Rigged Hilbert Space and Quantum Mechanics

Lecture Notes in Physics Edited by J. Ehlers, MCinchen,K. Hepp, ZiJrich R. Kippenhahn, MiJnchen, H. A. Weidenmi~ller,Heidelberg and J. Zittartz, KOln Managing Editor: W. Beiglb6ck, Heidelberg

78 A. B5hm

The Rigged HilbertSpace and Quantum Mechanics Lectures in Mathematical Physics at the University of Texas at Austin Edited by A. BShm and J. D. Dollard

Springer-Verlag Berlin Heidelberg New York 1978

Author A. BShm Physics Department University of Texas Austin, TX ?8712/USA Editors A. BShm J. D. Dollard Physics Department University of Texas Austin, TX ?8712/USA

ISBN 3-540-08843-1 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-08843-1 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 215313140-543210

TO FRITZ

ROHRLICH

PREFACE The purpose but to make tions

of this papem

a little-known

of q u a n t u m

Therefore

the discussion

will be limited is developed harmonic

mechanics

in Terms

in the

the purpose

statements,

of the physics

general

of This paper

precludes

to a wider

well-known

(Unfortunately,

still

of These mathematical

stand the content

example

notions

of the

have been

is compatible

with

in The second

of proving

hopefully,

subject

the mathematics

to an extent which,

but which,

audience. ~

The whole

form which

the p o s s i b i l i t y

founda-

and mathematics

and the m a t h e m a t i c a l

had to be a b b r e v i a t e d part,

accessable

of the simple,

least

new results,

area in the m a t h e m a t i c a l

to the bare essentials.

oscillator,

introduced

is not to present

The m a t h e m a t i c a l

allows

one to under-

statements).

After

one has once met these notions

for this well known particu-

lar ease,

absorb

form.

one will more

Most of The results

in much more general class

of enveloping

w i t h the symbol To m e n t i o n rial.

cians

easily

these

stated here are known

situations, algebras

in p a r t i c u l a r

of Lie groups.

o) have been inserted generalizations

The general

them in their

mathematical

~It is based on a series and physicists.

for a large

Throughout

background

of lectures

to be True

Remarks

and additional

general

(ending

the paper related

mate-

has been Treated

given

to m a t h e m a t i -

V!

in several monographs 1'2 w r i t t e n by the pioneers of this area, to which we r e f e r the interested reader. One can argue that the subject described here is very useful for physics, because rigorous and therewith ~o~ @l$~h@ physicists

it makes the Dirac formalism

gives a m a t h e m a t i c a l

mathematically

justification

undefined operations which

have been using for generations.

a slightly different

framework

the von Neumann axioms,

It also gives

for q u a n t u m mechanics

in which a physical state is de-

scribed by a collection of square integrable elements differ on a set of m e a s u r e more practical

than

zero.

functions whose

In a p h y s i c a l l y

d e s c r i p t i o n one would associate w i t h the physi-

cal state one test function

(an element of the Schwark space)

which is given by the resolution function of the experimental apparatus that is used in the p r e p a r a t i o n of the state. subject presented here provides

just such a description.

Thus the use of the m a t h e m a t i c a l

structure

set forth here

leads to a physically more practical and m a t h e m a t i c a l l y pler description,

The

sim-

even though it requires one to learn ini-

tially a little bit more mathematics

than that for the Hilbert

space formulation of quantum mechanics. But all this appears to me negligible main feature of this mathematical

compared t o the

structure,

its beauty.

The Rigged Hilbert Space has been one of the most beautiful pieces of art I have come across and has given me joy whenever I looked at it.

I hope that the simplified picture

which is shown in this paper still conveys enough of this

Vll

beauty to please even those physicists who avoid new mathematics because they see no need for complete mathematical rigour in physical calculations. In the following introductory section a general framework for the formulation of quantum mechanics in~ne Rigged Hilbert Space is given.

Though modifications of this frame-

work which incorporate the same mathematical structure are possible, it is within this framework that the subject is developed in the succeeding sections. In section I the algebraic representation space for the algebra generated by the momentum and position operator is constructed in a way very familiar to physicists~ the essential assumption is that there exists at least one eigenvector of the energy operator. In section II this algebraic space is equipped with two different topologies, the usual Hiibert space topology and a stronger, nuclear topology. a dual purpose:

This nuclear topology has

i.) the elements of the algebra are repre-

sented by continuous operators and 2.) there exist eigenvectors of the momentum and position operators,

(-as explained

in the following sections-) neither of which is the case for the usual Hilbert space formulation. In section III the spaces of antilinear functionals are described and the Rigged Hilbert Space is constructed.

VII1 Section IV gives the definition of generalized eigenvectons and a statement of the Nuclea~ Spectral Theorem, which guarantees the existence eigenvectors

of a complete

set of generalized

of momentum and position operators.

Realiza-

Tions of the Rigged Hilbert Space by spaces of functions and distributions

are described and in Appendix V, a simple

derivation of the Schrodinger realization

is given.

CONTENTS

O.

Introduction ........................................

I.

The Algebraic

II.

III. i17.

Structure

I

of the Space of States ..... 7

The Topological Structure of the Space of States ...................................

14

Conjugate

Space of ~ ..............................

28

Generalized Eigenvectors and N u c l e a r S p e c t r a l T h e o r e m ..................................

35

Appendix

I - Linear Spaces,

L i n e a r Operators ...... 47

A Dperzlix II - A l g e b r a ............................. A p p e n d i x I I I - L i n e a r ~Operators in H i l b e r t S~aces .................................

49

~.. 52

A p p e n d i x IV - Proof of Lemma (I 4) Section II by Induction ...........................

55

A p p e n d i x V ........................................

56

References

65

and Footnotes ..........................

~knowledgement

...................................

70

O.

To obtain matical

image

a physical

theory means

of a physical

for the domain Neumann3)~

INTRODUCTION

of quantum p h e n o m e n a

the set of operators

ease that n o s u p e r s e l e c t i o n systems)

the p h y s i c a l

("pure")

physical

all unit rays, operators This objects

rules

operators states

is, a c c o r d i n g

image

to yon

space.

are present

For the

(irreducible

are i d e n t i f i e d in the H i l b e r t

are identified

or equivalently,

a mathe-

The m a t h e m a t i c a l

in a H i l b e r t

observables

set of all self-adjoint the

system.

to obtain

w i t h the space

H and

with the set of

the one dimensional

projection

in H. one-to-one

correspondence

between

of a system and the operators

the physical

in a H i l b e r t

space

has two deficiencies: i) It does not contain

all the information

one has

in

q u a n t u m mechanics. 2) It contains

more

information

than one can ever obtain

in physics. To elaborate infinite are

dimensional)

-- r o u g h l y

means

on

I.):

It is w e l l known

Hilbert

speaking

that all physical

spaces

are isomorphic

-- their algebras systems

that all

(separable

and so

of operators.

w o u l d be equivalent,

This

which

is obviously not the ease ~nless one has only one physical system, which then would have to be the microphysical world). In every practical case one has to consider an isolated physical system which -- though it has to be large enough so that it does not become

trivial w

must be small enough

that one may command a view of its mathematical image. The various microphysical systems in nature have quite distinct properties and must therefore have quite distinct mathematical images.

Thus every particular physical system

has its algebra of operators and its linear space in which these operators act.

The algebra of observables is defined by

the algebraic relations and some additional conditions, like e.g., that a certain algebraic combination of operators be essentially self-adjoint.

A "larger" physical system can of

course be reduced into two "smaller" physical subsystems

(e.g.,

the quantum mechanical Kepler system can be reduced into the hydrogen atom and the electron-proton system).

For a particu-

lar subsystem some observables may be more physical than others and for certain subsystems some observables may be eompietely unphysical (e.g., for the hydrogen atom one can never prepare a state in which the electron position is in a narrow domain of space without destroying it). Thus for a particular physical system not all mathematical operators are physical observables.

Further, for

a particular physical system some observables appear to be more basic than others,and which observables are more basic depends upon the particular physical system.

Therefore, between

it appears

observables

An observable definition

is a physical

the quantity

for it in terms

can be physical state

observables, ment

which

are not p h y s i c a l l y

correspond

to 2.):

namely

topology,

number

of states

from this m e a s u r e Thus eigenstates

2.)--there

are vectors

These are the

but

vectors

of an observable

operator~

e.g.,

of infinite

the vector

energy.

space

dimensional

in a very p a r t i c u l a r

it is complete

of infinite

of one or more

The Hilbert

i.e. with respect

convergence

of

cannot be prepared.

of definition

to the state

sional and it is infinite sense,

the p r e p a r a t i o n

that are observables

realizable.

by an unbounded

expression

then not all

the m e a s u r e m e n t

operators

the domain

that is r e p r e s e n t e d

regard

because

this is c o n n e c t e d with

which

With

are observables,

are not observables

with

a prescription

quantities.

of the observables.

are c o n n e c t e d

which w o u l d

of e i t h e r

state that results

which

lie outside

and the physical

itself or a definite

states~

and a pure

Further--and

that

consists

constitutes

is an eigenstate

of operators

quantity

of other measurable

If not all operators

a physical

to make a d i s t i n c t i o n

and operators.

of an observable

for m e a s u r i n g

vectors

appropriate

w i t h respect

is infinite

to a very p a r t i c u l a r

to a very p a r t i c u l a r

sequences.

Since

can n e v e r be prepared,

dimen-

meaning

of

an infinite

physical

measurements

cannot tell us anything about infinite sequences, but at best give us information about aribtrarily large but finiie sequences.

Therefore, physics cannot give us sufficient

information to tell how to take the limit to infinity, i.e., how to choose the topology.

All that physics can tell us

is that we need a linear space sometimes finite but in general, of arbitrarily large dimensions. I believe that the choice of the topology will always be a mathematical generalization, but then one should choose the topology such that it is most convenient. At the time when quantum mechanics was developed the Hilbert space was the only linear topological space that had structures which were required by quantum mechanics. That is, it is linear to fulfill the superposition principle, and it could accommodate infinite dimensional matrices and differential operators.

Therefore it was natural that y o n

Neumann chose the Hilbert space when he wanted to give a mathematically rigorous formulation of quantum mechanics. With fbi's choice, though, he could not accommodate all the features of Dirac's formulation 4) of quantum mechanics, which von Neumann says is "scarcely to be surpassed in brevity and elegance" but "in no way satisfies the requirements of mathematical rigour". Since that time, stimulated by the Dirac formalism, a new branch of mathematics, the theory of distributions,

has been developed.

The abstraction of this theory provided

the mathematical tool for a mathematically rigorous formulation of quantum mechanics~ which not only overcomes the above mentioned shortcomings of von Neumann's axioms but also has all the niceties of the Dirac formalism. This mathematical tool is the Rigged Hilbert Space, which was introduced around 1960 by Gelfand I) and collaborators and Maurin 2) in connection with the spectral theory of selfadjoint operators.

Its use for the mathematical formulation

of quantum mechanics was suggested around 19655'6); the suggestion to choose the topology such that the observables are represented by continuous operators (for the canonical commutation relation) was made slightly earlier. 3i) We formulate the general framework by replacing the yon Neumann axiom which states the one-to-one correspondence between observables and self-adjoint operators in the Hilbert space by the following basic assumption 7) . A physical observable is represented by a linear operator in a linear space (space of states). The mathematical image of a physical system is an algebra A of linear operators in a linear scalar product space.

The

linear space is equipped with the weakest nuclear topology that makes this algebra an algebra of continuous operators. n Also, the operator A = ~ X i' 2 where Xi, i = 1,2,..n are the i=l generators of A, is essentially seif-adjoint (e.s.a.).

We emphasize again:

An observable is a physical quan-

tity which is defined by the prescription for its measurement. The algebra is the mathematical structure which is defined by the algebraic relations between its generators which represent some basic physical quantities and by other mathematical conditions, like the space in which the elements of the algebra act as operators. The connection between The quantities calculated in the mathematical image, such as the matrix elements

(~

A~) or Tr(AW)

and the quantities measured in the physical world, such as the expectation value (average value of a measurement) M(A), is given by M(A) = (~,A~) or in general M(A) = Tr(AW).

These

connections are stated by further basic assumptions which we will not discuss in this paper. Within the framework of the above stated basic assumption, The actual physical question is how to find the algebra. In non-relativistic quantum mechanics one can usually obtain this algebra from correspondence with the corresponding classical system.

When no corresponding classical system

exists one has to conjecture this algebra from the expertmental data.

I.

The Algebraic Structure of the Space of States We shall treat in detail one of the simplest physical

models, the one-dimensional harmonic oscillator, framework of the basic assumption section.

in the

stated in the preceeding

We will give a formulation which easily generalizes

to more complicated physical models~ and shall remark for which algebras these generalizations The algebra 9)

of operators

are already known.

8)

for the one-dimensional

harmonic oscillator is generated by the operators H representing the observable energy (i)

P representing the observable momentum Q representing the observable position

and the defining algebraic relations are (i)

PQ - QP

l

2 1 p2 m~ Q2 H = [m + --~-~ m, m are constants,

universal or characteristics

of the

system. A is an algebra of linear operators in a linear space, 4, with scalar productI0), (.,')(but ¢ is not a Hilbert space) Further: (2)

P,Q,H are symmetric operators,

i.e.

(P~,~) = (¢~P~) for all 4 , 4 e #. (i) and (2)

do not specify the mathematical

structure

8

completely.

The~e are many such linear spaces in which A

is an algebra of operators (whether the requirement that its topology be nuclear and the weakest puts sufficient restrictions is unknown).

Therefore we have to make one

further assumption to specify the algebra of operators, i.e. the representation of (I) and (2).

This requirement can be

formulated in two equivalent ways: (3a)H is essentially self a d j o i n ~ ( e . s . a . ) or (3b)There exists at least one eigenvector of H. We will use this requirement in the form (3b), as it is this form in which it is always used by physicists. Remark: We remark that either of these requirements leads to a representation of (I) which integrates to a representation of the group generated by P,Q,I (Weyl group); 3a)by a theorem of N e l s o n ) and 3b)because it leads to the well known ladder representation an ladder representations are always integrablelD(in this particular case one can easily see that H in the ladder representation will be e.s.a.). The generalization of (3) is the requirement that A = Z X. 2 be e.s.a. l

For the case that A is the enve~

loping algebra of a Lie group, this will always lead

to an i n t e g r a b l e

representstion

a representation

t h a t is c o n n e c t e d

of a g r o u p

though

(even

The p r o c e d u r e known

and w i l l

be

no

(Nelson

symmetry

m a y be i n v o l v e d ) , representation

only v e r y b r i e f l y :

One d e f i n e s

a = --

Q +

a+= I___ m / ~

Q

P

(4)

1 = ~

(5) N = a + a

These

~

ip

1 H - [ 1

operators

(~, a~)

_

clearly

fulfill

: (a+~,~)

fox e v e r y

: (N¢,¢)

for e v e r y

#,4 e

and

(¢, N¢)

~,@

as a c o n s e q u e n c e

of

As a c o n s e q u e n c e

of (I) a a n d a

e

(2). +

+ (6) a a (7a) As (Tb)

a = i

a consequence

or as

fulfill

+ - a

consequence

of of

(3a) (3b)

N

is

there

i.e.

with a representation

to find the l a d d e r sketched

t h e o r e m ) !2),

e.s.a.

exist

¢k such that _

o is w e l l

1o

(7b)

N¢ k : k¢ l F r o m the

c.r.

(6) then

+

follows

+

N(a¢ l) = a

a a el : (a a

- l) a ¢X = a ( a + a

= a(N-l)~l

(8) i.e.

a¢ l is an e i g e n v e c t o r or

Further

Ila + ~tl

a¢~ from 12

= a(l-l)¢l

with

- l)

¢k =

= (l-l)(a@k)

eigenva!ue

(X-l)

= 0 c.r.

(6)

follow

= (¢k,aa + ¢1) : (¢l,a + a ¢1) + (¢1' l¢l)

IIelll2

= IIa¢lll 2 +

o as ¢1 ¢ o

i.e. +

(9) a ¢1 ~ 0 always. Further

follows

from

+ (I0)

N(a

(6) +

¢i)

= (X+l)

a

+

a ¢1 is a l w a y s

i,eo

(k+l) .

Starting

can o b t a i n

with

an e i g e n v e c t o r CX' w h i c h

was a s s u m e d

eigenvectors m

Cx-m = a with

of N w i t h

eigenvalue N CX-m

m = 0,1,2,...

el

(l-m)

of N

= (l-m)

%k-m

eigenvalue

to exist,

one

11

A f t e r a finite n u m b e r

of steps am~

: 0

Proof: (¢l_m ,N ¢~_m ) : ( l - m ) ( ¢ l _ m , @ l _ m )

: (¢l_m,a+a¢~_m)

: ]la¢~_ml[ 2

therefore (ll)

(l-m)

=

IIa~ l _roll2 ......> 0

ll,x,mll2 i.e.

there exists

a #l-m such that

aCk_ m :

0

We call ¢o the n o r m a l i z e d

vector

¢o i.e.

#0 : i~1~011 ;

[12)

N¢ ° : 0

Then one defines

the states

¢o ].

Cz

:

,/z:

~

l

(13)

+ a

~_ o

+ 2

(a) ¢o t 1 l Cn

-

........ 1

(a+)n¢o

for w h i c h a¢ O : 0

12 which have the property

3) (¢n '¢m) =

(i~)

4)

~nm

For every Cn there exists a Cn+l ~ 0.

The linear space

spanned by ¢o'¢i''¢n ""

we call

~,i.e.

is the set of all vectors

c~'n'(

= ~ ] n=0 ~I, . , where ( n )

Cn

g B and m is a natural number which is arbitrarily

large but finite, We cal I R. the space spanned by ¢i' i,e. l

Rl" = {e¢iIse~}'(R'l are the energy eigenspaces), Using R i instead of ¢i we have a formulation applies when R i is not one dimensional.

which immediately

Then ~ is the algebraic

direct sum of the R.

1

:

R

a{g. i

In other words Y is the set of all sequences ¢:(¢o,¢i,¢2,..¢m,000...) where the algebraic

with ¢i ~ Ri

operations

of the linear space are

defined by

¢ + @ : (¢o + @o' ¢i + ~I' ~¢ = (~¢o' ~¢I

""

)

"'" )

13 The scalar product

is

m (#,~)

=

Z

(%i,~i)

i=o

And the n o r m is

ii ii2

m =

Z (Cirri) i=O

As the sum goes only up to an arbitrarily n u m b e r m the question (~i,¢i)

of convergence

large but finite

of the sum of numbers

does not arise.

is a linear now equip

space without

~ with a topology,

topological

space.

any t o p o l o g i c a l

structure.

i.e. we construct

a linear

We

II,

The Topological

Structure

A set ~ is a linear axiom of countability I) ~ is a linear

topological

of States

space(in

which the first

is fulfilled)' iff

space

II) For a sequence element

of The Space

of elements

is defined

of a convergent

of ~ the notion

(the limit

sequence

is unique

converges

of a limit

and a subsequence

to the same

limit

point) III) The algebraic i.e.

l)a)

operations

of the

linear

space

If # ~ ~n ÷ ~ ~ ~ then also ~ n

÷ ~¢

b) If ~ ~ a n ÷ ~g~ then also en~ ÷ ~

2)

is not the most

on also use spaces

bility

~g#

for

definition

of a linear topological

for our present

purpose

in w h i c h the first

(we will

axiom of counta-

is not fulfilled).

In a given in various

ways

linear leading

All the t o p o l o g i c a l edness,

for every

E ~; n : 1,2, ....

general

space but it is sufficient later

for e a

If ¢n+¢ and ~n ÷ ~ then Cn + ~n ÷ ¢+~

Cn,~n,¢,~ This

are continuous

closure,

space the convergence to various

notions

linear

topological

like continuity,

completeness...depend

can be defined spaces.

denseness,

bound-

then on the definition

of convergence. We shall

introduce

into ~ three

different

topologies.

IS

As a first example Hilbert

space

fulfills

we introduce

topology

T H (ane

the conditions

la)

into

~ the w e l l known

can check that this really

ib) and 2) of the definition. )

Def: 14) (i) qbT

÷ ~ fom Y ÷ ~ To undemstand

÷÷

the melation

space H let us intmoduce Def:

A sequence

sequence

w i t h respect

(2) an N such that Def:

A linear

every

Cauchy

an e l e m e n t

notion:

in

Space)

is called a Cauchy

to THif fore evemy

C > 0 there exists

l lXn - Xmll

< g for every m,n

> N.

topological

space

complete

sequence

R is called

{x } has a limit element n

all

is not complete

x which

limit elements

it can be completed

of Cauchy

It is easy to see that ~ is not Let us consider

the infinite

sequences TH-complete:

sequence

(3) h = (hl~ h 2, h 3 ... h i ...) with h i e R i h. ~ 0 for all i l 2

i=0

if

of the space R.

If a space adjoining

the following

(Nommed

and the H i l b e r t

between

{x n} of elements

a space with a n o r m

for y-~.

I I~y - ~II +0

by

to it.

is

16

F r o m this

follows

llh n II ÷ 0

z

llhiil

for n

2 ÷ o

÷

~

and

for n +

i=n+l

h is n o t an e l e m e n t Let us

of ~, b e c a u s e

consider SI,

it is an i n f i n i t e

sequence.

the s e q u e n c e

$2, S 3 ....

with

Si £

where

4) Sn = (hl, h 2,

As

S

n

- S

m

... hn,

= (0,0,

O, 0 .... )

... 0~ hn+l~

hn+ 2

..... hm,

00

..... )

we h a v e

IlSn-Sml i.e.

I = Ilhn+z+hn+2+hmll

{S } is a

÷ 0 for

n~

Cauchy sequence.

n

The T H - l i m i t

element

of S n is h:

TH S

because

n

÷h

I IS n - hl 12 =

I lhi

Z i:n+l

112

÷ 0 for n + ~

i

B u t h ~ •. If we a d j o i n

to

• all the

limit e l e m e n t s

of C a u e h y

sequences

17

i.e.

take

in addition

to the elements

= ( ¢ I ~ 2 ' .... ~n 0 0 0 .... )~ n a r b i t r a r i l y elements

h of the k i n d defined

in

(3),

large,

then we obtain

the

a com-

plete notched space in w h i c h the n o r m is given by the scalar product; space

this

is the H i l b e r t

H is the completion

defined by Def.

m)

space

H.

Thus ~ the H i l b e r t

of~ with ~espect

To The t o p o l o g y

(I).

A subspace

S of a complete

topological

space X is called

if S O {all limit points Thus

~ is a TH-dense The space

which

fulfill

subspace

this

H =

E i=c

and is called the H i l b e r t i a n direct

is written

as

l

direct

sum or orthogonal

sum of the R.. l

Thus the H i l b e r t i a n the a l g e b r a i c

direct

T@ is defined

direct

sum is the

sum w i t h respect

We n o w introduce T@~

X.

of all h = (ho,h I ...), h i c R i

space @R.

=

of H.

{4 is the space (3)~

of S}

We take the original

of

to TH.

into T a n e w topology

in the following

completion

which we call

way:

scalar p r o d u c t

( • , • ) and define

dense

18

the following quantities (6)

(¢,~)p = (¢,(N+I)P~)

and II $I Ip Remark: N~

p = 0,i,2~

...

= /(~,~)p

Before proceeding we note some properties of the operator

N is essentially self-adjoinT.

This is easily proved

using one of the criteria of essentially self-adjoinTness: Lemma:

An operator A is e.s.a, if (A+I) -I is continuous

and has a dense domain in H. !

The spectrum of (N+l) -I is n-~' consequently it is continuous.

Its domain is ~

quently N is e.s.a..

which is dense in H.

Conse-

As a consequence N+l is e.s.a.

Further (N+l) p for every p is e.s.a. 16) From the spectrum of (N+l) p one sees immediately that (N+l) p is positive definite.

From This it is easy to see

that (¢,~)p for every p fulfills the condition of a scalar product and

tloi 1t, Ill! tl, 1125 Remark:

.

.

.

Further, these norms are compaTible~ i.e., if a sequence

converges with respect to one norm and is a Cauchy sequence with respect to a/%other~ then it also converges with respect to This other norm. o A space in which a countable number of norms

(scalar-

19

products) are defined is called a countably normed 8~alar product) We now define

(countably

space.

T~ by:

Def: T~ (7) C y ÷ #

÷÷

lie T - ~II ÷ 0 P

From this definition (8)

•~ From Cy ÷ 0

Therefore

one immediately

(coarser)

A sequence

p and for every

sees

TH follows ~y + 0

T¢ is called stronger

called w e a k e r

Def:

÷

for every p.

than

but not vice versa.

(finer) than

TH, and

TH is

T#.

{x } is called a T¢-Cauchy sequence if for every Y e > 0 there exists an N = N(c~p) such that

x II

for every

>N.

P There are more quences because

TH-Cauchy sequences

for x H - Cauchy

than T#-Cauchy se-

sequences

(9) needs to be

fulfilled only for p = 0. We e o m p l e t 6 ~ limit

with respect to T~, i.e. adjoin to ~ the

points of all r#-Cauchy sequences.

cal space that we obtain by completing call 4. Then C ~,

• is ~ - d e n s e

in ~.

The linear topologi-

~ with respect to T# we

20

¢ is called a countably Hilbert are more TH-Cauchy sequences

space.

than T¢-Cauchy

As there

sequences

it

must be that (10) Thus

~ C ~ C H. ~ is T¢~dense in ¢ is

and

TH-dense in H

# is TH~ dense in H,

as already ~ is TH- dense in H •

To get a feeling for ¢ let us see which of the infinite quences

(3) that are elements

that h : (ho,hl,..) with

of H are also elements of ¢.

~ H be a limit point of a T¢-Cauchy

Sn = (h I h 2 ... hn 000)

seIn orde~

sequence

s

i.e. in order that: 3¢ S

n

÷h

one must have IIS

(Ii)

n

- hll ÷ 0 P

for every p , i.e.

((Sn-h),(N+i)P(Sn-h))

÷ 0

for every p,

i.e.~ one must have Z ( ~ ( N + I ) p h i ) = Z (i+l)Pllhill 2 ÷ 0 i=n+l i=n+l Therefore,

only those h a H are ~ -limit points of T¢-Cauchy

quences of elements

of ~ ~ t h a t

0 there exists an ( q ~ ) ( £ )

such that

(for every

IF($){

< £ for all

withl l¢l lq The set of antilinear functionals

on a space ~ may be

added and m u l t i p l i e d by numbers according to

: ~÷S (s) (~FI+~F 2) (¢) = eFI(¢)+SF2(¢)

The functional ~FI+SF 2 defined by (3) is again a lineam functional. Thus the set of a n T i l i n e a r forms a linear space.

functionals

on a linear space

This space is called the algebraic

dual or algebraic conjugate

space and denoted ~X.

29

Continuous antilinear funcTionals have not only to fulfill

(i) but also (2), therefore the set of continuous

antilinear functionals, which also forms a linear space, is smallem Than The set ~X and depends --like all Topological notions -- on The particular Topology that has been chosen. We shall denote the set of all T¢- continuous by ~X and the set of all TH-continuous As every sequence

{¢y} with

functionals

functionals by H X.

#y + 0 has the property ¢y ~H0

i.e. There are more TH=convergent

sequences than T#-convergenT

sequences and ~(#v ) ÷ F(0)) = 0 must be fulfilled for all TH-convergent

sequences

if F = F(~)E H ~ and for all T~-con~

vergent sequences if F = F (¢) e CX the set of F (H) e H X will be smaller Than the set of F(@)e CX (because There are more stringent conditions on elements of H x) .

Therefore:

HX

is the smallest set and ~X is the largest set of The sets H X, cX,

~X.

In a scalam product space each vector f e ~ defines an antilinear

functional by

(5):F(¢)

def (¢,f)

It is easy to see that F defined by the vector f as given in (5) fulfills the condition diTions

(i) if (¢ ~ f) fulfills the con-

for a scalar product.

30

Further if ~y + ¢ then for every f e • (even for every f e H) (~y~f) + (~ ~f) (every strongly convergent sequence converges also weakly).

Thus the functional defined by (5)

fulfills F(~y) ÷ F(~)~ i.e. it is a continuous antillnear functional.

Thus the scalar product defines a continuous

antilinear functional. In the Hilbert space the converse is also true~ i.e. any antilinear TH-continuous

functional

can be written as

a scalar product according to the F~echet-Riesz For every antilinear

TH-Continuous

theorem:

functional F (H)

there exists a unique vector f ~ H such that (6)

=F(H)(¢)=(¢,

f)

for all

¢ e H

Therefore we can identify the Hilbert space H and its conjugate H X by equating F ~ H X with the f e N given by (6). Then with (II~9) we have the situation: (7) ~ C # C H

= HX

For TH-continuous

functiona!s

F (H) the symbols

are the same~ after the identification (9)



and (,)

F (H) = f~

: (%,f)

However for the larger class of T#-continuous F the symbol is defined~ whereas

functionals

(%,F) is not unless

F ~ ~X is already e ~X in which case we identify:

(10)

= = (~,F)

31

In ~X one can introduce

various topologies

and there-

with various meanings of convergence of countable sequences. An example is the weak convergence convergence Def:

in H) :

A sequence of functionals

converges

(in analogy to the weak

{Fy} C ~ X

(weakly) to a functional F

(Ii) + ~

iff Y + ~

for every ~ E ~.

One can show that ~X is already complete with respect to The topology TX of this convergence. Remark:

~X is a space in which The topology cannot be com-

pletely described by the description limit of countable sequences

of the passage to the

(~X does not satisfy The fimst

axiom of countability and is, therefore, a more general topological space than those defined by our definition in II). Therefore we cannot attempt to prove the following statements.283o As #X is a linear topological space we can consider the T X -contanuous antilinear functionals

~ on ~X i.e. the set

#XX of all those functions on #X which satisfy

(i x ) ~(~Fl+SF 2) = ~ ~(F I) + [ ~(F2),FI,F2e~X; or in other notation

< Fl÷B 2r;> =

~,Se~

S2 and X (2 x) ~(F 7) ÷ ~(F) for every Fy ÷ F #XX is a linear topological

space if addition

and multiplica-

tion are defined by:



36

We shall use this definition only for the case that A is e.s.a.

(often one defines the generalized eigenvector

by F(A+~)

= %F($) where A + is the adjoint operator of A),

Let us assume A has generalized eigenvectors Hilbert space i.e. F in (I) is an element of HX.

in the Then (i)

reads (2)

~A~I~

= (A¢~f)

= (¢,A+f)

= ~(~,f)

for every ~ g ~ and consequently (3) A+f=~f

and if

A is e.s.a.

(since ~ is TH-dense

in H)

A+f = Af = ~f = ~f

Thus a generalized e i g e n v e c t o r which is an element of the H i l b e r t space is an ordinary e i g e n v e c t o r to the same eigenvalue it is an e i g e n v e c t o r

corresponding

(for the case That A is not e.s.a.

of A+).

Before we formulate The nuclear

spectral theorem let

us start with a few remarks. Every self-adjoint

operator A in a finite dimensional

Hilbert space has a complete there exists an orThonormal

(4) A hcl.)=lih(l.) l l

system of eigenvectors. set of eigenveetors

Iki)(lilh)

h(l.)=ll i) l

(hck.), h(k.))=(k i II i) = gi'i l l

such That for every h e H n (5) h = Z i=l

I.e.

n = dim H

H

87

The set of all eigenvalues spectrum

A = {~l,12,...In}

of A.

For an infinite is no more true,

dimensional

Hilbert

in H (e.g.

Q).

tor A in H is defined

as the set

A-A1 has no inverse.

The spectrum

is a subset

A in the infinite

ment

carries

dimensional

i.e.

oscillator

A operator

we restrict

A is cyclic

that Akf~ k = 0~I~2~...

2h ~

generalization

There exists

~)

ourselves

to cyclic

operators.

f ~ H such

space.

span the entire statement

operator with

spectral

an orthonorma!

discrete

It Qk~ = n H.

to the ease spectrum

theorem: set of eigenvectors

h~i = l~i ) ~ H

problems

of the o n e - d i m e n s i o n a l

is a vector

any n

state-

To avoid

for the p r i n c i p a l

problem

of the above

self adjoint

given by the following

case.

P and Q are cyclic because

n

of a cyc!ie

dimensional

if there

(a + a+) k # ~ k = 0,i~2..., The

then the above

span the entire Hilbert

is clear that the operator

operator

space H has only a dis-

if A = {If,12,...}

for the special

operator

If the self-adjoint

Hilbert

that

i for w h i c h

of a self-adjoint

which are inessential

and inapplicable

Def:

statement

of an opera-

A of all those

over to the infinite

complications

harmonic

The spectrum

of the real axis 28)

spectrum,

space this

in fact we know that there are operators

have no eigenveetor

crete

is called the

is

38

Ahl. = lihl., 1 i

k i e A,

(hk-'hl.) l 3

= ~I.I. 13

such that for every h e H

h = Z h%i(h%i,h) = Y.{%i )(%i{h) i:l i

(B)

~Ad~(l)!l)(Xlh)

:

The measure one point

is concentrated

p

sets have measure one~ i,e.

The operator

H is an example

remark that all eigenvectors position

eigenvectors statement

continuous in H.

holds

D{l. } = 1 l

of such an operator.

(7) of A appear

We

in the decom-

spectrum,

However

then there are no such

a generalization

of the above

in the form of the following N u c l e a ~

Spectral

29)

Let @ C H C e.s.a.

I. e A and l

(8),

If A has

Theorem:

on the eigenvalue

@)< be a rigged H i ! b e r t

T@-continuous

generalized

operator.

space and A a cyclic

Then there exists

a set of

eigenvectors

(9) (lO)

A×IX> =

XlX>,~eA

i.e. :

: k

for every ¢ e such that for every fined positive

~,~ e @ (~ e ~XX) and some uniquely

measure

~ on A

de-

39

(ll)

(~,,~) = /du(X) A

= /dv(X)<XI~> A

which is written formally

A

If we set ~ = ~ in (ii) then we see that from the vanishing of all components

of the spectral d e c o m p o s i t i o n

of

c o r r e s p o n d i n g to the operator A, it follows that II~II = 0 i.e. that ~ = 0.

Because of this property the set of gen-

eralized

eigenvectors

complete

(in analogy to the completeness

ordinary eigenvectors

II > occurring

in (Ii) or (12) is called of the system of

in (8)).

This spectral theorem assures the existence of a complete set of generalized eigenvectors

of the operator Q

(and P), i.e. of generalized vectors that fulfill: (13)

QXIx>

=

xlx>,

p X I p > = PlP>

and with the help of which every ~ e # can be written (14)

~ = / d~(X) x

Ix><xI¢ >,

¢

= fd~(p) Ip> spectrum P

where X =spectrum Q. The fact that there is a complete does not tell us what this spectrum is.

set of eigenveetors

40 It is widely known that the spectrum of Q and P is the real line.

However it is not so widely known that the deri-

vation 30) of this is far from trivial.

The reason for This

is that the problem of the physicist was the reverse of the problem described hez'e, namely to find the defining assumptions (I~l) and (I~3)~ from the spectrum of Q whioh was assumed to be the real line.

(of. remark on p. 49).

We will in The following only describe the results, the derivation of which will be given in the Appendix V. The set of generalized eigenvalues of Q and P is the complex plane.

The spectrum of Q and P is the following

subset of The set of generalized eigenvalues. spectrum Q = {xI-~<x

: ~(x'-x)

we have

(l~)

~ = { dxlx><xl,> --OO

41 Every vector ~ e ~ is fully characterized by its components

<xl~> with respect to the basis of g e n e r a l i z e d eigen-

vectors

Ix> (in the same way as every vector x in the 3-dimen-

-~

o

sional space can be equivalently

described by its components 3 i ÷ ÷ i ÷ x with respect to 3 basis vectors e i according to x = 7~ x e i i=l or a vector h ~ H by its components (hi~h) with respect to a complete basis

system h i i=i~2,...).

is called a realization tions

The function

<xl~>

of the vector ~ and the set of func-

<xI~> for every $ e ~ is called a r e a l i z a t i o n

of The

space ~. The realization differentiable

of ~ is the space S of all infinitely

functions

$(x) = <x l ~ w h i e h ,

their derivatives

Tend to zero for

any power of i~ I

( S chwar~z Spa ce).

IxI~

together with

more rapidly than

In this realization of ~ by S the o p e r a t o r Q is r e a l i z e d by the m u l t i p l i c a t i o n with x and the o p e r a t o r P is realized by

the d i f f e r e n t i a l

operator in S:

<xlQl*>

= x <xl*>

<xIPf*>

= ~_a

(17)

The elements polynomials

<xJ,>

of the algebra A are then r e a l i z e d by

d in x and ~

.

We can, according to (III,!2)

consider ~ in (14) or

42

(ll) as the functional on the space #X at the element F e ~X:

: : and write



= Idx <xl;> The l.h.s, of (187 is well defined and, consequently, the r.h.s.

so is

However one factor in the product of the inte-

grand~ has not been defined so far.

We define by

(!8)~