Schedule:
Monday 11-13 in Koll A4,
Friday    11-13 in Koll A4.

Prerequisites: Analysis 2.  Notes by Henrik Stetkær for Analysis 2 in the fall 2002 (denoted [St]) are available here  Analysis2,2002

Literature:
[D] E.B. Davies: "Spectral theory and differential operators", Cambridge University Press, 1995, 1. edition.
[Sk] E. Skibsted: "Spectral theory, spring 2003", notes  here
[Sk, Convolution]: Note on convolution  here
[Sk, Distributions]: Notes on distributions  here
[Sk,  Cayley transform]:  Notes   here

Exercises:

• (To [D, Chapter 1]):  ExerChap1
• (Partly to [D, Chapter 6]):  ExerChap6

• [D, Definition on p. 7] The adjoint operator is only an "operator" if it is densely defined.
• [D, proof of Theorem 1.2.10]: Liouville's theorem cannot be applied directly.
• [D, proof of Lemma 2.4.3, l. 3-5 on p. 35]: It is not clear that v belongs to L if L is cyclic with cyclic vector v.
• [D, l. 1-4 on p. 49]: There is no need to let beta be independent of alpha.
• [D, Lemma 3.2.4]: ?
• [D, Lemma 3.4.1]: The proof requires du Bois Reymond's lemma.
• [D, Lemma 3.2.5]: The proof relies on Lemma 3.4.1.
• [D, proof of Theorem 4.4.5]: Is Theorem 4.4.2 used to conclude that Qprime is closable?

• Feb4: [D, Def. p. 2, Def. p. 6]. Example 1.1.1; symmetry and existence of an ONB of eigenstates were shown, cf. Example 1.2.3. Three examples that call for spectral theory were discussed/mentioned: 1) The vibrating spring (or the elastic bar). In the separation of variable method for solving the corresponding wave equation a spectral problem for a certain Sturm-Liouville operator enters. (Example 1.1.1 deals with two examples of Sturm-Liouville operators.) 2) The membrane problem. It is an example of a Dirichlet problem and may be attacked by use of a Dirichlet Laplacian on a  region in the plane. 3) The Bohr-atom. Spectral theory is crucial at the fundamental level of quantum mechanics.
• Feb7: [D, p. 3-5]. We elaborated on the notion of analyticity of a Banach-valued function (without proofs). As a preliminary for Lemma 1.1.3 we intoduced a natural partial ordering of the set of operators on a given Banach space, and we characterized the subspaces of the product space that are given as the graph of an operator.
• Feb10: [D, p. 6-8 to Lemma 1.2.2]. We noticed that Riesz-Frechet [St, Theorem V1.1] provides an efficient formulation of the condition that a vector is in the domain of the adjoint operator. Various properties of the adjoint operator were mentioned for example the analogue of [St, Theorem VI.12]. We computed the adjoint of a maximal multiplication operator on  L^2 of a measure space (cf. [D, Section 1.3]) and noticed that multiplication by a  real-valued function is an example of a  self-adjoint operator.
• Feb14: [D, Lemma 1.2.2, Examples 1.2.3, 1.2.5]. We gave a different proof of Lemma 1.2.2 using that a maximal multiplication operator is self-adjoint (if the function is real-valued). We mentioned the "spectral theorem" [D, Theorem 2.5.1]. Did [ExerChap1, Exercise 1].
• Feb17:  We proved Lemma 1 and Corollary 3  in [Sk,  Cayley transform]. (As a substitute for  Lemma 1.2.6 and Theorem 1.2.7 we follow the notes [Sk,  Cayley transform].)
• Feb21:  We proved Lemma 4 in [Sk,  Cayley transform]. Did [ExerChap1, Exercise 3].
• Feb24:  We elaborate on Remarks 5  in [Sk,  Cayley transform]. Started on [Sk, Section 1].
• Feb28: [ExerChap1, Exercises 2,  4,  5].  Continued the proof of [Sk, Theorem 1.5].
• Mar3: Completed [Sk, Section 1]. Started on [Sk, Section 3].
• Mar7: Completed [Sk, Section 3]. The main result Theorem 4.1 is homework. (Notice that Exercise 4.2.1 is used.)
• Mar10: [D, Sections 3.1-2]. The needed convolution estimates for Lemmas 3.2.1 and 3.2.2 are given in the note  Convolution. As a  Corollary of Lemma 3.2.1 we proved the du Bois Reymond lemma.
• Mar14: The students were supposed to do [Sk, Exercises 3.6.1 and 4.2.1].
• Mar17: [D, Section 3.3]. We showed that the Fourier transform is one-one on L^1 and consistently defined on L^1\cap L^2, cf. [D, Lemma 3.4.1].
• Mar21: [D, Section 3.4]. We put emphasis on the concept of distribution. We proved D, Lemma 3.2.5]. The Fourier transform of a constant was computed. Theorems 3.4.3, 3.4.4 and Corollary 3.4.5 concern the question, when is the Fourier transform of a given function given by a function? We did not elaborate at this point. Filip did [Sk, Exercise 4.2.1].
• Mar24: [D, Section 3.5 minus Lemma 3.5.1 and Example 3.5.6]. We did not elaborate on the notion of ellipticity; Corollary 3.5.4 was discussed for the Laplacian only.
• Mar28: Filip and Erik did [Sk, Exercise 4.2.2].
• Mar31: Preparation for [D, Example 3.5.6]. Distributional characterization of the \barH in Theorem 3.5.3.
• Apr4: [D, Section 3.7].
• Apr7: We start on the notes  Distributions.
• Apr11: Lemma 8 in [Sk, Distributions]. Claus and Erik did [Sk, Exercise 3.6.3] and Exercise 1 in [Sk, Distributions].
• Apr14: We finished the notes [Sk, Distributions].
• Apr25: Claus did project on the Baire theorem and consequences.
• Apr28: Exercises 2, 4, 5 and 6 in [Sk, Distributions].
• May2: Claus completed project on the Baire theorem and consequences. Did [D, Lemma 7.1.1]. Defined the Dirichlet Laplacian on a general bounded domain in higher dimensions. Showed consistency with the old definition of Example 9 in [Sk, Distributions] in one dimension. Did [D, Lemma 6.2.1].
• May5: Did Exercise 6 C in [Sk,  Cayley transform] and started on [Sk, Section 5].
• May9: Did [Sk, Exercise 5.10.1] and completed [Sk, Section 5] (except for the proof of Theorem 5.7).
• May12: Proved [D, Corollary 4.2.3] and the min-max-principle of [ExerChap6, Exercise 5]. Proved that the Dirichlet Laplacian on any bounded domain has purely discrete spectrum. Home-work: [Sk, Exercise 5.10.4].
• May19: Did [Sk, Exercises 5.10.-2,3,5,6].
• May23: We proved [D, Theorem 6.3.1], and solved the membrane problem. Studied general stability of the essential spectrum (Weyl's criterion), exemplified by the Bohr-atom.