T − I) is a closed proper subspace of X. Call this Y1 := Im(T − I). Since T − I is injective, Y2 := (T − I)Y1 is a closed proper subspace of Y1. n Create a strictly decreasing sequence of Yn := Im(T − I) so Y1 ⊃ Y2 ⊃ · · · ⊃ Yn ⊃ · · · ⊃ Ym ⊃ ::: Now apply Reisz's Lemma for Y := Yn+1 and = 1=2: Choose yn 2 Yn so that jjynjj = 1 and d(yn;Yn+1) > 1=2: Also note that by compactness of T , fT yng must contain a norm convergent subsequence. 3 But for n < m, jjT yn − T ymjj = jj(T − I)yn + yn − (T − I)ym − ymjj and (T −I)yn −(T −I)ym −ym 2 Yn+1 and thus jjT yn −T ymjj > 1=2 which is our contradiction. Thus λ must be an eigenvalue. (2) This will also proceed by contradiction using Reisz' Lemma. Assume for the sake of contradiction that there are infinitely many distinct eigenvalues λn outside a ball centered at the origin of radius and each λn has associated eigenvector xn. Define Yn := spanfx1; x2; : : : ; xng. This gives a strictly increasing sequence Y1 ⊂ Y2 ⊂ · · · ⊂ Yn ⊂ · · · ⊂ Ym ⊂ ::: Now apply Reisz' Lemma for Y = Yn−1 and = 1=2: Choose yn 2 Yn so that jjynjj = 1 and d(yn;Yn−1) > 1=2. Again by compactness of T , fT yng must contain a norm convergent subsequence. But for n < m, jjT yn − T ymjj = jj(T − λn)yn + λnyn − (T − λm)ym + λmymjj Pm Pm−1 where (T − λn)yn + λnyn − (T − λm)ym 2 Ym−1 since ym = i=1 cixi where i=1 cixi 2 Ym−1 and (T − λm)xm = 0. Thus jjT yn − T ymjj > /2. This is a contradiction. Thus there are only finitely many eigenvalues outside a ball centered at 0. This gives us that zero is the only accumulation point. (3) From (2), [ fλng = fjλj > 1=ng n where each of these sets is finite. Thus σ(T ) is countable. Let T : X ! Y be a compact operator where X is a Banach space and Y is a Hilbert space, Then T is the limit (in operator norm) of a sequence of finite-dimensional oper- ators. More can be said about the operators on Hilbert spaces{especially when they are self-adjoint. For self-adjoint operators (or even symmetric) on Hilbert spaces all of the eigenvalues are real: Let T be a self-adjoint operator on a Hilbert space with eigenvalue λ so that T x = λx. Then λhx; xi = hλx, xi = hT x; xi = hx; T xi hT x; xi = hλx, xi = λhx; xi Thus λ = λ and so λ 2 R. 4 Spectral Theorem for Self-Adjoint Compact Operators on a Hilbert Space: Let T be a compact self-adjoint operator on a Hilbert space H, then L • H has an orthonormal basis vi of eigenvectors of T and H = the completion of λ Hλ. • λi ! 0 as i ! 1 (and this is the only accumulation point) • every eigenspace Hλ is finite-dimensional • (Ralleigh-Ritz) either ±|T jop is an eigenvalue. Recall that jT jop := inffc ≥ 0 j jT xj ≤ cjxj for all x 2 Xg. Example: Finite-dimensional Let T : X ! Y be a continuous linear mapping between normed spaces. If Ran(T ) has finite dimension then T is called a finite-dimensional operator. Finite dimensional operators are compact. For a bounded set B ⊆ X, T (B) is closed and bounded in the finite-dimensional subspaces Ran(T ) ⊆ Y . Heine-Borel says that T (B) is compact in Ran(T ). Example: Hilbert-Schmidt Hilbert Schmidt Theorem: Let X be a locally compact space endowed with a positive Borel measure and assume that L2(X) is a separable Hilbert space.
Load more

Recommended publications
  • Singular Value Decomposition (SVD)
    San José State University Math 253: Mathematical Methods for Data Visualization Lecture 5: Singular Value Decomposition (SVD) Dr. Guangliang Chen Outline • Matrix SVD Singular Value Decomposition (SVD) Introduction We have seen that symmetric matrices are always (orthogonally) diagonalizable. That is, for any symmetric matrix A ∈ Rn×n, there exist an orthogonal matrix Q = [q1 ... qn] and a diagonal matrix Λ = diag(λ1, . , λn), both real and square, such that A = QΛQT . We have pointed out that λi’s are the eigenvalues of A and qi’s the corresponding eigenvectors (which are orthogonal to each other and have unit norm). Thus, such a factorization is called the eigendecomposition of A, also called the spectral decomposition of A. What about general rectangular matrices? Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/22 Singular Value Decomposition (SVD) Existence of the SVD for general matrices Theorem: For any matrix X ∈ Rn×d, there exist two orthogonal matrices U ∈ Rn×n, V ∈ Rd×d and a nonnegative, “diagonal” matrix Σ ∈ Rn×d (of the same size as X) such that T Xn×d = Un×nΣn×dVd×d. Remark. This is called the Singular Value Decomposition (SVD) of X: • The diagonals of Σ are called the singular values of X (often sorted in decreasing order). • The columns of U are called the left singular vectors of X. • The columns of V are called the right singular vectors of X. Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/22 Singular Value Decomposition (SVD) * * b * b (n>d) b b b * b = * * = b b b * (n<d) * b * * b b Dr.
    [Show full text]
  • 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]
  • 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]
  • Notes on the Spectral Theorem
    Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner prod- uct space V has an adjoint. (T ∗ is defined as the unique linear operator on V such that hT (x), yi = hx, T ∗(y)i for every x, y ∈ V – see Theroems 6.8 and 6.9.) When V is infinite dimensional, the adjoint T ∗ may or may not exist. One useful fact (Theorem 6.10) is that if β is an orthonormal basis for a finite dimen- ∗ ∗ sional inner product space V , then [T ]β = [T ]β. That is, the matrix representation of the operator T ∗ is equal to the complex conjugate of the matrix representation for T . For a general vector space V , and a linear operator T , we have already asked the ques- tion “when is there a basis of V consisting only of eigenvectors of T ?” – this is exactly when T is diagonalizable. Now, for an inner product space V , we know how to check whether vec- tors are orthogonal, and we know how to define the norms of vectors, so we can ask “when is there an orthonormal basis of V consisting only of eigenvectors of T ?” Clearly, if there is such a basis, T is diagonalizable – and moreover, eigenvectors with distinct eigenvalues must be orthogonal. Definitions Let V be an inner product space. Let T ∈ L(V ). (a) T is normal if T ∗T = TT ∗ (b) T is self-adjoint if T ∗ = T For the next two definitions, assume V is finite-dimensional: Then, (c) T is unitary if F = C and kT (x)k = kxk for every x ∈ V (d) T is orthogonal if F = R and kT (x)k = kxk for every x ∈ V Notes 1.
    [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]
  • 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]
  • The Spectral Theorem for Self-Adjoint and Unitary Operators Michael Taylor Contents 1. Introduction 2. Functions of a Self-Adjoi
    The Spectral Theorem for Self-Adjoint and Unitary Operators Michael Taylor Contents 1. Introduction 2. Functions of a self-adjoint operator 3. Spectral theorem for bounded self-adjoint operators 4. Functions of unitary operators 5. Spectral theorem for unitary operators 6. Alternative approach 7. From Theorem 1.2 to Theorem 1.1 A. Spectral projections B. Unbounded self-adjoint operators C. Von Neumann's mean ergodic theorem 1 2 1. Introduction If H is a Hilbert space, a bounded linear operator A : H ! H (A 2 L(H)) has an adjoint A∗ : H ! H defined by (1.1) (Au; v) = (u; A∗v); u; v 2 H: We say A is self-adjoint if A = A∗. We say U 2 L(H) is unitary if U ∗ = U −1. More generally, if H is another Hilbert space, we say Φ 2 L(H; H) is unitary provided Φ is one-to-one and onto, and (Φu; Φv)H = (u; v)H , for all u; v 2 H. If dim H = n < 1, each self-adjoint A 2 L(H) has the property that H has an orthonormal basis of eigenvectors of A. The same holds for each unitary U 2 L(H). Proofs can be found in xx11{12, Chapter 2, of [T3]. Here, we aim to prove the following infinite dimensional variant of such a result, called the Spectral Theorem. Theorem 1.1. If A 2 L(H) is self-adjoint, there exists a measure space (X; F; µ), a unitary map Φ: H ! L2(X; µ), and a 2 L1(X; µ), such that (1.2) ΦAΦ−1f(x) = a(x)f(x); 8 f 2 L2(X; µ): Here, a is real valued, and kakL1 = kAk.
    [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]
  • 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]
  • 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]
  • ASYMPTOTICALLY ISOSPECTRAL QUANTUM GRAPHS and TRIGONOMETRIC POLYNOMIALS. Pavel Kurasov, Rune Suhr
    ISSN: 1401-5617 ASYMPTOTICALLY ISOSPECTRAL QUANTUM GRAPHS AND TRIGONOMETRIC POLYNOMIALS. Pavel Kurasov, Rune Suhr Research Reports in Mathematics Number 2, 2018 Department of Mathematics Stockholm University Electronic version of this document is available at http://www.math.su.se/reports/2018/2 Date of publication: Maj 16, 2018. 2010 Mathematics Subject Classification: Primary 34L25, 81U40; Secondary 35P25, 81V99. Keywords: Quantum graphs, almost periodic functions. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.math.su.se/ [email protected] Asymptotically isospectral quantum graphs and generalised trigonometric polynomials Pavel Kurasov and Rune Suhr Dept. of Mathematics, Stockholm Univ., 106 91 Stockholm, SWEDEN [email protected], [email protected] Abstract The theory of almost periodic functions is used to investigate spectral prop- erties of Schr¨odinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schr¨odingeroperators may have asymptotically close spectra if and only if the corresponding reference Lapla- cians are isospectral. Our result implies that a Schr¨odingeroperator is isospectral to the standard Laplacian on a may be different metric graph only if the potential is identically equal to zero. Keywords: Quantum graphs, almost periodic functions 2000 MSC: 34L15, 35R30, 81Q10 Introduction. The current paper is devoted to the spectral theory of quantum graphs, more precisely to the direct and inverse spectral theory of Schr¨odingerop- erators on metric graphs [3, 20, 24]. Such operators are defined by three parameters: a finite compact metric graph Γ; • a real integrable potential q L (Γ); • ∈ 1 vertex conditions, which can be parametrised by unitary matrices S.
    [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]