Hilbert space methods in quantum mechanics

1.1 Definition and elementary properties Throughout this text a Hilbert space 1neans a co1nplex linear vector space, equipped with a Hermitian scalar product, which is complete and adm

ILBERT SPACE METHODS QUANTUM MECHANICS

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Preface

This text is based on lectures given at the University of Geneva during the period 1994- 2005 These courses were intended for advanced undergraduate students who had completed a first course in quantum mechanics and were thus expected to be fa-miliar with the physical aspects and the basic 1nathernatical formalism of quantum theory Partly due to lack of time, quantum mechanics is often taught with no or little exposition of the mathematical questions arising through the introduction of infinite-dimensional Hilbert spaces I hope that the present volume will prove useful, espe-cially to somewhat theoretically minded students, for deepening their knowledge and understanding of the Hilbert space aspects of quantum mechanics, and prepare them for reading research papers

Mostly the lectures were organised as one-year courses (80 hours plus 25 hours of problem sessions) and covered essentially the contents of Chapters 1 - 5 and parts of Chapters 6 or 7 To offer a few more applications, some 1naterial has been added to the original lecture notes (Sections 5.8, 6.6, 6.7, 7.2 and 7.5) Of course a strict selection

of topics and applications to be treated had to be made from the outset The emphasis is placed on a certain number of basic mathematical techniques, usually without striving for the most general results For example there is no discussion of quadratic forms, and we have avoided the use of techniques from stochastic analysis However we give essentially complete proofs for all results involving Hilbert space objects Some of these proofs are collected in appendices to the various chapters Son1e acquaintance with measure theory is required: the essential facts are explained without detailed proofs

Chapter 1 gives the basic properties of Hilbert space and a description of the sary material from measure theory In Chapter 2 we present various classes of bounded linear operators and general notions on unbounded operators, including the invariance

neces-of self-adjointness under a class neces-of perturbations The problem neces-of self-adjointness is further investigated in Chapter 3 which contains the theory of extensions of symmetric operators, with applications to Sturm-Liouville and Schrodinger operators Chapter 4 deals with the spectral theory of self-adjoint operators, in particular with the spectral theorem and with the various spectral types In Chapter 5 we prove Stone's Theo-rem and then discuss the fundamental aspects of scattering theory: scattering states, asymptotic condition and wave operators, S-matrix, scattering cross sections Chapter

6 is devoted to the Mourre method for controlling the resolvent of self-adjoint ators near the real axis and to implications for their spectrum In the final Chapter 7

oper-we present stationary-state scattering theory and various applications of the results of Chapter 6: asymptotic completeness, properties of the S-matrix, time delay and the Flux-Across-Surfaces Theorem

Each chapter ends with some bibliographical notes and a selection of problems The bibliography consists mostly of books They are referred to for alternative or more advanced presentations of some material or for certain points not treated in the present text A very small number of original and review papers are cited for those interested

a deeper understanding of certain topics treated in the text In combination with tnodern electronic means these papers can also be useful as a basis for searching in the vast literature

The majority of the problems have been tested in class-room sessions Many of then1 are meant to help students to become familiar with the concepts discussed in the main body; son1e require a more detailed study of technical aspects of the text A few

of the more difficult problems are provided with a hint for the solution

As regards notations, it should be pointed out that some symbols have more than one 1neaning in the text In particular: the symbol II · II is used for various norms, the Greek letter CJ for spectra and for scattering cross sections, the letter P for projections and for mon1entum, and the letter R for resolvents and for S - I, where S is the scattering operator For the convenience of the reader we have included a Notation Index (page 385) Constants are often denoted generically by c, but in some proofs different constants are numbered as c1 , c2 , etc

is a great pleasure to thank all those who helped me in one way or another during the preparation of the lectures or of this volume The problems in the text are mostly due to Marius Mantoiu, Joachim Stubbe and Rafael Tiedra de Aldecoa; they assumed the responsibility for the problem sessions with much competence and devotion The continual interest of my students and their constructive comments on earlier versions

of the text have influenced a considerable number of details I received precious port from Philippe Jacquet, Andreas Malaspinas, Peter Wittwer and Luis Zuleta for coming to grips with TEX -related difficulties I thank Philippe Martin for proposing the publication of my lectures, the referee for pointing out some errors and for useful suggestions, and Fred Fenter from EPFL Pre,ss for advice and his very efficient man-agement of the publishing process Finally~' am indebted to the Physics Department

sup-of the University sup-of Geneva for its kind hospitality after my retirement

Werner Amrein

Geneva, Switzerland December 2008

Contents

1 Hilbert Spaces

1.1 Definition and elementary properties

1.2 Vector-valued functions

1.3 Subsets and dual of a Hilbert space

1.4 Measures, integrals and LP spaces

2.6 Resolvent and spectrum of an operator

2 7 Perturbations of self-adjoint operators

Problems

3 Symmetric Operators and their Extensions

3.1 The method of the Cay ley transform

4.3 Spectral parts of a self-adjoint operator

4.4 The spectral theorem The resolvent near the spectrum

Appendix: Proof of the Spectral Theorem

Problems

8

0 • • •

5 Evolution Groups and Scattering Theory 193

5.4 Simple scattering systems Scattering operator 218

CHAPTER!

Hilbert Spaces

Hilbert space sets the stage for standard quantum theory: the pure states of a physical system are identified with the unit rays of a Hilbert space 1{ and observables with self-adjoint operators acting in H In this initial chapter we present the essential concepts and prove the basic results concerning separable Hilbert spaces (Sections 1.1 - 1.3 ) In Section 1.4 we then introduce £2 spaces, which are of special importance for quantum mechanics This requires some familiarity with measure theory, and we include a short description of the necessary concepts from this theory

1.1 Definition and elementary properties

Throughout this text a Hilbert space 1neans a co1nplex linear vector space,

equipped with a Hermitian scalar product, which is complete and admits a countable basis More precisely a (separable) Hilbert space His defined by the four postulates

(HI)- (H4) stated below:

(Hl) H is a linear vector space over the field CC o.f co1nplex nu1nbers:

With each couple {f, g} of elements of H there is associated another element of H,

denoted f + g, and with each couple {a, f}, a E CC, f E H, there is associated an ment af ofH, and these associations have the following properties (where f, hE 1{

ele-and a, {3 E CC):

f+g==g+f a(f+g)==af+ag a({3f) == (a{3)f

f + (g + h) == (f +g) + h (a+(3)f-af+(3f

If- f

(1.1)

(1.2)

(1.3) Furthermore there exists a unique eletnent 0 E H (called the zero vector) such that 1

1

Here 0 denotes the complex number a = 0

1i is equipped vvith a strictly positive scalar product 2 :

With each couple {j, g} of elements ofH there is associated a complex number (f, g),

and this association has the following properties 3

or-The coefficient ak

this develop1nent off is given by O'k == (ek, f), and the series Lkakck converges

to f in the sense that II f - L~=l akck II -+ 0 as N -+ x

elements of 1i are called vectors; they will be denoted by f, g or h (sometimes

e vectors satisfying II ell == 1) The letters a and f3 will stand for complex bers reader should be fatniliar with finite-dimensional Hilbert spaces from Lin-ear Algebra However many problems in quantu1n mechanics involve Hilbert spaces infinite dimension We shall see that, when suitably interpreted, some properties of finite-ditnensional spaces have analogues in infinite-din1ensional spaces (for example spectral properties of compact operators are similar to those of tnatrices ), while

num-2

Abo called "inner product" by sotne authors

1 a denotes the complex conjugate of the cmnplex nutnber a

-+ N { 1, 2 ~3 }

5

i.e for each E > 0 there exists a number N N(s) > 0 such that llfn - fm II < E \In, m > N

DEFINITION AND ELEMENTARY PROPERTIES 3

other aspects of infinite-dimensional spaces do not appear in finite-dimensional ones (for example in finite-dimensional spaces there is no distinction between weak and strong convergence, and matrices do not have continuous spectrum)

Observe that, contrary to the convention adopted by mathematicians, the scalar product is linear in the second entry and anti-linear in the first one The non-negative number II f II defined by ( 1.8) is called the norm of the vector This norm function has the properties of a metric The quantity II!- g\1 can be interpreted as the distance

fromftog, andonehas llafll == \cvlllfll, in particular 11-fll 11(-l)fll == llfll· The scalar product can be expressed in terms of the norm of 1-i; this is the content of the

< 2\1!112

+ 2llg\\2

IIIJII-IIgll\ < llf- g\1

(1.12) (1.13)

(1.14)

( 1.15)

PROOF Iff== g, (1.12) is evident: (J,f) == llfl\2

Iff#- g, assume for example that g #- 0 Then, for any a E C:

0 <\If +ag\\2 == \f +ag,f ag) == 11!112

a(f,g) +a(g,f) la\2\lg\\2

which implies (1.12) upon multiplication by llgll2

For (1.13) we take a== 1 in (1.16):

o < llf- g\\2

== llfll2

- \f,g)- (g,.f) + llgll2

, hence

6 In ( 1.12), II f II · II g II denotes the product of the norms off and g In the remainder of the text a product

of two norms will be written simply as 11·1111·11· With this convention, (1.12) will read l(f, g)J:::;; llfllllgJJ

Upon insertion this inequality into ( 1.17) one obtains ( 1.14 ) Finally, to obtain (1.15), supposeforexan1plethat Jlgll < 11!11; then, by using (1.13):

11!11- llgll == II!- g +gil llgll < (II!- gil+ II gil)- llgll == II!- gil· o

1.1 2 There are two types of convergence in a Hilbert space, called strong gence and weak convergence, defined in terms of the norm and the scalar product respectively:

conver-Strong convergence: A sequence of vectors {fn}nEN in a Hilbert space His said to converge strongly to a vector f E H if llfn - !II converges to 0 when n ~ oo One then writes s-limn_,.oo fn == f

Weak convergence: A sequence of vectors {fn}nEN of H is said to converge weakly

to a vector f E H if, for each vector g E H, the sequence of complex numbers

{ \fn, g) }nEN converges to (f, g) when n ~ oo One then writes w-limn -7 00 fn ==f

The postulate (H3) is a Cauchy criterion for strong convergence: a strong Cauchy sequence has a li1nit (in H) The Cauchy criterion is also true for weak convergence: if

{ fn} n EN is weakly Cauchy (i.e such that, for each g E H, the sequence { (f n, g)} n EN

is a numerical Cauchy sequence), then there exists a vector f in H such that { f n} converges weakly to f By using ( 1 7) it is easy to prove the uniqueness of the limit of Cauchy sequences (see Proposition 1.3)

Proposition 1.1 (a) Let {fn }nEN be a sequence of vectors in H Then

s -liln J n == f ¢===:;> w - lim f n == f and li1n II fn II == II f II·

so w-limn ,00 .fn = f On the other hand, by using (1.15\ one obtains lllfn 11-11.!111 <

II fn - !II -+ 0 as n -+ oo, hence limn -7oo II fnll == II fll·

obtain the implication~ we observe that II fn- !112 == II fn 112 +II !112- (f n,

f)-\f~fn)· w-linln -7oofn == f andlimn -7oo llfnll == 11!11, the expression on the hand side converges to 11!112 + 11!112 - llfll2 - 11!112 == 0, so that s-lin1n -7oo fn ==f

right-(b) This follows from (1.12) and the result of (a):

.gn)- (J.g)l == 1\fn,gn- g) (fn- f,g)l < 1\fn,gn- g)l + 1\fn- f,g)l

Example 1.2 Let { e 77 } nEN be an infinite orthonormal sequence in a Hilbert space

H of infinite dimension, i.e satisfying (e1 , ek) == 61 k Then w-limn -7oo en == 0,

because for any vector gin H:

DEFINITION AND ELEMENTARY PROPERTIES 5

00

( 1.18) n=l

consequently limn-'roo (en, g) == 0 [the sequence {en} can be considered to be part of

an orthonormal basis of H, and for such a basis one has equality in ( 1.18) as explained afterEq (1.10)] Thus:

Every il1finite orthonormal sequence {en} converges weakly to the zero vector 0

On the other hand such a sequence cannot be strongly convergent (if one assumes that s-limn-'roo en== f, then w-linln-'roo en== f by Proposition 1.1(a), and by the

uniqueness of a weak limit, one must then have f == 0, i.e s -li1n 11 -'r()() e 11 == 0; this is impossible since lien- Oil == lien II == 1 for each n) This example shows in particular that in an infinite-dimensional Hilbert space the notion of weak convergence is really weaker than that of strong convergence

Proposition 1.3 The limit of a strong or weak Cauchy sequence is unique

PROOF If {fn} is a strong Cauchy sequence, it is also weakly Cauchy [see

Propo-sition 1.1 (a)]; thus it is enough to consider weak convergence So let us assume that w-lilTI11-'roo fn == h1 and w-lilnn-'roo fn == h2 Then one has for each g E H:

REMARK If {fn}nEN is a strong Cauchy sequence, then this sequence is bounded in

H, i.e supnEN II f n II < 00 This follows immediately from Proposition 1.1 (a) (because llfnll -+ 11!11) The same conclusion is true if one assumes only that {fn} is weakly

Cauchy, but the proof is longer This fact is often useful in the theory of Hilbert spaces, and we shall see in Chapter 2 other results of this type All these results can be deduced from a general theorem (the Uniform Boundedness Principle) which we state below for completeness (the proof can be found in most textbooks, see for example [K]) Uniform Roundedness Principle: Let A be a set and, for each element A of A, let

<p;> : H -+ [0, oo) be a continuous mapping 7 such that <p;> (f g) < <p;> (f) + <p;> (g)

for all J, g E H If for each fixed vector g the family { <p;> (g)} >-.EA is bounded (i.e

<p;> (g) < M \1 A E A, where M < oo is a constant depending only on g), then the family { cp >-.} is uniformly bounded on the unit ball of H, in other terms there exists a constant c < oo such that cp;> (h) < c for all hE H satisfying llhll < 1 and all A EA

To deduce for example the boundedness of a weakly convergent sequence { fn} nEN,

take A == N and cp >-.(g) cpn (g) == I (f n, g) I· For each g E H, the numerical sequence {I (f n, g) I} nEN is bounded because (f n, g) converges to a finite limit; then one uses the fact that llfn II== suphEH,Jihll=l 1\fn, h) I [see Eq (2.6)] to arrive at the inequality

II fn II == suphE'H.JJhlJ=l cpn (h) ~ c for all n

7

i.e cpA (f n) -+ cpA (f) when f n -+ f strongly

Vector-valued functions

Let A be a set and H a Hilbert space A vector-valued function on A is a mapping

f: A ~ H It associates a vector in H with each ele1nent of A Thus a sequence of vectors {fn}nEN can be viewed as a function f: N ~ H by setting f(n) == fn for

n EN The 1nost important case in our context is that where A is a (finite or infinite) interval ~l; we write f ( s) for the value of a function f: J ~ H at the point s E J So for each s E J, f ( s) is a vector in H Iff ( s) depends continuously on s, the family

{!( s) }sEJ describes a curve in H

As for nu1nerical functions, one can consider the notions of continuity or tiability and define integrals of such functions The derivative at a point of J and the integral of a vector-valued function will be vectors in H Since derivatives and inte-grals are limits, these notions may be defined in the strong sense or in the weak sense (for example a function f: J ~ H may be weakly differentiable but not strongly differentiable) Similarly one can define integrals in the sense of Riemann or in a

differen-In ore general sense, and one can integrate with respect to different measures on J (the concept of a measure will be discussed in Section 1.4 ) We restrict ourselves here to definitions in the strong topology of H, and we consider only integrals in the sense of Riemann (and with respect to Lebesgue measure)

A function f: J ~His strongly continuous if one has for each t E .1:

lirn II ! ( s) - ! ( t) II == o

A vector-valued function f on an open interval J is strongly differentiable at the point

s E tl if there is a vector g in H such that:

~-To II* [.f(s + T)- f(s)] - g II= 0 (1.20)

We shall write g == f' ( s) More generally, f is strongly differentiable in J iff is ferentiable at each point s E J, i.e if there exists a function : J ~ H such that

dif-~-=!~ II * [ f ( s + T) - .f ( s) J - f' ( s) II = 0\ v s E J ( 1.21) The function f' is called the (strong) derivative off, and one also writes

· · · < s N == band with Uk E ( s k- 1 , s k], will be called a partition of J (Figure 1.1) We set IITI == maxk=1, ,N isk- Sk-11·

Iff: J ~ H, let

N

L:(IT,f) == L (sk- sk-1)f(uk)· (1.23)

k=1

VECTOR-VALUED FUNCTIONS 7

This is a finite linear combination of vectors in H, hence it defines a vector I:(IT,f)

in H One chooses a sequence {I1r }rEN of partitions of J such that lin17'-""0C IITr I== 0 and defines

Figure 1.1 Partition of an interval ( o, b]

The integral of a vector-valued function on a finite open interval (a, b) or on a finite closed interval [a, b] is defined similarly by suitably adjusting the definition of

a partition of J: if J == (a, b), then one must assume that UN b, and if tl == [a, b],

then u1 == a is also admissible

If J is an infinite interval, one first defines the integral for each finite subinterval and then takes a sequence of finite subintervals converging to J in an appropriate way For exa1nple, J == (a, oo) for some finite a, one defines (if the limit exists)

by the norm llf(s)ll (details can be found for example in Section 4.4 of [AJS])

Proposition 1 4 Let (a, b] and (b, c] be finite or infinite intervals and suppose that all integrals below exist Then

( 1.26) ( 1.27) ( 1.28)

PROOF (i) Assume that a and bare finite Then (1.28) is obtained by using tion 1.1 (a), the triangle inequality (1.13) and the definition of the numerical Riemann integral:

Proposi-(ii) If for exa1nple a is finite and b == oo, one finds from Proposition 1.1 and (i) above that

Proposition 1.5 (a) ff [a, b] is a finite closed interval and f: [a, b] -+ H is strongly continuous, then ,[abf ( s) ds exists

(b) ff a and b are arbitrary (a < b), f: (a, b) -+ H is strongly continuous and

fab llf(s)llds < oo, then fabf(s)ds exists

(c) Iff is strongly differentiable on (a, b) and its derivative f' is strongly continuous and integrable on [a, b ], then

Notice that, iff is strongly continuous, then llf(s)ll depends continuously on s by Proposition 1.1 (a), so that ,[ab II f ( s) II ds can be defined as a numerical Riemann in-

tegral

Subsets and dual of a Hilbert space

We consider here certain types of subsets of a Hilbert space H A subset D of

H is dense in H if, given any vector f E H and E > 0, there exists an element g of

D such that II!- gJI < c Equivalently, Dis dense in H if, given any vector f E H,

is a sequence {fn }nEN in D such that liinn-+oo II/- fnll == 0 An example of

a dense set is given by the collection all finite linear combinations of vectors of

an orthononnal basis {en} of H In this case D is the set of all vectors of the form

1 CYk , with ak E C and N == 1, 2, 3, (N < oo); if one assumes in addition real part and the imaginary part each a k are rational nu1nbers, one obtains

linear manifold£ is a linear subset of H, i.e such that f, g d and c1: E C ====?

f cxg E £ (in other words each finite linear combination of elements of £ also belongs to£) An exa1nple is the already indicated set of all finite linear combinations

of of an orthonormal basis {en} of H; more generally, if N is an arbitrary subset H, the set all finite linear combinations vectors belonging to N is

a linear 1nanifold, called the linear manifold spanned by N If the linear manifold spanned by a subset N is dense in H, one says that N is a total set in H or a total

1-l An exa1nple a non-total subset is given by the vectors e2 , e4 , e 6 , of

an orthononnal basis {en}

linear 1nanifold £ in H is clearly a complex linear vector space equipped with a product (induced by the scalar product in H, i.e if J, g E £, then their scalar product (f g) E £ is sin1ply their scalar product (/,g) in H) In general £ will not

SUBSETS AND DUAL OF HILBERT SPACE 9

be complete in the norm 11·11, i.e the limit of a Cauchy sequence {f n}, with fn E £,will not belong to£ but only to H If£ is cornplete [i.e if£ satisfies all of the postulates

(Hl)- (H4)], then£ will be called a subspace of H In what follows we use the syrnbol

M for subspaces

A linear manifold£ determines uniquely a subspace£, called the closure of£: a vector f in H belongs to£ if and only if there exists a sequence {fn }nEN in £ that converges strongly to f as n -+ oo If£ is dense H, then£ == H, otherwise£ is a subspace of H that is strictly smaller than H

More generally, an arbitrary subset N of H defines uniquely a subspace M.!V,

called the subspace spanned by N: MN is the smallest subspace containing N, in other terms the closure £ of the linear manifold £ spanned by N

EXAMPLES OF SUBSPACES

(a) A subspace of dimension 0 is {0}, consisting of only the zero vector

(b) Each vector f 0 determines a one-dimensional subspace, the set { af \a E C},

i.e all multiples of the vector f

(c) A subspace of infinite dimension of an infinite-dimensional Hilbert space H is obtained by taking all (finite and infinite) linear combinations belonging to H of the vectors e2, , e6, of an orthonormal basis { en}nEN of H

REMARK The terminology concerning subspaces is not uniform; sotne authors say

"subspace" for a linear manifold (they have in mind a subspace of the vector space H)

and "closed subspace" for what we call a subspace

The scalar product leads to the notion of orthogonality in H Two vectors f and g

of H are said to be orthogonal if (j, g) == 0, and one then writes f _L g A sequence {!1 , f 2, j3 , } of vectors of His called orthonormal if (f~, fk) == 5.Jk·

If N is a subset of H, one defines its orthogonal comple1nent N _L as the set all vectors in H that are orthogonal to each vector inN:

Proposition 1.6 A subset N o.f H is total in H ~f and only iff N inlplies that

f == 0

PROOF N is total in H and f N, then f is orthogonal to H by ( 1.31 ) In ular f is then orthogonal to itself, i.e (f, f)== llfll2 == 0, hence f == 0 by (1.7)

partic-Conversely, suppose that f N implies that f == 0 By virtue of (1.31) this means that the orthogonal co1nplement of the subspace MN spanned by N is {0} Hence

Mjv == {O}_L == H (here one applies the Projection Theorem given below) D The next proposition generalises a well-known property of finite-dimensional vec-tor spaces equipped with a scalar product This is a fundamental result in the theory of Hilbert spaces, and it also holds in non-separable Hilbert spaces [the postulate (H4) not used in its proof]

Proposition 1.7 (Projection Theorem) Let M be a subspace of H, M_L its onal cornplement Then each vector f qf H admits a unique decomposition into a con1ponent in M and a conzponent in M _L:

orthog-( 1.32) PROOF We fix a vector f E H, and we denote by d the distance from f to M, i.e

d == infgE.A1 II f gjl We shall use the following identity: if h 1 and h2 are vectors in

then, by Eq ( 17) with f == h1 and g == -::r-h2:

( 1.33)

We first show that there is a unique vector g 0 E M such that d == II f - go II·

~~~ Choose a sequence {gk}kEN in M such that 1in1k +oo ll.f- gkli == d Let

us take h 1 - (f- g.7) /2 and h2 == (f - gk) /2 in (1.33) We obtain

Since g 1 + 9k EM, we have II f - (g.J gk) /2112

> d 2 Hence

~ llrh gkll2

<~III- gJII2

+~III- gkll2

- d 2 -'to asj, k -'too

{gk} is a Cauchy sequence H We denote its litnit by 9-D· Since M is a space, we have go EM Consequently II f- go II > d On the oth~r hand

sub-llf- 9oll <II!- 9kll ll9k- go II -+ d 0 == d ask -+ oo

have thus shown that II f - go II - d

Uniqueness Let us assume that there exist two vectors gg) and g~2) in M such that

/If- g~11)11 == II/- g62)11 ==d By taking h1 == - g~l))/2 and h2 == (f- g62

))/2 in ( 1.33) one obtains as in (i) that

1 II (I)

2 ) 11

SUBSETS AND DlJAL OF A HILBERT SPACE 1

(ii) prove (1.32) we set /1 == g 0 , where g 0 is the unique vector in M associated with f according to (i) Then f2 == f - g 0 , and we have to show that /2 E Mj_, in other terms that (h, /2 ) == 0 for each h E M So let h #- 0 be a fixed vector in M

and set a== (h, /2) Let g == /1 + ah/llhll 2 Clearly gEM, hence II!- c;ll > d and consequently

This implies that lal2 < 0, hence that a == 0

(iii) Finally let us check the uniqueness of the decomposition ( 1.32) Assume that g1, g2 E M and h1, h2 E M _i are such that g1 + h1 == 92 h2 We must show that g1 == .92 and h1 == h2 Now

0 == IIOII 2 == 11.91 + hl- (g2 + h2)11 2

because (g1 - g2, h1 - h2) == 0 Thus Jlgl - g21i == II h1 - h2ll == 0, i.e 91 == .92 and

The projection theorem shows that H may be viewed as the orthogonal (direct) sum

of two mutually orthogonal subspaces M and M _i We recall that the orthogonal sum

by all N-tuples {/1 , , }, where E Hk, with the following scalar product:

N == oo Again each Hk is a subspace of H (the set of all sequences {/1 , , } with

f7 - 0 for all j k), and the subspaces H.7 and Hk are mutually orthogonal if j #- k

As an example, suppose that each Hk is a one-dimensional space, i.e Hk == C for each k If N is finite, the orthogonal sum H == H1 E9 H2 E9 · · · E91iN is (isomorphic to) the space CN If N == oo one obtains the space denoted by I! 2 the Hilbert space of all infinite sequences { a1, a2, } of complex numbers satisfying ~ZO= 11 ak 12 < oo, with scalar product

00

k=1

Each (separable) infinite-dimensional Hilbert space His isomorphic toR 2 Indeed H

may be identified in a natural way with f 2 by choosing in H an orthonormal basis

{en} and by identifying the vector f == ~kakek (see Section 1.1) with the element

off 2 given by the sequence { a 1, a 2, } In the terminology that will be introduced

in §2.2.2, the correspondence H 3 f t-7 { a 1, a 2, } defines a unitary operator from

H to f 2

1.3.2 We end this section with a characterisation of the dual of a Hilbert space The

dual H* of H is by definition the set of all bounded linear functionals on H, i.e the set of all mappings <p: H -+ C having the following two properties:

cp(f ag) == cp(f) + acp(g)

Jcp(f)J < cJJfll \If E H,

(linearity) (boundedness)

where cis a constant (independent of f) The space H* is a complex linear vector space (for example the sum of two bounded linear functionals is again a bounded linear functional), and H* is also normed with the norm

(1.36)

Each vector g of H detennines an element <p 9 of H * by setting

"

Indeed, the scalar product (for fix~d g) is linear in f, and [use (1.12)]

In fact one has

(1.38)

because the supremum is attained for f == g: (g, g) /Jig II== jjgjj2 /II gil== llgJJ

We have thus shown that each vector of H determines an element of the dual space

H* It is an important fact that the converse is also true (so that one may identify H*

SUBSETS AND DUAL OF A HILBERT SPACE 13

(2) if {fn} is a sequence of vectors in M such that limm,n-r(X) llfnL- fnll == 0, then the sequence {fn} has a li1nit fin 7-i, and we must show that this vector f belongs to

M (completeness of M), i.e that cp(f) == 0 Now

because cp(fn) == 0 As llf- fnll ~ 0 when rt ~ oo and cp is bounded, one must

have cp(f) == 0

(ii) If cp 0 we take 9 == 0; indeed one then has 0 == cp(f) == (0, f) for each

f E 7-i If cp ¥::- 0 there is a vector h in 1i such that cp(h) i- 0 By the Projection Theorem we can write h == h 1 + h2 with h 1 E M and h2 E M _L Then

cp( h2) == cp( h h1) == cp( h) - cp(h1) == cp( h) i- 0,

~

0

in particular h 2 i- 0 (because cp(O) == 0)

Iff is an arbitrary vector in 7-i, let us consider the vector f- [cp(f)/cp(h2)] h 2 We then have

hence the vector f- [cp(f)/cp(h2)] h 2 belongs toM Since h 2 j_ M we obtain

Upon setting 9 == [ cp( h2) /II h2ll 2] h2 (which defines a vector in 1i because II h2ll i- 0), one obtains ( 1.39)

(iii) We finally show that 9 is unique If 91, 92 are two vectors such that cp(f) ==

(91, f) == (92, f) for all f E 7-i, then (91 -92, f) == 0 for all f E 7-i Hence 91 - 92 == 0

Let us add that the identification of 7-i* with 1i is not linear but anti-linear: if cp1

and C;J2 are elements of 7-i* such that cp1 (f) == (91, f) and cp2 (f) == (92, f) for all

f E 7-i, then

hence the vector in 1i representing the linear functional cp1 + (~cp 2 is the vector 91 +

a92·

1.4 Measures, integrals and LP spaces

rigourous treatlnent of the theory of L P spaces requires familiarity with measure theory and integration in the sense of Lebesgue We shall here present a description

of these spaces and of so1ne aspects of measure and integration theory, without giving the proofs The co1nplete theory is treated in numerous books

1.4 1 Consider a set 0 and a collection A of subsets of 0 having the structure of a

CT-algebra, i.e satisfying: ( 1) A contains the empty set 0 and 0 itself, (2) if V E A,

then its complementary set 0\ V also belongs to A, (3) A is stable under the operation taking countable unions in the sense of set theory: if Vk E A for k == 1, 2, 3, , then

U kVk E A If these conditions are satisfied, then A is also stable under the operation

of taking countable intersections, and the couple ( 0, A) is called a measurable space

We use the letter x for the elements of 0 (the "points") A measure m on ( 0, A) is

a mapping from A to [0, oo] that is a additive, i.e such that rn(Uk Vk) == I:km~(Vk)

for each countable family {Vk} of disjoint elements of A (i.e satisfying ~in Vk == 0

if j i-k ), and with rn( 0) == 0 A measurable space ( 0, A) together with a measure m~

defined on it is called a measure space, denoted by ( 0, A, rn )

We see that a measure rn on ( 0, A) assigns a "weight" (which may be infinite)

to each subset V of 0 belonging to A, and this assignment is additive (the weight

of a finite or countable union of disjoint subsets of 0 is the sum of the weights of these subsets) A set V of 1neasure zero will be called a null set with respect to m If

rn( 0) < oo, then 1n is called a finite measure If 0 can be expressed as the union of

a countable collection of elements of A each of which is of finite measure, then rn is

called aCT-finite measure In particular every finite measure is a finite An example

of a a finite n1easure that is not finite is given by~ Lebesgue measure on JR (see

§ 4.2) A finite measure with rn( 0) == 1 is a probability measure; in applications

the set 0 then represents the possible outcomes of an experiment, the elements of the a algebra A correspond to the events of interest in the experiment, and the measure

rn (V) is the probability that the event V will occur (an example in quantum mechanics will be discussed on page 168)

A function cp: 0 ~ JR assigns a real number cp(x) to each point x of 0 Given a a-algebra A of subsets of 0, such a function cp is said to be a measurable function

if for each interval J in IR, the set cp-1(.J) :== {x E 0 I cp(x) E J} belongs to the

a algebra A It is seen that measurability is not simply a property of the function cp but

depends very much on the a--algebra A cp may be measurable with respect to certain

a algebras and non-measurable with respect to other a algebras of subsets of 0

The important fact for integration theory in the sense of Lebesgue is that each n1easurable function is the limit of a sequence of simple measurable functions (one

could take this characterisation as the definition of a measurable function) A simple function is a function of the form cp == I:~= I akXv" with ak E JR and N < oo, where Xv,", is the characteristic function of the subset Vk of 0; a simple function is

MEASURES, INTEGRALS AND L P SPACES 15

measurable if E A for k == 1, , N We recall that the characteristic junction x v

of a set V is defined as

XV (X) == 1 if X E V, XV (X) == 0 if X Et V ( 1.41) Thus, if cp a measurable function, there exists a sequence { cpn} of measurable si1nple functions that cp(x) ==limn-too cpn (x) for each x E 0 Furthermore, if cp > 0, the

sequence { cpn} can be chosen non-decreasing, i.e such that 0 < cprz ( x) :::; cprn ( x) <

cp(x) each x E 0 and m > n

If cp is a measurable simple function, say cp == ~~=1 akXv", then the integral of cp

with respect to the 1neasure m on ( 0, A) is defined to be the real number

If cp is a general measurable function (not necessarily simple), one can define its

integral by approximating cp by a sequence of simple functions { 4?n} One considers

first measurable functions cp > 0 and defines

(1.43)

where { 4Jn} is such that 0 < cpn ( x) < cp( x) and lirr1 n -too cpn ( x) cp( 1;) for all

x E 0 This makes sense if each of the simple approximating functions 4?n is

rn-integrable and if the limit in (1.43) is finite If cp is not> 0, one applies the preceding

definition to its positive and its negative part, i.e one decomposes cp into cp

-with > 0 and cp_ > 0; the integral of and that of 4?- are defined as in (1.43), and cp is called m-integrable if each of these two integrals is finite Then one sets

If cp: 0 ~ CC is a complex function, it is rn-integrable if its real part and its inary part are rn-integrable in the sense of the preceding definition The integral of cp

imag-is obtained by integrating separately its real part and its itnaginary part

The Lebesgue type integral introduced above generalises the (proper) Riemann tegral on the real line As an illustration we shall co1npare in § 1.4.2 below the Le-besgue integral on IR with the Rietnann integral for continuous functions For measur-able functions without continuity properties, only the Lebesgue integral has a tnean-

in-Ing

In the situations considered in this text, the underlying set 0 will be a subset of IRn

( n) == 1, 2, 3, ) , often a finite open interval (a, b) or a semi-infinite interval (a CXJ),

or IRn itself In each of these situations the a algebra A will be the Borel a algebra

or the Lebesgue a algebra of the considered set 0 Mostly the measure rn will be the

Lebesgue 111easure m( dx) == dx [m( dx) == dnx if n > 2] In the next subsection we explain these tenns in somewhat more detail

1.4.2 B is an arbitrary collection of subsets of a set 0, there is a smallest a-algebra

A containing B (i.e such that, if A' is a a-algebra with B C A', then A C A') This 1ninimal a-algebra A, called the a-algebra generated by B, is simply the intersection

of all a-algebras containing B (this intersection is not empty because the collection of

all subsets of 0 is a a-algebra) In certain situations a measure m, on A is completely detern1ined if it is given on !3 An important example is the Borel-Lebesgue measure

on the real line which we shall now describe

So let us consider the case 0 == IR We take for !3 the set of all half-open intervals

(a, b] with -oo < a < b < +oo The a-algebra generated by the collection of all these intervals is called the Borel a-algebra of IR; we denote it by AB, and its ele-Inents will be called Borel sets 8 Let us take for the measure of an interval (a, b] its length R, i.e 112,( (a, b]) == R( (a, b]) b - a The thus defined mapping m: B ~

[0, oo) has an extension to a 1neasure on (IR, AB ), often called the Borel measure on

IR This 1neasure associates with each Borel set of IR a non-negative number (which can be infinite) in a a-additive way, and the number associated with an interval is its length The length of a general Borel set V is obtained in the following manner: one considers covers of V of the fonn V C U k Jk, where { Jk} kEN is a countable collection half-open intervals [Jk == (ak, bk], ak < bk], and one then defines

R(V) ==infLR(Jk) infL(bk -ak), (1.45)

where the infi1num is over all covers of V of the form specified above This length function has properties of a measure on (IR, AB ), viz) the Borel measure mB on the real line: 1nB (V) == R(V) for V E AB·

We 111ention some examples of null sets with respect to Borel measure (Borel sets length zero) The simplest example is a single point { x0 } This set is the intersec-tion of all intervals (x 0 - k-1, x 0 ], k == 1, 2, 3, The length of such an interval is

7n~ B ( (;:r0 - k-1, x 0 ]) == 1/ k; since { x0 } is contained in each of these intervals, its sure must be less than 1/ k for each k == 1, 2, 3, , hence equal to zero 9 A second ex-ample is that of a countable set { x 1 , x 2 , x 3 , } ; as m B is a -additive and the measure

mea-of each {xk} is zero, one has mB({x1,x2,x3, }) == LZ:1 n~B({xk}) == 0 There also exist Borel null sets that are not countable, for example the Cantor set <t which

is a subset of the interval [0, 1] obtained by removing from [0, 1] a countable union

8 The smne a-algebra is obtained by starting with the set of all open intervab or with that of all closed intervals Thus AB contain~ all open, closed and half-open intervals In fact each open and each closed

subset of IR is a Borel set (this is a consequence of the fact that each open subset of IR is the union of countably many disjoint open intervals), but there is no simple characterisation of individual Borel sets directly in tenns of unions and intersections of open and closed sets The use of intervals of the forn1 (a, b]

n1ay be ju~tified by the fact that ~uch intervals fit together in a nice way, and this will be crucial for the definition of the n1ore general Stiel~jes n1easures (see Ren1ark 4.6)

9

A basic property of a measure rn is that, if V, W E A are such that V ~ W, then m (V) :s; m(W),

which is a ~traightforward consequence of the fact that a 111easure is additive

MEASURES, INTEGRALS AND L P SPACES 17

of disjoint open subintervals of total length 1 This is done recursively in the ing manner (Figure 1.2): first one removes the open middle third (1/3, 2/3) from the interval [0, 1], next one removes the middle thirds (1/9, 2/9) and (7 /9~ 8/9) of the remaining intervals, and so on The Cantor set is a closed Borel set, it has Lebesgue measure zero and is uncountable The reader may find more details for example in Vol I of [RS] or in [GO]

etc

Figure 1.2 Construction of the Cantor set

As another illustration, let us compare the construction of the Riemann integral and of the Lebesgue integral for a continuous non-negative function f on some finite interval [a, b] Both are defined as limits of sequences of finite su1ns The difference resides in the fact that the Riemann integral is formulated in terms of partitions of the domain [a, b] off, whereas the Lebesgue integral uses partitions of the range off For

the Riemann integral, let IT be a partition of [a, b] as on page 6 The contribution to the Riemann sum~ (II, f) of an individual subinterval J of this partition is just f ( u')£ ( J),

as indicated in Figure 1.3(a) [f( u') is the value off at some point u' E J] The tegral is of course given by the area under the graph of f For the Lebesgue integral

in-one partitions the ordinate axis into disjoint intervals { ~{!} The contribution to the integral of a fixed one of these intervals, say of 6k, is fk£(Vk), where fk is so1ne

number in ~k and Vk - f- 1 (~k) is the Borel set on the x-axis formed by all points

x E [a, b] at which f assumes a value in ~k [Figure 1.3(b )] It is clear that, upon adding up the contributions from all intervals ~{! and passing to the limit where their length tends to zero, the value of the integral is again equal to the area under the graph

of f Observe that, for a continuous function f, the set Vk f- 1 (~k) is simply a finite union of intervals, whereas for an arbitrary measurable function on (IR, AB) it could be any Borel subset of IR

By prescription ( 1.45) one can assign a length to any subset V of JR (not only

to Borel sets) It seems that, by using this length function on the a algebra Amax

consisting of all subsets of JR, one might obtain a measure on this largest possible

o algebra, with the property that the measure of any interval coincides with the length of that interval As a 1natter of fact this maximal extension of the Borel measure in terms the length function is not a measure (it is not a additive on Amax) It is impossible

to extend the Borel measure to the class of all subsets of JR in such a way that the extension is still a measure which is invariant under translations 10

Of course there exist extensions of the Borel measure to a-algebras larger than

AB One such extension is frequently used in mathematics, namely the extension to

the Lebesgue a-algebra AL, called Lebesgue measure The elements V of AL have the fonn V == W U V0 , where W is a Borel set (WE AB) and Vo can be any subset

of some Borel set of measure zero 11 The Lebesgue measure of V is just the Borel Ineasure of W, the Lebesgue n1easure of V 0 is zero: m~L(V) == rnB(W) == £(W) So

A L contains, in addition to the Borel sets, all subsets of Borel sets of measure zero, in fact all subsets of JR of length zero This is not trivial, as shown for example by the fact that there exist subsets of the Cantor set which are not Borel sets (we recall that the Cantor set is a Borel set, and it is of measure zero) The measure space (JR, AL, mL)

is complete in the sense that the a algebra AL contains all subsets of sets of measure zero, vvhereas the measure space (JR, AB, m~B) is not complete Some statements in measure theory depend on the completeness of the underlying measure space

Fro1n now on we shall use the standard notation dx for the measure determined

by the length function R of Eq (1.45) and call this measure simply Lebesgue sure: rnB ( d:r) == 11l)L( dx) == dx The underlying a algebra will be either the Borel

mea-o algebra A B or the Lebesgue o algebra A L In contexts where it is necessary, we shall specify which of these two a algebras is involved

Above we have introduced the Lebesgue measure on the ~};line Its analogue on JRn is defined in much the same way by taking forB the collection of all subsets of the form { x E JRn j - oc < ak < Xk < bk < +oc, k == 1, , n} and for the measure of such a subset its n-dimensional volume [here we use the notation x == ( x1, x2, , Xn)

the individual points of JRn] A Borel set in JRn is an element of the a algebra erated by this collection B, and the Lebesgue measurable sets are obtained by adding

gen-to the Borel a algebra all subsets of Borel sets of measure zero

Other important measures in this text are Stieltjes measures on (JR, AB ), obtained

by considering weight functions different from the length function on the set B of half-open intervals In quantum mechanics, measures of this type occur in the charac-terisation of the spectral parts of observables; they are introduced in Section 4.1 and play an important role throughout Chapter 4

10 One uses the term outer 1neasure for the extension of a 1neasure given on some a-algebra A to the 1naximal a-algebra consisting of all subsets of 0 In general an outer measure does not satisfy all conditions in1posed on a n1easure, i.e an outer measure is not a 1neasure

11 So Vo is a subset of IR such that there exists a Borel set U satisfying P(U) = 0 and Vo C U

MEASURES, INTEGRALS AND LP SPACES 19

We next consider LP spaces on a general measure space ( 0, A, m,) Two surable functions (real or complex) cp1 and cp2 on 0 are said to be equivalent if they are equal rn-almost everywhere, i.e if they differ at most on a null set with respect to the measure rn Let W == { x E 0 \ cp1 ( x) =1- cp2 ( :x;)} be the set of all points x E 0

mea-at which these functions have different values; this set belongs to A, hence rn(W) is well defined, and cp1 and cp2 are equivalent if and only if rn(W) == 0 12

We now consider equivalence classes of measurable functions The important fact

is that, for any t > 0, the integral of I cp1 - cp2l t is zero if cp1 and cp2 are equivalent and that, one of these two functions ism-integrable, then

LP ( 0, m,) is a normed vector space, the norn1 being given by ( 1.46), and this space

is complete with respect to this norm (in other terms it is a Banach space) By

con-sidering equivalence classes of functions rather than individual functions as elements

of LP(O, m)), one ensures that the only element of LP(O, m,) with nonn zero is the equivalence class of the function f 0 (so the zero vector is unique, as in a Hilbert space)

When considering functions f: 0 ~ CC, one should each time specify whether f

is to be interpreted as an individual function or as an element of an LP space, i.e as an

equivalence class of functions In most situations encountered in this text the precise meaning of f will be clear from the context and there will be no need to mention it in detail

An important result in the context of LP spaces is the Dominated Convergence orein (also called Lebesgue Convergence Theorem) which gives a sufficient condition for interchanging limits and integrals This criterion is frequently used in our text

The-Proposition 1 9 (Dominated Convergence Theorem) Assun~e that

(i) fn, hE L1(0, rn),

(ii)foreachnEN, one has lfn(x)i < h(x) rn-a.e.,

(iii) limn-+oo fn(x) == f(x) m-a.e

Then the lin1it function f ism-integrable and one has

The following inclusion relations between LP spaces are useful:

To prove (1.48), let f E LP n Lq If r E [p, q ], then f> E Lr [by (1.50) with s == q and

t == r] and f< E Lr [by (1.51) with s == p and u == r] Hence f f> + f< E Lr

To verify (1.49) we observe that, under the conditions of (1.49), one has f> E Lr for

r < q [apply (1.50) as above], hence Xvf> E Lr On the other hand l(xvf<)(x)l <

xv(;:c), and Xv belongs to each LP if m(V) < oo (one has llxviiP == [ mj(V) J l/p) D Among the spaces LP( 0, rn ), only that for p == 2 is a Hilbert space (Problem 1.14) The scalar product in L2 ( 0, m) is given as follows:

By the Schwarz inequality this number is finite if f and g belong to L2 ( 0, m) (and the integral in ( 1.52) depends only on the equivalence classes of f and g) We use the simplified notation II · II for the norm II · 112 In quantum mechanics, when 0 is the configuration space of a physical system [typically 0 == JR or IR.n or an interval (a, b)],

the elements of L2 ( 0, 1n) (or the representative functions of an equivalence class) are usually called wave functions; we refer to Remark 2.32 for further considerations on this terminology

Before considering more explicit cases, let us mention the following generalisation (we restrict ourselves to p == 2): instead of looking at functions defined on 0 with values in CC, one may consider functions with values in a Hilbert space JC (vector-valued functions in the sense of Section 1.2) In analogy with the discussion above, one can introduce a Hilbert space L2 ( 0; JC, m); its elements are equivalence classes

of such functions, 14 with scalar product and norm given by

(f g)= l (f(x),g(.T))JCrn(dx) and [.£ II f ( x) II k rn ( d:r:) J 112, ( 1.5 3) where ( ·, ·) JC and II · II JC denote the scalar product and the norm in JC respectively For

J( == CC one regains the space L2 ( 0, mj) considered before

13 The space L00

( 0, rn) will be defined in Section 2.5 The proof of (1.48) is given here for q < oo

14 A function f: 0 ~ JC is measurable if for each g E JC the numerical function x ~ (g, f(x)) JC is measurable in the sense specified before

MEASURES, INTEGRALS AND L P SPACES 21

1 4 4 One may introduce £P spaces on the real line II{ in terms of Lebesgue sure, by taking either AB orAL for the a algebra Both definitions lead to the same space: £P(JR, AB, mB) == £P(JR, AL, mL) £P(JR, dx) This follows from the fact that null sets are irrelevant in an integral, by remembering that the elements of an

mea-£P space are equivalence classes of measurable functions [of course an equivalence class with respect to Lebesgue measure on (JR, A L) contains more functions than the corresponding equivalence class with respect to Lebesgue measure on (JR, AB),

since AB is strictly smaller than AL] Another way of understanding the relation

in terms of Riemann integrals without having recourse to measure theory, as explained towards the end of the present subsection

When the measure defining an £P space on a Borel subset V of II{ or of JR17

is the Lebesgue measure, we use the simplified notation £P(V) for the space £P(V, rn);

it is understood that the measure in L P (V) is the Lebesgue measure Also we then say "almost everywhere" instead of "m-almost everywhere" or of "Lebesgue ahnost everywhere"

If V is a Borel set in JR, for example an interval, the space £2 (V) is a Hilbert space, consisting of all equivalence classes of measurable functions f : V * C that are square-integrable over V (with respect to the Lebesgue 1neasure on which is the restriction of the Lebesgue measure on II{ to the collection of Borel sets contained in

V) The space L 2 (V) can also be considered in a natural way as a subspace of £2(JR), namely the subspace of all (equivalence classes of) functions f in £2 (JR) that are zero outside V, i.e which satisfy f ( x) == 0 if x tJ_ V (except possibly on a Borel set W of points x tJ_ V with IWI == 0, where IWI denotes the Lebesgue measure of W) In what follows we do not distinguish between these two meanings of £2 (V); it will each time

be clear from the context whether the space £2 (V) is considered as a Hilbert space without reference to L 2 (JR) or rather as a subspace of L 2 (JR)

If V and V' are two disjoint Borel subsets of JR, then £2 (V) and £2 (V ') are ally orthogonal as sub spaces of £2 (JR) Also £2 (JR \ V) is the orthogonal complement [in £2 (JR)] of the subspace £2 (V)

mutu-Examples of dense linear manifolds in the Hilbert space £2 (JR) are (modulo alence) the Schwartz space S (JR), consisting of infinitely differentiable functions of rapid decay, and the set C 0 (JR) of all infinitely differentiable functions of compact support A function f: JR * C belongs to S(JR) if it is infinitely differentiable and

equiv-if f and all its derivatives converge to zero at infinity more rapidly than any inverse power of I xI, i.e if

kldRf(x)l

A function f: JR * C belongs to C0 (JR) if it is infinitely differentiable and if there

is a finite interval J (depending on f) such that f ( x) == 0 for all r tJ_ tl Clearly

The density of the latter set is discussed in § 1.4.5

More generally, if -oo < a < b < +oo, a function f: (a, b) ~ C belongs to

C/)0

( (a~ b)) if j is infinitely differentiable and j (X) == 0 in son1e neighbourhood of

x ==a and in some neighbourhood of x == b The set C0 ( (a, b)) is dense in LP( (a, b))

for each p E [1, oo); In ore precisely the set of equivalence classes of the functions in

Si1nilar definitions and results apply in LP (IRn) C0 (IR11

) is the set of all infinitely differentiable functions f: IRn ~ C such that f(x) == 0 for lxl > NI for some number

) for each p E [1, oo ) The Schwartz space S (IRn) consists of infinitely differentiable functions satisfying

\1 k, £1, , fln == 0, 1, 2, (1.55)

is clear that C0 (IR72

L P ( (a, b)) if fab If (a;) IP dx < oo, where the integral can be interpreted as a Riemann integral One can then define LP((a, b)) as the nor1n closure of C0 ((a, b)), i.e one takes as elements of LP((a, b)) the (equivalence classes of) sequences {fn} such that each f n belongs to Co ( (a, b)) and limrn,n-+oo J~b If n ( x)-frn ( x) JP dx == 0 (Riemann integral!); two such sequences {fn} and {gn} are equivalent if their difference con-verges to zero in LP norm

For con1pleteness we recall that the support of a function f: 0 ~e, where 0

is an open subset of IR 11

( n == 1, 2, 3, ) , is the closure of the set of points x E 0

at which f ( x) =1- 0 So the support of f is a closed subset of IRn, which need not be contained in 0 because 0 is assumed to be open.15 If the support off is contained in

0 and bounded, 16 then f is said to have compact support in 0 (see the exa1nple of the functions in C0 ((a, b)) mentioned above)

1 4 5 We end with so1ne re1narks concerning the fact that C0 (IR) is dense in L P (IR)

1neasurable function f satisfying J IR If ( .:r) JP dx < oo is a limit (in L P norm) of

a sequence of bounded functions of compact support; (2) that a bounded function of co1npact support is a limit (in LP norm) of a sequence of continuous functions of com-support; and (3) that a continuous function of compact support is a 1i1nit (in L P

norm) of a sequence of functions belonging to C0 (IR) Some details on each of these three steps are given below Then one concludes the argument as follows Consider an

MEASURES, INTEGRALS AND L P SPACES 23

element P(JR) and let E > 0 Choose a representative f the considered lence class ( ) there is a bounded function of co1npact support 91 in P (IR) such that II f 91!!P < E /3 By (2) there is a continuous function of compact support 92 1n P(JR) such that 1191 - 92IIP < c/3 By (3) there exists 93 in C0 (IR) such that

equiva-1192 93IIP < c/3 Then llf- 93IIP < E, which proves the denseness of C0 (IR) in

£P(JR) (it is easily seen that the result is independent of the choice of a representative

in the equivalence class under consideration)

(1) Let f: IR -+ C satisfy J~!f(x)jPdx < oo For n/ E N define fn as follows:

fn(x) == f(x) if lf(x)l < nand j:x;l < n, and fn(:x:) == 0 otherwise Clearly fn is bounded (lfn(x)l < n for each x) and of compact support (i.e fn is zero outside some finite interval) Furthermore the sequence {fn} converges to fin P nor1n: one has f n E L P (IR) (because If n ( x) I < If ( x) I for all x) and

II!- fnllv = [hlf(.'£)- fn(x)IPdxr 1v

As n -+ oo the integrand converges to zero almost everywhere (for each x E IR for

which If ( x) I oo ), and for each n it is bounded by the function h( x) == If ( x) jP (because f(x)- fn(x) is equal to f(x) or to 0) Since hE L1(IR), the result of (1)

follows from the Dominated Convergence Theorem

(2) Here one uses the fact that the measurable space ( 0, A) is special: one has

0 - IR and A== AB In this situation one can approximate a measurable function by

a sequence of step functions; these are particular simple functions, namely functions

of the form rp == ~~=1 CYkXv", where ak are constants and Vk are intervals of the form (ak,bk], see Figure 1.4(a) If 9 is a measurable function with lg(:r)l <It! for all x, there exists a sequence { VJn} of step functions such that VJn ( x) -+ 9 ( ;x;) and

IVJn (x) I < J\;f for all x

So let 9 be a bounded function (I 9 ( x) I < 1\II) with support in an interval [-N, + N]

for some N E (0, oo) Then there exists a sequence {9n} of step functions, with port in the interval [-N, N], such that I 9n ( x) I < A1 and li1n n-'tcx; g 77 ( ;r;) == 9 ( x) for all x E JR By applying the Dominated Convergence Theorem as in (1) (observe that

sup-l9(x)- 9n(x)IP < (2Af)P and that the domain of integration [-N, N] is bounded)~

one sees that 119 - 9n liP -+ 0 Hence, given E > 0, one can choose a step tion h with support in [-N, N] such that 119 hi!P < c/2 It is clear that one can approximate h in P norm by a continuous function h 0 with support in the interval

func-[- N- 1, N 1] and with II h- h 0 liP < E /2 [slightly round off the real part and the imaginary part of the function h at its points of discontinuity, as illustrated in Figure

1.4(b) for a real-valued step function, or else use a regularisation as in (3) below] By the triangle inequality one then has 119- hol!p < 119- h!IP llh- ho!IP <E

(3) One now applies a method of regularisation 17

Choose a non-negative function 0 E C 0 (IR) such that

OCr) == 0 if lx I > 1 and J~ 1 0( x )dx - 1, for example

0 ( :r) == 0 if j.T I > 1 and 0 (X) == r exp [ -1 I ( 1 - X 2 )] if

I xi < 1, where the constant r is such that J~ 1 O(x )dx ==

1 For 0 < E < 1' set 0 c (X) == E - 1 0 ( :r; IE) Clear I y

Os E C0 (IR), Os(x) > 0, Os(x) == 0 if lxi > c and

-1 -E c 1

J ~oo 0 E ( x) dx == 1 A function f E L P (IR) is approximated by a family { f s} of smooth functions defined by

fs ( x) is so1ne sort of average of f over a small neighbourhood of x (over the interval

[x - E, T E]) For each fixed E the function fs is infinitely differentiable (one may interchange the derivatives with the integral, and the integrand is infinitely differen-tiable with respect to x) It can be shown that II!- is liP + 0 as E + 0

We consider the situation in which f is continuous and has compact support, say

f(x) == 0 if lx:l > N Then the support of is is contained in [N - E, N E], cause f(y) is zero if IYI > Nand Os(x - y) is zero if lx- Yl > E Let us show that

be-II f - is liP -7 0 as E -7 0 Since J ~00 Or:; (X) dx == 1, we have

fc(x)- f(x) =I:ec(x- y)f(y)dy -I:ec(z).f(x)dz

=I: Be (z) [f (x - z) - f(x)] dz

One may consider z f -7 f ( x- z) - f ( x) as a vector-valued function of z with values in

one finds that

ll.f- !cliP <f:ec(z)ll.f(- z)- !OIIpdz

(i) Let us first consider the integral with respect to dx (for fixed w) For each x the

integrand If ( x - EW) - f ( x) IP tends to zero as E + 0, because f is continuous

17 See for exan1ple Chapter II of [Adl

PROBLEMS 25

Furthermore the integrand is bounded: since If ( x) I < M < oo for all x E ~' one has lf(x- Ew)- f(x)IP < (2M)P, which is integrable on [-N- l,N 1] By Proposition 1.9 with h(x) == (2M)P we obtain

(ii) One can now apply Proposition 1.9 to the integral with respect to dw in ( 1.56) It

suffices to observe that, for each fixed w, the integrand converges to zero as E + 0 [by theresultof(i)] and that it is majorised by the function h(w) == 2M(2N 2)1/P()(w)

Bibliographical Notes

The books by Akhiezer and Glazman [AG], Riesz and Sz.-Nagy [RN] and Weidmann [W1] provide excellent presentations of the theory of Hilbert spaces and linear opera-tors There are numerous books on measure theory and integration; the following is a selective list: [B], [CK], [D], [H], [Na] [R], [Ra], [ST], Volume I of [RS] £P spaces are also discussed in many texts, see for example [CV] or [R]

Problems

1

1.1 Let H be a Hilbert space Verify the parallelogram identity

( 1.57) and the inequality

satisfy-LZ':1II fk 112 < oo

1.5 In Proposition 1.5( c), prove that the values of f at the endpoints a and b of the

open interval under consideration are well defined

equal to subspace spanned by

f 2 is a space

an orthonormal basis in f 2

) space of all bounded continuous functions f: IRn + CC

is a normed (even cotnplete) space for norm llflloo == supxEIRn lf(x)j Show

an exatnple a square-integrable continuous function f on IR does not satisfy lin11.rl-+oo f ( x) == 0

an example of a sequence {f n} of integrable functions on IR (with respect

converges almost everywhere to a function f but such not converge to f ( x) dx

a continuous function vanishing outside some finite interval

fn by fn ( x) - f ( x - n) using the fact that the set of

Pis not a

identity.]

the collection A L all

space the norm II· lip·

set of measure zero that A L a

quan-to multiplication operaquan-tors L 2 spaces Often quantum-mechanical operators are bounded Section 2.4 is concerned with some of the subtleties that one meets in situations, especially with the notion of closedness of an operator Section 2.6 we define the resolvent and the spectrum of closed operators

un-A very important class of linear operators are the self-adjoint operators, used to describe observables quantum 1nechanics unbounded operators the property

of being self-adjoint is rather subtle Some preliminary explanations, pointing out the difference between symmetric and self-adjoint operators, are given in Section 2.4, and Chapter 3 will be entirely devoted to a detailed study relations between symmetric and self-adjoint operators present chapter ends with some criteria the preservation of self-adjointness under perturbations (Section 2.7); particular we prove the self-adjointness of some Hamiltonians of non-relativistic quantu1n Inechan-ics as perturbations of the self-adjoint free Hamiltonian P2 /2rn

Let H be a Hilbert space We denote by B (H) the set of all bounded linear operators on H element of B(H) is a mapping that associates with each vector

f E another vector Af belonging to H, and this 1napping is linear and bounded, i.e

A( af g) == aAf + and there is a finite constant 1\;f (depending on A) such that

IIAfll < 1\IIIfll for each vector f The infimum of all possible numbers M

called the norm of the operator A and denoted by II A II :

II All == inf{ ME JR lilA! II < Mllfll V f E H}

= sup IIAJII = sup IIAgll·

0:/::fEH llfll gEH,jjgjj=l

(2.1)

The elements of B(H) will usually just be called bounded operators

Observe that (2.1) implies that

If A E B(H), then the itnage { Afn} of each strong Cauchy sequence {fn} is again a strong Cauchy sequence:

n -+c::x) n -+c::x)

This follows from (2.2) which implies that IIAJ- Afnll < IIAIIIIJ- !nil·

The following expression for the operator norm is sometimes useful: If V1 and V2 are arbitrary dense linear manifolds in H (in particular if V1 == V2 -H), then

!ED1, gED2,ilfll=llgll=l PROOF Let g be a vector in H with 11911 == 1 Let {gn} be a sequence in V2 con-verging strongly to g One may assume that ll9n II == 1 [if s-limn -+c::x) hn == h,

with h 0, then s-limn -+c::x)(hn/llhnll) - h/llhll, which is easy to see because llhnll -+ llhll by Proposition 1.1(a)] Then, by (2.3) and Proposition 1.1, IIAgll ==

lirn n -+c::x) II Agn I Hence, if we restrict the vectors g in the last expression in (2.1) to the set {g E V2 11911 1 }, we obtain the same supremum:

gED2,IIgll=l Next lethE H, h -::J 0 Then, if 11!11 == 1, one has l(f, h) I < llfllllhll == lihll, with equality for f - h/llhll· So llhll == sup!EH,II!II=ll(f,h)l By using an argument

as above (approximating a vector f E H, 11!11 - 1, by a sequence {fn} in V1, with

II frz II == 1), one sees that

llaA Bll < laiiiAII IIBII, (2.7)

THE ALGEBRA B(H) 29

because ll(aA B)fll < llaAfll IIBJII < laiiiAIIIIfll + IIBIIIIJII [we have used (1.13)] B(H) is an algebra, i.e a vector space equipped with an operation of multipli-cation The product of two bounded operators A and B is simply denoted by AB and

is just the composition of the two mappings: ( AB) f == A( B f) is the vector obtained from f by acting first with B on f and then with A on the image vector B f The mul-tiplication B (H) has the following properties:

To see that AB is bounded, observe that IIABJII < IIAllllBfll < IIAIIIIBIIIIJII as a consequence of (2.2), so that

IIABII < IIAIIIIBII· (2.8)

2.1.2 The algebra B(H) is involutive (called a *-algebra) The involution, denoted

by an asterisk, associates with each operator A E B(H) another element A* of B(H),

called the adjoint of A, such that (A*)* == A [one usually writes A** for (A*)*]

Given A, its adjoint A* is defined by the relation

This relation determines uniquely an element A* of B(H): for fixed f, the mapping

g ~ (f, Ag) is a bounded linear functional on H (it is linear in g, and I (f, Ag) I < IIJIIIIAIIIIgll) By virtue of the Riesz Lemma (Proposition 1.8) there exists a unique vector f * E H such that (f, Ag) == (f *,g) for each g E H One sets A* f == f *, which defines A* on each vector f of H We must check that the thus-defined mapping A*

is linear and bounded:

(i) One has

(A*(af h),g)==(af+h,Ag) a(f,Ag) (h,Ag)

== a(f*,g) + (h*,g) == a(A*j,g) (A*h,g)- (aA*f + A*h,g)

So A* ( af + h) a A* f - A* h is orthogonal to each g E H, hence equal to the zero vector (Proposition 1.6), i.e A* ( af + h) == aA *f + A* h

(ii) By using (2.4) one finds that

IIA*II == sup l(g,A*f)l == sup I(Ag,f)l == IIAII·

j, gEH, II !II= 11911=1 j, gEH, 11 fll=ll9ll=l Hence A* is bounded and

The definition (2.9) also implies that, if A,B E B(H), then 1

1 The relation A** = A is obtained from Proposition 1.6 by observing that (A** j, g) = (j, A* g) =

(Aj, g) for all j, g E 'H

Another useful relation is the following:

(2.12)

Indeed one has on the one hand IIA *All < IIA *II II All == IIAII2 and, on the other hand,

An operator A E B ( 1-i) is self-adjoint if == A, i.e

We set a== supJED,IIJII=ll (j, Af) I· Observe that I (h, Ah) I < ailhll2

for each

hE V (because h/JihJJ is a vector of norm 1 if h 0)

The inequality a < II A II follows from (2.4) To obtain the converse inequality we must that a> I (f, Ag) I for all j, g E V with 11!11 == 11.911 == 1 Now I (J, Ag) I ==

I for each real a, and for fixed f and g one can find a real number a such

(e~ 0 Ag) is real and non-negative Hence it suffices to show that a > (f, Ag)

f g E V with llfll == 11.911 - 1 and such that (f, Ag) E JR

we use the following generalisation of the polarisation identity ( 1.11 ):

- i(j ig, A(f ig)) i(f-ig,A(f-ig)) (2.15)

(h, real each h; E 1-i, the real part of 4(j, Ag) is given the su1n of first two terms on right-hand side of preceding equation So, iff and g are such that (f, E IR, one has [by also using (1 33)]

E B(H) that w-lin177~CX) ==A Analogous

II

proof of these statements (it involves Uniform

end Section 1.1)

Let B, , Bn E (H) be such that

isfies s -lirn n~oo Bn ==

II 1

nonn We Principle

n~oo

ciple (Section 1.1), a strongly (or weakly) convergent sequence of

one

lS

and sat-

bounded: there is a constant M < oo such that II II < JJI[ all n E N (in our

So let f E 1-{ Then