Topics in Spectral Theory

2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Complex measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Lebesgue-Radon-Nikodym theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Fourier transform of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 Differentiation of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

. The Nordfjordeid lectures were a natural continuation of the material covered in Grenoble, and [JKP] can be read as Section 6 of these lecture notes.
The subject of these lecture notes is spectral theory of self-adjoint operators and some of its applications to mathematical physics. This topic has been covered in many places in the literature, and in particular in [Da,RS1,RS2,RS3,RS4,Ka]. Given the clarity and precision of these references, there appears to be little need for additional lecture notes on the subject. On the other hand, the point of view adopted in these lecture notes, which has its roots in the developments in mathematical physics which primarily happened over the last decade, makes the notes different from most expositions and I hope that the reader will learn something new from them.
The main theme of the lecture notes is the interplay between spectral theory of self-adjoint operators and classical harmonic analysis. In a nutshell, this interplay can be described as follows. Consider a self-adjoint operator A on a Hilbert space H and a vector ϕ ∈ H. The function F (z) = (ϕ|(A − z) −1 ϕ) is analytic in the upper half-plane Im z > 0 and satisfies the bound |F (z)| ≤ ϕ 2 /Im z. By a well-known result in harmonic analysis (see Theorem 2.11) there exists a positive Borel measure µ ϕ on R such that for Im z > 0, The measure µ ϕ is the spectral measure for A and ϕ. Starting with this definition we will develop the spectral theory of A. In particular, we will see that many properties of the spectral measure can be characterized by the boundary values lim y↓0 F (x + iy) of the corresponding function F . The resulting theory is mathematically beautiful and has found many important applications in mathematical physics. In Section 4 we will discuss a simple and important application to the spectral theory of rank one perturbations. A related application concerns the spectral theory of the Wigner-Weisskopf atom and is discussed in the lecture notes [JKP].
Although we are mainly interested in applications of harmonic analysis to spectral theory, it is sometimes possible to turn things around and use the spectral theory to prove results in harmonic analysis. To illustrate this point, in Section 4 we will prove Boole's equality and the celebrated Poltoratskii theorem using spectral theory of rank one perturbations.
The lecture notes are organized as follows. In Section 1 we will review the results of the measure theory we will need. The proofs of less standard results are given in detail. In particular, we present detailed discussion of the differentiation of measures based on the Besicovitch covering lemma. The results of harmonic analysis we will need are discussed in Section 2. They primarily concern Poisson and Borel transforms of measures and most of them can be found in the classical references [Kat, Ko]. However, these references are not concerned with applications of harmonic analysis to spectral theory, and the reader would often need to go through a substantial body of material to extract the needed results. To aid the reader, we have provided proofs of all results discussed in Section 2. Spectral theory of self-adjoint operators is discussed in Section 3. Although this section is essentially self-contained, many proofs are omitted and the reader with no previous exposition to spectral theory would benefit by reading it in parallel with Chapters VII-VIII of [RS1] and Chapters I-II of [Da]. Spectral theory of rank one perturbations is discussed in Section 4.
Concerning the prerequisites, it is assumed that the reader is familiar with basic notions of real, functional and complex analysis. Familiarity with [RS1] or the first ten Chapters of [Ru] should suffice.
for every E ∈ F, µ(E) = inf{µ(V ) : E ⊂ V, V open} = sup{µ(K) : K ⊂ E, K compact}. Theorem 1.1 Let M be a locally compact metric space in which every open set is σ-compact (that is, a countably union of compact sets). Let µ be a Borel measure finite on compact sets. Then µ is regular.
The measure space we will most often encounter is R with the usual Borel σ-algebra. Throughout the lecture notes we will denote by 1l the constant function 1l(x) = 1 ∀x ∈ R.

Complex measures
Let (M, F) be a measure space. Let E ∈ F. A countable collection of sets for every E ∈ F and every partition {E i } of E. In particular, the series (1.1) is absolutely convergent. Note that complex measures take only finite values. The usual positive measures, however, are allowed to take the value ∞. In the sequel, the term positive measure will refer to the standard definition of a measure on a σ-algebra which takes values in [0, ∞].
The set function |µ| on F defined by where the supremum is taken over all partitions {E i } of E, is called the total variation of the measure µ. for all E ∈ F. The last relation is abbreviated dµ = hd|µ|.
A complex measure µ is called regular if |µ| is a regular measure. Note that if ν is a positive measure, f ∈ L 1 (M, dν) and The integral with respect to a complex measure is defined in the obvious way, f dµ = f hd|µ|. Notation. Let µ be a complex or positive measure and f ∈ L 1 (M, d|µ|). In the sequel we will often denote by f µ the complex measure Note that |f µ| = |f ||µ|.
Every complex measure can be written as a linear combination of four finite positive mea- This fact is known as the Hahn decomposition theorem.

Riesz representation theorem
In this subsection we assume that M is a locally compact metric space.
A continuous function f : M → C vanishes at infinity if ∀ > 0 there exists a compact set K such that |f (x)| < for x ∈ K . Let C 0 (M ) be the vector space of all continuous functions that vanish at infinity, endowed with the supremum norm f = sup x∈M |f (x)|. C 0 (M ) is a Banach space and we denote by C 0 (M ) * its dual. The following result is known as the Riesz representation theorem.
Moreover, there exists a unique f ∈ L 1 (R, dµ) such that ∀E ∈ F, The Radon-Nikodym decomposition is abbreviated as ν = f µ + ν s (or dν = f dµ + dν s ). If M = R and µ is the Lebesgue measure, we will use special symbols for the Radon-Nikodym decomposition. We will denote by ν ac the part of ν which is absolutely continuous (abbreviated ac) w.r.t. the Lebesgue measure and by ν sing the part which is singular with respect to the Lebesgue measure. A point x ∈ R is called an atom of ν if ν({x}) = 0. Let A ν be the set of all atoms of ν. The set A ν is countable and x∈Aν |ν({x})| < ∞. The pure point part of ν is defined by ν pp (E) = x∈E∩Aν ν({x}).
The measure ν sc = ν sing − ν pp is called the singular continuous part of ν.

Fourier transform of measures
Let µ be a complex Borel measure on R. Its Fourier transform is defined bŷ µ(t) is also called the characteristic function of the measure µ. Note that and so the function R t →μ(t) ∈ C is uniformly continuous.
The following result is known as the Riemann-Lebesgue lemma. The relation (1.2) may hold even if µ is singular w.r.t. the Lebesgue measure. The measures for which (1.2) holds are called Rajchman measures. A geometric characterization of such measures can be found in [Ly].
Recall that A ν denotes the set of atoms of µ. In this subsection we will prove the Wiener theorem: Theorem 1.6 Let µ be a signed Borel measure. Then Proof: Note first that Obviously, Since |K T (x, y)| ≤ 1, by the dominated convergence theorem we have that for all x, By Fubini's theorem, and by the dominated convergence theorem,

Differentiation of measures
We will discuss only the differentiation of Borel measures on R. The differentiation of Borel measures on R n is discussed in the problem set. We start by collecting some preliminary results. The first result we will need is the Besicovitch covering lemma.
In the sequel we will refer to {I i } and {I i,j } as the Besicovitch subcollections. Proof. |I| denotes the length of the interval I. We will use the shorthand I(x, r) = (x − r, x + r).
Setting r x = |I x |/2, we have I x = I(x, r x ).
Let d 1 = sup{r x : x ∈ A}. Choose I 1 = I(x 1 , r 1 ) from the family {I x } x∈A such that r 1 > 3d 1 /4. Assume that I 1 , . . . , I j−1 are chosen for j ≥ 1 and that A j = A \ ∪ j−1 i=1 I i is nonempty. Let d j = sup{r x : x ∈ A j }. Then choose I j = I(x j , r j ) from the family {I x } x∈A j such that r j > 3d j /4. In this way be obtain a countable (possibly finite) subcollection Suppose that j > i. Then x j ∈ A i and This observation yields that the intervals I( Since A is a bounded set and x j ∈ A, the disjointness of I j = I(x j , r j /3) implies that if the family {I j } is infinite, then lim j→∞ r j = 0. (1.4) The relation (1.4) yields that A ⊂ ∪ j I(x j , r j ). Indeed, this is obvious if there are only finitely many I j 's. Assume that there are infinitely many I j 's and let x ∈ A. By (1.4), there is j such that r j < 3r x /4, and by the definition of r j , x ∈ ∪ j−1 i=1 I i . Notice that if three intervals in R have a common point, then one of the intervals is contained in the union of the other two. Hence, by dropping superfluous intervals from the collection {I j }, we derive that A ⊂ ∪ j I j and that each point in R belongs to no more than two intervals I j . This proves (1).
To prove (2), we enumerate I j 's as follows. To I 1 is associated the number 0. The intervals to right of I 1 are enumerated in succession by positive integers, and the intervals to the left by negative integers. (The "succession" is well-defined, since no point belongs simultaneously to three intervals). The intervals associated to even integers are mutually disjoint, and so are the intervals associated to odd integers. Finally, denote the interval associated to 2n by I 1,n , and the interval associated to 2n + 1 by I 2,n . This construction yields (2). 2 Let µ be a positive Borel measure on R finite on compact sets and let ν be a complex measure. The corresponding maximal function is defined by .
The statement follows by taking a → −∞ and b → +∞. 2 In Problem 3 you are asked to prove: Proposition 1.9 Let A be a bounded Borel set. Then for any 0 < p < 1, We will also need: Proposition 1.10 Let ν j be a sequence of Borel complex measures such that lim j→∞ |ν j |(R) = 0. Then there is a subsequence ν j k such that Proof. By Theorem 1.8, for each k = 1, 2, . . . , we can find j k so that and so for µ-a.e. x, there is k x such that for k > k x , M ν j k ,µ (x) ≤ 2 −k . This yields the statement. 2 We are now ready to prove the main theorem of this subsection.
Proof. (1) We will split the proof into two steps.
This yields (2). 2 We finish this subsection with several remarks. If µ is the Lebesgue measure, then the results of this section reduce to the standard differentiation results discussed, for example, in Chapter 7 of [Ru]. The arguments in [Ru] are based on the Vitali covering lemma which is specific to the Lebesgue measure. The proofs of this subsection are based on the Besicovitch covering lemma and they apply to an arbitrary positive measure µ. In fact, the proofs directly extend to R n (one only needs to replace the intervals I(x, r) with the balls B(x, r) centered at x and of radius r) if one uses the following version of the Besicovitch covering lemma.

Theorem 1.12 Let
A be a bounded set in R n and, for each x ∈ A, let B x be an open ball with center at x. Then there is an integer N , which depends only on n, such that: Unfortunately, unlike the proof of the Vitali covering lemma, the proof of Theorem 1.12 is somewhat long and complicated. [1] Prove that the maximal function M ν,µ (x) is Borel measurable.
[5] Let µ be a complex Borel measure on R. Prove that |µ sing | = |µ| sing . Prove that if f n converges to zero in measure, then there is a subsequence f n j which converges to zero almost uniformly.
[8] State and prove the analog of Theorem 1.11 in R n .
[9] Let µ be a positive Borel measure on R finite on compact sets and f ∈ L 1 (R, dµ). Prove that lim Hint: You may follow the proof of Theorem 7.7 in [Ru].
[10] Let p ≥ 1 and f ∈ L p (R, dx). The maximal function of f , M f , is defined by (1) If p > 1, prove that M f ∈ L p (R, dx). Hint: See Theorem 8.18 in [Ru]. (2) Prove that if f and M f are in L 1 (R, dx), then f = 0.
[11] Denote by B b (R) the algebra of the bounded Borel functions on R. Prove that B b (R) is the smallest algebra of functions which includes C 0 (R) and is closed under pointwise limits of uniformly bounded sequences.

Preliminaries: harmonic analysis
In this section we will deal only with Borel measures on R. We will use the shorthand C + = {z : Im z > 0}. We denote the Lebesgue measure by m and write dm = dx. Let µ be a complex measure or a positive measure such that The Borel transform of µ is defined by The function F µ (z) is analytic in C + . Let µ be a complex measure or positive measure such that The Poisson transform of µ is defined by The function P µ (z) is harmonic in C + . If µ is the Lebesgue measure, then P µ (z) = π for all z ∈ C + . If µ is a positive or signed measure, then Im F µ = P µ . Note also that F µ and P µ are linear functions of µ, i.e. for λ 1 , λ 2 ∈ C, F λ 1 µ 1 +λ 2 µ 2 = λ 1 F µ 1 + λ 2 F µ 2 , P λ 1 µ 1 +λ 2 µ 2 = λ 1 P µ 1 + λ 2 P µ 2 , Our goal in this section is to study the boundary values of P µ (x + iy) and F µ (x + iy) as y ↓ 0. More precisely, we wish to study how these boundary values reflect the properties of the measure µ.
Although we will restrict ourselves to the radial limits, all the results discussed in this section hold for the non-tangential limits (see the problem set). The non-tangential limits will not be needed for our applications.

Poisson transforms and Radon-Nikodym derivatives
This subsection is based on [JL1]. Recall that I(x, r) = (x − r, x + r).

Lemma 2.2
Let ν be a complex and µ a positive measure. Then for all x ∈ R and y > 0, Proof. Since |P ν | ≤ P |ν| , w.l.o.g. we may assume that ν is positive. Also, we may assume that x ∈ supp µ (otherwise M ν,µ (x) = ∞ and there is nothing to prove). Since (2.11) The proof of this lemma is left for the problem set.

Lemma 2.4
Let µ be a positive measure and f ∈ C 0 (R). Then for µ-a.e. x, Remark. The relation (2.12) holds for all x for which (2.11) holds. For example, if µ is the Lebesgue measure, then (2.12) holds for all x. Proof. Note that Then .
Let x be such that (2.11) holds. The monotone convergence theorem yields that This yields the statement. 2 The main result of this subsection is: Theorem 2.5 Let ν be a complex measure and µ a positive measure. Let ν = f µ + ν s be the Radon-Nikodym decomposition. Then: In particular, ν ⊥ µ iff lim y↓0 P ν (x + iy) P µ (x + iy) = 0, for µ − a.e. x.
(2) Assume in addition that ν is positive. Then Proof. The proof is very similar to the proof of Theorem 1.11 in Section 1.
(1) We will split the proof into two steps. Step 1. Assume that ν µ, namely that ν = f µ. Let g n be a continuous function with compact support such that R |f − g n |dµ < 1/n. Set h n = f − g n . Then, It follows from Lemmas 2.2 and 2.4 that for µ-a.e. x, As in the proof of Theorem 1.11, there is a subsequence n j → ∞ such that g n j (x) → f (x) and M |hn|µ,µ (x) → 0 for µ-a.e. x, and (1) holds if ν µ.
In this subsection we prove Theorem 3.1 of [Si1]. ν is a complex measure, µ is a positive measure and ν = f µ + ν s is the Radon-Nikodym decomposition.
Theorem 2.6 Let A be a bounded Borel set and 0 < p < 1. Then (Both sides are allowed to be ∞). In particular, ν A ⊥ µ A iff for some p ∈ (0, 1), Proof. By Theorem 2.5, By Lemma 2.2, Hence, Proposition 1.9 and the dominated convergence theorem yield the statement. 2
Proof. Note that and so Clearly, sup y>0,t∈R |L y (t)| < ∞. By Lemma 2.4 and Remark after it, lim y↓0 L y (t) = 0 for all t ∈ R (see also Problem 2). Hence, the statement follows from the estimate (2.18) and the dominated convergence theorem. 2

Local L p -norms, p > 1
In this subsection we will prove Theorem 2.1 of [Si1].
Let ν be a complex or positive measure and let ν = f m + ν sing be its Radon-Nikodym decomposition w.r.t. the Lebesgue measure.
Let g be a continuous function with compact support contained in A and let q be the index dual to p, p −1 + q −1 = 1. Then, by Theorem 2.7, Hence, the map g → A g(x)dν(x) is a continuous linear functional on L q (A, dx), and there is a functionf ∈ L p (A, dµ) such that This relation implies that ν A is absolutely continuous w.r.t. the Lebesgue measure and that f (x) =f (x) for Lebesgue a.e. x. (1) and (2) follow. 2 Theorem 2.8 has a partial converse which we will discuss in the problem set.

Local version of the Wiener theorem
In this subsection we prove Theorem 2.2 of [Si1].

Theorem 2.9
Let ν be a signed measure and A ν be the set of atoms of ν. Then for any finite interval [a, b], .
Notice now that: (compute the integral using the residue calculus).
The result follows from these observations and the dominated convergence theorem.

Poisson representation of harmonic functions
Theorem 2.11 Let V (z) be a positive harmonic function in C + . Then there is a constant c ≥ 0 and a positive measure µ on R such that V (x + iy) = cy + P µ (x + iy).
The c and µ are uniquely determined by V .

Remark 1.
The constant c is unique since c = lim y→∞ V (iy)/y. By Theorem 2.7, µ is also unique. Remark 2. Theorems 2.5 and 2.11 yield that if V is a positive harmonic function in C + and dµ = f (x)dx + µ sing is the associated measure, then for Lebesgue a.e. x We shall first prove: Theorem 2.12 Let U be a positive harmonic function in D. Then there exists a finite positive Borel measure ν on Γ such that for all z ∈ D, Proof. By the mean value property of harmonic functions, for any 0 < r < 1, In particular, Each Φ r is a continuous linear functional on the Banach space C(Γ) and Φ r = U (0). The standard diagonal argument yields that there is a sequence r j → 1 and a bounded linear func- Obviously, Φ = U (0). By the Riesz representation theorem there exists a complex measure ν on Γ such that |ν|(Γ) = U (0) and (the proof of this relation is left for the problems-see Theorem 11.8 in [Ru]). Taking j → ∞ we derive Before proving Theorem 2.11, I would like to make a remark about change of variables in measure theory. Let (M 1 , This relation is easy to check if f is a characteristic function. The general case follows by the usual approximation argument through simple functions. If T is a bijection, then g ∈ L 1 (M 1 , dµ) iff g T −1 ∈ L 1 (M 2 , dµ T ), and in this case Proof of Theorem 2.11. We define a map S : C + → D by This is the well-known conformal map between the upper half-plane and the unit disc. The map S extends to a homeomorphism S : . Then there exists a positive finite Borel measure ν on Γ such that The map T : Γ \ {−1} → R is a homeomorphism. Let ν T be the induced Borel measure on R.

The Hardy class H ∞ (C + )
The Hardy class H ∞ (C + ) is the vector space of all functions V analytic in C + such that is a Banach space. In this subsection we will prove two basic properties of H ∞ (C + ).
A simple and important consequence of Theorems 2.13 and 2.14 is:

Theorem 2.15
Let F be an analytic function on C + with positive imaginary part. Then: (1) For Lebesgue a.e. x ∈ R the limit exists and is finite.
Proof. To prove (1), apply Theorem 2.13 to the function (F (z) + i) −1 . To prove (2), apply Theorem 2.14 to the function The map Φ y (f ) = R f dν y is a linear functional on L 1 (R, dt) and Φ y ≤ V . By the Banach-Alaoglu theorem, there a bounded linear functional Φ and a sequence y n ↓ 0 such that for all Taking n → ∞, we get and Theorem 2.5 yields the statement. 2 Remark 1. Theorem 2.13 can be also proven using Theorem 2.11. The above argument has the advantage that it extends to any Hardy class H p (C + ).

Remark 2.
In the proof we have also established the Poisson representation of V (the relation (2.24)).

Proposition 2.16 (Jensen's formula)
Assume that U (z) is analytic for |z| < 1 and that U (0) = 0. Let r ∈ (0, 1) and assume that U has no zeros on the circle |z| = r. Let α 1 , α 2 , . . . , α n be the zeros of U (z) in the region |z| < r, listed with multiplicities. Then Remark. The Jensen formula holds even if U has zeros on |z| = r. We will only need the above elementary version. Proof. Set Then for some > 0 V (z) has no zeros in the disk |z| < r + and the function log |V (z)| is harmonic in the same disk (see Theorem 13.12 in [Ru]). By the mean value theorem for harmonic functions, The substitution yields the statement. 2 Proof of Theorem 2.14. Setting U (e it ) = V (tan(t/2)), we have that Hence, it suffices to show that the integral on the r.h.s. is finite.
In the rest of the proof we will use the same notation as in the proof of Theorem 2.11. Recall that S and T are defined by (2.20) and (2.21). Let U (z) = V (T (z)). Then, U is holomorphic in D and sup z∈D |U (z)| < ∞. Moreover, a change of variables and the formula (2.24) yield that (The change of variables exercise is done in detail in [Ko], pages 106-107.) The analog of Theorem 2.5 for the circle yields that for Lebesgue a.e. θ The proof is outlined in the problem set.
We will now make use of the Jensen formula. If Fatou's lemma, the dominated convergence theorem and (2.27) yield that and the identity (2.26) yields the statement. 2

The Borel tranform of measures
Recall that the Borel transform F µ (z) is defined by (2.8).

Theorem 2.17
Let µ be a complex or positive measure. Then: (1) For Lebesgue a.e. x the limit exists and is finite.
has zero Lebesgue measure.
Remark. It is possible that x. We will prove the F. & M. Riesz theorem in Section 4. Proof. We will first show that .
If µ is a complex measure, we first decompose µ = (µ 1 − µ 2 ) + i(µ 3 − µ 4 ), where the µ i 's are positive measures, and then decompose Hence, (2.31) follows from the corresponding result for positive measures. Proof of (1): By Theorems 2.13 and 2.14, the limits R(x) = lim y↓0 R(x + iy) and G(x) = lim y↓0 G(x + iy) exist and G(x) is non-zero for Lebesgue a.e. x. Hence, for Lebesgue a.e. x, Proof of (2): F µ (x) is zero on a set of positive measure iff R(x) is, and if this is the case, and (2.28) follows from Theorem 2.14.

Theorem 2.18
Let ν be a complex and µ a positive measure. Let ν = f µ + ν s be the Radon-Nikodym decomposition. Let µ sing be the part of µ singular with respect to the Lebesgue measure. Then This theorem has played an important role in the recent study of the spectral structure of Anderson type Hamiltonians [JL2,JL3].
Poltoratskii's proof of Theorem 2.18 is somewhat complicated, partly since it is done in the framework of a theory that is also concerned with other questions. A relatively simple proof of Poltoratskii's theorem has recently been found in [JL1]. This new proof is based on the spectral theorem for self-adjoint operators and rank one perturbation theory, and will be discussed in Section 4. [1] (1) Prove Lemma 2.3.

Hint: To prove
[4] Prove the following converse of Theorem 2.8: If (1) and (2) [5] The following extension of Theorem 2.9 holds: Let ν be a finite positive measure. Then for any p > 1, Prove this and compute C p in terms of gamma functions. Hint: See Remark 1 after Theorem 2.2 in [Si1].
[7] Let µ be a complex measure. Prove that F µ ≡ 0 ⇒ µ = 0 if either one of the following holds: Let µ be a complex or positive measure on R and By Theorems 2.5 and 2.17, for Lebesgue a.e. x the limit exists and is finite.
The function H µ (x) is called the Hilbert transform of the measure µ (H f is called the Hilbert transform of the function f ).
(1) Prove that for Lebesgue a.e. x the limit [9] Let 1 ≤ p < ∞. The Hardy class H p (C + ) is the vector space of all analytic functions f on (1) Prove that · p is a norm and that H p (C + ) is a Banach space.
(2) Let f ∈ H p (C + ). Prove that the limit exists for Lebesgue a.e.
x and that f ∈ L p (R, dx). Prove that Hence, H 2 (C + ) can be identified with a subspace of L 2 (R, dx) which we denote by the same letter. Let [10] In this problem we will study the Poisson transform on the circle. Let Γ = {z : |z| = 1} and let µ be a complex measure on Γ. The Poisson transform of the measure µ is If we parametrize Γ by w = e it , t ∈ (−π, π] and denote the induced complex measure by µ(t), then Note also that if dµ(t) = dt, then P µ (z) = 2π. For w ∈ Γ we denote by I(w, r) the arc of length 2r centered at w. Let ν be a complex measure and µ a finite positive measure on Γ. The corresponding maximal function is defined by (1) Formulate and prove the Besicovitch covering lemma for the circle.
(2) Prove the following bound: For all r ∈ [0, 1) and θ ∈ (−π, π], You may either mimic the proof of Lemma 2.2, or follow the proof of Theorem 11.20 in [Ru]. (3) State and prove the analog of Theorem 2.5 for the circle.
(4) State and prove the analogs of Theorems 2.7 and 2.13 for the circle.
[11] In Part (4) of the previous problem you were asked to prove the relation (2.27). This relation could be also proved like follows: show first that and then use Problem 9 of Section 1.

[12] The goal of this problem is to extend all the results of this section to non-tangential limits.
Our description of non-tangential limits follows [Po1]. Let again Γ = {z : |z| = 1} and D = {z : |z| < 1}. Let w ∈ Γ. We say that z tends to w non-tangentially, and write z → w ∠ if z tends to w inside the region In the sector ∆ ϕ w inscribe a circle centered at the origin (we denote it by Γ ϕ ). The two points on Γ ϕ ∩ {z : Arg(1 − zw) = ±ϕ} divide the circle into two arcs. The open region bounded by the shorter arc and the rays Arg(1 − zw) = ±ϕ is denoted C ϕ w . Let ν and µ be as in Problem 10.
This is the key result which extends the radial estimate of Part (2)  (2) Let ν = f µ + ν s be the Radon-Nikodym decomposition. Prove that If ν is a positive measure, prove that (3) Extend Parts (3) and (4) of Problem 10 to non-tangential limits.
Using this observation extend all the results of this section to non-tangential limits.

Basic notions
Let H be a Hilbert space. We denote the inner product by (·|·) (the inner product is linear w.r.t. the second variable).

A linear operator on H is a pair (A, Dom (A)), where Dom (A) ⊂ H is a vector subspace and A : Dom (A) → H is a linear map. We set
We denote by B(H) the vector space of all bounded operators on H. B(H) with the norm (3.34) is a Banach space. If A is densely defined and there is a constant C such that for all ψ ∈ Dom (A), Aψ ≤ C ψ , then A has a unique extension to a bounded operator on H. An The graph of a linear operator A is defined by A is called closable if it has a closed extension. If A is closable, its smallest closed extension is called the closure of A and is denoted by A. It is not difficult to show that A is closable iff Γ(A) is the graph of a linear operator and in this case Γ(A) = Γ (A).
Let A be a densely defined linear operator. Its adjoint, A * , is defined as follows. Dom (A * ) is the set of all φ ∈ H for which there exists a ψ ∈ H such that Obviously, such ψ is unique and Dom (A * ) is a vector subspace. We set Theorem 3.1 Let A be a densely defined linear operator. Then: Let A be a closed densely defined operator. We denote by ρ(A) the set of all z ∈ C such that The set of all eigenvalues is called the point spectrum of A and is denoted by It is also possible that sp(A) = ∅. (For simple examples see [RS1], Example 5 in Chapter VIII).

Theorem 3.2 Assume that ρ(A) is non-empty. Then ρ(A) is an open subset of C and the map
The last relation is called the resolvent identity.

Digression: The notions of analyticity
Let Ω ⊂ C be an open set and X a Banach space. A function f : Ω → X is called norm analytic if for all z ∈ Ω the limit Obviously, if f is norm analytic, then f is weakly analytic. The converse also holds and we have:

Theorem 3.3 f is norm analytic iff f is weakly analytic.
For the proof, see [RS1]. The mathematical theory of Banach space valued analytic functions parallels the classical theory of analytic functions. For example, if γ is a closed path in a simply connected domain Ω, then γ f (z)dz = 0.
(3.35) (The integral is defined in the usual way by the norm convergent Riemann sums.) To prove (3.35), note that for x * ∈ X * , Since X * separates points in X, (3.35) holds. Starting with (3.35) one obtains in the usual way the Cauchy integral formula, Starting with the Cauchy integral formula one proves that for w ∈ Ω, where a n ∈ X. The power series converges and the representation (3.36) holds in the largest open disk centered at w and contained in Ω, etc.
In other words, A is symmetric if A ⊂ A * . Obviously, any symmetric operator is closable.

Theorem 3.4 Let
A be a symmetric operator on H. Then the following statements are equivalent:  (2) and (3) of Theorems 3.4 and 3.5 ±i can be replaced by z, z, for any z ∈ C \ R. Theorem 3.6 Let A be self-adjoint. Then:
(3) By replacing A with A − x, w.l.o.g. we may assume that x = 0. We consider first the case ψ ∈ Dom (A). The identity and (2) yield that iy(A − iy) −1 ψ + ψ ≤ Aψ /y, and so (3) holds. If ψ ∈ Dom (A), let ψ n ∈ Dom (A) be a sequence such that ψ n − ψ ≤ 1/n. We estimate (A). If A and B are bounded and self-adjoint, then obviously A ± B are also self-adjoint; A self-adjoint projection P is called an orthogonal projection. In this case H = Ker P ⊕ Ran P . We write dim P = dim Ran P .
Let A be a bounded operator on H. The real and the imaginary part of A are defined by Clearly, ReA and ImA are self-adjoint operators and A = ReA + iImA.

Direct sums and invariant subspaces
Let H 1 , H 2 be Hilbert spaces and A 1 , A 2 self-adjoint operators on H 1 , H 2 . Then, the operator This elementary construction has a partial converse. Let A be a self-adjoint operator on a Hilbert space H and let H 1 be a closed subspace of H. The subspace We will call A 1 the restriction of A to the invariant subspace H 1 and write Let Γ be a countable set and H n , n ∈ Γ, a collection of Hilbert spaces. The direct sum of this collection, is the set of all sequences {ψ n } n∈Γ such that ψ n ∈ H n and n∈Γ ψ n 2 Hn < ∞.
Let B n ∈ B(H n ) and assume that sup n B n < ∞. Then B{ψ n } n∈Γ = {B n ψ n } n∈Γ is a bounded operator on H and B = sup n B n .

Proposition 3.7 Let A n be self-adjoint operators on H n . Set
The proof of Proposition 3.7 is easy and is left to the problems. (1) For all n ∈ Γ there is a ψ n ∈ H n cyclic for A n .

Cyclic spaces and the decomposition theorem
(2) H = ⊕ n H n and A = ⊕ n A n .
Proof. Let {φ n : n = 1, 2, · · · } be a given cyclic set for A. Set ψ 1 = φ 1 and let H 1 be the cyclic space generated by A and ψ 1 (H 1 is the closure of the linear span of the set of vectors {(A − z) −1 ψ 1 : z ∈ C \ R}). By Theorem 3.6, ψ 1 ∈ H 1 . Obviously, H 1 is invariant under A and we set A 1 = A H 1 . We define ψ n , H n and A n inductively as follows. If H 1 = H, let φ n 2 be the first element of the sequence {φ 2 , φ 3 , · · · } which is not in H 1 . Decompose φ n 2 = φ n 2 + φ n 2 , where φ n 2 ∈ H 1 and φ n 2 ∈ H ⊥ 1 . Set ψ 2 = φ n 2 and let H 2 be the cyclic space generated by A and ψ 2 . It follows from the resolvent identity that H 1 ⊥ H 2 . Set A 2 = A H 2 . In this way we inductively define ψ n , H n , A n , n ∈ Γ, where Γ is a finite set {1, · · · , N } or Γ = N. By the construction, {φ n } n∈Γ ⊂ ∪ n∈Γ H n . Hence, (1) holds and H = ⊕ n H n .
To prove the second part of (2), note first that by the construction of A n , IfÃ = ⊕A n , then by Proposition 3.7,Ã is self-adjoint and (Ã − z) −1 = ⊕ n (A n − z) −1 . Hence A =Ã. 2

The spectral theorem
We start with: Dom ( Then: The proof of this theorem is left to the problems. The content of the spectral theorem for self-adjoint operators is that any self-adjoint operator is unitarily equivalent to A f for some f . Let H 1 and H 2 be two Hilbert spaces. A linear bijection U :  We will prove the spectral theorem only for separable Hilbert spaces. In the literature one can find many different proofs of Theorem 3.10. The proof below is constructive and allows to explicitly identify M and f while the measure µ is directly related to (A − z) −1 .

Proof of the spectral theorem-the cyclic case
Let A be a self-adjoint operator on a Hilbert space H and ψ ∈ H.

Theorem 3.11
There exists a unique finite positive Borel measure µ ψ on R such that µ ψ (R) = ψ 2 and

37)
The measure µ ψ is called the spectral measure for A and ψ.
The first relation yields that c = 0. The second relation and the dominated convergence theorem yield that µ ψ (R) = ψ 2 . The functions and U (z) are analytic in C + and have equal imaginary parts. The Cauchy-Riemann equations imply that F µ ψ (z) − U (z) is a constant. Since F µ ψ (z) and U (z) vanish as Im z → ∞, F µ ψ (z) = U (z) for z ∈ C + and (3.37) holds. 2 Corollary 3.12 Let ϕ, ψ ∈ H. Then there exists a unique complex measure µ ϕ,ψ on R such that Proof. The uniqueness is obvious (the set of functions {(x − z) −1 : z ∈ C \ R} is dense in C 0 (R)). The existence follows from the polarization identity: In particular, The main result of this subsection is:

Theorem 3.13 Assume that ψ is a cyclic vector for A. Then
A is unitarily equivalent to the operator of multiplication by x on L 2 (R, dµ ψ ). In particular, sp(A) = supp µ ψ .
Proof. Clearly, we may assume that ψ = 0. Note that By a limiting argument, the relation holds for all z, w ∈ C \ R. Hence, the map (3.42) extends to a unitary U : H → L 2 (R, dµ ψ ). Since is unitarily equivalent to the operator of multiplication by (x − z) −1 on L 2 (R, dµ ψ ). For any φ ∈ H, and so A is unitarily equivalent to the operator of multiplication by x. 2 We finish this subsection with the following remark. Assume that ψ is a cyclic vector for A and let A x be the operator of multiplication by x on L 2 (R, dµ ψ ). Then, by Theorem 3.13, there exists a unitary U : H → L 2 (R, dµ ψ ) such that

Proof of the spectral theorem-the general case
Let A be a self-adjoint operator on a separable Hilbert space H. Let H n , A n , ψ n , n ∈ Γ be as in the decomposition theorem (Theorem 3.8). Let U n : H n → L 2 (R, dµ ψn ) be unitary such that A n is unitarily equivalent to the operator of multiplication by x. We denote this last operator bỹ A n . Let U = ⊕ n U n . An immediate consequence of the decomposition theorem and Theorem 3.13 is Theorem 3.14 The map U : H → n∈Γ L 2 (R, dµ ψn ) is unitary and A is unitarily equivalent to the operator n∈ΓÃ n . In particular, Note that if φ ∈ H and U φ = {φ n } n∈Γ , then µ φ = n∈Γ µ φn .
Theorem 3.10 is a reformulation of Theorem 3.14. To see that, choose cyclic vectors ψ n so that n∈Γ ψ n 2 < ∞. For each n ∈ Γ, let R n be a copy of R and You may visualize M as follows: enumerate Γ so that Γ = {1, . . . , N } or Γ = N and set R n = {(n, x) : x ∈ R} ⊂ R 2 . Hence, M is just a collection of lines in R 2 parallel to the y-axis and going through the points (n, 0), n ∈ Γ. Let F be the collection of all sets F ⊂ M such that F ∩ R n is Borel for all n. Then F is a σ-algebra and is a finite measure on M (µ(M ) = n∈Γ ψ n 2 < ∞). Let f : M → R be the identity function (f (n, x) = x). Then Set These subspaces are invariant under A. Moreover, ψ ∈ H ac iff the spectral measure µ ψ is absolutely continuous w.r.t. the Lebesgue measure, ψ ∈ H sc iff µ ψ is singular continuous and ψ ∈ H pp iff µ ψ is pure point. Obviously, When we wish to indicate the dependence of the spectral subspaces on the operator A, we will write H ac (A), etc.

Harmonic analysis and spectral theory
Let A be a self-adjoint operator on a Hilbert space H, ψ ∈ H, and µ ψ the spectral measure for A and ψ. Let F µ ψ and P µ ψ be the Borel and the Poisson transform of µ ψ . The formulas provide the key link between the harmonic analysis (the results of Section 2) and the spectral theory. Recall that µ ψ,sing = µ ψ,sc + µ ψ,pp .
exists and is finite and non-zero. (2) Assume that for some p > 1 Let H ψ be the cyclic subspace spanned by A and ψ. W.l.o.g. we may assume that ψ = 1. By Theorem 3.13 there exists a (unique) unitary U ψ : H ψ → L 2 (R, dµ ψ ) such that U ψ ψ = 1l and U ψ AU −1 ψ is the operator of multiplication by x on L 2 (R, dµ ψ ). We extend U ψ to H by setting U ψ φ = 0 for φ ∈ H ⊥ ψ . Recall that The interplay between spectral theory and harmonic analysis is particularly clearly captured in the following result.

Spectral measure for A
Let A be a self-adjoint operator on a separable Hilbert space H and let {φ n } n∈Γ be a cyclic set for A. Let {a n } n∈Γ be a sequence such that a n > 0 and n∈Γ a n φ n 2 < ∞.
The spectral measure for A, µ A , is a Borel measure on R defined by µ A (·) = n∈Γ a n µ φn (·).
Obviously, µ A depends on the choice of {φ n } and a n . Two positive Borel measures ν 1 and ν 2 on R are called equivalent (we write ν 1 ∼ ν 2 ) iff ν 1 and ν 2 have the same sets of measure zero.  (A).
The proofs of these two theorems are left to the problems.

The essential support of the ac spectrum
Let B 1 and B 2 be two Borel sets in R. Let B 1 ∼ B 2 iff the Lebesgue measure of the symmetric difference (B 1 \ B 2 ) ∪ (B 2 \ B 1 ) is zero. ∼ is an equivalence relation. Let µ A be a spectral measure of a self-adjoint operator A and f (x) its Radon-Nikodym derivative w.r.t. the Lebesgue measure (dµ A,ac = f (x)dx). The equivalence class associated to {x : f (x) > 0} is called the essential support of the absolutely continuous spectrum and is denoted by Σ ess ac (A). With a slight abuse of terminology we will also refer to a particular element of Σ ess ac (A) as an essential support of the ac spectrum (and denote it by the same symbol Σ ess ac (A)). For example, the set is an essential support of the absolutely continuous spectrum. Note that the essential support of the ac spectrum is independent on the choice of µ A .
Theorem 3.20 Let Σ ess ac (A) be an essential support of the absolutely continuous spectrum. Then cl(Σ ess ac (A) ∩ sp ac (A)) = sp ac (A).
The proof is left to the problems. Although the closure of an essential support Σ ess ac (A) ⊂ sp ac (A) equals sp ac (A), Σ ess ac (A) could be substantially "smaller" than sp ac (A); see Problem 6.

The functional calculus
Let A be a self-adjoint operator on a separable Hilbert space H. Let U : H → L 2 (M, dµ), f , and A f be as in the spectral theorem. Let B b (R) be the vector space of all bounded Borel functions on R. For h ∈ B b (R), consider the operator A h•f . This operator is bounded and   (1) The map Φ is an algebraic * -homomorphism.
The map Φ is uniquely specified by (1)-(4). Moreover, it has the following additional properties: We remark that the uniqueness of the functional calculus is an immediate consequence of Problem 11 in Section 1. Let For any Borel function h : R → C we define h(A) by (3.44). Of course, h(A) could be an unbounded operator. Note that h 1 (A) In fact, to define h(A), we only need that The two classes of functions, characteristic functions and exponentials, play a particularly important role.
Let F be a Borel set in R and χ F its characteristic function. The operator χ F (A) is an orthogonal projection, called the spectral projection on the set F . In these notes we will use the notation 1 F ( the Stone formula follows from Theorem 3.21. 2 Another important class of functions are exponentials. For t ∈ R, set U (t) = exp(itA). Then U (t) is a group of unitary operators on H. The group U (t) is strongly continuous, i.e. for all ψ ∈ H, lim s→t U (s)ψ = U (t)ψ.
On the other hand, if the limit on the l.h.s. exists for some ψ, then ψ ∈ Dom (A) and (3.46) holds.
The above results have a converse:

Theorem 3.23 (Stone's theorem) Let U (t) be a strongly continuous group on a separable
Hilbert space H. Then there is a self-adjoint operator A such that U (t) = exp(itA).

The Weyl criteria and the RAGE theorem
Let A be a self-adjoint operator on a separable Hilbert space H. Assume that e ∈ sp (A). Let ψ n ∈ Ran 1 (e−1/n,e+1/n) (A) be unit vectors. Then, by the functional calculus, On the other hand, assume that there is a sequence ψ n such that (3.47) holds and that e ∈ sp (A). Then and so 1 = ψ n → 0, contradiction. 2 The discrete spectrum of A, denoted sp disc (A), is the set of all isolated eigenvalues of finite multiplicity. Hence e ∈ sp disc (A) iff 1 ≤ dim 1 (e− ,e+ ) (A) < ∞ for all small enough. The essential spectrum of A is defined by Hence, e ∈ sp ess (A) iff for all > 0 dim 1 (e− ,e+ ) (A) = ∞. Obviously, sp ess (A) is a closed subset of R. Proof.
(1) First, recall that any compact operator is a norm limit of finite rank operators. In other words, there exist vectors φ n , ϕ n ∈ H such that K n = n j=1 (φ j |·)ϕ j satisfies lim n→∞ K − K n = 0. Hence, it suffices to prove the statement for the finite rank operators K n . By induction and the triangle inequality, it suffices to prove the statement for the rank one operator K = (φ|·)ϕ. Thus, it suffices to show that for φ ∈ H and ψ ∈ H cont , w.l.o.g we may assume that φ ∈ H cont . Finally, by the polarization identity, we may assume that ϕ = ψ. Since for ψ ∈ H cont the spectral measure µ ψ has no atoms, the result follows from the Wiener theorem (Theorem 1.6 in Section 1).
(2) Since Dom (A) ∩ H cont is dense in H cont , it suffices to prove the statement for ψ ∈ Dom (A) ∩ H cont . Write and use (1). 2

Stability
We will first discuss stability of self-adjointness-if A and B are self-adjoint operators, we wish to discuss conditions under which A + B is self-adjoint on Dom (A) ∩ Dom (B). One obvious sufficient condition is that either A or B is bounded. A more refined result involves the notion of relative boundedness.
Let A and B be densely defined linear operators on a separable Hilbert space H. The operator B is called A-bounded if Dom (A) ⊂ Dom (B) and for some positive constants a and b and all ψ ∈ Dom (A), The number a is called a relative bound of B w.r.t. A.

Theorem 3.27 (Kato-Rellich) Suppose that A is self-adjoint, B is symmetric, and B is A-
bounded with a relative bound a < 1. Then: (2) A + B is essentially self-adjoint on any core of A.
(3) If A is bounded from below, then A + B is also bounded from below.
Proof. We will prove (1) and (2); (3) is left to the problems. In the proof we will use the following elementary fact: if V is a bounded operator and V < 1, then 0 ∈ sp(1 + V ). This is easily proven by checking that the inverse of 1 + V is given by 1 + ∞ k=1 (−1) k V k . By Theorem 3.4 (and the Remark after Theorem 3.5), to prove (1)   The proof of (2) is based on Theorem 3.5. Let D be a core for A. Then the sets (A ± iy)(D) = {(A ± iy)ψ : ψ ∈ D} are dense in H, and since 1 + B(A ± iy) −1 are bijections, are also dense in H. 2 We now turn to stability of the essential spectrum. The simplest result in this direction is: Proof. By symmetry, it suffices to prove that sp ess (A + B) ⊂ sp ess (A). Let e ∈ sp ess (A + B) and let ψ n be an orthonormal sequence such that lim n→∞ (A + B − e)ψ n = 0.
Since ψ n converges weakly to zero and B is compact, Bψ n → 0. Hence, (A − e)ψ n → 0 and e ∈ sp ess (A). 2 Section XIII.4 of [RS4] deals with various extensions and generalizations of Theorem 3.28. The proof of this proposition is simple and is left to the problems (see also [RS3]). Let H be a separable Hilbert space and {ψ n } an orthonormal basis. A bounded positive self-adjoint operator A is called trace class if

Scattering theory and stability of ac spectra
The number Tr(A) does not depend on the choice of orthonormal basis. More generaly, a bounded self-adjoint operator A is called trace class if Tr(|A|) < ∞. A trace class operator is compact.
Concerning stability of the ac spectrum, the basic result is: For the proof of Kato-Rosenblum theorem see [RS3], Theorem XI.7. The singular and the pure point spectra are in general unstable under perturbations-they may appear or dissapear under the influence of a rank one perturbation. We will discuss in Section 4 criteria for "generic" stability of the singular and the pure point spectra.

Notions of measurability
In mathematical physics one often encounters self-adjoint operators indexed by elements of some measure space (M, F), namely one deals with functions M ω → A ω , where A ω is a self-adjoint operator on some fixed separable Hilbert space H. In this subsection we address some issues concerning measurability of such functions.
Let (M, F) be a measure space and X a topological space. A function f : M → X is called measurable if the inverse image of every open set is in F.
Let H be a separable Hilbert space and B(H) the vector space of all bounded operators on H. We distinguish three topologies in B(H), the uniform topology, the strong topology, and the weak topology. The uniform topology is induced by the operator norm on B(H). The strong topology is the minimal topology w.r.t. which the functions B(H) A → Aψ ∈ H are continuous for all ψ ∈ H. The weak topology is the minimal topology w.r.t. which the functions B(H) A → (φ|Aψ) ∈ C are continuous for all φ, ψ ∈ H. The weak topology is weaker than the strong topology, and the strong topology is weaker than the uniform topology.
A function f : M → B(H) is uniform/strong/weak measurable if it is measurable with respect to the uniform/strong/weak topology. Obviously, uniform measurability ⇒ strong measurability ⇒ weak measurability. Note that f is weakly measurable iff the function M ω → (φ|f (ω)ψ) ∈ C is measurable for all φ, ψ ∈ H.

Theorem 3.31 A function f : M → B(H) is uniform measurable iff it is weakly measurable.
The proof of this theorem is left to the problems. A function f : M → B(H) is measurable iff it is weakly measurable (which is equivalent to requiring that f is strongly or uniform measurable).
Let ω → A ω be a function with values in (possibly unbounded) self-adjoint operators on H. We say that A ω is measurable if for all z ∈ C \ R the function Until the end of this subsection ω → A ω is a given measurable function with values in self-adjoint operators.

Theorem 3.32 The function ω → h(A ω ) is measurable for all
Note also that if h n ∈ T , sup n,x |h n (x)| < ∞, and h n (x) → h(x) for all x, then h ∈ T . Hence, by Problem 11 in Section 1, From this theorem it follows that the functions ω → 1 B (A ω ) (B Borel) and ω → exp(itA ω ) are measurable. One can also easily show that if h : R → R is an arbitrary Borel measurable real valued function, then ω → h(A ω ) is measurable.
We now turn to the measurability of projections and spectral measures.
Proof. Let {φ n } n∈N be an orthonormal basis of H and let P n be the orthogonal projection on the subspace spanned by {φ k } k≥n . The RAGE theorem yields that for ϕ, ψ ∈ H, (the proof of (3.52) is left to the problems), and the statement follows. 2
Let M (R) be the Banach space of all complex Borel measures on R (the dual of C 0 (R)).
We denote by µ ω ψ the spectral measure for A ω and ψ.

Proof. Since for any Borel set
, the statement follows from Propositions 3.34 and 3.35. 2 Let {ψ n } be a cyclic set for A ω and let a n > 0 be such that n a n ψ n 2 < ∞ . We denote by µ ω = n a n µ ω ψn the corresponding spectral measure for A ω . Proposition 3.36 yields Let Σ ess,ω ac be the essential support of the ac spectrum of A ω . The map does not depend on the choice of representative in Σ ess,ω ac .

Proposition 3.38 The function (3.53) is weakly measurable, namely for all
Proof. It suffices to prove the statement for h(x) = (1 + x) 2 χ B (x) where B is a bounded interval. Let µ ω be a spectral measure for A ω . By the dominated convergence theorem and the statement follows. 2

Non-relativistic quantum mechanics
According to the usual axiomatization of quantum mechanics, a physical system is described by a Hilbert space H. Its observables are described by bounded self-adjoint operators on H. Its states are described by density matrices on H, i.e. positive trace class operators with trace 1. If the system is in a state ρ, then the expected value of the measurement of an observable A is The variance of the measurement is The Cauchy-Schwarz inequality yields the Heisenberg principle: For self-adjoint A, B ∈ B(H), Of particular importance are the so called pure states ρ = (ϕ|·)ϕ. In this case, for a selfadjoint A, where µ ϕ is the spectral measure for A and ϕ. If the system is in a pure state described by a vector ϕ, the possible results R of the measurement of A are numbers in sp(A) randomly distributed according to dµ ϕ (recall that µ ϕ is supported on sp(A)). Obviously, in this case A ρ and D ρ (A) are the usual expectation and variance of the random variable R.
The dynamics is described by a strongly continuous unitary group U (t) on H. In the Heisenberg picture, one evolves observables and keeps states fixed. Hence, if the system is initially in a state ρ, then the expected value of A at time t is In the Schrödinger picture, one keeps observables fixed and evolves states-the expected value of A at time t is Tr([U (t) * ρU (t)]A). Note that if ρ = |ϕ)(ϕ|, then The generator of U (t), H, is called the Hamiltonian of the system. The spectrum of H describes the possible energies of the system. The discrete spectrum of H describes energy levels of bound states (the eigenvectors of H are often called bound states). Note that if ϕ is an eigenvector of H, then AU (t)ϕ 2 = Aϕ 2 is independent of t.
By the RAGE theorem, if ϕ ∈ H cont (H) and A is compact, then Compact operators describe what one might call sharply localized observables. The states associated to H cont (H) move to infinity in the sense that after a sufficiently long time the sharply localized observables are not seen by these states. On the other hand, if ϕ ∈ H pp (H), then for any bounded A, The mathematical formalism sketched above is commonly used for a description of nonrelativistic quantum systems with finitely many degrees of freedom. It can be used, for example, to describe non-relativistic matter-a finite assembly of interacting non-relativistic atoms and molecules. In this case H is the usual N -body Schrödinger operator. This formalism, however, is not suitable for a description of quantum systems with infinitely many degrees of freedom like non-relativistic QED, an infinite electron gas, quantum spin-systems, etc.
[6] Let 0 < < 1. Construct an example of a self-adjoint operator A such that sp ac (A) = [0, 1] and that the Lebesgue measure of Σ ess ac (A) is equal to .
[13] Let M ω → A ω be a function with values in self-adjoint operators on H. Prove that the following statements are equivalent: is measurable for all Borel sets B.
[16] Recall that M (R) is the Banach space of all complex measures on R. Assume that ω → µ ω ∈ M (R) is measurable. Prove that this function is also measurable w.r.t. the uniform topology of M (R).
The next set of problems deals with spectral theory of a closed operator A on a Hilbert space H.
[18] Let F ⊂ sp(A) be an isolated, bounded subset of sp (A). Let γ be a closed simple path in the complex plane that separates F from sp(A) \ F . Set (1) Prove that 1 F (A) is a (not necessarily orthogonal) projection.
(2) Prove that Ran 1  Prove that the operator (3.56) is closed and that its spectrum is sp(A) \ F .
Remark. The set of z 0 ∈ sp(A) which satisfy (5) is called the discrete spectrum of the closed operator operator A and is denoted sp disc (A). The essential spectrum is defined by sp ess (A) = sp(A) \ sp disc (A) [19] Prove that sp ess (A) is a closed subset of C.
[20] Prove that z → (A − z) −1 is a meromorphic function on C \ sp ess (A) with singularities at points z 0 ∈ sp disc (A). Prove that the negative coefficents of of the Laurent expansion at z 0 ∈ sp disc (A) are finite rank operators. Hint: See Lemma 1 in [RS4], Section XIII.4. For additional information about numerical range, the reader may consult [GR].
[22] Let z ∈ sp (A). A sequence ψ n ∈ Dom (A) is called a Weyl sequence if ψ n = 1 and (A − z)ψ n → 0. If A is not self-adjoint, then a Weyl sequence may not exist for some z ∈ sp (A). Prove that a Weyl sequence exists for every z on the topological boundary of sp (A). Hint: See Section XIII.4 of [RS4] or [VH].
[23] Let A and B be densely defined linear operators. Assume that B is A-bounded with a relative bound a < 1. Prove that A + B is closable iff A is closable, and that in this case the closures of A and A + B have the same domain. Deduce that A + B is closed iff A is closed.
[24] Let A and B be densely defined linear operators. Assume that A is closed and that B is A-bounded with constants a and b. If A is invertible (that is, 0 ∈ sp(A)), and if a and b satisfy a + b A −1 < 1, prove that A + B is closed, invertible, and that Hint: See Theorem IV. 1.16 in [Ka].
[25] In this problem we will discuss the regular perturbation theory for closed operators. Let A be a closed operator and let B be A-bounded with constants a and b. For λ ∈ C we set A λ = A + λB. If |λ|a < 1, then A λ is a closed operator and Dom (A λ ) = Dom (A). Let F be an isolated, bounded subset of A and γ a simple closed path that separates F and sp(H) \ F .
(6) Prove that the differential equation U λ = [P λ , P λ ]U λ , U 0 = 1, (the derivatives are w.r.t. λ and [A, B] = AB − BA) has a unique solution for |λ| < Λ, and that U λ is an analytic family of bounded invertible operators such that U λ P 0 U −1 λ = P λ . Hint: See [RS4], Section XII.2. (7) SetÃ λ = U −1 λ A λ U λ and Σ λ = P 0Ãλ P 0 . Σ λ is a bounded operator on the Hilbert space Ran P 0 . Prove that sp(Σ λ ) = F λ and that the operator-valued function λ → Σ λ is analytic for |λ| < Λ. Compute the first three terms in the expansion (3.57) The term Σ 1 is sometimes called the Feynman-Hellman term. The term Σ 2 , often called the Level Shift Operator (LSO), plays an important role in quantum mechanics and quantum field theory. For example, the formal computations in physics involving Fermi's Golden Rule are often best understood and most easily proved with the help of LSO.
(8) Assume that dim P 0 = dim Ran P 0 < ∞. Prove that dim P λ = dim P 0 for |λ| < Λ and conclude that the spectrum of A λ inside γ is discrete and consists of at most dim P 0 distinct eigenvalues. Prove that the eigenvalues of A λ inside γ are all the branches of one or more multi-valued analytic functions with at worst algebraic singularities.
(9) Assume that F 0 = {z 0 } and dim P 0 = 1 (namely that the spectrum of A inside γ is a semisimple eigenvalue z 0 ). In this case Σ λ = z(λ) is an analytic function for |λ| < Λ. Compute the first five terms in the expansion z(λ) = ∞ n=0 λ n z n .

Spectral theory of rank one perturbations
The Hamiltonians which arise in non-relativistic quantum mechanics typically have the form where H 0 and V are two self-adjoint operators on a Hilbert space H. H 0 is the "free" or "reference" Hamiltonian and V is the "perturbation". For example, the Hamiltonian of a free non-relativistic quantum particle of mass m moving in R 3 is − 1 2m ∆, where ∆ is the Laplacian in L 2 (R 3 ). If the particle is subject to an external potential field V (x), then the Hamiltonian describing the motion of the particle is (4.59) where V denotes the operator of multiplication by the function V (x). Operators of this form are called Schrödinger operators. We will not study in this section the spectral theory of Schrödinger operators. Instead, we will keep H 0 general and focus on simplest non-trivial perturbations V . More precisely, let H be a Hilbert space, H 0 a self-adjoint operator on H and ψ ∈ H a given unit vector. We will study spectral theory of the family of operators (4.60) This simple model is of profound importance in mathematical physics. The classical reference for the spectral theory of rank one perturbations is [Si2]. The cyclic subspace generated by H λ and ψ does not depend on λ and is equal to the cyclic subspace generated by H 0 and ψ which we denote H ψ (this fact is an immediate consequence of the formulas (4.62) below). Let µ λ be the spectral measure for H λ and ψ. This measure encodes the spectral properties of H λ H ψ . Note that H λ H ⊥ ψ = H 0 H ⊥ ψ . In this section we will always assume that H = H ψ , namely that ψ is a cyclic vector for H 0 .
The identities Note that if z ∈ C + , then F λ (z) is the Borel transform and Im F λ (z) is the Poisson transform of µ λ . The second identity in (4.62) yields and so (4.64) These elementary identities will play a key role in our study. The function will also play an important role. Recall that G(x) = ∞ for µ 0 -a.e. x (Lemma 2.3).
In this section we will occasionally denote by |B| the Lebesgue measure of a Borel set B.

The spectral theorem
By Theorem 3.13, for all λ there exists a unique unitary U λ : H ψ → L 2 (R, dµ λ ) such that U λ ψ = 1l and U λ H λ U −1 λ is the operator of multiplication by x on L 2 (R, dµ λ ). In this subsection we describe U λ .
For φ ∈ H and z ∈ C \ R let whenever the limits exist. By Theorem 2.17 the limits exist and are finite for Lebesgue a.e. x. For consistency, in this subsection we write F 0 (x + i0) = lim y↓0 F 0 (x + iy).
Combining this relation with (4.63) and (4.64) we derive (4.68) (1) follows from the identity (4.67) and Part 1 of Theorem 3.17. Since µ λ,sing is concentrated on the set {x : lim y↓0 F 0 (x + i0) = −λ −1 }, the identity (4.68) and Part 2 of Theorem 3.17 yield (2). 2 Note that Part 2 of Theorem 4.2 yields that for every eigenvalue x of H λ (i.e. for all x ∈ T λ ), This special case (which can be easily proven directly) has been used in the proofs of dynamical localization in the Anderson model; see [A, DJLS]. The extension of (4.69) to singular continuous spectrum depends critically on the full strength of the Poltoratskii theorem. For some applications of this result see [JL3].

Spectral averaging
In the sequel we will freely use the measurability results established in Subsection 3.16. Let where B ⊂ R is a Borel set. Obviously, µ is a Borel measure on R. The following (somewhat surprising) result is often called spectral averaging: The measure µ is equal to the Lebesgue measure and for all f ∈ L 1 (R, dx), Proof. For any positive Borel function f , (both sides are allowed to be infinity). Let where y > 0. Then R f (t)dµ λ (t) = Im F λ (x + iy) = Im F 0 (x + iy) |1 + λF 0 (x + iy)| 2 .
By the residue calculus, and so the Poisson transform of µ exists and is identically equal to π, the Poisson transform of the Lebesgue measure. By Theorem 2.7, µ is equal to the Lebesgue measure. 2 Spectral averaging is a mathematical gem which has been rediscovered by many authors. A detailed list of references can be found in [Si3].
The proof of Theorem 4.6 is left to the problems. Theorem 4.4 is the celebrated result of Simon-Wolff [SW]. Although Theorems 4.5 and 4.6 are well known to the workers in the field, I am not aware of a convenient reference.

Some remarks on spectral instability
By the Kato-Rosenblum theorem, the absolutely continuous spectrum is stable under trace class perturbations, and in particular under rank one perturbations. In the rank one case this result is also an immediate consequence of Theorem 4.1.
The situation is more complicated in the case of the singular continuous spectrum.
There are examples where sc spectrum is stable, namely when H λ has purely singular continuous spectrum in (a, b) for all λ ∈ R. There are also examples where H 0 has purely sc spectrum in (a, b), but H λ has pure point spectrum for all λ = 0.
A. Gordon [Gor] and del Rio-Makarov-Simon [DMS] have proven that pp spectrum is always unstable for generic λ. Assume that (a, b) ⊂ sp(H 0 ) and that for Lebesgue a.e. x ∈ (a, b), G(x) < ∞. Then the spectrum of H λ in (a, b) is pure point for Lebesgue a.e. λ. However, by Theorem 4.7, there is a dense G δ set of λ's such that H λ has purely singular continuous spectrum in (a, b) (of course, H λ has no ac spectrum in (a, b) for all λ).

Boole's equality
So far we have used the rank one perturbation theory and harmonic analysis to study spectral theory. In the last three subsections we will turn things around and use rank one perturbation theory and spectral theory to reprove some well known results in harmonic analysis. This subsection deals with Boole's equality and is based on [DJLS] and [Po2].
Let ν be a finite positive Borel measure on R and F ν (z) its Borel transform. As usual, we denote F ν (x) = lim y↓0 F ν (x + iy).
The following result is known as Boole's equality: Proposition 4.8 Assume that ν is a pure point measure with finitely many atoms. Then for all t > 0 Proof. We will prove that |{x : F ν (x) > t}| = ν(R)/t. Let {x j } 1≤j≤n , x 1 < · · · < x n , be the support of ν and α j = ν({x j }) the atoms of ν. W.l.o.g. we may assume that ν(R) = j α j = 1. Clearly, is strictly increasing on (x j , x j+1 ), with vertical asymptots at x j , 1 ≤ j ≤ n. Let r 1 < · · · < r n be the solutions of the equation F ν (x) = t. Then On the other hand, the equation Since {r j } are all the roots of the polynomial on the l.h.s., n j=1 r j = −t −1 + n j=1 x j and this yields the statement. 2 Proposition 4.8 was first proven by G. Boole in 1867. The Boole equality is another gem that has been rediscovered by many authors; see [Po2] for the references.
The rank one perturbation theory allows for a simple proof of the optimal version of the Boole equality.
Theorem 4.9 Assume that ν is a purely singular measure. Then for all t > 0 Proof. W.l.o.g. we may assume that ν(R) = 1. Let H 0 be the operator of multiplication by x on L 2 (R, dν) and ψ ≡ 1. Let H λ = H 0 + λ(ψ| · )ψ and let µ λ be the spectral measure for H λ and ψ. Obviously, µ 0 = ν and F 0 = F ν . Since ν is a singular measure, µ λ is singular for all λ ∈ R. By Theorem 4.1, for λ = 0, the measure µ λ is concentrated on the set {x : Then for λ = 0, By the spectral averaging, A similar argument yields that |{x : The Boole equality fails if ν is not a singular measure. However, in general we have Theorem 4.10 is due to Vinogradov-Hruschev. Its proof (and much additional information) can be found in the paper of Poltoratskii [Po2].

Poltoratskii's theorem
This subsection is devoted to the proof of Theorem 2.18. We follow [JL1].
This proves the Poltoratskii theorem in the special case where ν s = 0, µ is compactly supported, and f ∈ L 2 (R, dµ) is real-valued.
We now remove the assumptions f ∈ L 2 (R, dµ) and that f is real valued (we still assume that µ is compactly supported and that ν s = 0). Assume that f ∈ L 1 (R, dµ) and that f is positive. Set g = 1/(1 + f ) and ρ = (1 + f )µ. Then for µ sing -a.e. x. By the linearity of the Borel transform, for µ sing -a.e.
Since [a, b] is arbitrary, we have removed the assumption that µ is compactly supported. Finally, to finish the proof we need to show that if ν ⊥ µ, then lim y↓0 F ν (x + iy) F µ (x + iy) = 0 (4.81) for µ sing -a.e. x. Since ν can be written as a linear combination of four positive measures each of which is singular w.r.t. µ, w.l.o.g. me may assume that ν is positive. Let S be a Borel set such that µ(S) = 0 and that ν is concentrated on S. Then lim y↓0 F χ S (µ+ν) (x + iy) F µ+ν (x + iy) = χ S (x), for µ sing + ν sing -a.e. x. Hence, lim y↓0 F ν (x + iy) F µ (x + iy) + F ν (x + iy) = 0 for µ sing -a.e. x, and this yields (4.81). The proof of the Poltoratskii theorem is complete. The Poltoratskii theorem also holds for complex measures µ: Theorem 4.11 Let ν and µ be complex Borel measures and ν = f µ+ν s be the Radon-Nikodym decomposition. Let |µ| sing be the part of |µ| singular with respect to the Lebesgue measure. Then lim y↓0 F ν (x + iy) F µ (x + iy) = f (x) for |µ| sing − a.e. x.
Theorem 4.11 follows easily from Theorem 2.18.

F. & M. Riesz theorem
The celebrated theorem of F. & M. Riesz states: Theorem 4.12 Let µ = 0 be a complex measure and F µ (z) its Borel transform. If F µ (z) = 0 for all z ∈ C + , then |µ| is equivalent to the Lebesgue measure.
In the literature one can find many different proofs of this theorem (for example, three different proofs are given in [Ko]). However, it has been only recently noticed that F. & M. Riesz theorem is an easy consequence of the Poltoratskii theorem. The proof below follows [JL3]. Proof. For z ∈ C \ R we set F µ (z) = R dµ(t) t − z and write F µ (x ± i0) = lim y↓0 F µ (x ± iy).
By Theorem 2.17 (and its obvious analog for the lower half-plane), F µ (x ± i0) exists and is finite for Lebesgue a.e. x.
[5] Prove the Poltoratskii theorem in the case where ν and µ are positive pure point measures.
[6] In the Poltoratskii theorem one cannot replace µ sing by µ. Find an example justifying this claim.
The next set of problems deals with various examples involving rank one perturbations. Note that the model (4.60) is completely determined by a choice of a Borel probability measure µ 0 on R. Setting H = L 2 (R, dµ 0 ), H 0 = operator of multiplication by x, ψ ≡ 1, we obtain a class of Hamiltonians H λ = H 0 + λ(ψ| · )ψ of the form (4.60). On the other hand, by the spectral theorem, any family Hamiltonians (4.60), when restricted to the cyclic subspace H ψ , is unitarily equivalent to such a class.
[8] Assume that µ 0 = µ C . Prove that for all λ = 0, H λ has only pure point spectrum. Compute the spectrum of H λ . Hint: This is Example 1 in [SW]. See also Example 3 in Section II.5 of [Si2].
[9] Let µ n = 2 −n 2 n j=1 δ(j/2 n ), and µ = n a n µ n , where a n > 0, n a n = 1, n 2 n a n = ∞. The spectrum of H 0 is pure point and equal to [0,1]. Prove that the spectrum of H λ in [0, 1] is purely singular continuous for all λ = 0. Hint: This is Example 2 in [SW]. See also Example 4 in Section II.5 of [Si2].