DOCSLIB.ORG

Math 356 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 356 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps MATH 356 LECTURE NOTES INTRO TO PDES EIGENFUNCTION EXPANSIONS FOR IBVPS J. WONG (FALL 2019) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes ◦ Separation of variables • The wave equation ◦ Standard solution, standing waves ◦ Example: tuning the vibrating string (resonance) • Interpreting solutions (what does the series mean?) ◦ Relevance of Fourier convergence theorem ◦ Smoothing for the heat equation ◦ Non-smoothing for the wave equation 3 1. The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. The procedure here for the heat equation will extend nicely to a variety of other problems. For now, consider an initial boundary value problem of the form ut = −Lu + h(x; t); x 2 (a; b); t > 0 hom. BCs at a and b (1.1) u(x; 0) = f(x) We seek a solution in terms of the eigenfunction basis X u(x; t) = cn(t)φn(x) n by finding simple ODEs to solve for the coefficients cn(t): This form of the solution is called an eigenfunction expansion for u (or `eigenfunction series') and each term cnφn(x) is a mode (or `Fourier mode' or `eigenmode'). Part 1: find the eigenfunction basis. The first step is to compute the basis. The eigenfunctions we need are the solutions to the eigenvalue problem Lφ = λφ, φ(x) satisfies the BCs for u: (1.2) By the theorem in ??, there is a sequence of eigenfunctions fφng with eigenvalues fλng that form an orthogonal basis for L2[a; b] (i.e. one with all the required properties). 1 2 J. WONG (FALL 2019) If possible, we compute solutions explicitly via the standard procedure. Note that the BCs imposed on the eigenvalue problem must be homogeneous. Now at each fixed time t, the function u(x; t) is a function of x defined on [a; b]. It fol- lows that there are coefficients cn(t) such that X u(x; t) = cn(t)φn(x): (1.3) n For each t,fcn(t)g is the set of coefficients for expressing u(x; t) in terms of the basis fφng: Part 2: get equations for the coefficients: Our objective now is to reduce the prob- lem to ODEs for coefficients cn(t): The order of steps can be changed here. Step 2a (Write known functions in the basis): We express every known function in the problem (PDE and ICs) in the eigenfunction basis using the orthogonality rule (??). In this case, there are two such functions, the source term and the initial condition: X hf; φni f(x) = f φ (x); f = ; (1.4) n n n hφ ; φ i n n n 1 X hh(x; t); φn(x)i h(x; t) = h (t)φ (x); h (t) = : (1.5) n n n hφ ; φ i n=0 n n R b 2 Here hf; gi = a f(x)g(x) dx is the L inner product in x; often, it is not worth calculating the integrals explicitly (they can be a mess). Note that in (1.5), the t-variable is not part of the integral, e.g. Z b h(x; t) = tx =) hh; φni = t xφn(x) dx: a Step 2b (Plug series in to the PDE): We now `plug the series in' to the PDE - which expands the PDE in terms of the eigenfunction basis. Taking this one part at a time1, ! @ X u = c (t)φ (x) t @t n n n X 0 = cn(t)φn(x); (t-derivs. only act on coeffs.) n X Lu = cn(t)Lφn n X = λncn(t)φn(x)(φn is an eigenfunction) n . Plugging this into the PDE, we get ut = −Lu + h(x; t) X 0 X X =) cn(t)φn(x) = − λncn(t)φn(x) + hn(t)φn(x): n n n 1After getting used to the process, you can shortcut the work here and skip some steps. They tend to be the same for most problems, but you should be careful and recognize when the steps must be changed. EIGENFUNCTION EXPANSION 3 The eigenfunction property means that the coefficient of φn in Lu depends only on cn(t). That is, the n-th term in the series is independent of the others. Now since fφng is a basis, the coefficients on the left and right sides must be equal term-by- term. Alternatively, X 0 X (cn(t) + λncn(t) − hn(t))φn(x) = 0 = 0 · φn n n and since zero must have a unique representation in the basis, 0 cn(t) + λncn(t) = hn(t) for all n: (1.6) The difficult part is now over. Step 2c (solve the ODE): Next, solve the ODE (1.6) for each n. Note that sometimes, this will involve some case work (as we saw with Fourier series). P The solution will have some arbitrary constants. At this point, the series cn(t)φn(x) is a `general solution' to the PDE with the given BCs with the initial condition not yet imposed. Step 2d (apply initial conditions): This can be done earlier. We turn the ODE into an IVP with a unique solution by applying the initial conditions (any `leftover' equations not yet used). As in the PDE, plug in the series: X X u(x; 0) = f(x) =) cn(0)φn(x) = fnφn(x) n which gives initial conditions for the cn's: cn(0) = fn: (1.7) Solving (1.6) with (1.7) yields unique solutions for cn(t), and we are done. Step 3 (summarize the solution): As the steps can be long, it is useful to make the solution clear. You should check that all functions and constants are well-defined by some formula. We have obtained a solution X u(x; t) = cn(t)φn(x) n to the IBVP (1.1) where the cn's and φn's are given by cn(t) = ··· ; φn = ··· (replace with formula computed in the procedure) and fn; hn(t) are given by (1.4) and (1.5). Note that it is often best to leave expressions in terms of explicit integrals e.g. like Z π hf; φni 2 2 fn = = x sin nx dx hφn; φni π 0 when computing would not give any new insight. The inner product shorthand can be used, but its exact definition should be clear (the interval etc.). 4 J. WONG (FALL 2019) 1.1. Example 1: no source; Dirichlet BCs. The simplest case. We solve ut =uxx; x 2 (0; 1); t > 0 (1.8a) u(0; t) = 0; u(1; t) = 0; (1.8b) u(x; 0) = f(x): (1.8c) The eigenvalues/eigenfunctions are (as calculated in previous sections) 2 2 λn = n π ; φn = sin nπx; n ≥ 1: (1.9) Assuming the solution exists, it can be written in the eigenfunction basis as 1 X u(x; t) = cn(t)φn(x): n=0 R 1 2 There is only one function to write in the eigenfunction basis (note that 0 φn dx = 1=2 here) 1 Z 1 X hf; φni f(x) = f φ (x); f = = 2 f(x) sin nπx dx: (1.10) n n n hφ ; φ i n=1 n n 0 00 Substitute into the PDE (1.8a) and use the fact that −φn = λnφ to obtain 1 X 0 (cn(t) + λncn(t))φn(x) = 0: n=1 Doing the same for the initial condition, we get the IC for cn: 1 1 X X u(x; 0) = f(x) =) cn(0)φn(x) = fnφn(x) =) cn(0) = fn: n=1 n=1 Equating coefficients of each basis function (to zero), we find that 0 cn(t) + λncn(t) = 0; cn(0) = fn: (1.11) We are now done. The solution to the IBVP (1.8) is 1 X −n2π2t u(x; t) = bne sin nπx with bn given by (1.13): (1.12) n=1 Alternatively, we could summarize by writing that the solution is 1 X −λnt u(x; t) = fne φn(x) n=1 with eigenfunctions/values φn; λn given by (1.9) and fn by (1.10). Long-time behavior: Note that every term in the solution (1.12) has a negative exponential (since all the eigenvalues are positive). Furthermore, terms further down in the series decay much faster since λn grows quadratically with n. It follows (informally) that lim u(x; t) = 0 t!1 EIGENFUNCTION EXPANSION 5 independent of the initial condition f(x) (which just affects the bn's and not the eigenvalues). Moreover, by approximating u by its first term, −λ1t −λ1t u(x; t) ≈ f1e φ1(x) =) decays to zero at least as fast as f1e : Thus, the smallest eigenvalue of a non-zero term gives the convergence rate for the solution (which is exponential). As an explicit example, suppose the initial condition is f(x) = x(1 − x): After some laborious integration by parts, we get ( 2(1 − (−1)n) 0 n even bn = 3 3 = 2 (1.13) π n π3n3 n odd: The first few terms of the solution are 2 2 2 2 u(x; t) = e−π t sin πx + e−9π t sin 3πx + ··· π3 27π3 Note that u(x; t) is missing a φ2 = sin 2πx term; this does not affect the convergence rate. However, suppose instead that f(x) = x − 1=2: Then we have Z 1 Z 1 b1 = 2 (x − 1=2) sin πx dx = 0; b2 = 2 (x − 1=2) sin 2πx dx = −1=pi: 0 0 the n = 1 term vanishes, so 1 2 u(x; t) ≈ − e−λ2π tφ (x) + ··· as t ! 1 π 2 2 and the convergence rate is given by the second eigenvalue λ2 = 4π (faster convergence!).
Load more

Recommended publications
  • Math 353 Lecture Notes Intro to Pdes Eigenfunction Expansions for Ibvps
    Math 353 Lecture Notes Intro to PDEs Eigenfunction expansions for IBVPs J. Wong (Fall 2020) Topics covered • Eigenfunction expansions for PDEs ◦ The procedure for time-dependent problems ◦ Projection, independent evolution of modes 1 The eigenfunction method to solve PDEs We are now ready to demonstrate how to use the components derived thus far to solve the heat equation. First, two examples to illustrate the proces... 1.1 Example 1: no source; Dirichlet BCs The simplest case. We solve ut =uxx; x 2 (0; 1); t > 0 (1.1a) u(0; t) = 0; u(1; t) = 0; (1.1b) u(x; 0) = f(x): (1.1c) The eigenvalues/eigenfunctions are (as calculated in previous sections) 2 2 λn = n π ; φn = sin nπx; n ≥ 1: (1.2) Assuming the solution exists, it can be written in the eigenfunction basis as 1 X u(x; t) = cn(t)φn(x): n=0 Definition (modes) The n-th term of this series is sometimes called the n-th mode or Fourier mode. I'll use the word frequently to describe it (rather than, say, `basis function'). 1 00 Substitute into the PDE (1.1a) and use the fact that −φn = λnφ to obtain 1 X 0 (cn(t) + λncn(t))φn(x) = 0: n=1 By the fact the fφng is a basis, it follows that the coefficient for each mode satisfies the ODE 0 cn(t) + λncn(t) = 0: Solving the ODE gives us a `general' solution to the PDE with its BCs, 1 X −λnt u(x; t) = ane φn(x): n=1 The remaining coefficients are determined by the IC, u(x; 0) = f(x): To match to the solution, we need to also write f(x) in the basis: 1 Z 1 X hf; φni f(x) = f φ (x); f = = 2 f(x) sin nπx dx: (1.3) n n n hφ ; φ i n=1 n n 0 Then from the initial condition, we get u(x; 0) = f(x) 1 1 X X =) cn(0)φn(x) = fnφn(x) n=1 n=1 =) cn(0) = fn for all n ≥ 1: Now everything has been solved - we are done! The solution to the IBVP (1.1) is 1 X −n2π2t u(x; t) = ane sin nπx with an given by (1.5): (1.4) n=1 Alternatively, we could state the solution as follows: The solution is 1 X −λnt u(x; t) = fne φn(x) n=1 with eigenfunctions/values φn; λn given by (1.2) and fn by (1.3).
    [Show full text]
  • Sturm-Liouville Expansions of the Delta
    Gauge Institute Journal H. Vic Dannon Sturm-Liouville Expansions of the Delta Function H. Vic Dannon [email protected] May, 2014 Abstract We expand the Delta Function in Series, and Integrals of Sturm-Liouville Eigen-functions. Keywords: Sturm-Liouville Expansions, Infinitesimal, Infinite- Hyper-Real, Hyper-Real, infinite Hyper-real, Infinitesimal Calculus, Delta Function, Fourier Series, Laguerre Polynomials, Legendre Functions, Bessel Functions, Delta Function, 2000 Mathematics Subject Classification 26E35; 26E30; 26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30. 1 Gauge Institute Journal H. Vic Dannon Contents 0. Eigen-functions Expansion of the Delta Function 1. Hyper-real line. 2. Hyper-real Function 3. Integral of a Hyper-real Function 4. Delta Function 5. Convergent Series 6. Hyper-real Sturm-Liouville Problem 7. Delta Expansion in Non-Normalized Eigen-Functions 8. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & ab==0 9. Fourier-Cosine Expansion of Delta Associated with 1 yx"( )+=l yx ( ) 0 & ab==2 p 10. Fourier-Sine Expansion of Delta Associated with yx"( )+=l yx ( ) 0 & a = 0 11. Fourier-Bessel Expansion of Delta Associated with n2 -1 ux"( )+- (l 4 ) ux ( ) = 0 & ab==0 x 2 12. Delta Expansion in Orthonormal Eigen-functions 13. Fourier-Hermit Expansion of Delta Associated with ux"( )+- (l xux2 ) ( ) = 0 for any real x 14. Fourier-Legendre Expansion of Delta Associated with 2 Gauge Institute Journal H. Vic Dannon 11 11 uu"(ql )++ [ ] ( q ) = 0 -<<pq p 4 cos2 q , 22 15. Fourier-Legendre Expansion of Delta Associated with 112 11 ymy"(ql )++- [ ( ) ] (q ) = 0 -<<pq p 4 cos2 q , 22 16.
    [Show full text]
  • Lecture 15.Pdf
    Lecture #15 Eigenfunctions of hermitian operators If the spectrum is discrete , then the eigenfunctions are in Hilbert space and correspond to physically realizable states. It the spectrum is continuous , the eigenfunctions are not normalizable, and they do not correspond to physically realizable states (but their linear combinations may be normalizable). Discrete spectra Normalizable eigenfunctions of a hermitian operator have the following properties: (1) Their eigenvalues are real. (2) Eigenfunctions that belong to different eigenvalues are orthogonal. Finite-dimension vector space: The eigenfunctions of an observable operator are complete , i.e. any function in Hilbert space can be expressed as their linear combination. Continuous spectra Example: Find the eigenvalues and eigenfunctions of the momentum operator . This solution is not square -inferable and operator p has no eigenfunctions in Hilbert space. However, if we only consider real eigenvalues, we can define sort of orthonormality: Lecture 15 Page 1 L15.P2 since the Fourier transform of Dirac delta function is Proof: Plancherel's theorem Now, Then, which looks very similar to orthonormality. We can call such equation Dirac orthonormality. These functions are complete in a sense that any square integrable function can be written in a form and c(p) is obtained by Fourier's trick. Summary for continuous spectra: eigenfunctions with real eigenvalues are Dirac orthonormalizable and complete. Lecture 15 Page 2 L15.P3 Generalized statistical interpretation: If your measure observable Q on a particle in a state you will get one of the eigenvalues of the hermitian operator Q. If the spectrum of Q is discrete, the probability of getting the eigenvalue associated with orthonormalized eigenfunction is It the spectrum is continuous, with real eigenvalues q(z) and associated Dirac-orthonormalized eigenfunctions , the probability of getting a result in the range dz is The wave function "collapses" to the corresponding eigenstate upon measurement.
    [Show full text]
  • Positive Eigenfunctions of Markovian Pricing Operators
    Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing Likuan Qin† and Vadim Linetsky‡ Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Sciences Northwestern University Abstract This paper develops a spectral theory of Markovian asset pricing models where the under- lying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process (BRP)) and the stochastic discount factor (SDF) is a positive semi- martingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability mea- sure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman (2009) factorization of the Markovian SDF corresponding to the recurrent eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time, finite state irreducible Markov chains to recurrent BRPs, and obtain the long- maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically arXiv:1411.3075v3 [q-fin.MF] 9 Sep 2015 going to infinity or to a zero lower bound trap without possibility of escaping.
    [Show full text]
  • Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
    Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces Stefan Klus1, Ingmar Schuster2, and Krikamol Muandet3 1Department of Mathematics and Computer Science, Freie Universit¨atBerlin, Germany 2Zalando Research, Zalando SE, Berlin, Germany 3Max Planck Institute for Intelligent Systems, T¨ubingen,Germany Abstract Transfer operators such as the Perron{Frobenius or Koopman operator play an im- portant role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose kernel transfer operators, which extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of con- ditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume ex- isting data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel- based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigende- compositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis. 1 Introduction Transfer operators such as the Perron{Frobenius or Koopman operator are ubiquitous in molecular dynamics, fluid dynamics, atmospheric sciences, and control theory (Schmid, 2010, Brunton et al., 2016, Klus et al., 2018a).
    [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]
  • On the Eigenvalues of Integral Operators with Difference Kernels
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 53, 554-566 (1976) On the Eigenvalues of Integral Operators with Difference Kernels R. VITTAL RAO Department of Applied Mathematics, Indian Institute of Science, Bangalore 560012, India Submitted by G. Milton Wing 1. INTRODUCTION Integral equations with difference kernels on finite intervals occur in the study of a wide class of transport phenomena such as neutron transport [l], radiative transfer [2], linearized gas dynamics [3, 41, and electromagnetic wave refraction [5], as well as in the study of fields such as stochastic processes, elasticity, etc. By such an equation, we mean an integral equation of the form fW =g@) + J1’K(~ - Y)f(Y) dy. (1.1) The spectral analysis of these integral operators plays an important role in the determination of the critical dimensions of nuclear reactors in which the energy of the neutrons is considered constant (that is, monoenergetic neutron transport). The spectral analysis has also received attention from a different context, namely, these integral operators can be regarded as continual analogs of Toeplitz matrices [6-91. In this paper we will be studying some properties of the spectrum of these integral operators. We prove three theorems. Of these the first one is valid even without the assumption that the kernel is a difference kernel. We denote by where we make the following assumptions on the kernel: q, Y) = qy, 4, (1.3) K(X, y) is locally square integrable. (1.4) 554 Copyright Q 1976 by Academic Press, Inc.
    [Show full text]
  • EIGENFUNCTION EXPANSIONS in Rn 1. Introduction and Statement Of
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 12, December 2011, Pages 4361–4368 S 0002-9939(2011)11022-0 Article electronically published on April 5, 2011 EIGENFUNCTION EXPANSIONS IN Rn TODOR GRAMCHEV, STEVAN PILIPOVIC, AND LUIGI RODINO (Communicated by Richard Rochberg) Abstract. The main goal of this paper is to extend in Rn a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in Rn a selfadjoint, globally elliptic Shubin type differential operator with spec- trum consisting of a sequence of eigenvalues λj ,j ∈ N, and a corresponding 2 n sequence of eigenfunctions uj ,j ∈ N, forming an orthonormal basis of L (R ). Elements of Schwartz S(Rn), resp. Gelfand-Shilov S1/2 spaces, are character- 1/2 ized through expansions j aj uj and the estimates of coefficients aj by the power function, resp. exponential function of λj . 1. Introduction and statement of the result We shall give a version in Rn of some results, already known on compact mani- folds, concerning eigenfunction expansions. Broadly speaking, the aim is to relate the regularity of a function with the decay properties of the sequence of the Fourier coefficients. More precisely, we want to reproduce in Rn the classical results of [15], Section 10, for Sobolev regularity and [16] for analytic functions on a compact manifold taking into account Weyl asymptotics for eigenvalues. Our basic example of an operator will be the harmonic oscillator appearing in Quantum Mechanics, (1.1) H = − + |x|2, whose eigenfunctions are the Hermite functions −|x|2/2 n (1.2) hα(x)=Hα(x)e ,α=(α1, ..., αn) ∈ N , where Hα(x)istheα-th Hermite polynomial; cf.
    [Show full text]
  • 1 Function Spaces, Linear Operators, and Green's Functions
    1 1 Function Spaces, Linear Operators, and Green’s Functions 1.1 Function Spaces Consider the set of all complex-valued functions of the real variable x, denoted by f (x), g(x), ..., and defined on the interval (a, b). We shall restrict ourselves to those functions which are square-integrable.Definetheinner product of any two of the latter functions by b ∗ (f , g) ≡ f (x) g (x) dx, (1.1.1) a in which f ∗(x) is the complex conjugate of f (x). The following properties of the inner product follow from definition (1.1.1): (f , g)∗ = (g, f ), ( f , g + h) = ( f , g) + ( f , h), (1.1.2) ( f , αg) = α( f , g), (αf , g) = α∗( f , g), with α a complex scalar. While the inner product of any two functions is in general a complex number, the inner product of a function with itself is a real number and is nonnegative. This prompts us to define the norm of a function by 1 b 2 ∗ f ≡ ( f , f ) = f (x) f (x)dx , (1.1.3) a provided that f is square-integrable , i.e., f < ∞. Equation (1.1.3) constitutes a proper definition for a norm since it satisfies the following conditions: scalar = | | · (i) αf α f , for all comlex α, multiplication (ii) positivity f > 0, for all f = 0, = = f 0, if and only if f 0, triangular (iii) f + g ≤ f + g . inequality (1.1.4) Applied Mathematical Methods in Theoretical Physics, Second Edition. Michio Masujima Copyright 2009 WILEY-VCH Verlag GmbH & Co.
    [Show full text]
  • Eigenfunction Expansions Associated with Singular Differential Operatorso
    EIGENFUNCTION EXPANSIONS ASSOCIATED WITH SINGULAR DIFFERENTIAL OPERATORSO BY JOANNE ELLIOTT 1. Introduction. In this paper we shall prove some expansion theorems connected with pairs of so-called adjoint differential operators reducible to the form d2 d (1.1) U = — + b(x)- + c(x) dx' dx and dY d "1 (1.2) 0* = - \--Hx) \ + c(x), dx \_dx J on an interval — oo^n^x^r2^oo; the coefficient b(x) is assumed continu- ous, but not necessarily bounded, in (ru r2), and c(x) is assumed continuous in [fi, r2]. We shall impose other conditions of a global character on b(x). (See §2.) Motivated by applications to diffusion theory, we shall consider (1.1) in the Banach space C[ri, r2] oi functions continuous on [ru r2] and (1.2) in the space L(ri, r2) of functions integrable on (ri, r2). More specifically, fi will be considered as an operator from C[ri, r2] to itself and fi* as an operator from T.(ri, ri) to itself. For a more detailed discussion of the domains of fi and fi*, see the end of §2 and §3. The method we use here, which is entirely real, may be outlined as fol- lows: we start with (1.1) under boundary conditions of the type (3.3)-(3.5). Under our restrictions on b(x), we find that the eigenfunctions of fi under the boundary conditions contain a set orthonormal and complete in a certain weighted L2 space (§4). To bridge the gap between the Hilbert space and the Banach spaces we use the theory of semi-groups.
    [Show full text]
  • Eigenfunction Expansions
    276 Chapter 12 Eigenfunction Expansions The Sturm-Liouville theory provides an infinite sequence of eigenvalue-eigen- function pairs. Among the triumphs of nineteenth-century mathematics was the realization that these sequences of eigenfunctions can be used to 1 1 represent arbitrary functions via infinite series. Thus if fungn=0 is the sequence of eigenfunctions of a Sturm-Liouville problem on an interval [a; b] of the real axis, and f is any function defined on this interval, one seeks a representation of f in the form 1 X f (x) = cnun (x) ; (12.1) 0 where the constants fcng must characterize the function f. This is the subject of the present chapter. 12.1 Kinds of Convergence For a sequence of functions ffkg, where each function fk (x) is defined on an interval I of the real axis, there is a variety of possibilities where convergence is concerned. Some examples are 1. The sequence converges to a function f at some points of the interval but not at others; 2. The sequence converges to a function f at each point of the interval; 3. The sequence converges to a function f uniformly on the entire interval. 1Well, almost arbitrary. 277 278 These examples make different mathematical demands on the convergence. The last kind of convergence, uniform convergence, demands that the pre- ceding kind (pointwise convergence) hold, and more. We have encountered it previously, in Chapter 6, and shall encounter it again below. What, if anything, can we make of convergence number 1 above? If the subset of points of the interval where convergence fails is not too large, there is a natural way to make sense of this situation.
    [Show full text]
  • Eigenfunction Expansions 2 1. Notation: the Kronecker Delta Is {0
    Eigenfunction Expansions 2 1. Notation: The Kronecker delta is 0 for i 6= j δ = ij 1 for i = j. 2. Definition: Let H be a Hilbert space with inner product hu , vi and norm kwk = phw , wi. A set {vk} of vectors in H is orthogonal if hvi , vji = 0 for i 6= j. 3. Definition: A set {uk} in H is orthonormal if hui , uji = δij. Thus {uk} is orthonormal if it is orthogonal and the uk are unit vectors. 4. Note that if {vk} is an othogonal set of nonzero vectors, then {vk/kvkk} is orthonormal. 5. Definition: The set {ek} in H is an orthonormal basis of H if a. it is orthonormal and b. every vector v ∈ H can be represented as a linear combination of the ek: X v = ckek, (1) k for scalars ck. 6. Example: The set {~i,~j,~k} is a an orthormal basis of R3. The set {~i,~j} in R3 is orthonormal, but is not a basis. 7. If the sum in (1) is infinite, then equality is understood to be in the sense of norm convergence: N X v − ckek → 0 as N → ∞. k=1 8. Let v lie in H, and O = {ek} be an orthonormal basis of H. Then for scalars ck, X v = ckek. k By taking inner products, we see that cm = hv , emi. Thus, X v = hv , ekiek. (2) k The sum (2) is called the Fourier series of v with respect to O. The numbers vˆk = hv , eki are the Fourier coefficients.
    [Show full text]