Open quantum systems i; the hamiltonian approach

It can be considered as a part ofstatistical mechanics at the interface with the ergodic theory of stochastic processesand dynamical systems.. The paradigm of this statistical approach t

Lecture Notes in Mathematics

BC

Systems I

Open Quantum

S Attal • A Joye • C.-A Pillet (Eds.)

The Hamiltonian Approach

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3-540-30991-8 Springer Berlin Heidelberg New York978-3-540-30991-8 Springer Berlin Heidelberg New YorkDOI 10.1007/b128449

Universit Claude Bernard Lyon 1

21 av Claude Bernard

France

69622 Villeurbanne Cedex

Alain JoyeInstitut FourierUniversit de Grenoble 1

BP 74France

Closed vs Open Systems

By denition, the time evolution of aclosedphysical systemS is deterministic It

is usually described by a differential equationxt = X (xt) on the manifoldM of

possible congurations of the system If the initial congurationx0 M is known

then the solution of the corresponding initial value problem yields the conguration

xtat any future timet This applies to classical as well as to quantum systems In theclassical caseM is the phase space of the system andxt describes the positions andvelocities of the various components (or degrees of freedom) ofS at timet In the

quantum case, according to the orthodox interpretation of quantum mechanics,M is

a Hilbert space andxt a unit vector — the wave function — describing the quantumstate of the system at timet In both cases the knowledge of the statext allows

to predict the result of any measurement made onS at timet Of course, what wemean by the result of a measurement depends on whether the system is classical

or quantum, but we should not be concerned with this distinction here The onlyrelevant point is thatxt carries the maximal amount of information on the systemS

at timet which is compatible with the laws of physics

In principle any physical systemS that is not closed can be considered as part

of a larger but closed system It sufces to consider withS the setR of all systemswhich interact, in a way or another, withS The joint systemS R is closed andfrom the knowledge of its statext at time t we can retrieve all the information onits subsystemS In this case we say that the systemS is openand thatR is its envi-

ronment.There are however some practical problems with this simple picture Sincethe joint systemS R can be really big (e.g.,the entire universe) it may be dif-cult, if not impossible, to write down its evolution equation There is no solution to

VI Preface

this problem The pragmatic way to bypass it is to neglect parts of the environment

R which, a priori, are supposed to be of negligible effect on the evolution of thesubsystemS For example, when dealing with the motion of a charged particle it isoften reasonable to neglect all but the electromagnetic interactions and suppose thatthe environment consists merely in the electromagnetic eld Moreover, if the parti-cle moves in a very sparse environment like intergalactic space then we can considerthat it is the only source in the Maxwell equations which governs the evolution of

R Assuming that we can write down and solve the evolution equation of the jointsystemS R we nevertheless hit a second problem: how to choose the initial cong-uration of the environment ? IfR has a very large (e.g.,innite) number of degrees

of freedom then it ispractically impossible to determine its conguration at someinitial time t = 0 Moreover, the dynamics of the joint system is very likely to bechaotic,i.e., to display some sort of instability or sensitive dependence on the initialcondition The slightest error in the initial conguration will be rapidly amplied andruin our hope to predict the state of the system at some later time Thus, instead ofspecifying a single initial conguration ofR we should provide a statistical ensem-ble of typical congurations Accordingly, the best we can hope for is a statisticalinformation on the state of our open systemS at some later timet The resulting

theory of open systems is intrinsically probabilistic It can be considered as a part ofstatistical mechanics at the interface with the ergodic theory of stochastic processesand dynamical systems

The paradigm of this statistical approach to open systems is the theory of ian motion initiated by Einstein in one of his celebrated 1905 papers [3] (see also [4]for further developments) An account on this theory can be found in almost anytextbook on statistical mechanics (see for example [9]) Brownian motion had a deepimpact not only on physics but also on mathematics, leading to the development ofthe theory of stochastic processes (see for example [12])

Brown-Open systems appeared quite early in the development of quantum mechanics.Indeed, to explain the nite lifetime of the excited states of an atom and to computethe width of the corresponding spectral lines it is necessary to take into accountthe interaction of the electrons with the electromagnetic eld Einsteins seminalpaper [5] on atomic radiation theory can be considered as the rst attempt to use aMarkov process — or more precisely a master equation — to describe the dynamics

of a quantum open system The theory of master equations and its application toradiation theory and quantum statistical mechanics was subsequently developed byPauli [8], Wigner and Weisskopf [13], and van Hove [11] The mathematical theory

of the quantum Markov semigroups associated with these master equations started

to develop more than 30 years later, after the works of Davies [2] and Lindblad [7]

It further led to the development of quantum stochastic processes

To illustrate the philosophy of the modern approach to open systems let us sider a simple, classical, microscopic model of Brownian motion Even though thismodel is not realistic from a physical point of view it has the advantage of beingexactly solvable In fact such models are often used in the physics literature (see[10, 6, 1])

con-Preface VII

Brownian Motion: A Simple Microscopic Model

In a cubic crystal denote byqx the deviation of an atom from its equilibrium position

x N = { N, , N }3 Z3and bypx the corresponding momentum Supposethat the inter-atomic forces are harmonic and only acts between nearest neighbors ofthe crystal lattice In appropriate units the Hamiltonian of the crystal is then

p2 x

and Dirichlet boundary conditions are imposed by settingqx = 0 for x Z3\ N

If the atom at sitex = 0 is replaced by a heavy impurity of massM 1 then theHamiltonian becomes

H

p2x2mx

N , p = ( px)x

N, Q = q0, P = p0 For x Z3

we denote byx the Kronecker delta function atx and by|x| the Euclidean norm of

x We also set = |x |=1 x The motion of the joint systemS R is governed bythe following linear system

H0= 1

2 (p, p) + ( q,

2q)

VIII Preface

This ensemble is given by the Gaussian measure

d = Z 1e H0 ( q,p)dqdp,whereZ is a normalization factor and = 1 /kBT with kB the Boltzmann constant

At time t = 0 we release the impurity The subsequent evolution of the system

is determined by the Cauchy problem for Equ (1) The evolution of the environmentcan be expressed by means of the Duhamel formula

q(t) = cos( 0t)q(0) + sin( 0t)

0

p(0) +

t 0

K (t s)Q(s) ds + (t), (2)where the integral kernelK is given by

of the state of the environment The dissipative and the uctuating forces are related

by the so calleductuation-dissipation theorem

The solutionzt = ( Q(t), P (t)) of the random integro-differential equation (2)denes a family of stochastic processes indexed by the initial conditionz0 Theseprocesses provide a statistical description of the motion of our open system An in-variant measure for the processzt is a measure onR3 R3such that

f (zt) d (z0) = f (z) d (z),

The Hamiltonian Approach

Remark that in our example, such a steady state fails to exist since the motion ofthe joint system is clearly quasi-periodic However, in a real situation the number ofatoms in the crystal is very large, of the order of Avogadros numberNA 6 • 1023

In this case the recurrence time of the system becomes so large that it makes sense totake the limitN In this limit 2becomes the discrete Dirichlet Laplacian

on the innite lattice Z3\ { 0} This is a well dened, bounded, negative operator onthe Hilbert space2(Z3) Thus, Equ (2),(3), (4) and (5) still make sense in this limit

In the sequel we only consider the resulting innite system

We distinguish two main approaches to the study of open systems The rst one,the Hamiltonian approach, deals directly with the dynamics of the joint systemS R

We briey discuss the second one, the Markovian approach, in the next paragraph

In the Hamiltonian approach we rewrite the equation of motion (1) as

Z = i Z,where 2= m 1/ 2 2m 1/ 2with m = I + ( M 1) 0( 0, • ) the operator of multi-plication bymx and 2is the discrete Laplacian onZ3 The complex variableZ is

given byZ = 1/ 2m1/ 2q+ i 1/ 2m 1/ 2p andq = ( qx)x Z3, p = ( px)x Z3 It

fol-lows from elementary spectral analysis that forM > 1 the operator is self-adjoint

with purely absolutely continuous spectrum( ) = ac( ) = [0 , 2 0] on 2(Z3)

A simple argument involving the scattering theory for the pair2 2/M , 2shows

that the systemS has a unique steady statesuch that (6) holds for all0which are

absolutely continuous with respect to Lebesgue measure Moreover,is the marginal

on S of the innite dimensional Gaussian measureZ 1e H dpdqdP dQwhich

de-scribes the thermal equilibrium state of the joint system at temperatureT = 1 /kB

This is easily computed to be the Gaussian measure

(dP, dQ) = N 1e ( P2/ 2M + 2Q2/ 2)dP dQ,whereN is a normalization factor and

( 0, 2

0).

X Preface

The Markovian Approach

A remarkable feature of Equ (2) is the memory effect induced by the kernelK As aresult the processzt is non-Markovian,i.e., for s > 0, zt + sdoes not only depend on

zt and{ (u) | u [t, t + s]} but also on the full history{ zu| u [0, t]} The onlyway to avoid this effect is to haveK proportional to the derivative of a delta function

By Relation (5) this means thatshould be a white noise This is certainly not thecase with our choice of initial conditions However, as we shall see, it is possible

to obtain a Markov process in some particular scaling limits This is not a uniquelydened procedure: different scaling limits correspond to different physical regimesand lead to distinct Markov processes

As a simple illustration let us consider the particular scaling limit

KM(t s)QM(s) ds + M(t),where

KM(t) M1/ 2K (M1/ 2t),and the scaled processM(t) M1/ 4 (M1/ 2t) has covariance

L =2

2

P P • Q + 20Q • P

It is a simple exercise to show that the unique invariant measure of this process is theLebesgue measure Moreover, one can show that for any initial condition(Q0, P0)and any functionf L1(R3 R3, dQdP) one has

lim

t E(f (Q(t), Q(t))) = f (Q, P ) dQdP,

a scaled version of return to equilibrium

Preface XI

It is worth pointing out that in many instances of classical or quantum open tems the dynamics of the joint systemS R is too complicated to be controlledanalytically or even numerically Thus, the Hamiltonian approach is inefcient andthe Markovian approximation becomes the only available option The study of theMarkovian dynamics of open systems is the main subject of the second volume inthis series The third volume is devoted to applications of the techniques introduced

sys-in the rst two volumes It aims at leadsys-ing the reader to the front of the current search on open quantum systems

re-Organization of this Volume

This rst volume is devoted to the Hamiltonian approach Its purpose is to developthe mathematical framework necessary to dene and study the dynamics and ther-modynamics of quantum systems with innitely many degrees of freedom

The rst two lectures by A Joye provide a minimal background in operator ory and statistical mechanics The third lecture by S Attal is an introduction to thetheory of operator algebras which is the natural framework for quantum mechanics

the-of many degrees the-of freedom Quantum dynamical systems and their ergodic theoryare the main object of the fourth lecture by C.-A Pillet The fth lecture by M.Merkli deals with the most common instances of environments in quantum physics,the ideal Bose and Fermi gases Finally the last lecture by V Jaksi«c introduces one

of the main tool in the study of quantum dynamical systems: spectral analysis

Lyon, Grenoble and Toulon, St«ephane Attal

Claude-Alain Pillet

XII Preface

References

1 Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion ica A 121(1983), 587

Phys-2 Davies, E.B.: Markovian master equations Commun Math Phys.39 (1974), 91

3 Einstein, A: Uber die von der molekularkinetischen Theorie der W‹arme geforderte wegung von in ruhenden Fl‹ussigkeiten suspendierten Teilchen Ann Phys.17 (1905),549

Be-4 Einstein, A:Investigations on the Theory of Brownian Movement.Dover, New York1956

5 Einstein, A: Zur Quantentheorie der Strahlung Physik Zeitschr.18 (1917), 121

6 Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled tors J Math Phys.6 (1965), 504

oscilla-7 Lindblad, G.: Completely positive maps and entropy inequalities Commun Math Phys

40 (1975), 147

8 Pauli, W.: Festschrift zum 60 Geb‹urtstage A Sommerfeld S Hirzel, Leipzig 1928

9 Reif, F.: Fundamentals of Statistical and Thermal Physics.McGraw-Hill, New York1965

10 Schwinger, J.: Brownian motion of a quantum oscillator J Math Phys.2 (1961), 407

11 Van Hove, L.: Master equation and approach to equilibrium for quantum systems InFundamental Problems in Statistical Mechanics.E.G.D Cohen ed., North Holland, Am-sterdam 1962

12 Wax, N (Editor):Noise and Stochastic Processes.Dover, New York 1954

13 Weisskopf, V., Wigner, E.: Berechnung der nat‹urlichen Linienbreite auf Grund der schen Lichttheorie Zeitschr f‹ur Physik63 (1930), 54

Introduction to the Theory of Linear Operators

Alain Joye 1

1 Introduction 1

2 Generalities about Unbounded Operators 2

3 Adjoint, Symmetric and Self-adjoint Operators 5

4 Spectral Theorem 13

4.1 Functional Calculus 15

4.2 L2Spectral Representation 22

5 Stones Theorem, Mean Ergodic Theorem and Trotter Formula 29

6 One-Parameter Semigroups 35

References 40

Introduction to Quantum Statistical Mechanics Alain Joye 41

1 Quantum Mechanics 42

1.1 Classical Mechanics 42

1.2 Quantization 46

1.3 Fermions and Bosons 53

2 Quantum Statistical Mechanics 54

2.1 Density Matrices 54

3 Boltzmann Gibbs 57

References 67

Elements of Operator Algebras and Modular Theory St«ephane Attal 69

1 Introduction 70

1.1 Discussion 70

1.2 Notations 71

2 C -algebras 71

2.1 First denitions 71

2.2 Spectral analysis 73

XIV Contents

2.3 Representations and states 79

2.4 CommutativeC -algebras 83

2.5 Appendix 84

3 von Neumann algebras 86

3.1 Topologies onB(H ) 86

3.2 Commutant 89

3.3 Predual, normal states 90

4 Modular theory 92

4.1 The modular operators 92

4.2 The modular group 96

4.3 Self-dual cone and standard form 100

References 105

Quantum Dynamical Systems Claude-Alain Pillet 107

1 Introduction 107

2 The State Space of aC -algebras 110

2.1 States 110

2.2 The GNS Representation 119

3 Classical Systems 123

3.1 Basics of Ergodic Theory 123

3.2 Classical Koopmanism 127

4 Quantum Systems 130

4.1 C -Dynamical Systems 132

4.2 W -Dynamical Systems 139

4.3 Invariant States 141

4.4 Quantum Dynamical Systems 142

4.5 Standard Forms 147

4.6 Ergodic Properties of Quantum Dynamical Systems 153

4.7 Quantum Koopmanism 161

4.8 Perturbation Theory 165

5 KMS States 168

5.1 Denition and Basic Properties 168

5.2 Perturbation Theory of KMS States 178

References 180

The Ideal Quantum Gas Marco Merkli 183

1 Introduction 184

2 Fock space 185

2.1 Bosons and Fermions 185

2.2 Creation and annihilation operators 188

2.3 Weyl operators 191

2.4 TheC -algebrasCARF(H), CCRF(H) 194

2.5 Leaving Fock space 197

Contents XV

3 The CCR and CAR algebras 198

3.1 The algebraCAR( D ) 199

3.2 The algebraCCR(D ) 200

3.3 Schr‹odinger representation and Stone — von Neumann uniqueness theorem 203

3.4 Q—space representation 207

3.5 Equilibrium state and thermodynamic limit 209

4 Araki-Woods representation of the innite free Boson gas 213

4.1 Generating functionals 214

4.2 Ground state (condensate) 217

4.3 Excited states 222

4.4 Equilibrium states 224

4.5 Dynamical stability of equilibria 228

References 233

Topics in Spectral Theory Vojkan Jaksi«c 235

1 Introduction 236

2 Preliminaries: measure theory 238

2.1 Basic notions 238

2.2 Complex measures 238

2.3 Riesz representation theorem 240

2.4 Lebesgue-Radon-Nikodym theorem 240

2.5 Fourier transform of measures 241

2.6 Differentiation of measures 242

2.7 Problems 247

3 Preliminaries: harmonic analysis 248

3.1 Poisson transforms and Radon-Nikodym derivatives 249

3.2 LocalLpnorms,0 < p < 1 253

3.3 Weak convergence 253

3.4 LocalLp-norms,p > 1 254

3.5 Local version of the Wiener theorem 255

3.6 Poisson representation of harmonic functions 256

3.7 The Hardy classH (C+) 258

3.8 The Borel transform of measures 261

3.9 Problems 263

4 Self-adjoint operators, spectral theory 267

4.1 Basic notions 267

4.2 Digression: The notions of analyticity 269

4.3 Elementary properties of self-adjoint operators 269

4.4 Direct sums and invariant subspaces 272

4.5 Cyclic spaces and the decomposition theorem 273

4.6 The spectral theorem 273

4.7 Proof of the spectral theoremthe cyclic case 274

4.8 Proof of the spectral theoremthe general case 277

XVI Contents

4.9 Harmonic analysis and spectral theory 279

4.10 Spectral measure forA 280

4.11 The essential support of the ac spectrum 281

4.12 The functional calculus 281

4.13 The Weyl criteria and the RAGE theorem 283

4.14 Stability 285

4.15 Scattering theory and stability of ac spectra 286

4.16 Notions of measurability 287

4.17 Non-relativistic quantum mechanics 290

4.18 Problems 291

5 Spectral theory of rank one perturbations 295

5.1 Aronszajn-Donoghue theorem 296

5.2 The spectral theorem 298

5.3 Spectral averaging 299

5.4 Simon-Wolff theorems 300

5.5 Some remarks on spectral instability 301

5.6 Booles equality 302

5.7 Poltoratskiis theorem 304

5.8 F.& M Riesz theorem 308

5.9 Problems and comments 309

References 311

Index of Volume I 313

Information about the other two volumes Contents of Volume II 318

Index of Volume II 321

Contents of Volume III 323

Index of Volume III 327

List of Contributors

St·ephane Attal

Institut Camille Jordan

Universit«e Claude Bernard Lyon1

21 av Claude Bernard

69622 Villeurbanne Cedex

France

email: attal@math.univ-lyon1.fr

Vojkan Jak si·c

Department of Mathematics and

McGill University

805 Sherbrooke Street WestMontreal, QC, H3A 2K6Canada

email: merkli@math.mcgill.caClaude-Alain Pillet

CPT-CNRS (UMR 6207)Universit«e du Sud Toulon-Var

BP 20132

83957 La Garde CedexFrance

email: pillet@univ-tln.fr

Introduction to the Theory of Linear Operators

Alain Joye

Institut Fourier, Universit«e de Grenoble 1,

BP 74, 38402 Saint-Martin-dH‘eres Cedex, France

e-mail: alain.joye@ujf-grenoble.fr

1 Introduction 1

2 Generalities about Unbounded Operators 2

3 Adjoint, Symmetric and Self-adjoint Operators 5

4 Spectral Theorem 13

4.1 Functional Calculus 15

4.2 L2Spectral Representation 22

5 Stones Theorem, Mean Ergodic Theorem and Trotter Formula 29

6 One-Parameter Semigroups 35

References 40

1 Introduction

The purpose of this rst set of lectures about Linear Operator Theory is to provide the basics regarding the mathematical key features of unbounded operators to readers that are not familiar with such technical aspects It is a necessity to deal with such operators if one wishes to study Quantum Mechanics since such objects appear as soon as one wishes to consider, say, a free quantum particle inR The topics covered

by these lectures are quite basic and can be found in numerous classical textbooks, some of which are listed at the end of these notes They have been selected in order

to provide the reader with the minimal background allowing to proceed to the more advanced subjects that will be treated in subsequent lectures, and also according to their relevance regarding the main subject of this school on Open Quantum Systems Obviously, there is no claim about originality in the presented material The reader is assumed to be familiar with the theory of bounded operators on Banach spaces and with some of the classical abstract Theorems in Functional Analysis

2 Alain Joye

2 Generalities about Unbounded Operators

Let us start by setting the stage, introducing the basic notions necessary to studylinear operators While we will mainly work in Hilbert spaces, we state the generaldenitions in Banach spaces

If B is a Banach space overC with norm • andT is a bounded linear operator

on B, i.e T : B B , its norm is given by

a sequence of normalized functionsn L2(R), n N, such that q n

as n , and, there are functions ofL2(R) which are no longerL2(R) whenmultiplied by the independent variable We shall adopt the following denition of(possibly unbounded) operators

Denition 2.1 A linear operatoron B is a pair (A, D ) whereD B is a denselinear subspace ofB andA : D B is linear

We will nevertheless often talk about the operatorA and call the subspaceD thedomain ofA It will sometimes be denoted by Dom(A)

Denition 2.2 If (A, D ) is another linear operator such thatD D andA = A

for all D , the operatorA denes an extensionof A and one denotes this fact by

A A

That the precise denition of the domain of a linear operator is important for thestudy of its properties is shown by the following

Example 2.3.: Let H be dened on L2[a, b], a < b nite, as the differential operator

H (x) = (x), where the prime denotes differentiation Introduce the dense sets

The notion of continuity naturally associated with bounded linear operators isreplaced for unbounded operators by that of closedness

Denition 2.4 Let (A, D ) be an operator onB It is said to beclosedif for anysequencen D such that

n B and A n B ,

it follows that D andA =

Introduction to the Theory of Linear Operators 3

Remark 2.5 i.In terms of thegraphof the operatorA, denoted by (A) and givenby

(A) = { (, ) B B | D, = A } ,

we have the equivalence

A closed (A) closed (for the norm (, ) 2= 2+ 2)

ii If D = B, thenA is closed if and only ifA is bounded, by the Closed GraphTheorema

iii If A is bounded and closed, thenD = B so that it is possible to extendfl A to thewhole ofB as a bounded operator

iv If A : D D B is one to one and onto, thenA is closed is equivalent

to A 1 : D D is closed This last property can be seen by introducing theinverse graphof A, (A) = { (x, y ) B B | y D, x = Ay } and noticingthatA closed iff (A) is closed and (A) = (A 1)

The notion of spectrum of operators is a key issue for applications in QuantumMechanics Here are the relevant denitions

Denition 2.6 The spectrum (A) of an operator(A, D ) on B is dened by itscomplement (A)C = (A), where theresolvent setof A is given by

(A) = { z C | (A z) : D B is one to one and onto, and

(A z) 1: B D is a bounded operator.}

The operatorR(z) = ( A z) 1is called theresolventof A

Actually, A z is to be understood asA z1l, where1l denotes the identityoperator

Here are a few of the basic properties related to these notions

Proposition 2.7.With the notations above,

i If (A) = C, thenA is closed

ii If z (A) and u C is such that|u| < R(z) 1, thenz + u (A) Thus,(A) is open and (A) is closed

iii The resolvent is an analytic map from(A) to L (B), the set of bounded linearoperators onB, and the following identities hold for anyz, w (A),

Tn = (1l T ) 1 where T : B B is such that T < 1, (4)

shows that the natural candidate for(A z u) 1is R(z)(1l uR(z)) 1: B D Then one checks that onB

(A z u)R(z)(1l uR(z)) 1= (1l uR(z))(1l uR(z)) 1= 1l

and that onD

R(z)(1l uR(z)) 1(A z u) = (1l uR(z)) 1R(z)( A z u)

= (1l uR(z)) 1(1l uR(z)) = 1lD,where1lD denotes the identity ofD

iii) By (4) we can write

dx on L2[0, 1] on the followingdense sets IfAC2[0, 1] denotes the set of absolutely continuous functions on[0, 1]whose derivatives are inL2[0, 1], (hence inL1[0, 1]), set

Introduction to the Theory of Linear Operators 5

D1= { | AC2[0, 1]} , D0= { | AC2[0, 1] and (0) = 0 }

Then, one checks that(T, D1) and(T, D0) are closed and such that1(T ) = C and

0(T ) = (with the obvious notations)

To avoid potential problems related to the fact that certain operators can beapriori dened on dense sets on which they may not be closed, one introduces thefollowing notions

Denition 2.9 An operator(A, D ) is closableif it possesses a closed extension(A, D )

Lemma 2.10.If (A, D ) is closable, then there exists a unique extension( flA, flD )

called theclosureof (A, D ) characterized by the fact thatflA A for any closedextension(A, D ) of (A, D )

Proof Let

fl

D = { B | n D and B with n and A n } (5)

For any closed extensionA of A and any D , we havefl D andA = is

uniquely determined by Let us dene ( flA, flD ) by flA = , for all D Then flfl A

is an extension ofA and any closed extensionA A is such thatflA A The graph( flA) of flA satises ( flA) = (A), so that flA is closed

Note also that the closure of a closed operator coincide with the operator itself.Also, before ending this section, note that there exist non closable operators Fortu-nately enough, such operators do not play an essential role in Quantum Mechanics,

as we will shortly see

3 Adjoint, Symmetric and Self-adjoint Operators

The arena of Quantum Mechanics is a complex Hilbert spaceH where the notion

of scalar product• | • gives rise to a norm denoted by• Operators that areself-adjoint with respect to that product play a particularly important role, as theycorrespond to the observables of the theory We shall assume the following conven-tion regarding the positive denite sesquilinear form• | • on H H : it is linear inthe right variable and thus anti-linear in the left variable We shall also always as-sume that our Hilbert space is separable, i.e it admits a countable basis, and we shallidentify the dualH of H with H , since l H , ! H such thatl (•) = | •

Let us make the rst steps towards self-adjunction

Denition 3.1 An operator(H, D ) in H is said to besymmetricif , D H

|H = H | For example, the operators( d2

dx 2, DD) and( d2

dx 2, DN) introduced above aresymmetric, as shown by integration by parts

6 Alain Joye

Remark 3.2.If H is symmetric, its eigenvalues are real

The next property is related to an earlier remark concerning the role of non able operators in Quantum Mechanics

clos-Proposition 3.3.Any symmetric operator(H, D ) is closable and its closure is metric

sym-This Proposition will allow us to consider that any symmetric operator is closedfrom now on

Proof Let flD D as in (5) and D , D We compute for any such,fl

H Let us nally check the symmetry property Ifn D is such that n D ,flwith H n and D , (6) saysfl

|H n = H |fl n Taking the limitn , we get from the above

Denition 3.4 Let (A, D ) be an operator onH The adjoint of A, denoted by(A , D ), is determined as follows:D is the set of H such that there exists

Let us proceed with some properties of the adjoint

Introduction to the Theory of Linear Operators 7

Proposition 3.5.Let (A, D ) be an operator onH

i The adjoint(A , D ) of (A, D ) is closed If, moreover,A is closable, thenD isdense

ii If A is closable, flA = A

iii If A B , thenB A

Proof i) Let (, ) D H belong to (A ) This is equivalent to saying

|A = | , D,which is equivalent to(, ) M , where

M = { (A, ) H H , | D } ,with the scalar product ( 1, 2)|( 1, 2) = 1| 1 + 2| 2 As M isclosed, (A ) is closed too Assume nowA is closable and suppose there exists

H such that | = 0 , for all D This implies that(, 0) is orthogonal to(A ) But,

(A ) = M = M Therefore, there existsn D , such that n 0 andA n As A is closable,

iii) Follows readily from the denition

WhenH is symmetric, we get from the denition and properties above thatH

is a closed extension ofH This motivates the

Denition 3.6 An operator(H, D ) is self-adjointwhenever it coincides with its joint (H , D ) It is therefore closed

ad-An operator (H, D ) is essentially self-adjointif it is symmetric and its closure( flH, flD ) is self-adjoint

Thedeciency indices are the dimensions ofL–, which can be nite or innite.

To get an understanding of these names, recall that one can always write

H = Ker(H i ) Ran(H – i ) L– Ran(H – i ) (7)Note that the denitions ofL– is invariant if one replacesH by its closureflH

For (H, D ) symmetric and any D observe that

(H + i ) 2= H 2+ 2= (H i ) 2= 0

This calls for the next

Denition 3.8 Let (H, D ) be symmetric TheCayley transformof H is the isometricoperator

U = ( H i )( H + i ) 1: Ran(H + i ) Ran(H i )

It enjoys the following property

Lemma 3.9.The symmetric extensions ofH are in one to one correspondence withthe isometric extensions ofU

Proof Let (H, D ) be a symmetric extension of(H, D ) andU be its Cayley form We have

trans-Ran(H – i ) D D such that = ( H – i ) = ( H – i ),

hence Ran(H – i ) Ran(H – i ), and

U = ( H i )( H + i ) 1 = U , Ran(H – i ) (8)Conversely, letU : M+ M , be a isometric extension ofU, where Ran(H – i )

M– We need to construct a symmetric extension ofH whose Cayley transform is

U Algebraically this means, see (8),

Introduction to the Theory of Linear Operators 9

We can now state the

Theorem 3.10.If (H, D ) is symmetric onH , there exist self-adjoint extensions ofH

if and only if the deciency indices are equal Moreover, the following statements areequivalent:

1 H is essentially self-adjoint

2 The deciency indices are both zero

3 H possesses exactly one self-adjoint extension

Proof 1) 3): Let J be a self-adjoint extension ofH Then H J = J and

J H HenceJ = Jfl Hfl = flH , so thatJ = flH

1) 2): We can assume thatH is closed soH = flH = H For any L– =Ker(H i ),

0 = (H i ) 2= (H i ) 2= H 2+ 2 2, L– = { 0} (10)2) 1): Consider(H + i ) : D Ran(H + i ) By (10) above, this operator is one

to one, and we can dene(H + i ) 1 : Ran(H + i ) D By the same estimate itsatises

(H + i ) 1 2 (H + i )( H + i ) 1 2= 2

As H can be assumed to be closed (i.e.H = H ) and Lfl + = { 0} , we get thatRan(H + i ) is closed so thatH = Ran(H + i ), due to (7) Therefore, for any D ,there exists a D such that(H + i ) = ( H + i ) As H H ,

(H + i )( ) = 0 , i.e Ker (H + i ) = { 0} ,

we get that D andH = H , which is what we set out to prove

3) 2): if K is a self-adjoint extension ofH , its deciency indices are zero (by 2)).Therefore, (see (7)), its Cayley transformV is a unitary extension ofU, the Cayleytransform ofH In particular,V |L+ : L+ L is one to one and onto, so that thedeciency indices of H are equal That yields the rst part of the Theorem Nowassume these indices are not zero By the preceding Lemma, there exist an innitenumber of symmetric extensions ofH , parametrized by all isometries fromL+ to

L In particular, there exist extensions with zero deciency indices, which by 2) and1) are self-adjoint, contradicting the fact thatK is the unique self-adjoint extension

of H

10 Alain Joye

Remark 3.11.It is a good exercise to prove that in case(H, D ) is symmetric and

H 0, i.e |H 0 for any D , then H is essentially self-adjoint iffKer(H + 1) = { 0}

As a rst application, we give a key property of self-adjoint operators for therole they play in the Quantum dogma concerning measure of observables It is thefollowing fact concerning their spectrum

Theorem 3.12.Let H = H Then, (H ) R and,

(H z) 1 1

| z|, if z R. (11)Moreover, for anyz in the resolvent set ofH ,

(H z)fl 1= (( H z) 1) (12)Proof Let D , D being the domain ofH andz = x + iy , with y = 0 Then

(H x iy ) 2= (H x) 2+ y2 2 y2 2 (13)This implies

Ker(H z) = Ker(H z) = { 0} i.e Ran(H z) = H ,

andH z is invertible on Ran(H z) (13) shows that(H z) is bounded withthe required bound, and as the resolvent is closed, it can be extended onH with thesame bound Equality (12) is readily checked

As an application of the rst part of Theorem 3.10, consider a symmetric operator(H, D ) which commutes with aconjugationC More precisely:

C is anti-linear,C2 = 1l and C = Hence | = C |C Moreover,

so thatC + Ran(H i ) = Ker(H + i ) = L In other words, one has

C : L+ L , and one shows similarly thatC : L L+ As C is isometric, thedimensions ofL+ andL are the same

A particular case where this happens is that of the complex conjugation and adifferential operator onRn, with real valued coefcients

Introduction to the Theory of Linear Operators 11

An example of direct application of this criterion is the following Consider thesymmetric operatorH = i on the domainC0 (0, ) L2(0, ) A vec-tor D iff there exists L2(0, ) such that |H = | , for all

C0 (0, ) Expressing the scalar products this means

(x) fl(x)dx = i (x) fl (x)dx = iDxT ( fl),where T denotes the distribution associated with In other words, we have

W1,2(0, ) = D andH = i in the weak sense Elements of Ker(H i )satisfy

H = – i = – (x) = ce– x L

2(0, )

L2(0, )Hence there is no self-adjoint extension of that operator If it is considered on

C0 (0, 1) L2(0, 1), the above shows that the deciency indices are both 1 andthere exist innitely many self-adjoint extensions of it

Specializing a little, we get a criterion for operators whose spectrum consists ofeigenvalues only

Corollary 3.13 Let (H, D ) symmetric onH such that there exists an orthonormalbasis { n}n N of H of eigenvectors ofH satisfying for anyn N, n Dand H n = n n, with n R ThenH is essentially self-adjoint and( flH ) ={ n| n N}

Proof Just note that any vector in L– satises in particular

at , which yields the result

As a rst example of application we get thatdxd22 on C2(a, b) (or C (a, b))with Dirichlet boundary conditions is essentially self-adjoint with spectrum

n2 2(b a)2 n N ,

as the corresponding eigenvectors

12 Alain Joye

are known to form a basis ofL2[a, b] by the theory of Fourier series

Another standard operator is the harmonic oscillator dened onL2(R) by thedifferential operator

b = ( x x)/ 2, b = ( x + x)/ 2 to show that the solutions of

Hosc n(x) = n n(x), n N,are given by n = n + 1 / 2 with eigenvector

n(x) = cnHn(x)e x2/ 2, with Hn(x) = ( 1)nex2 d

n

dxne x2,andcn = (2nn! ) 1/ 2 These eigenvectors also form a basis ofL2(R), so that thisoperator is essentially self-adjoint with spectrumN + 1 / 2 Note that we cannot work

on C0 to apply this criterion here

Another popular way to prove that an operator is self-adjoint is to compare it

to another operator known to be self-adjoint and use a perturbative argument to getself-adjointness of the former

Let (H, D ) be a self-adjoint operator onH and let(A, D (A)) be symmetric withdomainD (A) D

Denition 3.14 The operatorA has arelative bound 0 with respect toH ifthere existsc < such that

The inmum over such relative bounds is the relative bound ofA w.r.t H

Remark 3.15.The denition of the relative bound is unchanged if we replace (14) bythe slightly stronger condition

A 2 2 H 2+ c2 2, D

Lemma 3.16.Let K : D H be such thatK = H + A If 0 <

1, is the relative bound ofA w.r.t H , K is closed and symmetric Moreover,A(H + i ) 1 < 1, if R has large enough modulus

Proof The symmetry ofK is clear Let us considern D such that n and

K n Then, by assumption,

H n H m K n K m + A n A m

K n K m + H n H m + c n m ,

Introduction to the Theory of Linear Operators 13

The proof of the statement concerning the resolvent reads as follows LetH ,

= ( H + i ) 1 and0 < < 1 Then, for| | > 0 large enough

A 2 ( H + c )2 2 H 2+ 2 2

= 2 (H + i ) 2= 2 2.Hence A(H + i ) 1

This leads to the

Theorem 3.17.If H is self-adjoint andA is symmetric with relative bound< 1w.r.t H , thenK = H + A is self-adjoint on the same domain as that ofH

Proof Let | | be large enough From the formal expressions

4 Spectral Theorem

Let us start this section by the presentation of another example of self-adjoint ator, which will play a key role in the Spectral Theorem, we set out to prove here.Before getting to work, let us specify right away that we shall not provide here afull proof of the version of the Spectral Theorem we chose Some parts of it, of apurely analytical character, will be presented as facts whose detailed full proofs can

oper-be found in Daviess book [1] But we hope to convey the main ideas of the proof inthese notes

ConsiderE RN a Borel set and a Borel non-negative measure onE Let

H = L2(E, d ) be the usual set of measurable functionsf : E C such that

(Af )( x) = a(x)f (x), f D.

Lemma 4.1.(A, D ) is self-adjoint and ifL2

c denotes the set of functions ofH whichare zero outside a compact subset ofE , thenA is essentially self-adjoint onL2

c.Proof A is clearly symmetric Ifz R, the bounded operatorR(z) given by

So thatA is closed and essentially self-adjoint, hence self-adjoint

Concerning the last statement, we need to show thatA is the closure of its striction toL2

Lemma 4.2.The spectrum and resolvent ofA are such that

(A) = essential range ofa = { R | ({ x | |a(x) | < } ) > 0, > 0}

If (A), then

(A ) 1 = 1

dist(, (A)).

Introduction to the Theory of Linear Operators 15

Proof If is not in the essential range ofa, it is readily checked that the cation operator by(a(x) ) 1 is bounded (outside of a set of zeromeasure).Also one sees that this operator yields the inverse ofa for such s, which, con-

multipli-sequently, belong to(A) Conversely, let us take in the essential range ofa and

show that (A) We dene sets of positive measures by

Sm = { x E | | a(x)| < 2 m} Let m be the characteristic function ofSm, which is a non zero element of

(a(•) ) 1 = essential supremum of(a(•) ) 1,

where we recall that for a measurable functionf

Let us now come to the steps leading to the Spectral Theorem The general setting

is as follows One has a self-adjoint operator(H, D ), D dense in a separable HilbertspaceH We rst want to dene a functional calculus, allowing us to take functions

of self-adjoint operators IfH is a multiplication operator by a real valued function

h, as in the above example, thenf (H ), for a reasonable functionf : R C, is

easily conceivable as the multiplication byf h We are going to dene a function

of an operatorH in a quite general setting by means of an explicit formula due toHelffer and Sj‹ostrand and we will check that this formula has the properties we ex-pect of such an operation Finally, we will also see that any operator can be seen as

a multiplication operator on someL2(d ) space

Let us introduce the notation< z > = (1 + |z|2)1/ 2and the set of functions wewill work with Let R andS be the set of complex valuedC (R) functions

such that there exists acn so that

|f( n )(x)| = d

n

dxnf (x) cn < x > n, x R n N

This set of functions enjoys the following properties:

A is an algebra for the multiplication of functions, it contains the rational functionswhich decay to zero at and have non-vanishing denominator on the real axis.Moreover, it is not difcult to see that

1 f ! f (H ) is linear and multiplicative, (i.e.fg ! f (H )g(H ))

2 flf (H ) = ( f (H )) , f E

3 f (H ) f , f E

4 If w R andrw(x) = ( x w) 1, thenrw(H ) = ( H w) 1

5 If f C0 (R) such that supp(f ) " (H ) = thenf (H ) = 0

For f C , we dene its quasi-analytic extensionf : C C by

in general, but it isC Its support is conned to the set|y| 2 < x > due to thepresence of Also, the projection on thex axis of the support off is equal to thesupport off The choice of andn will turn out to have no importance for us.Explicit computations yield

Introduction to the Theory of Linear Operators 17

Denition 4.4 For any f A and any self-adjoint operatorH on H the Sj‹ostrand formulafor f (H ) reads

Helffer-f (H ) = 1

C flf (z)( H z)

1dxdy L (H ) (16)Remark 4.5.This formula allows to compute functions of operators by means of theirresolvent only Therefore it holds for bounded as well as unbounded operators More-over, being explicit, it can yield useful bounds in concrete cases Note also that it islinear inf

We need to describe a little bit more in what sense this integral holds

Lemma 4.6.The expression (16) converges in norm and the following bound holds

f (H ) cn f n +1, f A and n 1 (17)Proof The integrand is bounded andC on C \ R, therefore (16) converges in norm

as a limit of Riemann sums on any compact ofC \ R It remains to deal with the limitwhen these sets tend to the whole ofC Let us introduce the sets

We need a few more properties regarding formula (16) before we can show itdenes a functional calculus

It is sometimes easier to deal withC0 functions rather then with functions ofA The following Lemma shows this is harmless

18 Alain Joye

Lemma 4.7.C0 (R) is dense inA for the norms• n

Proof We use the classical technique of molliers Let 0 be smooth withthe same conditions of support as Set m(x) = (x/m ) for all x R and

fm = mf Hence,fm A and support considerations yield

The next Lemma will be useful several times in the sequel

Lemma 4.8.If F C0 (C) andF (z) = O(y2) asy 0 at xed real x, then

so thatdflz # dz = 2 idx # dy = 2 idxdy Moreover, since fl(H z) 1 = 0 byanalyticity,

d(F (z)( H z) 1dz) =

z (F (z)( H z)

1)dz # dz+

Introduction to the Theory of Linear Operators 19

|I | lim

0

12

N N

1dz

Neglecting support considerations, iff was analytic, this is the way we would rally dene f (H )

natu-We can now show a comforting fact about our denition (16) off (H )

Lemma 4.10.If f A andn 1, thenf (H ) is independent of andn

Proof By density ofC0 in A for the norms• n and Lemma 4.6, we can assume

f C0 Let f 1,n andf 2,n be associated with1and 2 Then

is identically zero fory small enough, so Lemma 4.8 applies Similarly, ifm > n

1, with similar notations,

and Lemma 4.8 applies again

We are now in a position to show that formula (16) possesses the properties of afunctional calculus

Proposition 4.11.With the notations above,

a If f C0 and supp(f ) " (H ) = , thenf (H ) = 0

K flf (z)

dxdy

w z = f (w),the equivalent one forg and one gets, changing variables toz,

c) The rst point follows from (H z) 1 = ( H z)fl 1, the convergence innorm of (16) and the fact thatflf (z) = f (flz) if is even, which we can alwaysassume For the second point, takef A andc > 0 such that f c Den-ing g(x) = c (c2 | f (x)|2)1/ 2, one checks thatg A as well The identity

f flf 2cg + g2= 0 in the algebraA implies with the above

f (H )f (H ) 2cg(H ) + g(H )g(H ) = 0

f (H ) f (H ) + ( c g(H )) (c g(H )) = c2.Thus, for any H , it follows

f (H ) 2 f (H ) 2+ (c g(H )) 2 c2 2,

wherec f is arbitrary

d) Let us taken = 1 and assume w > 0 We further choose

(x, y ) = (y/ < x > ),where 1 will be chosen large enough so thatw does not belong to the support ofand then kept xed in the rest of the argument The sole effect of this manipulation

is to change the support of, but everything we have done so far remains true for

Introduction to the Theory of Linear Operators 21

> 1 and xed Let us dene, for m > 0 large,

by support considerations and elementary estimates on the different pieces ofm.Admitting this fact we have

rw(H ) = lim

m

12i m rw(z)( H z)

1dz

= res(rw(z)( H z) 1)|z= w = ( H w) 1,due to the analyticity of the resolvent insidem

We can now state the rst Spectral Theorem for the setC (R) of continuousfunctions that vanish at innity

on the set of functionsR

R = {

n

i =1

irw i, where i C, wi R}

22 Alain Joye

But, it is a classical result also that the setR satises the hypothesis of the Weierstrass TheorembandR = C , so that the two functional calculus must coin-cide everywhere

Stone-We shall pursue in two directions Stone-We rst want to show that any self-adjointoperator can be represented as a multiplication operator on someL2space Then weshall extend the functional calculus to bounded measurable functions

4.2 L2 Spectral Representation

Let (H, D ) be self-adjoint onH

Denition 4.13 A closed linear subspaceL of H is said invariant under H if(H z) 1L L for anyz R

Remark 4.14.It is an exercise to show that if L " D , thenH L , as expected.Also, for any R, Ker(H ) is invariant If it is positive, the dimension ofthis subspace is called the multiplicity of the eigenvalue

Lemma 4.15.If L is invariant underH = H , thenL is invariant also Moreover,

f (H )L L , for all f C (R)

Proof The rst point is straightforward and the second follows from the tion of the integral representation (16) off (H ) for f A by a norm convergentlimit of Riemann sums and by a density argument forf C (R)

approxima-Denition 4.16 For (H, D ) self-adjoint onH , thecyclic subspace generated by thevectorv H is the subspace

L = span{ (H z) 1v, z R} Remark 4.17 i.Cyclic subspaces are invariant underH , as easily checked

ii If the vectorv chosen to generate the cyclic subspace is an eigenvector, then, thissubspace isCv

iii If the cyclic subspace corresponding to some vectorv coincides withH , we saythatv is a cyclic vector forH

iv In the nite dimensional case, the matrixH has a cyclic vectorv iff the spectrum

of H is simple, i.e all eigenvalues have multiplicity one

These subspaces allow to structure the Hilbert space with respect to the action

of H

Lemma 4.18.For (H, D ) self-adjoint onH , there exists a sequence of orthogonalcyclic subspacesLn H with cyclic vectorvn such thatH = N

n =1Ln, with Nnite or not

bLet X be locally compact and considerC (X ) If B is a subalgebra ofC (X ) thatseparates points and satisesf B ffl B , thenB is dense inC (X ) for •

Introduction to the Theory of Linear Operators 23

Proof As H is assumed to be separable, there exists an orthonormal basis{ fj}j Nof

H Let L1be the subspace corresponding tof1 By induction, let us assume onal cyclic subspacesL1, L2, • • • , Ln are given Letm(n) be the smallest integersuch thatfm ( n ) L1 • • • Ln and letgm ( n ) be the component offm ( n ) orthog-onal to that subspace We letLn +1 be the cyclic subspace generated by the vector

orthog-gm ( n ) Then we haveLn +1 $ Lr, for all r n andfm ( n ) L1• • • Ln Ln +1.Then either the induction continues indenitely andN = , or at some point, such

a m(n) does not exist and the sum is nite

The above allows us to consider eachH |Ln, n = 1 , 2 • • • , N separately Note,however, that the decomposition is not canonical

Theorem 4.19.Let (H, D ) be self-adjoint onH , separable LetS = (H ) R.Then there exists a nite positive measure on S N and a unitary operator

U : H L2 L2(S N, d ) such that if

h : S N R(x, n ) ! x,then H belongs toD if and only ifhU L2 Moreover,

U HU 1 = h, U(D ) L2(S N, d ) and

U f (H )U 1 = f (h), f C (R), L2(S N, d )

This Theorem will be a Corollary of the

Theorem 4.20.Let (H, D ) be self-adjoint onH and S = (H ) R Further sume thatH admits a cyclic vectorv Then, there exists a nite positive measure

as-on S and a unitary operatorU : H L2(S, d ) L2such that if

h : S R

x ! x,then H belongs toD if and only ifhU L2and

U HU 1 = h U(D ) L2(S, d )

Proof of Theorem 4.20.Let the linear functional : C (R) C be dened by

(f ) = v|f (H )v The functional calculus shows that( flf ) = (f ) And if 0 f C (R), then,with g = f , we have

(f ) = g(H )v 2 0, i.e is positive

Thus, by Riesz-Markov Theoremc, there exists a positive measure onR such that

c

If X is a locally compact space, any positive linear functionall on C (X ) is of the form

l (f ) = f d , where is a (regular) Borel measure with nite total mass

= v|g (H )f (H )v H = g(H )v|f (H )v H.Dening

M = { f (H )v H | f C (R)} ,

we have existence of an onto isomorphismU such that

U : M C (R) L2 such thatU f (H )v = f

Now, M is dense inH sincev is cyclic andC (R) is dense inL2, so thatU admits

a unitary extension fromH to L2

Let f1, f2, f C (R) and i = fi(H )v H Then

where f denotes the obvious multiplication operator In particular, iff (x) =

rw(x) = ( x w) 1, we deduce that for all L2and allw R

U rw(H )U 1 = U(H w) 1U 1 = rw (20)Thus,U maps Ran(H w) to Ran(rw), i.e D and{ L2| x (x) L2} are inone to one correspondence

If L2 and = rw , then D (h), whereD (h) is the domain of themultiplication operator byh : x ! x Then, with (20)

U HU 1 = U HU 1rw = U Hrw(H )U 1 = wrw + = xrw = h

Proof of Theorem 4.19.We knowH = nLnwith cyclic subspacesLnwith vectors

vn We will assume vn = 1 / 2n, n N By Theorem 4.19, there existn ofmassSd n = vn 2= 2 2n and unitary operatorsUn : Ln L2(S, d n) such