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Spectral Theory and Asymptotic Methods Lecture Notes of The Spectral theory and asymptotic methods Lecture notes of the course given at the University of Oldenburg in April-July 2020 Lecturer: Konstantin Pankrashkin Contents Notation 1 1 Unbounded operators 3 1.1 Closed and adjoint operators . 3 1.2 Differential operators and Sobolev spaces . 9 2 Operators and forms 16 2.1 Operators defined by forms . 16 2.2 Semibounded operators and Friedrichs extensions . 20 3 Spectrum: first observations 26 3.1 Definitions and examples . 26 3.2 Basic facts on the spectra of self-adjoint operators . 29 3.3 Compactness and spectra . 34 4 Spectral theorem 41 4.1 Continuous functional calculus . 42 4.2 L2 spectral representation . 47 4.3 Some direct applications of the spectral theorem . 52 4.4 Spectral projections . 53 4.5 Tensor products (and operators with separated variables) . 55 5 Perturbations 59 5.1 Kato-Rellich theorem . 59 5.2 Essential self-adjointness of Schr¨odingeroperators . 61 5.3 Discrete and essential spectra . 67 5.4 Weyl criterion and relatively compact perturbations . 68 5.5 Essential spectra for Schr¨odingeroperators . 71 6 Variational principle for eigenvalues 73 6.1 Min-max principle . 73 6.2 Existence of negative eigenvalues for Schr¨odingeroperators . 77 7 Laplacian eigenvalues for bounded domains 82 7.1 Dirichlet and Neumann eigenvalues . 82 7.2 Weyl asymptotics . 84 8 Schr¨odingeroperators: more on eigenvalues and eigenfunctions 89 8.1 Finiteness/infiniteness of the discrete spectrum . 89 8.2 Weyl-type asymptotics . 92 8.3 Decay of eigenfunctions . 98 8.4 Strong coupling asymptotics . 106 9 Laplacians in unbounded domains 115 9.1 Bottom of the essential spectrum, decay of eigenfunctions . 115 9.2 Domains with cylindrical ends . 123 Notation Here we list some conventions used throughout the text. The symbol N denotes the sets of the natural numbers starting from 1. If (M; µ) is a measure space and f : M ! C is a measurable function, then we denote n o essµ ran f := z 2 C : µ m 2 M : z − f(m) < " > 0 for all " > 0 ; n o essµ sup jfj := inf a 2 R : µ m 2 M : f(m) > a = 0 : 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 \H" appears without explanations, it denotes a certain Hilbert space. If H is a Hilbert space and x; y 2 H, then by hx; yi we denote the scalar product of x and y. If there is more than one Hilbet space in play, we use the more detailed notation hx; yiH. We assume that the scalar product is linear with respect to the second argument and as anti-linear with respect to the first one, i.e. that for all α 2 C we have hx; αyi = hαx; yi = αhx; yi. This means, for example, that the scalar product in the standard space L2(R) is defined by Z hf; gi = f(x)g(x) dx: R If A is a finite or countable set, we denote by `2(A) the vector space of the functions x : A ! C with X 2 ξ(a) < 1; a2A and this is a Hilbert space with the scalar product X hx; yi = x(a)y(a): a2A If H and G are Hilbert spaces, then by L(H; G) and K(H; G) we denotes the spaces of the linear operators and the one of the compact operators from H and G, respectively. Furtheremore, L(H) := L(H; H) and K(H) := K(H; H). If Ω ⊂ Rd is an open set and k 2 N, then Hk(Ω) denotes the kth Sobolev space, i.e. the space of L2 functions whose weak partial derivatives up to order k are also 2 k 1 in L (Ω), see Section 1.2, 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 0 d functions is denoted as C (Ω). This should not be confused with C0(R ) which is the d set of the continuous functions f on R vanishing at infinity: limjxj!1 f(x) = 0. The subindex comp or c means that we only consider the functions with compact supports in the respective space (i.e. the functions vanishing outside a compact set). E.g. 1 d 1 d Hcomp(R ) is the set of the functions from H (R ) having compact supports. 1 Recommended books During the preparation of the notes I used a part of the text by Bernard Helffer which is available online: B. Helffer: Spectral theory and applications. An elementary introductory course. Available at http://www.math.u-psud.fr/~helffer/. An extended version of the above text was published as a book: B. Helffer: Spectral theory and its applications. Cambridge Studies in Applied Mathematics, 2012. Other recommended books are G. Teschl: Mathematical methods in quantum mechanics. With applications to Schr¨odingeroperators. AMS, 2009. Available from http://www.mat.univie. ac.at/~gerald/ B. Helffer: Semi-Classical Analysis for the Schr¨odingerOperator and Appli- cations. Lecture Notes in Mathematics. Springer, 1988 (mostly for more ad- vanced material related to the asymptotic analysis). Additional references on particular topics will be given during the course. At many points we will be obliged to use some facts on distributions and Sobolev spaces. I tried to include some elementary facts in these notes and I hope that it will be sufficient. An excellent introduction to the theory of distributions (which contains all necessary information on the Sobolev spaces) can also be found in these lecture notes by Patrick G´erard: https://www.imo.universite-paris-saclay.fr/~pgerard/Distributions2019_Chap1,2,3.pdf 2 1 Unbounded operators 1.1 Closed and adjoint operators A linear operator T in H is a linear map from a subspace (the domain of T ) D(T ) ⊂ H to H. The range of T is the set ran T := fT x : x 2 D(T )g. We say that a linear operator T is bounded if the quantity kT xk µ(T ) := sup x2D(T ) kxk x6=0 is finite. In what follows, the word combination \an unbounded operator" should be understood as \an operator which is not assumed to be bounded". If D(T ) = H and T is bounded, we arrive at the notion of a continuous linear operator in H; the space of such operators is denoted by L(H). This is a Banach space equipped with the norm kT k := µ(T ). 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 H, it can be uniquely extended to a continuous linear operator. Let us discuss a similar idea for unbounded operators. The graph of a linear operator T in H is the set gr T := (x; T x): x 2 D(T ) ⊂ H × H: For two linear operators T1 and T2 in H we write T1 ⊂ T2 if gr T1 ⊂ gr T2. I.e. T1 ⊂ T2 means that D(T1) ⊂ D(T2) and that T2x = T1x for all x 2 D(T1); the operator T2 is 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 H is called closed if its graph is a closed subspace in H × H. A linear operator T in H is called closable, if the closure gr T of the graph of T in H × H is still the graph of a certain operator T . This operator T with gr T = gr T is called the closure of T . The following proposition is obvious: Proposition 1.2. A linear operator T in H is closed if and only if the three condi- tions xn 2 D(T ), xn converge to x in H, T xn converge to y in H 3 imply the inclusion x 2 D(T ) and the equality y = T x. Definition 1.3 (Graph norm). Let T be a linear operator in H. Define on D(T ) a new scalar product by hx; yiT = hx; yi + hT x; T yi. The associated norm p p 2 2 kxkT := hx; xiT = kxk + kT xk is called the graph norm for T . The following assertion is also evident. Proposition 1.4. Let T be a linear operator in H. T is closed iff D(T ) is complete in the graph norm. If T is closable, then D(T ) is exactly the completion of D(T ) with respect to the graph norm. Informally, one could say that D(T ) consists of those x for which there is a unique candidate for T x if one tries to extend T by density. I.e., a vector x 2 H belongs to D(T ) iff: there exists a sequence (xn) ⊂ D(T ) converging to x, their exists the limit of T xn, this limit is the same for any sequence xn satisfying the previous two properties. Let us consider some simple examples. More sophisticated examples involving dif- ferential operators will be discussed later in Section 1.2. Example 1.5 (Bounded linear operators are closed). By the closed graph theorem, a linear operator T in H with D(T ) = H is closed if and only if it is bounded.
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