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Introduction to the Spectral Theory Lecture Notes of the Course Given At Introduction to the spectral theory Lecture notes of the course given at the University Paris-Sud (Orsay, France) in September-December 2012 within the master programs “Partial Differential Equations and Numerical Analysis” and “Analysis, Arithmetics and Geometry” Lecturer: Konstantin Pankrashkin Webpage of the course: http://www.math.u-psud.fr/~pankrash/2012spec/ Contents Notation 1 1 Unbounded operators 3 1.1 Closedoperators ............................. 3 1.2 Adjointoperators............................. 6 2 Operators and forms 9 2.1 Operatorsdefinedbyforms. 9 2.2 Semibounded operators and Friedrichs extensions . ..... 14 3 Spectrum and resolvent 16 3.1 Definitions................................. 16 3.2 Examples ................................. 17 3.3 Basic facts on spectra of self-adjoint operators . 19 4 Spectral theory of compact operators 23 4.1 Integral and Hilbert-Schmidt operators . 25 4.2 Operators with compact resolvent . 28 4.3 Schr¨odinger operators with growing potentials . ....... 29 5 Spectral theorem 31 5.1 Continuous functional calculus . 32 5.2 Borelian functional calculus and L2 representation . 37 6 Some applications of spectral theorem 44 6.1 Spectralprojections............................ 44 6.2 Generalized eigenfunctions . 47 6.3 Tensorproducts.............................. 48 7 Perturbations 50 7.1 Kato-Rellichtheorem. 50 7.2 Essential self-adjointness of Schr¨odinger operators . ....... 52 7.3 Discreteandessentialspectra . 55 7.4 Weyl criterion and relatively compact perturbations . ........ 56 7.5 Essential spectra for Schr¨odinger operators . 59 7.6 Perturbationsofdiscretespectra. 60 8 Variational principle for eigenvalues 63 8.1 Max-minandmin-maxprinciples . 63 8.2 Negative eigenvalues of Schr¨odinger operators . ...... 65 9 Laplacian eigenvalues for bounded domains 68 9.1 Dirichlet and Neumann eigenvalues . 68 9.2 Weylasymptotics ............................. 71 9.3 Simplicity of the lowest eigenvalue . 74 10 Self-adjoint extensions 77 10.1 Description of self-adjoint extensions . 77 10.2Spectralanalysis ............................. 82 References 88 Notation Here we list some conventions used throughout the text. The symbol N denotes the sets of the natural numbers starting from 1 (contrary to the French tradition where one starts with 0). If (M,µ) is a measure space and f : M C is a measurable function, then we denote → ess ran f := z C : µ m M : z f(m) <ε > 0 for all ε> 0 , µ ∈ ∈ − n o ess sup f := inf a R : µ m M : f(m) >a = 0 . µ | | ∈ ∈ n o If the measure µ is uniquely determined by the context, then the index µ will be sometimes omitted. In what follows the phrase “Hilbert space” should be understood as “separable complex Hilbert space”. Most propositions also work in the non-separable case if reformulated in a suitable way. If the symbol “ ” appears without explanations, it denotes a certain Hilbert space. If is a HilbertH space and x, y , then by x, y we denote the scalar product of x andH y. If there is more than one∈H Hilbet spaceh ini play, we use the more detailed notation x, y . We assume that the scalar product h iH is linear with respect to the second argument and as anti-linear with respect to the first one, i.e. that for all α C we have x, αy = αx, y = α x, y . This means, for example, that the scalar∈ product in theh standardi h spacei L2(Rh) isi defined by f,g = f(x)g(x) dx. h i R Z If A is a finite or countable set, we denote by ℓ2(A) the vector space of the functions x : A C with 2 → ξ(a) < , ∞ a A X∈ and this is a Hilbert space with the scalar product x, y = x(a)y(a). h i a A X∈ If and are Hilbert spaces, then by ( , ) and ( , ) we denotes the spaces of theH linearG operators and the one of the compactL H G operatorsK H froG m and , respectively. Furtheremore, ( ) := ( , ) and ( ) := ( , ). H G L H L H H K H K H H If Ω Rd is an open set and k N, then Hk(Ω) denotes the kth Sobolev space, i.e. the⊂ space of L2 functions whose∈ partial derivatives up to order k are also in 2 k L (Ω), and by H0 (Ω) we denote the completion of Cc∞(Ω) with respect to the norm of Hk(Ω). The symbol Ck(Ω) denotes the space of functions on Ω whose partial derivatives up to order k are continuous; i.e. the set of the continuous functions 0 d is denoted as C (Ω). This should not be confused with C0(R ) which is the set 1 d of the continuous functions f on R vanishing at infinity: lim x f(x) = 0. The | |→∞ subindex comp means that we only consider the functions with compact support in 1 Rd 1 Rd the respective space. E.g. Hcomp( ) is the set of the functions from H ( ) having compact supports. 2 1 Unbounded operators 1.1 Closed operators A linear operator T in is a linear map from a subspace (the domain of definition H of T ) dom A to . The range of T is the set ran T := T x : x dom T . We say that a linear⊂H operatorH T is bounded if the quantity { ∈ } T x µ(T ) := sup k k x dom T x ∈x=0 k k 6 is finite. In what follows, the word combination “an unbounded operator” should be under- stood as “an operator which is not assumed to be bounded”. If dom T = and T is bounded, we arrive at the notion of a continuous linear operator in ; theH space of such operators is denoted by ( ). This is a Banach space equippedH with the norm T := µ(T ). L H k k During the whole course, by introducing a linear operator we always assume that its domain is dense, if the contrary is not stated explicitly. If T is a bounded operator in , it can be uniquely extended to a continuous linear H operator. Let us discuss the question on the continuation of unbounded operators. The graph of a linear operator T in is the set H gr T := (x, T x) : x dom T . ∈ ⊂H×H For two linear operators T1 and T2 in we write T1 T2 if gr T1 gr T2. I.e. T T means that dom T dom T andH that T x = T⊂x for all x dom⊂ T ; T is 1 ⊂ 2 1 ⊂ 2 2 1 ∈ 1 2 then called an extension of T1 and T1 is called a restriction of T2. Definition 1.1 (Closed operator, closable operator). A linear operator T in is called closed if its graph is a closed subspace in • . H H×H A linear operator T in is called closable, if the closure gr T of the graph of • T in is still theH graph of a certain operator T . This operator T with gr T =H×Hgr T is called the closure of T . The following propositions are obvious Proposition 1.2. A linear operator T in is closed if and only if the three condi- tions H x dom T , • n ∈ x converge to x in , • n H 3 T x converge to y in • n H imply the inclusion x dom T and the equality y = T x. ∈ Proposition 1.3. Let T be a linear operator in . Equip dom T with the scalar H product u, v T = u, v, + Tu,Tv . h i h iH h iH T is closed if and only if (dom T, , ) is a Hilbert space. • h· ·iT If T is closable, then dom T is exactly the completion of dom T with respect to • , . h· ·iT Consider some examples. Example 1.4 (Bounded linear operators are closed). By the closed graph theorem, a linear operator T in with dom T = is closed if and only if it is bounded. In this course we considerH mostly unboundedH closed operators. Example 1.5 (Multiplication operator). Take again = L2(Rd) and pick f d H ∈ L∞ (R ). Introduce a linear operator M in as follows: loc f H dom M = u L2(Rd) : fu L2(Rd) and M u = fu for u dom M . f { ∈ ∈ } f ∈ f Let us show that this operator is closed. Note first that one can construct an orthonormal basis (em)m N of consisting of ∈ H functions with compact support, and then one has clearly em dom Mf . 2 ∈ 2 Let (u ) dom M be a sequence such that u L u and M u = fu L v. For n ⊂ f n −→ f n n −→ any fixed basis vector em we have then: n n e , u →∞ e , u and e , M u →∞ e ,v . h m ni −−−→h m i h m f ni −−−→h n i On the other hand, using the integral expression for the scalar product, n e , M u = fe , u →∞ fe , u , which implies the equalities h m f ni h m ni −−−→h m i f(x)em(x)u(x)dx = em(x)v(x)dx m N. Rd Rd ∀ ∈ Z Z By the Parseval equality, 2 2 f(x)em(x)u(x)dx em(x)v(x)dx < , Rd Rd ∞ m N Z m N Z X∈ X∈ which shows that fu L2(Rd). This gives u dom M and, finally, v = fu = M u. ∈ ∈ f f An interested reader can generalize this example by considering multiplications op- erators in measure spaces. 4 d 2 d Example 1.6 (Laplacians in R ). Take = L (R ) and denote by T0 the operator H Rd acting as T0u = ∆u with the domain dom T0 = Cc∞( ). We are going to show − Rd that this operator is not closed.
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