Spectral theory (spring 2003)
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
[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
[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
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
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.