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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. X Ax = λkhx; ekiek (1.1) k from there it is possible to form the inverse, build a functional calculus, analyze the stability of the operator in numeric situations, determine whether the operator is positive or negative definite etc. They play a role even in non-linear situations like the classification of quadratic forms. In quantum mechanics they correspond to the stationary states of the system. For normal operators with a continuous spectrum there is a diagonalization using a projection valued spectral measure Eλ but the eigenvectors do not exist in the space the operator is defined in. Hence it is necessary to consider larger spaces and to replace the sum in (1.1) with an integral (since the spectrum of the operator is not discrete). Thus the generalization of the above formula would look like: Z Ax = λhx; ek(λ)iek(λ)dλ (1.2) σ(A) − where the vectors ek(λ) are defined in a generalized space X (implying that the vectors x must be in a smaller space X+ in the above formula). An idea which may already have been anticipated by Dirac in his bra and ket notation and for which a rigorous mathematical foundation has been developed in the 1950's by Y. Berezansky, L. Garding, I. Gelfand, G. Kats, A. Kostyuchenko and K. Maurin.1 The framework in which this is possible is the Rigged Hilbert space (also called Gelfand triplet):2 X+ ⊂ H ⊂ X− where X− is the space of continuous antilinear functionals on X+. Several condi- tions on the rigging have been established that lead to an eigenfunction expansion (1.2), the most basic one beeing that the embedding X+ into H is Hilbert-Schmidt, provided that X+ itself is a Hilbert space. If X+ is only a Banach space, the nu- clearity of the embedding is also a sufficient condition. In [BSU96] and [PSW89] also conditional riggings3 are considered, which require for a given operator A only 1[de 05], [BSU96], [Mau68] 2Usually denoted by Φ ⊂ H ⊂ Φ×. 3In [BSU96] they are called non-nuclear riggings. 5 1 Introduction that f(A): X+ ! H is Hilbert-Schmidt, provided that f is bounded away from zero. There is an explicit eigenfunction expansion for the Laplace operator A := −∆ n n on the euclidean space L2(R ) which implies that for any x 2 L2(R ) there are locally integrable functionsx ^(λ) s.t. Z 1 f(A)x = f(λ)^x(λ) dλ 0 where the generalized eigenfunctionsx ^(λ) are in the weighted spaces n n s n L2;s(R ) := ff 2 L1;loc(R ) j ((1 + k · k ) f(·)) 2 L2(R )g −1 for s > 2 . If the function f itself is in L2(R) then the above is a Bochner integral. At first sight there is no reason to expect that this result4 does not fit into the general framework above. However we discovered that - this is not the case, in fact the only map n f(A): L ( n) ! L ( n) for s < 2;s R 2 R 2 n−1 which is Hilbert-Schmidt is the zero map. To close this gap (of 2 ) we have investigated a different approach, based on an operator relation of the following form BA − AB = h(A) leading to an eigenfunction expansion on riggings:5 + × − X ⊂ [D(B); H]1=2;1 ⊂ H ⊂ [D(B); H]1=2;1 ⊂ X where X+ and (its antidual) X− are reflexive Banach spaces. In the case of homogeneous self-adjoint elliptic differential operators (e.g. the Laplace operator) this leads to an eigenfunction expansion on the rigging L2;s ⊂ H ⊂ L2;−s for all 1 6 s > 2 . The common denominator between the latter result and the classic results for (conditional) Hilbert-Schmidt and nuclear riggings is the existence of a σ− finite measure s.t. the following estimate holds 2 2 kχE(A)xkH ≤ µ(E)kxk+ For example in the case of Hilbert-Schmidt-Riggings the measure can be defined by µ(E) = hχE(A)xk; xki 4It can be derived from the Plancharel formula in [Str89][p. 64]. 5Theorem 4.4.4 6Theorem 4.6.3 6 + where xk is an orthonormal system of X . On the other in the case of the operator relation Z µ(E) = jh−1(λ)jj2 + h0(λ)jdλ. E The latter means that (outside of the zeros of h) the measure µ and hence the generalized eigenfunctions themselves are absolutely continuous with respect to the Lebesgue measure on σ(A). 7 2 Eigenfunction expansions on Hilbert spaces The first two sections introduce some preliminaries, namely spectral measures for normal operators and the Phillips theorem. The main object of our investigation will be rigged Hilbert spaces X+ ⊂ H ⊂ X−, i.e. Hilbert spaces which have a dense linear subset of test vectors X+ and a space X− of generalized functions (defined as antil-linear functionals on X+). The triple of spaces (X+; H; X−) is sometimes also called Gelfand triple. In general the spaces X+ and X− are topological vector spaces. However we will only consider the case where X+ and X− are reflexive Banach spaces. We will introduce and investigate them in section 2.3. Provided there is a spectral measure µ for a self adjoint operator A s.t. the estimate 2 2 + kχE(A)vkH ≤ kvk+µ(E) for all v 2 X (2.1) holds we will see that there is a complete system of generalized eigenvectors. If the operator A is closable on X− then the generalized eigenvectors are in fact real eigenvectors of the closure A−. The last section contains a summary with the results to which we will refer in the next chapters. 2.1 Spectral Measures The spectral theorem for normal operators gives a measure on the spectrum of the operator, which we call spectral measure. But it is not necessarily unique (w.r.t to the operator). As mentioned before we will find spectral measures with the special condition (2.1). Thus it is reasonable to have a precise definition of them, which we will do using the spectral theorem (cf. [Wer05, p. 350, VII.3.1]): Theorem 2.1.1 (Spectral theoerem) For A self-adjoint operator on a separable complex Hilbert space H there is a Borel set Ω ⊂ C, a σ-finite measure ν on the Borel-σ-Algebra of Ω, a continuous function m :Ω ! R and a unitary operator U : L2(Ω; ν) ! H s.t. the operator M on L2(Ω; ν) defined by (Mx)(!) := m(!)x(!) with D(M) := fx 2 L2(Ω; µ) j Mx 2 L2(Ω; ν)g is a representation of A i.e. D(A) = UD(M) and AUx = UMx for all x 2 D(M). 9 2 Eigenfunction expansions on Hilbert spaces Definition 2.1.2 A σ-finite measure µ on the Borel-σ-Algebra of σ(A) is called spectral measure for the operator A if and only if µ(E) = 0 , ν(m−1(E)) = 0 for all Borel sets E ⊂ σ(A). There is allways such a measure. We can define one for example by using a countable disjoint decomposition Ek of Ω s.t. ν(Ek) < 1. The measure µ defined by −1 X ν(m (E) \ Ek) µ(E) := k 2 ν(Ek) k is an (actually finite) spectral measure for A. There is an algebra homomorphism from the bounded µ-measurable functions to the bounded operators commuting with A, which we call the functional calculus of A: Proposition 2.1.3 For a self-adjoint operator A on a separable Hilbert space, and a spectral measure µ, there is a continuous map ΦA : L1(σ(A); µ) !B(H) −1 s.t. for all µ 2 ρ(A) it is true that Φ(rλ) = (λI − A) where −1 rλ(t) := (λ − t) . Using the notation f(A) for ΦA(f); for all f; g; fn 2 L1(σ(A); µ) it is true that • χσ(A)(A) = I, • (fg)(A) = f(A)g(A), ∗ • f(A) = (f(A)) and • kf(A)k = kfk1.
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