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The Weyl functional calculus Cressman, Ross Eric 1974

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THE WEYL FUNCTIONAL CALCULUS by ROSS ERIC CRESSMAN B.Sc. (Hon.), University of Toronto, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1974 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a i ABSTRACT The Weyl f u n c t i o n a l calculus for a family of n s e l f - a d j o i n t oper-ators acting on a H i l b e r t space provides a map from spaces of functions on R n into the set of bounded operators. The calculus i s not m u l t i p l i c a t i v e under point-wise m u l t i p l i c a t i o n of functions unless the s e l f - a d j o i n t operators commute. However, i f the operators happen to generate a strongly continuous unitary representation of a L i e group, we can hope to define a "skew product" on the function spaces under which the calculus i s m u l t i p l i c a t i v e . In part I, we show that, f o r exponential groups, a natural skew product e x i s t s by using the exponential map to p u l l the convolution on the group back to the L i e algebra. Moreover, whenever a skew product i s defined i n part I, i t depends only on the underlying L i e group and not on the p a r t i c -u l a r representation. We then examine when the skew product of two functions i s again i n the o r i g i n a l function space. For compact L i e groups, the theory becomes more complex. A skew product i s constucted but by a rather a r t i f i c i a l method. The e x p l i c i t c a l c u l a t i o n s f o r SU(2) demonstrates the d i f f i c u l t i e s . In part I I , a unique skew product i s developed for the p o s i t i o n and momentum operators of one dimensional quantum mechanics. The dynamics of quantum mechanics on phase space can be formulated through t h i s skew product whenever the underlying Hamiltonian corresponds to a tempered d i s t r i b u t i o n on the plane. The r e s u l t i n g evolution operator on phase space i s shown to be equivalent to the d i f f e r e n c e of two " s i n g u l a r " i n t e g r a l operators obtained from the usual configuration space formulation. The evolution and configuration operators are then bounded with appropriate domains for the same set of tempered d i s t r i b u t i o n s . The skew product on t h i s set of d i s t r i b u t i o n s i s i i used to define noncommutative Banach algebras and to determine the m u l t i p l i e r s on these spaces. For r e a l , compactly supported d i s t r i b u t i o n s , i t i s shown that the phase space formulation has a unique s o l u t i o n i f and only i f there i s a unique s o l u t i o n on configuration space. On the other hand, we observe that the symmetries of the evolution operator seem to imply that the two formulations of quantum mechanics are not equivalent for a l l r e a l tempered d i s t r i b u t i o n s . i i i TABLE OF CONTENTS Page INTRODUCTION 1 PART I THE MULTIPLICATIVE STRUCTURE Chapter One Preliminaries 5 Chapter Two Exponential L i e Groups 7 Chapter Three Compact L i e Groups 16 Chapter Four SU(2) 22 PART II THE EVOLUTION EQUATION Chapter Five Preliminaries and Generalizations 33 Chapter S'ix The Weyl Correspondence and Evolution Operator 36 Chapter Seven Bounded Operators 42 Chapter Eight M u l t i p l i e r s 49 Chapter Nine Real Tempered D i s t r i b u t i o n s 55 BIBLIOGRAPHY 66 ACKNOWLEDGEMENTS I wish to express my appreciation to my supervisor, Dr. R.F.V. Anderson, for his o r i g i n a l suggestion of my thesis topic as we l l as for his p a r t i c i p a t i o n i n many stimulating discussions during the preparation of t h i s t h e s i s . I would also l i k e to mention that the foundation of the f u n c t i o n a l calculus used i n the thesis was developed by Dr. Anderson i n h i s own research. The f i n a n c i a l support of the Univ e r s i t y of B r i t i s h Columbia and the National Research Council of Canada i s g r a t e f u l l y acknowledged. \ 1 INTRODUCTION The usual von Neumann fu n c t i o n a l calculus for a s i n g l e s e l f - a d j o i n t operator assigns, through the s p e c t r a l r e s o l u t i o n , an operator to every Borel-measurable complex-valued function defined on the r e a l l i n e . R. Anderson [1] has extended this to the Weyl f u n c t i o n a l calculus for c e r t a i n f a m i l i e s of s e l f - a d j o i n t operators and functions on R n. An n-tuple of s e l f - a d j o i n t operators A= (A^,...,A n) on a H i l b e r t space H i s c a l l e d a s e l f - a d j o i n t  n-tuple i f ; when the operators are r e s t r i c t e d to t h e i r common domain, any r e a l l i n e a r combination pf them i s e s s e n t i a l l y s e l f - a d j o i n t . For any s e l f - a d j o i n t n-tuple A, the Weyl fu n c t i o n a l calculus T(A) maps various subspaces S of functions on R n into the set of bounded operators B(H) on the H i l b e r t space. A m u l t i p l i c a t i v e structure on S w i l l mean a map S x S -»• S given by (f,g) ->• f * g # that s a t i s f i e s T(A) f * g = T(A)f T(A)g. The function f * g i s c a l l e d the skew product of f and g. This m u l t i p l i c a t i v e structure for s p e c i a l n-tuples plays an important r o l e i n phase space quantum mechanics as i s explained l a t e r i n the introduction. Our purpose i s to define and study the skew product f o r various s e l f - a d j o i n t n-tuples. The thesis i s divided, into two parts. The f i r s t examines necessary and s u f f i c i e n t conditions on the s e l f - a d j o i n t n-tuple f o r a skew product to e x i s t on various spaces of functions. The second studies an evolution equation that a r i s e s n a t u r a l l y from the skew product for one of our s p e c i a l n-tuples. In part I, we w i l l always assume that the s e l f - a d j o i n t n-tuple A comes from the set of generators of a strongly continuous unitary representa-t i o n of a r e a l , connected L i e group G. In f a c t , i A = ( i A ^ , . . . , i A n ) w i l l be a representation of the L i e algebra T of G. 2 If G i s n i l p o t e n t , a continuous skew product has already been developed i n [2] for these n-tuples on the Schwartz class S(R n). Unfortun-ate l y , t h i s r e s u l t cannot be extended to other groups for the space of functions S(R n). Hence, we consider the other three spaces of Fourier transforms that are introduced i n Chapter One (namely; F ( L 1 ( R n ) ) , F ( C o ( R n ) ) , CO ^ and F(C Q(R )) ). In Chapter Two, we show that for exponential groups there — . . . . i s a unique continuous skew product on these spaces for the n-tuple associated to the l e f t regular representation. Moreover, t h i s skew product holds for any strongly continuous unitary representation of the exponential group. The example at the end of the chapter demonstrates that G must be nilpotent i n order to expect a skew product on S(R n). In Chapters Three and Four, we attempt to carry the program of Chapter Two oyer to compact groups. Again, there i s a continuous skew product on S =F(L^(R n)) that holds for a l l our representations but i t i s no longer unique. As the exponential map i s no longer a diffeomorphism, the existence of a skew product for general compact groups on the Fourier transform of a space of continuous functions seems p a r t i c u l a r l y d i f f i c u l t to e s t a b l i s h . However, with considerable e f f o r t , a skew product i s produced for 5U(2). In part I I , we w i l l be dealing p r i m a r i l y with the p a i r of s e l f -2 adjoint operators (Q,cP) on L (R) that denote m u l t i p l i c a t i o n by x and the d i f f e r e n t i a l operator ~ i c ^ " r e s p e c t i v e l y . The constant c i s always a p o s i t i v e number. With c thought of as Planck's constant and h i n S(R n), T(Q,cP)h i s nothing but the quantization procedure suggested by Weyl [18 ; page 275] that i n t e r p r e t s c l a s s i c a l q uantities on phase space as quantum 2 mechanical operators (henceforth c a l l e d Weyl operators A c(h) ) on L (R). 2 A unique continuous skew product * i s provided on the Schwartz class 5(R ) i n Chapter Six. If h i s regarded as a f i x e d Hamiltonian, the evolution equation on phase space ^ ~ - = i ( h * f - f * h ) i s equivalent to the Schrodinger equation on configuration space as explained i n [2]. However, most i n t e r e s t i n g Hamiltonians dp not come from functions 2 i n S(R ). Fortunately, the above paragraph can be reasonably extended to 2 ..define a Weyl operator A £ on S(R) and an evolution operator on S(R ) 2 for any h i n the set of tempered d i s t r i b u t i o n s S'(R ). The greater portion of part II studies the equivalence of the r e s u l t i n g evolution equation | ^ = H f and the Weyl equation ^ - = i A <f>. In Chapter Six, our main theorem shows that the evolution operator i s equivalent to the di f f e r e n c e of two kernel operators on the plane. These kernel operators are i n t r i n s i c a l l y r e l a t e d to the Weyl operator of the o r i g i n a l d i s t r i b u t i o n . 2 2 In Chapter Seven, H and A are regarded as operators on L (R ) c c 2 and L (R) re s p e c t i v e l y . From t h i s point of view, H i s bounded with 2 domain S(R ) i f and only i f A i s bounded with domain S(R). As any bounded 2 operator on L (R) i s an extension of a Weyl operator, the above set of 2 d i s t r i b u t i o n s i s quite a large subset of S'(R ) and, i n f a c t , includes L 2 ( R 2 ) and FO^CR 2)). 2 2 The m u l t i p l i c a t i v e structure on the Banach spaces L (R ) and 1 2 F(L (R )) defined through the bounded Weyl operators of Chapter Seven pro-duces two noncommutative Banach algebras. In Chapter Eight, we study the 2 2 m u l t i p l i e r s on these spaces. The m u l t i p l i e r s on L (R ) simply correspond to the d i s t r i b u t i o n s with bounded Weyl operators. Furthermore, a v a r i a t i o n of 1 2 Wendel's Theorem [6], proves the m u l t i p l i e r s on F(L (R )) come from the Fourier transform of f i n i t e Radon measures. In the l a s t chapter, we return to the viewpoint of Chapter Seven to 4 study the r e l a t i o n between H and A for r e a l tempered d i s t r i b u t i o n s . c c Most Schrodinger operators of a one dimensional p a r t i c l e are extensions of these Weyl operators. One important r e s u l t of Chapter Nine states that H c has dense domain i f and only i f A has dense domain. Moreover for d i s t r i b u -c tions with compact support (or whose Fourier transform has compact support), ^he—evolution equation has a unique s o l u t i o n i f and only i f the Weyl equation has a unique s o l u t i o n . In other words, the two equations are equivalent i n t h i s case. F i n a l l y , the geometric i n t u i t i o n present i n phase space i s used to suggest that they are not always equivalent. 5 PART I THE MULTIPLICATIVE STRUCTURE CHAPTER ONE PRELIMINARIES The functional analysis notation used i n t h i s thesis i s as i n K. Yosida [19] unless otherwise s p e c i f i e d . The following four spaces of functions with the given topology w i l l be considered throughout. DEFINITION 1.1: Lzt F be thz TouhleA tha.ns{ohm dz{lnzd on L 1 ( R n ) accondlng to thz {onmula n e 1 5 ' X f ( x ) d x {OK ZVZAy f E L ^ R " ) . Ff (O = (2TT) N / 2 This opeAaton Induces a topology on thz {unction spaces bzlow. 1. T0}(^))'= {Ff : f e L 1 ( R n ) } li> a Banach spacz with nonm ||Ff || - ||f || 1 2. F ( C Q ( R n ) ) ={Ff : f i s a continuous function with compact support}. C Q(R n) has tliz topology o{ unl{onm convzngzncz on compact subsets. 3. F(C^(R n)) ={Ff : f i s a C° function with compact support}. Thz topology Is Induced. {nom C^(R n) consldzuzd as thz locally convex lineax. topolo-gical spacz o{ test {unctions {on. distributions on R n. 4. S(R ) Is thz spacz o{ njxpldly dzcAzaslng {unctions with usual topology. Thz Tounlzn tnans{onm li a homzomonphlsm on this spacz [19 ; chapter VI]. DEFINITION 1.2: (The Weyl Functional Calculus) SuppoSZ A = (A ,...,An) Is a Szl{-adj'olnt n-tuplz o{ opzhatons on H. let S bz onz o{ thz {oun. abovz spaces. Let e - ± ^ * A bz thz unltaAy opznaton associated to thz esszntially szl{-adjolnt opznaton -(t^A^ + . . . + S nA n) through Stone's Theorem. Then T(A) : S—>B(H) .is dz{lnzd by (1.1) T(A)f=(27r) -n/2 Ff(£) e ~ i ? ' A d 5 &0K 2V2JUJ f e S, u)heAz the. Zntngtial on the. fvight AJ> tka. BochneA JLnX.ZQh.oJL [19 ; page 132]. For part I, l e t us assume that i A = (LA,,...,1A ) i s a set of I n generators for a strongly continuous unitary representation U (henceforth c a l l e d a representation U) on a r e a l , connected Lie group G. Without loss of generality, G i s simply connected because the representation can be l i f t e d to the u n i v e r s a l covering group of G without a f f e c t i n g the generators. Formula (1.1) may be rewritten i n terms of t h i s representation. If exp denotes the exponential map from the L i e algebra r to G, then there i s a i f • A basis for V corresponding to A such that U(exp?) =e . With t h i s basis (1.2) T(A)f=(2Tr) -n/2 R n =r F f ( 0 U(exp ( - C ) ) dt. We w i l l be p a r t i c u l a r l y i n t e r e s t e d i n the l e f t regular representation R on G. This i s constructed through l e f t t r a n s l a t i o n on the group. If u i s 2 2 a l e f t invariant Haar measure on G, l e t R(a) : L ( G , u ) —>L (G,y) by (1.3) ( R ( aH ) (p) = ij; ( a _ 1 p ) for ^ e L 2 ( G , y ) and a,peG. This defines a representation on G that i s generated by n l e f t i nvariant 2 vector f i e l d s (iA^, , . .. ,iA^) of skew-adjoint operators on L (G,y). For a l l groups considered i n t h i s t h e s i s , the skew product for the generators of any representation on G, w i l l depend only on the l e f t regular representation. CHAPTER TWO EXPONENTIAL LIE GROUPS DEFINITION 2.1: A Lte. gn.oup G called exponential the exponential map ektahllAheb an analytic dl^eomonplvum o{, V onto G. Exponential groups have been examined by a number of authors (eg. [4] and [13]). This class of Lie groups may be studied through their Lie algebras as there is a purely algebraic criterion on the Lie algebra to determine i f the group is exponential. In particular, a l l connected, simply connected nilpotent Lie groups are exponential while a l l exponential groups are solvable - each inclusion being proper. The following lemma is needed to study the l e f t regular representa-tion on these groups. The result should be obvious for those familiar with the theory of Lie groups [5 ; page 364]. LEMMA 2.2: 1^ G ti an exponential Qiowp, the. measuAe tnduced on G by LebeAgue measwie on r thn.ou.gh the. exponential map t6 o& the. ^onm x(a)dy(a) mhzn.lL 1(a) U> a positive analytic function. 1^ G Jj> nllpotent, then E IA a constant. PROOF: Let us construct the l e f t invariant Haar measure u on G. The l e f t Invariant vector fields iA^,...,±A of the l e f t regular representation form a basis for the tangent space at each point of G. Define n 1-forms pn G by co (iA.) =6, . for l<k,j <n. These are clearly l e f t invariant and the form J K J - -co = co"'" A . . . A to11 i s a l e f t invariant n-form that is non-degenerate at each point of G. By using exponential coordinates with respect to our chosen basis 8 f o r G, cu can be pul l e d back to a non-degenerate n-form v on T. But the n-form dx^ A ... A dx^ corresponding to Lebesgue measure i s also non-degenerate and so v=A(x, ,...,x )dx, A . . . A dx where A : Rn—>-R i s non-zero. By the 1 n 1 n choice of coordinates A i s obviously a n a l y t i c and A ( 0 , . . . , 0 ) =1. Thus A i s a p o s i t i v e a n a l y t i c function. ^ - i - - - - - ' Define the measure u for f eC Q(G) by u ( f ) = f (exp(x)) A(x) dx. r I t i s easy to check u Is a p o s i t i v e measure on G that i s l e f t i n v a r i a n t since i t corresponds to the n-form c u . The d e f i n i t i o n of u states that u induces the measure A(x)dx on T. A l t e r n a t i v e l y , Lebesgue measure induces the measure E ( a ) d u ( a ) on G where E ( a ) = 1/A(exp " ^ a ) . Therefore E has the desired properties. If *G i s ni l p o t e n t , an appropriate basis for F may be chosen using the Campbell-Baker-Hausdorff Theorem so that exp 1{exp(C 1,... ,C n) exp(n 1,...,n )} = C^ + r^, £ 2 + n 2 + a polynomial i n ( t ^ . r ^ ) , 5 N + 1 n + a polynomial i n (Sp.n^, . . . »? n_i> n n_i' ) • With these coordinates, Lebesgue measure i s c l e a r l y l e f t i n v a r i a n t and t h i s insures that E i s a constant. THEOREM 2.3: T<J A <u> the. beJL{-adjoint n~tuple. associated to the. le.it  tie.gu.laA. tizpttesentatlon o& an exponential, gioup G,the.n theAt lb a unique, continuous Akeio pioduct on S = F ( L 1 ( R n ) ) , S=F ( c (R n)) and S = F(C°°(R n)). O 0 In {act, l{ r = R n hoi, the. basis cotftzsponding to A, then F ( f * g) = K {on. f, g e s wheAz K ( Q - ( 2 T T ) _ n / 2 f F f ( e x p - 1 ( e x p ( - r 1 ) e x P 5 ) ) F g ( T i ) L ^ ( ~ ^ 1 ^ . J p j i L{ exp a,; PROOF: We w i l l determine the skew product on F ( L 1 ( R N ) ) . Define a new function on the group for any f E F ( L 1 ( R 1 1 ) ) as follows f(a) = Ff(exp ""a)-11(a). By formula (1.2), we have for every I | I E L (G,y) (T(A)f) (ty) = (2ir) -n/2 R' n F f ( 0 R(exp(-5 ) ) i J ; de; = (2ir) -n/2 Ff(exp 1a) R(a  1)ty E(a)dy(a) = (2TT) -n/2 f ( a ) R(tr )^  du(a), Since f eL 1(G,y), the l a s t i n t e g r a l i s defined as a convolution on G. of an 1 2 L function and an L function. Let f and g be two a r b i t r a r y functions i n F ( L 1 ( R n ) ) . Then (T(A)f T(A)g)ty = (2TT) -n/2 f(a) R(a  X) 4 ( 2 7 r ) ~ n / 2 G g(d) R ( 5 1 ) 4 » dy(<5) V dy(a) = (2TT) -n G J f(a) g ( 6 ) R(a 16 ^ dy(6) dy(a) Interchange the order of i n t e g r a t i o n (legitimate since these are convolutions) and replace a by 6 ( r e c a l l : y i s l e f t i nvariant) = (2TT) -n f ( 5 1a) g(6) R(a ^ty du(a) du(6) GxG - (2ir) -n/2 (2TT) -n/2 f (fi 1a) g ( 6 ) dy(6) I R(a dy(a) 10 { (2ir) n / 2 i * f ( o ) 1/Z'a) } R(o I ) ^ Z(a)du(cr) G G where * means the usual convolution on a group [6] G* K(£) R(exp ( - 5 ) ) * de: R n where K i s the function K ( 0 = ( 2 , ) - n / 2 ggfCexpe:) - ( 2 . ) " n / 2 R n Ff (exp- 1(exp ( - n)expe:))Fg ( n ) ^ f ^ p 5 1 ^ K i s a function i n L"*"(Rn) because g * f e (G,y) and t h e i r norms s a t i s f y the equality ||K|| = (2T T ) ~ 11 g * f j| ^  . This l a s t equation also shows that the skew product i s continuous with respect to the induced topology on F(L (R n)) when f * g i s set equal to F K. Furthermore, the continuity of the skew product on F ( C Q ( R n ) ) and F(C™(R n)) i s a d i r e c t consequence of 00 the c o n t i n u i t y of the convolution on G for C Q ( G ) and C o(G). The skew product for a l l these spaces i s unique since T(A)f i s 2 e s s e n t i a l l y a convolution operator on L (G,y) with kernel uniquely deter-mined by f. From the proof, i t i s seen that the skew product for exponential groups i s r e a l l y convolution on the group p u l l e d back to the L i e algebra. In order to generalize t h i s method to other groups, i t seems e s s e n t i a l that the exponential map i s onto the group. In the next chapter we look at another case of s u r j e c t i v e exponentials; namely, the compact groups. The space 5(R n) i s conspicuous by i t s absence i n Theorem 2.3. 11 We w i l l see i n part II that a skew product on S(R n) should e x i s t to study the m u l t i p l i c a t i v e structure further. For nil p o t e n t L i e groups, t h i s i s indeed the case (Corollary 2.4). Unfortunately, even for the simplest exponential group that i s not irilpotent we have no skew product on the Schwartz cla s s (Theorem 2.7). ' ' C O R O L L A R Y 2.4: IjJ G is nilpotent In addition to the hypothesis o{ The.on.em 2..3, then theJte Is a unique, continuous skew product on s=S(R n) {on. the le{t n.egulan nepnesentation. PROOF: See [2 ; section 2]. Remark: In the s p e c i a l case of abelian groups, the skew product i s exactly what i s expected - i t i s simply the point-wise product of functions (that i s f *g(£) =f(£)g(£)). The s e l f - a d j o i n t n-tuple i s composed of operators whose commuting s p e c t r a l f a m i l i e s enable the von Neumann f u n c t i o n a l calculus to be immediately extended. The l e f t regular representation plays such an important r o l e because of the following. THEOREM 2.5: let G be an exponential gnoup. The skew pnoduct o{ Theonem 2.3 holds {on the Weyl calculus o{ the geneXatons o{ any hepnesenta.tlon o{ G. PROOF: If iA = (iA , . . . , i A n ) . . are'the generators of the representation U, then there i s a basis of T such that (1.2) holds. As i n the proof of Theorem 2.3, we have T(A)f T(A)g = (2ir) n Ff(C) Fg(n) U(exp(-5)exp(-n)) dndt; R nxR n 12 (2,rn f(a) g(6) U(cr 16 1 ) dy(6)dy(a) GxG ( 2 T T ) - n / 2 ( 2 7 r ) - n / 2 ( 2 T T ) - n / 2 f(6 Xa) ~g(a) dy(<5) U(a 1 ) dy(a) R n F ( f * g ) a ) U ( e x p ( - t : ) ) d£. where f * g i s given i n Theorem 2.3. The r e s t of t h i s chapter c a l c u l a t e s the skew product for one s p e c i f i c exponential group i n order to demonstrate that there i s more to th i s structure than i s f i r s t apparent. EXAMPLE: Suppose G i s the L i e group c o n s i s t i n g of points on the plane together with the m u l t i p l i c a t i o n (t,x)(s,y) = (t + s, x + e~y) . Let us cal c u l a t e the* exponential map on G. We must determine the d i f f e r e n t i a b l e one parameter subgroup Y : R —>• G given by y ( s ) = (<x(s),B(s)) that s a t i s f i e s the three conditions i ) Y ( 0 ) = (0,0) i i ) Y ' ( 0 ) = ( t Q , x o ) i i i ) Y ( s + t ) = Y ( s ) Y ( t ) . When we f i n d such a curve, then e x p ( t Q , x o ) =Y(1)? Writing these conditions out i n terms of a and 3 , we r e a l l y have two d i f f e r e n t i a l equations whose solutions are (st ) , t , e o X e~"*x a(s) = t s and g(s) = — — — x where — 7 — = 1 at t=0. o t o t o e — 1 Therefore, exp(t,x) = ( t , — - — x ) . The exponential obviously establishes an a n a l y t i c diffeomorphism 2 between T=R and G. With the coordinate system on the L i e algebra given 13 by the basis {(1,0) , (0,1)}, one can show the m u l t i p l i c a t i o n on T comes from the bracket [(1,0) , (0,1)] = (0,1). The o r i g i n a l coordinates on G are not exponential coordinates. In order to f i n d the skew product according to Theorem 2.3, the following r e s u l t s are required. LEMMA 2.6: let G be the above gfioup and let Y have the Indicated baA-u. t s Then a) exp (exp (t,x)exp (s,y) ) = (t + s y~~^—i~—x + e t ( e s )y } ) e -1 l - e ~ S and b) dy(s,y) = — — ds dy ti, Uaah. meaMiAe In exponential coofidlnateA. 1- ~ s (that IA, A(s,y) = — ~ wheAe A ti> a-t, In the ph-oofa oi lemma 2.2) PROOF: a) On the one hand, by d e f i n i t i o n e ' - l e S - l exp(t,x)exp(s,y) = ( t , — — x ) ( s , — — y ) 9 . , e -1 , t ,e -l s s = (t + s,——-x+ e ( — — ) y ) . On the other hand, e x p ( t + s > _ ^ _ { e ^ l x + e t ( e ^ l ) y } ) = ( t + s > e ^ l x + e t ( e ^ l ) y } > e -1 b) To show y i s l e f t i n v a r i a n t , the equation below must be v e r i f i e d when exponential coordinates are used i n the i n t e g r a l s . -s _ f ((t,x)(s,y)) i - ^ - d s d y R^ 3 By part a), we have 1 - ~ S f ( ( t , x ) ( s , y ) ) ^ f - d s d y = ^R -s . , f(s,y) i - ^ - d s d y for a l l f eC (R 2), R Z 3 t s —s f ( t + s , - - ^ — { — — x+e (~7T-)y } ) — dsdy R e ' -1 14 f ( s s { e ^ x + e t ( e T ^ l ) y } ) k £ L d a d y R e -1 Interchange the order of i n t e g r a t i o n and change the v a r i a b l e y to obtain s 1 i t-s r / \ e -1 - t s-t 1-e , , , f ( B . y ) — e - — - ds dy R e -1 s . ^ -s , s tN ct \ e ~1 -t e (e -e ) , , n 2 f ( s - y ) V e -t, s t T d s d y R e (e -e ) l - e _ s , f (s,y) — — ds dy. R 2 3 THEOREM 2.7: li G <t6 .trie above gn.oup and A -ci -trie i>eJLi-ad joint n-tuple. oAAociated to the, le.it n.e.guloji n.epn.eJ>entation with the. Indicated ba&iA ion. r , 2 then S=5(R ) f ioi no ifeew pnodact. 2 PROOF: Assume that a skew product does e x i s t . For f,g e 5(R ), the function F(f *g) has the following form according to Theorem 2.3 and Lemma 2.6. F ( f * g ) ( t , x ) - £ f 2 F f ( t - s ) - ^ { V i y + e - S ( 4 : i x ) }) Fg(s,y) ^ ds dy . J R e -1 1-e It i s apparent that F ( f *g) i s defined everywhere and i s i n f a c t a continuous s—t f unction (since the factor (t-s)/(1-e ) i s a polynomially bounded a n a l y t i c function of s ) . As there i s a skew product, F ( f *g) as defined above i s i n S(R 2). To obtain a contr a d i c t i o n , we have only to ex h i b i t two functions 7 2 f and g i n S(R ) such that F ( f *g) i s not i n S(R ). To t h i s end, suppose Ff = Qxty and F g = a * 8 (that i s , Ff(u,v) = <f>(u)iKv) etc.) where $,ip,a,.B are a l l non-negative functions i n 5(R) to be chosen below. With t h i s decomposition 15 F( f *g)(0,x) = 2TT F f (_ s >_=s__ (^ll y + e sx ) ) F g ( s , y ) — ^ dsdy R e " S - l S l - e S _1_ 2TT <K-s)a(s) R 1-e"" <K-y ^ r x ) B(y) dy \ ds. R l - e S Define ty and B so that they are non-negative functions i n S ( R ) and s a t i s f y i ) T^-(y) = 1 for | y | < l and i i ) B ( y ) E e y for y < l . Since s / ( l - e ) <0 for a l l s, we have f or any x<0 F(f *g)(0,x) > y - <j)(-s)a(s) R -s 1-e' ( l - s x / ( l - e s ) ) ( - l - s x / ( l - e S ) ) y e dy ds 1 -1 e -e 2TT <K-s)a(s) — — e s x / ( 1 e } ds. R l - e S Now define § and a so that they are non-negative, i n S ( R ) , and s a t i s f y i i i ) <f>(-s) = - s / ( l - e s ) for s>0 and iv) <x(s) E - - ^ - { - s / ( l - e S ) } for s > 0. Substituting these functions into our equation for F(f * g ) , we obtain -oo . t i v F ( f * g ) ( 0 , x ) >~ ( ^ 7 ) 2 (-•^••{-s/d-e-8)}) e ~ S x / ( 1 _ e ' ds for any x<0 ' o 1-e 1 r 2 sx , - r -s e ds 4 Jl = T { e X/x - 2e X/x 2 + 2e X/x 3 - 2/x3 } for x < 0. Obviously, F ( f *g)(0,x) does not belong to S ( R ) . Therefore, there i s no skew product on S ( R ) 16 CHAPTER THREE COMPACT LIE GROUPS 0 If G i s a compact group, .the technique of Theorem 2.3 must be al t e r e d because the exponential map i s not a diffeomorphism. Fortunately, the theory of Riemannian geometry i s applicable and supplies the necessary jv» -f a c t s concerning the exponential i n place of Lemma 2.2. LEMMA 3.1: Let G be a compact gnoup with n dimensional Lie algebra. a) The set C o{ singular points o{ the. exponential map Is a closed set In R n with measuAe zero. b) TheAe Is an open, nelatlvely compact neJ.ghbon.hood E o{ 0 e r such that i ) exp li, a dl{{eomorphlsm o{ E onto expE i i ) E -is the longest connected nelghbonhood with this pnopenty i i i ) exp(&) =G wheAe E is the closune o{ E. c) With w the le{t tnvanlant n-{orm on G and A(x) defined thnough the exponential as In the proo{ o{ Lemma 2.2, i ) Haan measure is given by <Ka) 'du(a) = G <J>(exp£) A(£) d? {on all continuous § on G E i i ) When f Is a continuous {unction on R n with suppont wheAe exp is a dl{{eomonphlsm (so exp 1 is de{lned) n no dc = R n -1 1  f ( e x p a ) |A(exp _ 1a)| d y ( a ) exp(supp f ) PROOF: a) Since C = {£ e T | exp :T—*G i s not a l o c a l d i f f eomorphism at £ } , i t i s c l e a r l y a closed set. I f the exponential i s composed with an a n a l y t i c coordinate chart, C i s l o c a l l y the set of points where the determinant of an a n a l y t i c map i s zero. The c a l c u l a t i o n of the determinant i s an a n a l y t i c 17 operation so, l o c a l l y , C i s the set of zeroes of an a n a l y t i c real-valued map. If t h i s a n a l y t i c map vanished i d e n t i c a l l y on any non-empty open set i n T, then exp would be singular everywhere by extending the chart to an a n a l y t i c a t l a s . But exp i s not always singular, hence, C i s the set of zeroes of a not i d e n t i c a l l y zero a n a l y t i c function. I t i s an easy exercise -to--show that such a set has measure zero. b) This i s the part i n which the Riemannian metric plays a leading r o l e . The d e t a i l s of i t s construction and c e r t a i n theorems concerning Riemannian manifolds w i l l not be proved because the Riemannian structure of our groups i s o v e r a l l of secondary importance. For a rigorous presentation of the theory, the reader i s referred to [9] - e s p e c i a l l y chapters IV and VIII. For our purposes, the most important aspect of the metric i s that the one parameter subgroups obtained from the exponential map are the geodesies of the manifold that pass through the i d e n t i t y e of G. Since G i s compact, i t i s a complete Riemannian manifold. As such, any point i n G can be joined to e by a geodesic that minimizes arclength [9 ; chapter IV, page 172]. Define the set E of the theorem as follows E = {t~er : the curve Y^Ct) =exp(tn) minimizes arclength from e to exp(n) for a l l n i n some neighborhood of £} . Then E i s an open, r e l a t i v e l y compact neighborhood of e = 0 i n V and exp : E—>-expE i s a d i f feomorphism onto an open set of G. In addition, G i s the d i s j o i n t union of expE and e x p ( E ~ E ) . See [9 ; chapter VIII, page 100]. c) The' t i g h t side of the equation i n i ) i s a l e f t i n v a r i a n t p o s i t i v e measure because A(£) i s p o s i t i v e on the connected set E and the measure of the boundary of E i s zero. I t should be noted that A ( C ) i s not always p o s i t i v e 18 on the enti r e L i e algebra. The second statement i s simply the construction of the Haar measure from the l e f t i n v a r i a n t n-form co as i n Lemma 2.2. The absolute value appears on the i n t e g r a t i o n factor because the exponential map need not be o r i e n t a t i o n preserving at a l l points where i t i s a l o c a l diffeomorphism. With the above r e s u l t , a skew product may be developed for general compact groups. T H E O R E M 3.2: Itf A U> the 4ell--adjoint n-tuple. aaociated to the le{t neaulaA. n.eph.uentation oi a compact gscoup G, then thexe it> a continuous Akeiv pfiodact on S = F ( L 1 ( R n ) ) . , • • PROOF: With C and E as i n Lemma 3.1, assume that Ff £ C Q ( R n •• C) (that i s , Ff i s a continuous function on R n with compact support outside C) and define a function on the group analogous to that i n Theorem 2.3 by the formula f(a) = i f^ffffr} for 0 e G exp? = a £ esupp(Ff) where A i s taken from the previous lemma. For fi x e d a e G , f(o) i s a f i n i t e sum of f i n i t e numbers, The numbers are f i n i t e since A(£) ^0 f o r £ esupp Ff . If the number of terms were not f i n i t e , there would be a sequence £ n with a l i m i t point £Q i n supp Ff such that expt~n = a and thus exp would be singular at a point outside C. This y i e l d s a co n t r a d i c t i o n . I t i s easy to see that f i s a c t u a l l y a continuous function on the group. Furthermore, 1^! = I f Ca) I du(a) G 19 expg = o" £ £ supp Ff Ff ( g ) dy(a) G expg = cr E, E supp Ff Ff ( g ) A(£) dy(a) Ff(£)| dg by Lemma 3.1 c) 11) = l l F f l l , With t h i s d e f i n i t i o n of f, the Weyl calculus has the form (see Theorem 2.3) T(A)f (i|0= (2TT) n / 2 f(o) R(a du(a) for a l l ^ e L 2 ( G , y ) and T(A)f T (A) g Op)' = (2ir) -n g * f (a) R(a hi> dy(a) where f , g e F ( C Q ( R n - C)). The convolution present i n the l a s t i n t e g r a l may be pu l l e d back to a function on R n as follows (3.1) K(g) = (2TT) n / 2 A(g) i * f ( e x p £ ) for g e E 0 i f g ^ E . It i s obvious that K s a t i s f i e s the r e l a t i o n for the product of operators; thus, (2TT) -n/2 K(g) R(exp ( - 0 ) 1 » dg = T(A)f T(A)g (<J0 R Let us examine t h i s new function more c l o s e l y . Since g * f i s a continuous function on G, therefore K i s continuous on E. Moreover, K£L^"(R n) and, i n f a c t , there i s an estimate f or i t s norm given below. ( 2 l r ) n / 2 HKI^- U g g f H ^ IIS f| -L H f H ^ H F f H J l F g l ^ . 20 The d e f i n i t i o n f * g = F provides a m u l t i p l i c a t i v e structure by giving a map F ( C 0 ( R n ~ C ) ) x F ( C Q ( R N ~ C ) ) >- FQ^CR 1 1)) that i s contin-uous, by means of the norm i n e q u a l i t y , when F ( C Q ( R N ~ C ) ) has the r e l a t i v e topology induced from FCL^CR 1 1)). AS C has measure zero, F(CQ(R n~C)) i s dense i n F(L (R ) ) . Theref ore, there i s a unique continuous extension of the above map that y i e l d s a continuous skew product on F(L"*"(R n)). The analogue of Theorem 2.5 i s proved using the Peter-Weyl Theorem. THEOREM 3.3: Let G be. a compact gfioup. The, &keu) product oi The.oA.em 3.2 hold* ion, the. We,gZ cat.ciUuA conAeApondlng to any n.e,pn.ej>e,ntatlon oi G. PROOF: F i r s t suppose that V : a—*-V(a) i s a subrepresentation of the l e f t 2 regular representation. Then there i s a p r o j e c t i o n operator P^ on L (G,u) such that V(a) = R(a) P . Let A^ denote the generators of the representa-• I n t i o n corresponding to a fi x e d basis of T. For f i n F(L (R ) ) , V i s rel a t e d to R by T ( A y ) f (ty) = (2TT) n / 2 j ( 2 T T ) " n / 2 J Ff(£) V(exp ( - O ) i f i d£ for ty z P,,(L 2(G,y)) R n V F f ( 0 R(exp(-t:))P ty d£ R n V = T(A)f (ty) since ?^ty = ty. From t h i s equation, i t i s clear that the skew product f o r the l e f t regular representation i s v a l i d for any of i t s subrepresentations. Now suppose that W : a—*-W(a) i s an i r r e d u c i b l e representation. By the Peter-Weyl Theorem [6 ; page 24], W i s equivalent to an i r r e d u c i b l e subrepresentation V of the l e f t regular representation. Hence, there i s a l i n e a r isometry B between the underlying H i l b e r t spaces such that the 21 equality W(cr) = B 1V(a)B i s true. I f H i s the H i l b e r t space for W, then K A ^ f ( I , ) = ( 2 i r ) n / 2 ( 2 , ) " n / 2 Ff(g) W(exp(-g))ip dg for a l l ^ e rf R n „ Ff(g) B 1V(exp(-g))B^ dg R n = ( 2 T 7 ) " n / 2 B _ 1 - Ff(g) V(exp(-g))B^ dg R = B ~ 1 ( T ( A v ) f ) (B*). Writing out the product of two such operators, a t r i v i a l c a n c e l l a t i o n shows that the skew product remains v a l i d f o r i r r e d u c i b l e representations. Any representation for a compact group i s the d i r e c t sum of irreduc-i b l e representations. As the skew product holds for a l l the summands, i t w i l l also hold f o r the d i r e c t sum. Remark: The skew product of Theorem 3.2 i s c l e a r l y not unique. A s p e c i f i c set E was chosen i n the proof that behaved n i c e l y under the exponential map. There are c e r t a i n l y other sets that would do equally as w e l l . Since the function K given by (3.1) i s usu a l l y discontinuous at the boundary of E, our skew product w i l l not s u f f i c e for the other three function spaces introduced i n Chapter One. However, i t i s i n t e r e s t i n g to question the existence of some m u l t i p l i c a t i v e structure on these spaces. The d i f f i c u l t i e s encountered i n Chapter Four f o r the group SU(2) discour-ages us from looking at the general case. 22 CHAPTER FOUR SU (21 Let G be the r e a l , simply connected L i e group 5U(2) of 2x2 unitary matrices of determinant one. Since G i s the two-fold u n i v e r s a l covering group of the r o t a t i o n group SO(3), any skew product defined through G w i l l , a p r i o r i , e s t a b l i s h a skew product for S0(3). The L i e algebra Y i s the set of 2x2 skew-hermitian matrices of trace zero. Choose the following matrices as our basis for Y. i 0 ' 0 i I r o i 0 9 i 0 J • -i o J As always, we need some fa c t s about the exponential map before producing a skew product. LEMMA 4.1: a) With the above basis, exp : r — > G is given by s i n |.£ exp(£ 1,£ 2,? 3) = c o 8 | ? | + 1 ± f t f 5 1 i ^ - L ( ? 2 i - 5 3 ) whexe | ? | = ^ + K 2 2 + K \ ) 1 1 2 and ^ ^ = 1 at |?|=0. b) The exponential. Is periodic o{ period 2TT along any tine through the origin. Vor any non-negative Integer n, the set {£ : nir < |g| < (n+l)iT} parameterizes the group G and exp Is a dl{{eomorphism on the Interior o{ any o{ these sets. The set E o{ lemma 3.1 is o{ the above {orm with n=0. c) J{ o denotes group composition on one o{ these parameterlzations> then cos | n "^ o £ = cos n | cos | £ | + s i n | n | s i n | £ | T^V^ d) With the notation o{ lemma 3.1, A(g) = ( s i n 15 |)/ (15 |) ' 23 PROOF: a) and b) are easy exercises using the exponential of a matrix. c) This i s an immediate consequence of matrix m u t i p l i c a t i o n and the fact that cos |?| = y trace(exp£)• d) A s l i g h t modification of the Weyl Integration Formula using the roots of G produces t h i s i n t e g r a t i n g f a c t o r . For a more constructive proof, we r e f e r 'the reader to [12 ; page 220]. THEOREM 4.2: Ton G = SU(2) , theAe. <U a Akew product on the, Apace. o& 3 lunctxonA s = F ( c Q ( R ) ) . PROOF: We have only to consider the l e f t regular representation of. G due to Theorem 3.3. Let D ( I T , 2 I T ) be the set {£ : IT < | g \ < 2TT> i n 3-space. By Lemma 4.1, D(TT,2TT) may be used as a coordinate system for G under the exponential map. The inner and outer boundaries of t h i s set correspond to the matrices - I and I of the group r e s p e c t i v e l y . 3 Define a function on the group for a l l f G F ( C Q ( R ) ) , with respect to the above coordinates, by expa .= £ I F f a + 2 f ^ ) s i n £ n=-°° 1 2nrr£ 2 by the p e r i o d i c i t y of the exponential (Lemma 4.1). We set f equal to 0 at the boundary of D(iT,2Tr). 3 Following Theorem 3.2, the skew product of f , g e F ( C 0 ( R )) should be provided through the function K defined on the next page. 24 Ul2 f ( n 1 o g ) g ( n ) d y ( n ) ' D ( T T , 2 T T ) for £ i n the i n t e r i o r of D ( T T , 2 T T ) . Of course, we do not as yet know i f K i s defined point-wise because the singular set of the exponential was not avoided as i t was i n Theorem 3.2. Let us rewrite K i n order to examine the 3 l a s t i n t e g r a l more c l o s e l y . F i r s t , for f E F ( C Q ( R ) ) , define CO f'(?) = - ^ I Ff(?+^f) r 2 L ^ £ E, n=-°° 1 1 2 for a l l K e D ( T T , 2T V ) . With t h i s notation, K becomes (4.1) k"(5) = ( 2 u ) - 3 / 2 s l n |g< f. f ^ o C ) g ' ( n ) — ^ - - ^ L | C | D (IT , 2TT) s i n | n o £ by changing Haar measure into Lebesgue measure. It should be evident why D ( T r,2ir) was chosen for our parameter-i z a t i o n instead of the set E as i n Lemma 3.1. On D ( T T , 2T T ) , f and g' 2 —1 2 are uniformly continuous and the expressions j£| and |n o £ | are never zero. Our proof consists of three steps concerning the properties of K. Af t e r these are performed, i t w i l l only remain to "round o f f the edges" of K at the boundary of our coordinate system. The steps are: Step 1. K(£) e x i s t s f or each £ i n the i n t e r i o r of D ( T T , 2 T T ) . Step 2. K(£) i s continuous at each of these points. Step 3. K(g) can be extended to a continuous function on D ( T T,2T T ) that i s constant on each boundary. Step 1. F i x g so that TT < |g| < 2TT. Then, for some constant c , depending on c|K(0 I < sup|f' (TI) I sup|g' (n) - n n -1 _,2 n 0 ? l - d n 2 -1 D( T T , 2TT) s i n n o £ 25 where c i s greater than zero and the sup i s taken over D(Tf,2Tr). The problem i n showing the integrand i n the l a s t i n t e g r a l i s summable i s that the denominator i s sometimes zero. In f a c t , {n : s i n 2 | n _ 1 o £ | = 0} = f n : | r f 1 o ? | E { f r , 2T T } } = { n : r f 1 o £ =±1} = ( t , C o (-1)}. Since neither of these points are on the boundary of D(-rr,2Tr) , we can choose b a l l s and around £ and £ o .(-I) r e s p e c t i v e l y , whose closures stay away from the boundary. Let us s p l i t up the i n t e g r a l f o r the estimate of K into an i n t e g r a l over M , over N , and over the remainder. The Haar measure gives a bound for the f i r s t and second i n t e g r a l ; namely, dn = -1 - 2 n o £ M s i n 2 | n ^ o £ n 1 o z\2 |n| 2 dy(n) s i n 2 | n o £ | s i n 2 | n K- C l -1 _|2 n 0 5 1 - dy(n) E, s i n n o E, 2 2 where |n| / s i n |n| i s bounded by c^ on M£ C l r o M < c^ • volume (D (IT , 2TT) ) . Likewise, the i n t e g r a l over i s bounded by c^'volume(D(ir, 2 I T ) ) . The i n t e g r a l over the remainder i s also bounded since the integrand i s a uniformly continuous function on t h i s set. Therefore, formula (4.1) does produce a candidate f o r the skew product. 26 Step 2. This is essentially a refinement of the argument used in step 1. 2 2 As sin | g | / | g | is continuous near a fixed point g between ir and 2TT , we need only show the integral in (4.1) is continuous. To this end, suppose that g^ is close to g but is not ±1. The difference of the integrals is -1 ,i2 I -1 r |2 In o? 1| D(TT,2TT) sin^ l n ^ o g | sin 2 | n "*"o£.jJ f ' ( n 1 o £ ) g f (n) — — f ' ( n ^e ^ g ' C n ) dn D (TT , 2TT ) Is1Cn) i o I -1 _ 2 r f / - 1 r\ n O g j . , , -1 . 1 f ' ( n og) — ^ — f (n o g ^ ) — 2 — - j -sin |n og| sin |n og^| d n. The last integral w i l l again be s p l i t up as in step 1 in order to obtain an estimate. Instead of writing the expression inside the long absolute value signs of the last integral each time, i t w i l l be denoted by fi whenever i t i s used. Let us f i r s t calculate a bound over M . M, |g' (n) | |n|dr, < su P|f•(n) | suplg' ( n) In ogj -1 r 12 In og 1| sin 2 | n *^og| sin 2 | n "*"og^ | dn < c3*{volume(g o^M^ .) + volume (g-^oM^) } i f g^ is close enough to g. This inequality results by changing to Haar measure and then back again as we did in the calculation on page 25. The 2 2 constant c^ Is a bound for the expression sup|f'|•sup|g' | •|n | /sin |n| when n is restricted to l i e in M^ . .-1 Suppose a>0 is given. Choose M so that volume(g oM^ ) <a/8c^. Then there is a &^>0 such that |g-g.J < 6^  =*> volume(g^oM) < ct/6c3 and 6^ . is less than half the radius of M^ . With this restriction for g, we have g'(n)||fi|dn < c„-{a/8c~+ a/6c,} < a/3. Likewise, choose N and then 6^ such that 6 2 is less than half the 27 the distance from £ to-the boundary of N^o(-I) and |S _£-jJ < ^2 ^ |g' (n)I|n|dn < a / 3 . We have only to estimate the i n t e g r a l over the remainder. Define a function u : {D (TT , 2TT) - (M U N) } x {E,± : | \ < minimum^, 6 £) } — ( f , by u ( n , c : 1 ) = f' ( r i ~ 1 o t : 1 ) ' | n ~ 1 o t ; 1 | 2 / s i n 2 | n ~ 1 o c : 1 | where (f denotes the complex numbers. This function i s uniformly continuous since n remains away from ±1 on t h i s set. Therefore, there i s a 6 < minimum (<5^ , 5 2) such that |£-£^| < 6 => |u(n,£^) - u(n»£) I < a/3sup|g'| f o r a l l n where u i s defined. Putting a l l t h i s together, we conclude that | | <5 implies |g'(n)I|n| n < a. D(T T , 2 T T ) This completes step 2. Step 3. I t w i l l only be proved here that K(£) can be extended continuously to a constant on the sphere | £ | =2TT. The analogous proof f o r the other boundary i s l e f t to the reader. We must show that f o r a given a>0 there i s a 6>0 such that 2TT-<5 < |£ I < 27T for j = l , 2 implies | K ( £ 1 ) - K ( ? 2 ) | < a. The demonstration of t h i s f a c t rests heavily on the following statement: Given a>0 there e x i s t s 5^ > 6.2 > 0 such that 2TT - $2 < | 5 | < 2TT s i n 2 ] KI dn + [ sin 2l-g. 2i -1 _ i „/„ „ „ s . '21 -1 dn '< ot. JD(ir ,T r+6 1) s i n | n oC | ' P ( 2 I T - 6 1 , 2TT) s i n |n o?| The notation D(a,b) means the set {£ : a< |£| <b}. The proof of (4.2) w i l l 28 be postponed u n t i l the end of the theorem for fear of l o s i n g the flow of step 3 i f given at t h i s time. Let us continue assuming (4.2). By formula (4.1), (2T T) 3 / / 2 | K ( ? ) - K ( ? 2 ) | i s bounded by |g'(n) D(TT,2TT) , |n ^og. | Z« sin^g.. | |n  1oK?\ 2' s i n 2 I? 1 i _ 2 .2 -1 _ 2 12 , 2 -1 £ 1 «sin n ot:i £ [ «sin n og. dn. This i n t e g r a l w i l l be s p l i t up again and the expression i n s i d e the long absolute value signs w i l l be denoted by 0. — 1 2 2 Suppose a>0 i s given. Notice that |n og| /|g| < 4 for our parameterization. Choose 6 ^ > > 0 so that the i n t e g r a l i n (4.2) i s bounded by a/8• sup | f' | • sup | g' | . Then, i f 2TT-6„ < | g . | < 2TT for j=l,2 | g ' ( n)||0 | d n + f |g '(n) | |0|dn < a/2. J D(TT,TT+6 1) j D ^ T T - S ^ T T ) Let* us estimate the i n t e g r a l over the remainder. As i n step 2, introduce a function u : D ( T T + 6 1 , 2 H - S ^ ) X D(2TT-6 2,2TT) > (f through • , r N , w r - l ... |n 1og | 2* sin^ J g I u(n,g) = f ' ( n og) — 2 — L — 2 — • |g| • s i n " |n ogI This function i s uniformly continuous on i t s domain and, i n f a c t , s a t i s f i e s the following statement since the second v a r i a b l e i s r e s t r i c t e d to l i e i n a neighbourhood of the i d e n t i t y matrix I i n G. Given a>0 there i s a p o s i t i v e 6^ < 5 2 such that ^(n.g-^) - u ( n , £ 2 ) | < a whenever 2TT—5^ < |g..| < 2TT for a l l n where u i s defined, Thus, for small enough 6 , we have Is* Cn) 1 10.1 dn < a/2. D(TT+5 1,2ir-6 1) ' This completes step 3. The above three steps prove that K(g) i s a uniformly continuous 29 function on D(TT,2TT) that i s constant on each boundary. In order to exhibit V 3 a skew product of F(C n(R )) , l e t us change K into a continuous function 3 with compact support i n R while r e t a i n i n g the m u l t i p l i c a t i v e property. Define 3 '• R—*R by the formula 3TT 3(t) = \ 1 - i IT t 2 for TT/2 < t <_ 5TT/2 0 otherwise. 3(t) i s a continuous function with compact support, maximum value of 1 attained at t=3-rr/2 , and decreases l i n e a r l y to zero at t= T r/2 and t=5Tr/2 In addition, i t s a t i s f i e s the equation 00 I 3 ( 11 + 2niT |) = 1 for every t . n = - o o F i n a l l y , the stage i s set to introduce the skew product of the 3 two functions f ,g eF(Cg(R' )) by means of 2 F ( f *B)(0 = I B d S l ) - ^ K(C + 2-j|4) f o r a l l Z. 3 Then F(f * g ) e C Q(R ) and has support i n D (TT/2, 5TT/2) . Going back to the constuction of the function f on page 23, we can see immediately that t h i s d e f i n i t i o n y i e l d s a skew product. Moreover, the skew product i s a l i n e a r 3 map that i s continuous with respect to the topology of F(C Q(R )) . Proof of (4.2): I t w i l l only be shown that the i n t e g r a l near 2TT i n (4.2) may be made small uniformly for £ near 2IT . The method described e a s i l y extends to the other i n t e g r a l and together they imply (4.2). If 6^^ i s a small p o s i t i v e number to be r e s t r i c t e d l a t e r (always assume that i t i s l e s s than TT/2 ) , then the i n t e g r a l i s estimated as follows 30 s i n 2 1 5 D ( 2 i r r 6 l t 2 i r ) s i n 2 | r f •'p? ; dn 2 s i n |g| dn by Lemma 4.1 -D (2 i r-5 1 ,2 i r ) 1 - {cos | n | cos | £ | + s i n | n | s i n | £ | (n?ll \ n | | 5 | ) } 2 Change to polar coordinates with £ along the p o s i t i v e z-axis and | c|=r 2 p s i n <j> d<()dpdG •2TT •2lT riT . 2 s i n r • . 0 • 2ir-6 J 2 0 1 - {cos p cos r + s i n p s i n r cos <J> } Let = cos p cos r + s i n p s i n r cos ty = 2ir r2ir fcos(p+r) 2 . 2 , . -p s i n r , , K — dvdp J 2 , - f i J cos(p-r) s i n p s i n r (1-v ) = TTsinr 2TT 2TT-6, p1 , I 1 + cos(p-r) 1 - cos(p+r) s i n p 8 ( 1 - cos(p-r) 1 + cos(p+r) dp The proof depends on an estimate of t h i s i n t e g r a l . Since 5^<TT/2 and £ i s taken such that 2ir - 6^ < | £ | < 2tr , the function s i n r i s negative and so i s w(p) below w ( p ) = _1_ l o J 1 + cos ( p-r) 1 - cos(p+r) s i n p °l 1 - cos ( p-r) 1 + cos (p+r) 2TT - 61< p < 2TT In f a c t , we claim that w increases from -°° i n the i n t e r v a l r < p < 2TT . I t s u f f i c e s to show that the d e r i v a t i v e of w i s non-negative i n t h i s i n t e r v a l . Only a sketch of t h i s r e s u l t w i l l be provided here. The v e r i f i c a t i o n of each step i s l e f t to the reader. If we set x(p) = ( s i n 2 p /cos p ) ~ > then dp 9 2 x(2ir)=0 and dx/dp = {sin p s i n r s i n 2p }/{sin ( p - r ) s i n (p+r)} which i s l e s s than zero i n our i n t e r v a l . Thus x(p) i s p o s i t i v e i n the i n t e r v a l and so i s dw/dp . We are now ready to estimate the i n t e g r a l at the top of the page. I t i s s p l i t into an i n t e g r a l f o r 2ir - 8^ < p < — (2-rr+r) and then for -jj^tr+r) < p < 2 ir . The second i n t e g r a l i s bounded by the expression , 3 . , 0 2 T r + r v 4TT sxn r (2TT 2 — ) s i n ( -y-) log ,, / 2 T r - r N , / 2 T f+3r N ( l + C O S ( —7T- ) l - C O S ( ^ ) 1 , 2 T r - r N .2Tr+3r N l - c o s ( — 2 — ) l + c o s ( — 2 ^ — ) since w i s increasing on t h i s i n t e r v a l . Write the l a s t part of t h i s expression as l o g ( t ) . We have the following l i m i t s . 31 ,. s i n r , lim K~~T~ = 2 „ . ,2-rr+r. r->2ir s i n ( — 2 ~ - ) l i m l o g ( t ) = log 9 r+2Tr Therefore, i f a>0 i s fi x e d , there i s a ^ 2 ^1 s u c " 1 that 2 1 T -62 < |g| < 2TT implies that 9 I I + I F I Son 2 |f 1 dn < a / 2 . D(2-2±i£J,2,r) sin 2 | n " V | To 'estimate the i n t e g r a l over the f i r s t i n t e r v a l , expand the functions l-cos(p-r) and l-cos(p+r) about p=r and p+r=4Tr respec-t i v e l y by means of Taylor's Formula. The expansion up to second order provides the following approximations i f <5^  i s small enough. 2 2 l-cos(p-r) > (p-r) /4 and l-cos(p-tr) < (p+r-4i0 As these are the only factors i n the Integral that cause problems over t h i s i n t e r v a l , there i s a constant c such that 2 s i n g dn s i n £ —J I — < C — J — D(2T T-6 1, 2 j I^^I ) s i n 2 |n  1oZ,\ s i n ( 2-rr-r 2TT+ £ f 2 ^ I) J 2TT-6. 2 ' v l Again, there i s a l i m i t f o r t h i s i n t e g r a l as below r2ir l o g ( ^ P + l j | - ^ dp l i m 6^0 j 2 T T - 6 1 2 l o g 4(p+|g|-4TQ P-HT dp = 0 . 32 Of course i n th i s l a s t l i m i t , the E, was r e s t r i c t e d to have absolute value between 2TT-5^ and 2ir . Combining a l l these estimates, we have shown statement (4.2). 33 PART II THE EVOLUTION EQUATION CHAPTER FIVE PRELIMINARIES AND GENERALIZATIONS The s e l f - a d j o i n t p a i r considered i n t h i s part i s provided through a representation of the Heisenberg group. The Heisenberg group G(l) i s the 2 subgroup of unitary operators on L (R) that have the form U(p,t)<j> (x) = e±v(-K\(x+t) f o r every <f>eL2(R) where p(x) i s a real-valued polynomial of degree at most 1. Under the usual product for operators, G(l) becomes a nilpotent L i e group. The gener-ators of t h i s s e l f - r e p r e s e n t a t i o n are the operators ( i d , iQ, icP) where c i s a p o s i t i v e constant, I i s the i d e n t i t y operator, and the other two operators are as i n the introduction. The only non-vanishing bracket of t h i s basis f o r the L i e algebra i s [iQ , icP] = - i d . The skew product * of Chapter Six evolves from the above n i l p o t e n t group and the given representation. As explained i n the introduction, the Weyl f u n c t i o n a l calculus applied to the p a i r (Q,cP) i n t e r p r e t s c l a s s i c a l q u antities on phase space 2 as Hamiltonians on L (R). However, quantum mechanics may also be formulated on phase space. This aspect has been studied by a number of authors; notably, J . E. Moyal i n h i s 1949 paper [11] and also J . Jordan and E. Sudarshan [8]. The evolution equations that appear i n these papers and the one that i s developed here do not seem to be the same because the methods of formulation 34 vary widely. The equivalence of these formulations i s revealed most s u c c i n c t l y i n [16]. Before studying the evolution equation, i t must be emphasized that most of the r e s u l t s of the ensuing four chapters immediately generalize to the phase space formulation of a system with n degrees of freedom. In t h i s -ca'se, the operators ( i l , iQ^, i P ^ , . . . , iQ n» i ^ n ) form a basis for the n i l p o t e n t L i e algebra where and iP.^ are the obvious s e l f - a d j o i n t th 2 ri operators acting on the j v a r i a b l e of functions i n L (R ). The brackets for t h i s system are of the form [iQj , i P j ] = - i l . The skew product and evolution equation can be r e a d i l y defined by comparison with the Heisenberg group. The generalizations of the theorems are l e f t to the inter e s t e d reader. Reference [10] provides d i f f e r e n t aspects of t h i s theory. In passing, i t would be negligent not to mention that the important r e s u l t (Theorem 6.5) can be generalized i n yet another d i r e c t i o n . Let G(m) be the group s i m i l a r to G(l) except the polynomial i s of degree at most m. The L i e algebra of G(m) has basis ( i d , iQ^,..., iQ^, icP) where i s m u l t i p l i c a t i o n on L 2(R) by x^. A skew product * e x i s t s f or the s e l f -adjoint (m+l)-tuple (Q ,;... .Q^cP) on the space S ( R m + 1 ) [2 ; page 430]. The generalization states that there i s a unitary operator on L2(Rm+"'") that i s a homeomorphism of 5(Rm+''") and s a t i s f i e s for f , g e S ( R m + ^ ) U(f *U > 1g)(x 1,...,x m,y) = (2rrc) 1 / 2 Uf(z y) g(x 1,...,x m,z) dz R In other words, the skew product i s equivalent to the point-wise m u l t i p l i c a t i o n of functions except i n the f i r s t and l a s t v a r i a b l e s . The author has performed the e x p l i c i t c a l c u l a t i o n of t h i s unitary operator for m < 4 and f i r m l y believes the equation i s true f o r every G(m) . Of course, with these two generalizations, one could form d i r e c t sums of the L i e algebras considered on the l a s t page and so obtain deeper knowledge of the skew product on many ni l p o t e n t L i e groups. This program w i l l not be c a r r i e d out here. 36 CHAPTER SIX THE W E Y L C O R R E S P O N D E N C E AND E V O L U T I O N E Q U A T I O N The following operators w i l l be used throughout. I t should be 2 2 noted that the f i r s t three operators a l l extend to unitary operators on L (R ) • D E F I N I T I O N 6.1: Let 5'(R 2) have the strong dual topology induced by the bounded subsets o{ S(R ) . The operators below axe homeomorpltisms o{ S ( R 2 ) and S'(R 2) . Suppose that f belongs to S(R 2) . 1 f (izx ) 7. (Vartial Vourler Transform) F f(x.,,x 2) = . e v 2' fCx-^z) dz . R F^ Is defined similarly. 2. (Twisting Operator) S c f ( x 1 5 x 2 ) = /c f ( x 1 - ^ c x ^ x ^ +-|cx2) . S ~ 1 f ( x 1 , x 2 ) = l / ^ c f ( ( x 1 + x 2 ) / 2 , ( x 2 - x 1 ) / c ) . 3. (Translation) x ( x ; L , x 2 ) f (y1,'y2) = f ( x + y 1 , x 2 + y 2 ) . 4. (Rotation and dilation) c may be negative {or this opexaton.. V f ( x r x 2 ) = f (cx 2 / 2,-cx 1 / 2 ) . The Weyl operator has a p a r t i c u l a r l y simple form i n terms of these d e f i n i t i o n s . P R O P O S I T I O N 6.2: l{ h e S ( R 2 ) , T(Q,cP)h Is the bounded Integral operator on L ( R ) with kernel l//2irc s F.h . That Is, c 2 (T(Q,cP)h)u (x) = l/Z^irc MOPE: See [1 ; page 264]. S - 1 F h(y,x) u(y) dy f o r a l l u e L 2 ( R ) , R c 37 The skew product of Coro l l a r y 2.4 f o r the s e l f - a d j o i n t t r i p l e 2 (d,Q,cP) can be r e s t r i c t e d to a unique continuous skew product * on S(R ) c Let h[f] or h ( x ^ , x 2 ) [ f ( x ^ , x 2 ) ] denote the ac t i o n of a tempered d i s t r i b u -2 t i o n h on a function f eS(R ) . Through d u a l i t y , * i s extended to a c separately continuous map * : S'(R 2)x5(R 2) >-S'(R2) given by c h * f [g] = h[f * g] for heS'(R 2) and f , g e 5 ( R 2 ) . Let us c o l l e c t these f a c t s . 3 PROPOSITION 6.3: a) The Akeu) pnoduct {on (cI,Q,cP) on S(R ) Xi -3/2 F ( f * g ) ( x ( ) , x 1 , x 2 ) = (2TT) f * g ( x 0 , x 1 , x 2 ) = (2TT) 3 F f ( x 0 - y 0 + R i(tu+sv) y l x 2 " X l y 2 > x 1 - y 1 , x 2 - y 2 ) Fg(y) dy OK f(x 0,x 1+sx 0/2,x 2+u)g(x 0,x 1-tx 0/2,x 2+v) dudvdsdt whene f ,g eS(R ) . 3 b) J{ f AJ> defined by f (x 1,x 2) = f ( c , x 1 , x 2 ) {on f eS(R ) , then 2 T(cl,Q,cP)f = T(Q,cP)f . Thai, the tkew pnoduct {on f,ge5(R ) AJ> (6.1) f * g ( x r x 2 ) - C2ir)' R e 1 < t u + s v > f ( x +sc/2,x 2+u)g(x -tc/2,x2-Hv) dudvdsdt _1_ 2TT R 2 ^Vc T ( x i » x 2 ) f ^ y l ' y 2 ^ ( F T ( x i ' x 2 ) g ^ y i ' y 2 ^ d y i d y 2 " J{ complex conjugation ti> denoted by , then f * g = g * f c) Ton h e 5 ' ( R 2 ) and f eS(R 2), define (6.2) h * f ( X ; L , x 2 ) = ~ (V T ( x 1 , x 2 ) h ) [ F x ( x 1 , x 2 ) f ] 38 Then h * f is a C {unction with. polynomially bounded derivatives o{ alt c 2 orders that AatU{ies {on g e S ( R ) (6.3) h * ftg] = h[f *g] (duality) h * ( f *g) = ( h * f ) * g (asAociativtty) d) By the last statement o{ b) and duality, the evolution equation o{ the . . i - 1 • — introduction extends to — = H f where dt c (6.4) H c f W A ^ { V T(x)h[Pt(x)f] - V C T ( X ) F [ F T ( X ) T ] } ^ h e S ' ( R ) , f e S ( R ^ ) = ~ { V c x ( x ) h [ F T ( x ) f ] - V_ C T(x)h[Fx(x)f]} . Notice that the dependence o{ u.^ on the dlstribtition is AuppreSAed Aince h is regarded as a {ixed Hamtltontan. PROOF: See [2 ; section 3]. We are almost ready to express (6.4) i n the form of an equivalent singular kernel operator on the plane. But f i r s t , a s i m i l a r r e s u l t must be 2 proved f o r the test functions S(R ) . LEMMA 6.4: 1{ h e S ( R 2 ) , let H' : 5(R 2) -> S(R 2) be the operator S ' ^ H ^ F " ' 1 ^ de{ined through the homeomonphisms at the beginning o{ the 2 chapter. Writing tills in integral {orm, we obtain, {or f eS(R ) H (;f(x 1,x 2) = i/y^rc" { S c 1 F 2 h ( y , x 2 ) f ^ . y ) - S ^ h ^ y ) f ( y , x 2 ) } dy PROOF: We w i l l show only one h a l f of the formula; namely, that i U ( h * U 1f)(x 1,x„) = if/lire c 1 z Uh(y,x ) f ( x ,y) dy where U = S 1 F R C ~ 39 The proof of t h i s i s s t r i c t l y a computation. We have i h * f ( y r y 2 ) = 1/2TT R 2 V c T - y l ' y 2 ^ h ^ z l , Z 2 ^ F x ^ i ' y 2 ' ) f ( ' Z 1 ' Z 2 ' ) d z i d z 2 b y 1/2TT j ( F 1 V c T ( y 1 , y 2 ) h ) ( z 1 , z 2 ) ( F 2 x ( y ^ ^ f ) ( z ^ d Z j d z ^ _ i _ 2 2ir c J R" i ± ( 2 z l y 2 / c ) F 9 h ( y 1 + c z ? / 2 ) - 2 z 1 / c ) e 1 ( y 2 z 2 } F 2 f ( y 1 + z 1 > z £ dz-jdz. i_ T 2 { e ± y 2 ( 2 z l / c z2 )Uh(y 1+z 1+cz 2/2,y 1-z 1+cz 2/2) Uf(y 1+z 1-cz 2/2,y 1+z 1+cz 2/2)} d z 1 d z 2 Let u^ = z^ + cz 2/2 and u 2 = z^-cz^/2 A. TTC R 2 e i ( 2 y 2 u 2 / c ) U h ( y 1 + u 1 , y 1 - u 2 ) Uf(y 1+u 2,y 1+u 1) d u ^ Therefore, F_ (ih *U f)(x,,x„) i s equal to J. c l z i / T r c v ^ r r f e i ( x 2 y 2) R ; i ( 2 y 2 u 2 / c ) u h ( u 1 , x 1 - u 2 ) f ( x ^ + i ^ . u ^ d u ^ u ^ y . i / / 2 7 T Uh(u^,x^+cx 2/2) f(x^-cx 2/2,u^) du^ Thus, iU(h * U _ 1 f ) ( x i s x 2 ) = i//2rrc R Uh(s,x 2) f(x^,s) ds THEOREM 6.5: IQ h e 5 ' (R2) , Itt H' : S(R 2) -->• S' (R2) bd thu opUuvtoH. —1 —1 2 Sc F 2 H c F 2 Sc ' ^ e n ' f » g e 5 ( R ) we. naue. £he tqiuvalzYit {onmiilas H c'f( X ; L,x 2) - 1//2TTC { S ^ " 1 F 2 h ( - , x 2 ) [ f ( x 1 , - ) ] - S c X F 2 h ( X ; L , - ) [ f ( ' , x 2 ) ] } -1. OK. H*f[g] = i / v ^ r c S c 1 F 2 h ( y 1 , y 2 ) [ | f ( z . y ^ g C z . y ^ d z -R f ( y 2 , z ) g ( y 1 , z ) d z ] 40 PROOF: Define a j o i n t l y continuous map o : S(R 2) *S(R 2) —»- S(R 2) by (6.5) f o g(x 1,x 2) R f ( z , x 2 ) g(x ] L,z) dz The proof follows immediately from the d u a l i t y r e l a t i o n (6.3) and the commutativity of the diagram below by Lemma 6.4, S(R 2) x S(R 2) A C c 2 c 2 c 2 S(R 2) x S ( R 2 ) 1 / / 2 ^ ° > S(R 2) The s i m i l a r i t y between the quantum mechanical operator of Proposition 6.2 and the operator i n Theorem 6.5 suggests a natural extension of the Weyl quantization procedure to tempered d i s t r i b u t i o n s . With t h i s extension, Theorem 6.5 i s interpreted as a separation of v a r i a b l e s f o r the evolution equation. Indeed, Weyl operators corresponding to the o r i g i n a l tempered d i s t r i b u t i o n h act on each v a r i a b l e separately (see Theorem 6.7). MAIN DEFINITION 6.6: (The Weyl Correspondence) Suppose h is a tempered distribution on the plane. Ve{i.ne a map A (h) : S(R) —»• S'(R) by ,-1 (6.6) I (A (h)) (<jj) [ip] = 1//2TTC S F„h[<j>xijj] {or cf>,ipeS(R) or equ-Lvalently C C i. , . (A (hH)(x) =l//27c s'h9h(-,x)tK-)] . c c z A c is a continuous Linear operator called the Weyl operator corresponding to the Hamiltonian h . J{ the distribution is clear {rom the context, the dependence o{ A c on h is o{ten suppressed. 41 Suppose that B.. : X..—>-Y_. ,j=l,2 are two l i n e a r operators with domains ^ ( ^ j ) and that ^ - j ' ^ j a r e complex function spaces. Let the tensor product, B^ 8 B 2 be the operator with domain a l l f i n i t e l i n e a r combin-ations of functions of the form f,xf_ f. E P ( B . ) and defined as follows: 1 2 j J B l 8 B2 n k=l k k n = ak <Vi > x ( B 2 f 2> °k e (f k=l k k Also, define the complex conjugate operator B^ by B^f = (B^f) f o r f et?(B^) Adopting the above notation, we have a reformulation of Theorem 6.5. THEOREM 6.7: H)_ is the unique extension o{ i { l 8 A £ (h) - A^Ch) 8 1} to a 2 9 continuous linear, operator, {rom S(R ) to S' (R") . PROOF: I t i s easy to check the following important equality of d i s t r i b u t i o n s . (6.7) S c 1'F 2h(x 1,x 2) = S c 1 F 2 h ( x 2 , x 1 ) . 2 Since the l i n e a r span of 5(R) x S ( R ) i s dense i n S ( R ) , we have only to v e r i f y the two operators i n the theorem are equal f o r functions of the form cpxuV f o r <J>,^ES(R) . H^(^x^)( X ; L,x 2) = i//2^c {S~ 1F 2h(-,x 2)[<()(x 1)^(-)] - S c" 1F 2h(x 1 , - ) [< f»(')^(x 2)]} = l <|>(x 1)(A c(h.)t|»)(x 2) - i / v ^ c S c 1F 2h(.,x 1)t()>(-)^(x 2)] by (6.7) i{<f>xA (h)*.- A (h)<j>x^}(x ,x„) by (6.6) c c I I i ( { I % A (h) - A (h) 8 I } ()»xl|,)(x1,x2) 42 CHAPTER SEVEN BOUNDED OPERATORS One method of solving the evolution equation of (6.4) i s to regard 2 2 2 H c and A^ as operators on L (R ) and L (R) r e s p e c t i v e l y , with domains 0 ( A ) = {* eS(R) : A<j>eL2(R)'} and P<H) = {f e 5(R 2) : H^f e L 2 ( R 2 ) } . 2 In p a r t i c u l a r , i f H c i s a bounded operator with P(H C) =S(R ) , then the usual power serie s expansion of the exponential provides a group of operators i f t H ) e c that immediately solves the evolution equation. If we adopt t h i s point of view, Theorem 6.5 establishes a unitary equivalence between H and H' when P(H') ={f E S ( R 2 ) : H'f e L 2 ( R 2 ) } . c c c c Although H' remains an extension of 1{I ® A (h) - A (h) 8 1} which i s c c c now defined on l i n e a r combinations of functions i n £>(A (h)) xP(A (h)) , c c we have no guarantee that i s contained i n the closure of t h i s operator. Indeed, a p r i o r i , H could be densely defined while A c i s not (or v i c e versa). The r e l a t i o n between H and A i s studied i n t h i s chapter and • c c again i n Chapter Nine. Here, the case of bounded operators i s examined. As a s t a r t , we have the following. 2 THEOREM 7.1: H c lj> bounded with t?(H ) =5(R ) l{ and only l{ A^ li> bounded with V(A ) =S(R) . c PROOF: It c l e a r l y s u f f i c e s to prove the statement for H^ i n place of H c Assume 'A (h) i s bounded by b and V(A^(h)) =S(R) . Then, for 2 a l l < M e $ ( R ) , |Ac(h)cf)[^] | <b||<j>||2 ||T|»|| where ||-|| i s the L -norm. By (6.7), |Ac(hH[<Jj] | = |Ac(h)[^x<l>] | <b||i|)|| 2 ||«|»||2. Thus A„ (h) i s 43 bounded by b and V(A (h))=S(R) . For f eS(R*") , define two fa m i l i e s of functions i n S(R) by fx ) f ^ x j ( z ) = f ( x 1 , z ) and fK 2' (z) = f (z ,x 2) . Theorems 6 . 5 and 6 . 7 y i e l d H c f ( x l ' x 2 ) = i ( A c ( h ) f ( x 1 ) ) ( x 2 ) " i ( A c ( h ) f ( x 2 ) ) ( x 1 ) Thus, we have the following estimate f o r the norm: l l H ; f h 2 < 1/2 U 2 | (A c (h)f ( x 2 ) ) ( x 1 ) | 2d X ; Ldx 2 1/2 U R K ^ l ^ )m + \ j / " f ( X 2 ) | l 2 d x 2 1/2 by hypothesis = 2b R 2 | f ( x 1 , x 2 ) | dx 1dx 2 1/2 = 2b f 2 ' Hence, H^ i"s bounded by 2b and P(H^) =S(R/') . 2 To prove the converse, assume that H' i s bounded and P(H') = S ( R ) . c c Let I J J ES(R) be a r b i t r a r y and choose <J > Q , P Q £ S ( R ) such that o P 0 ^ ^ ' Consider the two i n e q u a l i t i e s i ) |H^((fr 0x^)[ P ( )xe]| < b ||P 0|| 2 ||91| 2 for a l l 6 eS(R) ( b i s a bound of H^) i i ) |-H^(4>0xiJ>)[p0xe]| = |Ac(h)*[e]'/<fr0p0 - • A c ( h ' ) * 0 [ p 0 ] - / * 8 | by Theorem 6 . 7 . Thus, |Ac(hMe]|< a j|p0||2||e||2+ |Ac(h)<(,0tp0]-/*e|} < b' ||e||2 for some constant b' by Cauchy-Schwarz. As ty was a r b i t r a r y , the domain of A c i s a l l of S(R) . By the same argument, A (h) has the same domain. We have only to show that A i s c c 44 bounded to complete the proof. • If A were not bounded, there would be a sequence d> eS(R) such c n that ||Ac(h)<j>n || ^ —• °° but ||<f>n|| = l . For t h i s sequence, l|H^C*1xd>Ti) || 2 = ||<^xA c(h)* n - ^ ) \ ^ J \ 2 > ll* 1lM|A c(h)* n|| - ||Ac(h)i1|||Un|| >• °° as n >• 0 0 . This provides a con t r a d i c t i o n and so A^ i s bounded. Remark on the proof: I f the operator norm i s denoted by II * l Dp » *-*ie s u f f i c i e n c y part of the proof insures that 11*^ 1^  1 ^  l l ^ l ^ p • However, there i s no such estimate i n the other d i r e c t i o n . In f a c t , when A =1 , c the evolution operator i s the zero operator. Since h can be thought of as the Hamiltonian of the system, we are p a r t i c u l a r l y interested i n r e a l tempered d i s t r i b u t i o n s (that i s , h [f] i s r e a l f o r a l l real-valued test f u n c t i o n s ) . In t h i s case, the l a s t r e s u l t may be strengthened to obtain Proposition 7.3 and Theorem 7.4. DEFINITION 7.2: Let K be. a. tineoJt operator on a Hilbert space. H . 7. K Is {ormally skew-adjoint (skew-henmttian) l{ (Ku,v) =-(u,Kv) {on. oXl u,v zV(X) • 2. K Is j>kew-symmetrlc l{ t?'(K) Is dense, and K is {ormally skew-adjoint. 3. K Is essentially skew-adjoint l{ the closure o{ K Is skew-adjoint. 4. K Is skew-ad joint l{ K Is skew-symmetnlc and R ( K ± I ) = H . ( R means the range o{ an operator) Vor these de{lnltions without the adjective "skew", see [19 ; chapter XI]. 45 PROPOSITION 7.3: Lzt h bz a nzal tzmpzrzd distribution on thz planz. Thzn H~c and ±A^ anz {onmatty 6km-adjoint. 1{ zithzn. onz iA dznizly dz{inzd, tt is Akzw-hymmztxiz. PROOF: Let us show i s formally skew-adjoint. The unitary equivalence then implies that H i s formally skew-adjoint. c (H'f.g) - J 2 H j f ( X ; L , x 2 ) g ( x l f x 2 ) dx xdx 2 for a l l f,geP(H^) R = H j f [ g ] = i//2^c S 1F.h(x 1,x„)[ c / 1 I f(z,x-)g(z,x„)dz - f ( x ? , z ) g ( x , , z ) d z ] R JR Use (6.7) and the f a c t h i s r e a l Slk S c l F 2 h ( x 2 ' x l ) [ g(z,x_)f(z,x )dz -R g(x 1,z)f(x„,z)dz ] R } Hlgtf] = - (f.H^g) . By a s i m i l a r argument and formula (6.7), i s formally skew-adjoint. The l a s t statement i n the theorem i s true by d e f i n i t i o n . THEOREM 7.4: H^ ti, a boundzd zAAZntiaXLy skew-adjoint opzhaloh. with domain 2 P(H c) =S(R ) i{ and only i{ i A c is a boundzd zi>i>zntialZy Akzw-adjoint opznaton. with ~V(h ) = S(R) . PROOF: One uses the fac t that a bounded skew-symmetric operator with dense domain i s e s s e n t i a l l y skew-adjoint ( i t s closure i s defined on the whole H i l b e r t space). The theorem i s then a d i r e c t consequence of Theorem 7.1 and Proposition 7.3. 46 As yet, we only know that the Weyl operators of functions i n 2 the Schwartz class on phase space are bounded operators on L ( R ) . The remainder of t h i s chapter i s intended to remove t h i s gap. P R O P O S I T I O N 7.5: for any bounded (respectively, bounded A, eZ{-adjoint) operator A defined on all o{ L ( R ) , there is a unique tempered distri-bution (respectively, real- tempered distribution) whose Weyl operator agrees with A on S ( R ) . P R O O F : I f A : L 2 ( R ) —> L 2 ( R ) i s bounded, then A S ( R ) : S ( R ) — 5' ( R ) i s a continuous operator. By the Schwartz Kernel Theorem [17 ; page 531], there 2 i s a unique h 1 eS'(R ) that s a t i s f i e s A<j>[ijj] = h1 [tyxty] f o r every cj),iJ;eS(R) . Introduce the tempered d i s t r i b u t i o n h as h = /2ifc ( S 1 h ' = v^rTc F „ 1 S h' . c z Z c T r i v i a l l y , by D e f i n i t i o n 6.6, A (h)<|> = Ac|> , for every <j> e S ( R ) . I f , i n addition, A i s s e l f - a d j o i n t , then the inner product equality shows that h'(x^,X2) = h'(x2>x^) . The converse of (6.7) implies that h i s r e a l . The preceeding propo s i t i o n asserts that there are many tempered d i s t r i b u t i o n s that have a corresponding bounded Weyl operator, but i t does not give any i n d i c a t i o n of the d i s t r i b u t i o n s i n t h i s set. Theorem 7.6 r e c t i f i e s the s i t u a t i o n . 4 7 THEOREM 7 . 6 : The Weyl operators o{ the {allowing distributions are continuous linear operators on L (R) with domain S(R) and bound specl{ied below. 2 2 a) l{ h belongs to L (R ) , A c is actually a Hilbert-Schmidt operator. on L (R) with kernel 1 / / 2 T T C F 2 h . Therefore, II A J ^ < H/llTc ||h||2. b) 1{ h is a {inite Radon measure [ 1 7 ; chapter 2 1 ] with total variation HhJ^, then H A J ^ < ( 1 / C ) ( / 2 7 T T ) Uhl^. In particular, this is true {or l?'-{unctions. c) 1{ h is, such a distribution, so is FV ch with ||A c(FV ch) |^  = 2/c | | A C | [ 3 P . PROOF: a) An obvious extension of Proposition 6 . 2 . b) For f eS(R 2) , i t i s easy to check s ' ^ h f f ] = F 2 h [ S c f ] = h f F ^ f ] . Since h i s a f i n i t e Radon measure, we have I A $[ty]\ = -|l / / 2 i f c S~1F0h[<f>x^] | f o r a l l <j>,^eS(R) c c t. = 1 / / 2 7 G " |h[F2Sc<f>xt(,] | < l//2~rrc ||h|| 1sup{|F 2S c<J.x^(x 1,x 2)| : ( x ^ x ^ e R }. But sup |F2 Sc (j)X<() (x ^ - . X j , ) | = sup x^,x 2 e R x l ' x 2 e R i(x„z) , cz, . , ,cz, , e 2 'tyix^-j ^ ^ ( x ^ - J d z R sup x^ e R Vc" (2/c) |<))(x -z) | |i|)(x.+z) |dz R < 2//c ||<r||2l|if'||2 b y Cauchy-Schwarz. By these two i n e q u a l i t i e s , | A <|>[ij>]| < ( l / c ) (/l/rr) ||h|| || <)> || 2 || if) || 2.. Hence, P(A c)=S(R) and || A C 1^  < ( l / c ) ( / 2 A r r ) ||h|| ± . c) Let us f i r s t demonstrate ( 7 . 1 ) given below. Let 6^  be the Dirac d i s t r i b u t i o n i n 5 ' ( R 2 ) (that i s , 6 Q [ f ] = f ( 0 ) ) . By formula ( 6 . 2 ) , one cal c u l a t e s that 2TT<5„ * f ( x ) = FV f(x) f o r each f e5(R 2) . Through the 0 c c a s s o c i a t i v i t y i n (6 .3 ) , FV (f * g) = ( F V c f ) * g for a l l f,geS(R 2) . The extension through d u a l i t y y i e l d s (7.1) FV £ (h * f) = FV h * f where f e S ( R 2 ) , heS'(R 2). Part c) w i l l now be shown. Let U = S ^F 2 . By Theorems 6.5 and 6.7, ~^lTxA c(FV chH = U{(FV ch) £ t f 1 (<f>xi|,) } when <j>,<lieS(R) = UFVc {h * U _ 1 } by (7.1) = U F V c U - 1 { ( J ) x A c ( h ) t } . Since U and F are unitary operators while 1 11 2 = l l ^ l ^ ' ||<J)xAc(FVch)^||2= (2/c) ||+xAc(h)ip||2. ' Therefore, V(A (FV h)) = S (R) and ||A (FV h) |l _ = (2/c) ||A (h) |l . c c " c c 'bp " c 'op Sections a) and b) of the above theorem were previously known, though i n a d i f f e r e n t s e t t i n g . They were communicated to the author i n the form of an unpublished paper by R. Anderson. 49 CHAPTER EIGHT MULTIPLIERS The bounded Weyl operators of Chapter Seven form an algebra under operator products. Through the Weyl correspondence, a m u l t i p l i c a t i v e structure i s established for the set of d i s t r i b u t i o n s with bounded Weyl operators. Certain subspaces S of d i s t r i b u t i o n s w i l l be i n v a r i a n t under t h i s m u l t i p l i c a t i o n and w i l l , perhaps, form noncommutative Banach algebras with respect to some norm. Following M. R i e f f e l [15], i f S i s a Banach space and * i s a continuous skew product on S, then those bounded operators M on S that s a t i s f y M(f *g) = Mf * g for a l l elements i n S are c a l l e d m u l t i p l i e r s on S. Theorem 8.1 demonstrates that the m u l t i p l i e r s on the Banach spaces we have been^considering must be tempered d i s t r i b u t i o n s . 2 2 THEOREM 8.1: a) Alt continuous lincan. opeAalpns M : 5(R ) — • 5' (R ) 2 that satls{y M(f og) = (Mf) og {on. all {unctions in S(R ) an.e. o{ the. 2 {onm Mf = hof {on. some heS'(R ) and conversely. 2 2 b) Alt continuous linejxn. operatons M : S(R ) —*• S' (R ) that satis {y 2 M(f *g) = (Mf) * g {on. alt {unctions in S(R ) are. o{ the. {onm Mf = h * f c c c 2 {on. some. heS'(R ) and conversely. PROOF: b) This follows immediately from a) by l e t t i n g MT = UMU 1 where U i s the usual operator S "hi^ and using the commutative diagram i n Theorem 6.5. The converse statement i s j u s t (6.3). a) If Mf = h o f , the equality (h o f) og .= h o (f o g) i s quickly v e r i f i e d by formula (6.5). 50 Conversely, assume that M(f og) = (Mf) o g . The Schwartz Kernel Theorem states that there i s an LeS'(R^) such that Mf[k] = L[kxf] f o r f , k e S ( R ) = L ( x 1 , x 2 , z 1 , z 2 ) [ k ( x 1 , x 2 ) f ( z , z 2 ) ] Since M permutes with the m u l t i p l i c a t i v e structure, L ( x 1 , x 2 , z 1 , z 2 ) [ k ( x 1 , x 2 ) R f(y»z 2)g(z 1»y)dy ] = M ( f o g ) [ k ] for f,g,keS(R z) by (6.5) = (Mf) o g[k] = M f ( x 1 9 x 2 ) [ g(y,x 1)k(y,x 2)dy ] L ( x ^ , x 2 , z ^ , z 2 ) [ R g(y,x 1)k(y,x 2)dy f ( z 1 > z 2 ) ] Into the above expression, substitute the functions f = ctxg , g = px8 , and k = <f>xip with the assumption that a, 3,p , 6, < j > a l l belong to S(R) . Then L ( x 1 , x 2 , z 1 , z 2 ) [(()'(x 1)ij)(x 2)p(z 1)e(z 2) jaB ] = L ( x 1 , x 2 , z ; ] ,z 2) [6(x 1)4)(x 2)a(z ] L)6(z 2) /pcj) ] or by extension, L(x 1,x 2,z 1,z 2)[<Kx 1 ) p(z 1)q(x 2,z 2) juQ ] =• L ( x 1 , x 2 . z 1 , z 2 ) [ 6 ( x 1 ) a ( z 1 ) q ( x 2 , z 2 ) /pcj) ] for qe5(R ) . In order to obtain the d i s t r i b u t i o n h , f i x an element q^ eS(R") and define D(q Q) : S(R ) <p by D(q Q) [p] = L ( x 1 , x 9 , z 1 , z 2 ) [ p ( x 1 , z 1 ) q 0 ( x 2 , z 2 ) ] f o r p e S ( R ) Then ^(QQ ) i - s a tempered d i s t r i b u t i o n that s a t i s f i e s , f o r a l l test functions on R, the equality ( J a 9)D(q Q) [<f>xp] = (/R*)D(qQ) [6x a] . 51 Suppose that ^CQQ) ^ s n o t t n e zero d i s t r i b u t i o n , then choose test functions ^Q ^O s u c ^ t n a t D ( l g ) [<(>QXPQ] ^ 0. We see immediately that (JPQ<)>Q/D(q^) [^^xp^]) D(q^) i s a p o s i t i v e tempered d i s t r i b u t i o n and hence a Radon measure. This measure m s a t i s f i e s a(y)9(y)dy = JR J R 2 J ( x 1 ) a ( x 2 ) dm(x 1,x 2) . Therefore, m i s the measure that assigns each point on the l i n e x ^ = x 2 unit mass ( i n standard notation, m = 6 ( x ^ - x 2 ) ). Set h(q Q) =D(q Q) [<j)0xpo]/Jpo(j)o for q Q as i n the l a s t paragraph. If D(q^) =0 , set Mqg) = 0. I t i s obvious that h does not depend on the p a r t i c u l a r choice of test functions on R and that, i n f a c t , h i s a tempered d i s t r i b u t i o n i t s e l f . This i s a d i r e c t consequence of the equation L ( x 1 , x 2 , z 1 , z 2 ) [ p ( x 1 , z 1 ) q ( x 2 , z 2 ) ] = h(q) J p(y,y)dy Perhaps we should change notation from h(q) to h[q] . Now choose a function p such that p(y>y)dy = l . From the R l a s t equation, we have L(x^,x 2, z^, z,,) = h ( x 2 , z 2 ) 6 ( x ^ - z^) . Thus Mf[k] = L ( x 1 , x 2 , z 1 , z 2 ) [ k ( x 1 , x 2 ) f ( z 1 , z 2 ) ] h ( x 2 , z 2 ) [ k ( y , x 2 ) f ( y , z 2 ) d y ] = h o f [k] where h(a,b)=h(b,a) . This completes the c h a r a c t e r i z a t i o n of the operators M 52 2 2 If f and g belong to L (R ) , then f *g i s defined as the unique tempered d i s t r i b u t i o n of Proposition 7.5 whose Weyl operator corres-ponds to the bounded operator A (f)A (g) . Due to the r e l a t i o n between 2 2 2 L (R ) and Hilbert-Schmidt operators on L (R) (Theorem 7.6 a ) ) , f * g i s c 2 2 2 i n L (R ) and the skew product Is continous with respect to the L -norm. - _ i _ 2 Any m u l t i p l i e r M on t h i s space i s c l e a r l y i n S'(R ) because 2 M r e s t r i c t e d to 5(R ) s a t i s f i e s the hypothesis of Theorem 8.1 b). If the operator f —y H^f i s replaced by f —>• h * f i n Theorem 7.1, the m u l t i p l i e r s are seen to correspond to bounded Weyl operators. Recording t h i s , we have: THEOREM 8.2: The multipliers on ( L 2 ( R 2 ) , *) are o{ the {Ohm Mf = h * f 2 {oh. some tempered distribution h whose Weyl operator, -is bounded on L (R) with domain S(R) and conversely. 1 2 The skew product on F(L (R )) may be defined as at the top of the page because Theorem 7.6 confirms that these d i s t r i b u t i o n s have bounded Weyl operators. However, i t i s not e n t i r e l y obvious that the skew product i s 1 2 again i n F(L (R )) and Is continous. Lemma 8.3 d e t a i l s a more constructive method of obtaining the skew product. With a l i t t l e more e f f o r t than needed i n Theorem 8.2, the m u l t i p l i e r s on t h i s space are characterized i n Theorem 8.4. 1 2 L E M M A 8.3: There -is a continous skew product on the Banach space F ( L (R ) ) 1 2 that has norm Induced {rom L (R ) . l 2 PROOF: Take f,g i n F ( L (R ) ) . By comparison to formula (6.1), l e t f * g ( x r x 2 ) R 2 f(x 1+cy 2/2,x 2-cy 1/2)e i ( x l ' x 2 ) ' ( y l ' y 2 ) F g ( y 1 , y 2 ) d y ]dy. 1 2 As f i s a uniformly continous, bounded function and Fg eL (R ) , the expression for f *g c e r t a i n l y produces an everywhere defined, continous 2 function. In addition, t h i s d e f i n i t i o n extends the skew product f o r S(R ) . Also, formula (7.1) may be rewritten for our s i t u a t i o n i n the form; FV, (f * g) = ( F V , f ) * g for a l l f,g i n F ( L 1 ( R 2 ) ) . Thus, " " ' l f C ^ ' C F L 1 ) = l l F ( f * S ) H i » H F V f c g ) l l l . = ! l ( F V c f ) c 8 " l 1 2TT R 2 2FV c f ( X ]+ cy?_/ 2, x 2- c y 2 ) e 1 ( X 1 ' X 2 } ' ( y l ' y 2 } Fg ( Y ] ,y 2) d y ; Ldy 2 R dx^dx2 < „- FV f ^ ||Fg||^ by Fubini's Theorem 2TT H f I' (FL 1) 'I 8 II (FL 1) ' 1 2 Therefore, f * g e F(L (R )) and the skew product i s continuous. THEOREM 8.4: The multipliers on ( F ( L 1 ( R 2 ) ) , *) are o{ the {onm Mf = h * f {on. some, tempered tiistribution h whose Fourier trans {onm is a {inite Radon measure and conversely. 1 2 • PROOF: Suppose that Mf = (Fm) * f for a l l functions i n F(L (R )) where m i s a f i n i t e Radon measure with v a r i a t i o n m . An easy modification of the argument i n Lemma 8.3 shows that HFCFm^f)!^ < ||m|| |[ f | j In addition, (Fm * f ) * g = Fm * (f '* g) f o r a l l f,g e F ( L 1 ( R 2 ) ) since a l l these d i s t r i b u t i o n s correspond to bounded Weyl operators by Theorem 7.6. 1 2 On the other hand, suppose that M Is a m u l t i p l i e r on F(L (R )) . 2 2 Theorem 8.1 implies there i s an heS'(R ) such that, for f,geS(R ) j i ) Mf = h * f and c i i ) . || F Cbi * g) J j 1 < b IJFgl^ for some constant b 54 Comparing the skew product given i n Lemma 8.3 to the usual convolution on oo 2 the plane, l e t us choose a sequence a e C„(R ) such that a —> 2TT6„ n 0 n 0 2 i n S'(R ) and ||a 1^ =2^  (see [19 ; page 157]). The Fourier transform 2 sequence has the l i m i t F a R —»- 1 i n S'(R ) and by an argument s i m i l a r 2 to the usual convolution, Fa i s an approximate i d e n t i t y f or S(R ). 2 (that -is, Fa * g —*- g i n S(R ) as n —*- ~ ). Using t h i s approximate i d e n t i t y , we have F V h [ f * g ] = lim FV h[Fa * (f * g) ] for f,geS(R 2) c c c n c c n-Ko = l i m (FV h * F a ) [f *g] by the d u a l i t y i n (6.3). c c n c But ||FV ch*Fa n|| 1 = j|FV c(h*Fa n)|| 1. by (7.1) < b ||F(Fa n)||^ by property i i ) of h = 2irb f or a l l n . Thus, | F V h [ f * g ] | < 2 7 T b || f ^  S Noo w h e r e l l k I L =sup{|k(x)| : x e R 2 } . This l a s t statement insures that there i s a f i n i t e Radon measure ? m that s a t i s f i e s FV ch[f *g] = m[f * g] for a l l f,geS(R ) . By ass o c i a -t i v i t y , (FV h) * f [ g ] = m * f [ g ] and so (FV h) * f = m * f . I t i s easy to c c c c c c see by (6.2) that FV^h = m. Hence, Mf = (Fm1) * f for some f i n i t e Radon measure m' . The reader f a m i l i a r with harmonic analysis w i l l notice the connection of t h i s theorem and proof with Wendel's Theorem [6 ; page 376]. 55 CHAPTER NINE REAL TEMPERED DISTRIBUTIONS In t h i s chapter, we w i l l examine further the equivalence of the evolution and Weyl operators. Only r e a l tempered d i s t r i b u t i o n s h w i l l be considered. By Proposition 7.3, H and iA are formally skew-adjoint on t h e i r respective domains. As i n Chapter Seven, these operators are regarded as acting on H i l b e r t space. However, H c and i A ^ are no longer required to be bounded. The cases when the two operators have skew-adjoint extensions w i l l be of p a r t i c u l a r i n t e r e s t as a means o f . s o l v i n g the evolution and Weyl equations. F i r s t , l e t us demonstrate that most Schrodinger operators associated to a p a r t i c l e moving i n one dimension are extensions of Weyl operators. The r e s u l t i s the analogue of Proposition 7.5. PROPOSITION 9.1: Tor any symmetric operator A whose domain includes 5(R) , there -is a'unique leal tempered d i s t r i b u t i o n such that A agrees with the corresponding Weyl operator on a l t o{ S(R) . PROOF: For <peS(R) with ||<j>||? < 1, define a map T^ : S(R) -> (f by T± (ip) = (Acp.ip) . If <p i s f i x e d i n S(R) , then {T ( ip)} i s bounded i n $ since sup{|T 0J))|} = sup{| (Acf . ,^ ) |} = sup{|(<|>,Ai|))|} < 11Asp where the sup i s taken over the above cp with norm at most one. By the uniform boundedness theorem [19 ; page 68], T^ (ip) goes to zero uniformly i n cp as ip —*- 0 i n 5(R) . In p a r t i c u l a r , i f B i s any 56 bounded set i n S(R) , then sup { | T Op) | : i|i E B , cp e S (R) , ||<p||2 < D i s f i n i t e . Thus, for any a>0 , there e x i s t s a 6>0 such that ||<p||2< ^ ^ p_(<p) = sup{ I Acj> [ ip ] I : ip e B} < ct . JD Since the family of semi-norms p w define the strong dual topology on S(R) , the map <p —>-Acp i s continuous from S(R) to S'(R) . We now continue as i n the proof of Proposition 7.5 to obtain the associated r e a l tempered d i s t r i b u t i o n h . Proposition 7.3 can be improved to show the domains of the evolution and Weyl operators are c l o s e l y r e l a t e d . As we are now dealing with unbounded operators, the proof of the following theorem i s much more d e l i c a t e than that i n Theorem 7.1. The r e s u l t i s not true for general tempered d i s t r i b u t i o n s . -1 2 For example, when F^h = u ^ X u 2 where u^ belongs to L (R) but u 2 does not, then V(k (h)) i s a l l of S(R) while V(k (h)) i s not dense 2 2 2 i n L (R) and ^ ( H c ) i ' s n o t dense i n L (R ) . Of course, h i s not a r e a l tempered d i s t r i b u t i o n . T H E O R E M 9.2: Let h be a real tempered distribution. H C is densely defined i{ and only i{ k^ is densely defined. In other, words, H C is skew-symmetric. i{ and only i{ ±k^ is skew-symmetric. PROOF: As always, i t s u f f i c e s to prove the statement with H ^ i n place of H . c Assume f i r s t that A i s densely defined. Since h i s r e a l , 57 Theorem 6.7 states that H^cpx^) = t^xA^ - A^x^) for a l l cfi,ip e T7(A ) . Therefore, the domain of H' includes a l l f i n i t e l i n e a r combinations of c functions i n P(A ) x£>(A ) . As V(A ) i s dense i n L (R) , P(H') w i l l c c c c 2 2 be dense i n L (R ) . Now assume that H/ i s densely defined. Let fy^ e5(R) and • j f ( H ' ) be a r b i t r a r y . I t w i l l be shown that the domain of A includes 0 c J c R f Q ( y , x ) <f>Q(y) dy . For every \p e S (R) , we have iA R f Q ( y , x ) <()0(y) dy ,-1, = 1//2TTC S F . h ( X l , x _ ) [ C 2. X L R f Q ( y , x 1 ) <j>0(y) dy ^(x 2> ] = H'f0f<j>0x,j,] + i / / 2 ^ c S c 1 F 2 h ( x 1 , x 2 ) [ j An easy c a l c u l a t i o n gives the d i s t r i b u t i o n a l e q u a l i t y : f 0(x 2,y)i|»(y)* 0(x 1)dy] « J S ^ M x ^ ) [ f Q ( x 2 > y ) ^ -( X ]) ]t|) (y)dy f Q ( x 2 , y ) <Ky) dy <(.0(x1) ] S c 1 F 2 h ( x 1 , x 2 ) t The expression F 2h(x^,x 2) [f ^ (x2,y)cf>g (x^) ] i s c l e a r l y i n S(R) as a 2 function of y . Let the L -norm of t h i s function be bounded by b . Then m • iA c f 0 ( y , x ) ( f ) 0 (y)dy < !|H;f 0|| 2||* 0|| 2IMI 2 + b / ^ c ||*||2 = b' ||^||2 for some constant b' . Thus, the domain of A contains a l l such functions. c 2 Suppose that the functions of t h i s form are not dense i n L (R), There e x i s t s some test function <j)^  eS(R) and a p o s i t i v e number a such that, f o r a l l f eP(H') , c i ) || * Q || 2 = 1 and i l ) R f(y,x)<j) (y)dy Hence, a < R f (y,x)<j>0(y)dy - tyQ (x) dx <j>0(x) > a R {f(y,x) - 4>0(y)<!>0(x)}<f>0(y)dy dx by i ) 2 2 <f>0l| |f(y,x) - (j>0(y)<()0(x) | dydx by Cauchy-Schwarz = P - V * 0 H 2 2 . 2 2 This cannot happen because P(H^) I s dense i n L (R ) . Therefore, A c densely defined. If . h i s r e a l , the proof of Theorem 7.1 can be adapted to show that H i s bounded on a dense set i f and only i f iA i s bounded on a c J c dense set. Since a skew-symmetric operator bounded on a dense set i s nece s s a r i l y bounded on i t s e n t i r e domain, an immediate consequence of Theorem 9.2 i s the following simpler version of Theorem 7.4. COROLLARY 9.3: ti, a bounded essentiaJULy skew-adjoint opoAatofi i{ and only l{ i A c ti, also. The next two r e s u l t s demonstrate that, f o r c e r t a i n d i s t r i b u t i o n s the evolution operator i s indeed equivalent to the Weyl operator. The condition on the d i s t r i b u t i o n h that produces the Hamiltonian of the system i s seen to be p h y s i c a l l y s i g n i f i c a n t . 59 PROPOSITION 9.4: Let h = h^+Fh^ be a real tempered distribution where h.^ and h 2 are (not necessarily real) distributions with compact support. Then a) V(R ) = S(R 2) and V(k ) = 5(R) c c and b) H/ ii> in the closure o{ i { l 8 A £ - A c ® 1} as operators. PROOF: The proposition w i l l follow from formula (7.1) and the proof of -the "statement for h having compact support but perhaps not r e a l . a) By the l o c a l structure of compactly supported d i s t r i b u t i o n s [17 ; page 256], there i s a p o s i t i v e integer k and an r such that h[f ] < b sup{ |D mf (y) | : 0<|m|<k, |y|<r} for a l l f i n S(R 2) where b i s some constant depending on h and the notation m= (m^,m2) , |m| = m^+m2 i s the usual multiindex notation for d e r i v a t i v e s . Therefore, | h * f ( x 1 , x 2 ) | = | i h ( y i , y 2 ) [ e - 1 ( x r X 2 ) - ( - 2 V C ' 2 V C ) F f 2 ( x 2 - y 2 ) j 2 ( y 1 - x 1 ) < b' sup D(V) f"1 '*2> ' ('2Yl/C'2Yl/C) Ff (2 ( X 2 " y 2 > / c . 2 . ( 7 ^ ) / c ) } where D™^ means the d e r i v a t i v e with respect to the y va r i a b l e s and the sup Is taken over the same set as i n the l o c a l structure of h . The f i r s t l i n e i n the expansion of h * f i s j u s t (6.2) with the various operators removed. The derivatives with respect to y w i l l produce polynomials i n x mu l t i p l i e d by derivatives of Ff evaluated at a translated point. Because Ff e S ( R ) , |h * f (x) | w i l l d ecrease fa s t e r than any polynomial as |x| — 2 2 2 In f a c t , as f approaches 0 i n S(R~) , h * f w i l l approach 0 i n L (R ). — — 2 The same argument applied to h * f demonstrates that ^(H-c) =S(R ) . A s l i g h t v a r i a t i o n of Theorem 9.2 implies P(A (h)) =fl(A c(h)) =S(R) . 60 b) Let f be an a r b i t r a r y element i n the domain of . Since the domains of A c(h) and A c ( b ) are a l l of S(R) , there i s a sequence f i n = 2 V(l 0 A c(h) T A c(h) 0 I) that converges to f i n the topology of S(R ) . 2 2 This being the case, we know that H^f n converges to H/f i n L (R ) by the proof of a). As H/f n = i { l 0 A c(h) - A c(h) 0 I}f , section b) i s shown. Remark: I t should be pointed out at t h i s time that there i s a product r u l e for the d e r i v a t i v e of the skew product. The ru l e i s (compare, [19 ; page 156]) • !x\?f + hSfx\ f o r f ^ ( R 2 ) , h s S ' C R 2 ) , i = l , 2 . 3 3 1 2 2 With t h i s d e r i v a t i o n , one can show that H c : 5(R ) —> S(R ) i s a continuous map when h has compact support. T H E O R E M 9.5: . let h = h 1 + F h 2 + h 3 be a real tempered distribution where. h^ and h 2 have, compact support and corresponds to a bounded Weyl operator with domain S ( R ) . Then a) H c is essentially skew-adjoint i{ and only i{ i A ^ is essentially skew-adjoint. b) The closure o{ i A c generates a strongly continuous unitary group V(t) 2 Id) on L ( R ) that solves the Weyl equation = iA <f> i{ and only i{ the ' C l t c closure o{ H generates a strongly continuous unitary group w(t) on 2 2 d f L ( R ) that solves the evolution equation ™ = H o f . Moreover, w(t) is the closure o{ F 2 1 s c ( v(t) 0 v(t ) ) s c xF 2 PROOF: Proposition 9.4 remains v a l i d with the ad d i t i o n of the d i s t r i b u t i o n that has an associated bounded Weyl operator with domain 5(R) . a) Suppose i A ^ i s e s s e n t i a l l y skew-adjoint. Obviously, iA i s also 61 e s s e n t i a l l y skew-adjoint. By using the r e s o l u t i o n of the i d e n t i t y f o r the operators i A c and iA , » Ju. Berezanskii [3 ; VI §4 of the English t r a n s l a t i o n ] proves that t h i s separation of va r i a b l e s produces an e s s e n t i a l l y skew-adjoint operator I 8 i A c + i A c 8 I . This proof i s not reproduced here as many new concepts would have to be introduced. Since H' i s an extension of t h i s c tensor product operator, i t i s e s s e n t i a l l y skew-adjoint. For further r e s u l t s along t h i s l i n e that apply to tempered d i s t r i -butions which are not r e a l , the reader i s encouraged to see [7] or [14]. For the converse, assume i s e s s e n t i a l l y skew-adjoint while iA ^ i s not. By the theory of defi c i e n c y i n d i c e s , one of the subspaces - 1 2 R(LA ±1) = {ueL (R) : ( ( i A ±I)<J>,u)=0 for a l l <J>eP(iA)} i s n o n - t r i v i a l . Without loss of gener a l i t y , assume that u^ i s a non-zero element i n R(iA +1) . Let <|>,ipeP(iA ) be a r b i t r a r y . Then • c c ( ( I T +21)<j)xtp,u0xu0) = ( { I x ( l A c + I ) + ( IA C + I) x D H . y V = (cf,,u0)((iAc+I)<j,,u0) + (u 0,(iA c+lH)(,f,,u 0) « 0 . Since H' i s i n the closure of I 8 iA + iA 8 1 , u„x u. i s i n the set c c c 0 0 _L R(H' +21) . This contradicts the e s s e n t i a l skew-adiointness of H . c c b) I f h i s a d i s t r i b u t i o n as i n the statement of the theorem, then, by Stone's Theorem, the closures of the two operators generate stongly contin-uous unitary groups f o r the same set of d i s t r i b u t i o n s . Only the r e l a t i o n between W(t) and V(t) i s i n doubt. Let W'(t) be the closure of the operator V ( t ) 8 V ( t ) defined 2 2 on L (R ) . W'(t) i s a strongly continuous unitary group with domain a l l 62 2 2 of L (R ) . The generator of W (t) i s an extension of the tensor product i { I 8 A - A 8 1} because c c l l m ^ L m ^ ^ ± . l i m v ( t ) ^ v ( t ) t - w f o r a l l ^ e t 7 ( A ) t-K) C t-K) 11 C = l i m ^ ^ H ~ ^ x V C t H + <|> x (V(t)i|< -ip) t - 0 t = llm m&zA x l i m V ( t ) ^ + • x l i m v(t)»-» t-»-o t->o t+o c = iA ()> x \p + (j, x iA I|J . Thus, the closure of IV generates W (t) . The unitary equivalence established i n Theorem 6.5 implies the closure of generates F ^ S W' ( t ) S " 1 F . Z C C 2. So f a r , we have seen only those r e s u l t s that support the equivalence of H c and i A £ . These l a s t few pages are an attempt to j u s t i f y the study of the evolution equation on i t s own. The geometry of the plane i s used to point out how the equivalence may break down. No s p e c i f i c counterexample i s provided and, indeed, our e f f o r t to produce one has been unsuccessful to t h i s time. The unpublished paper of R. Anderson r e f e r r e d to on page 48 must be acknowledged at t h i s moment. In i t , the evolution operator of a r e a l , compactly supported d i s t r i b u t i o n was shown to have dense domain (compare to Proposition 9.4 a)) and, i n the case of odd d i s t r i b u t i o n s , to have a skew-adjoint extension (Theorem 9.7 a ) ) . The statement of Lemma 9.6 on the next page i s reproduced almost verbatim from the paper. 63 L E M M A 9.6: J{ a skew-symmetric operator K is unltarlly equivalent to -K, then K has a skew-adjoint extant,ion. -L J_ P R O O F : I t s u f f i c e s to prove that dim{R(K + I) } = dim{R(K-I) } by the theory of def i c i e n c y indices [19 ; page 349]. Let UKU = -K be the 2 2 assumed unitary equivalence. Then, with f ^ an a r b i t r a r y element of L ( R ) , ( ( K - I ) g , U _ 1 f 0 ) = ( U ( K - I ) g , f 0 ) f o r a l l gefl(K) = ( ( U K U - 1 - I ) U g , f 0 ) = ( ( - K - I ) U g , f Q ) = -((K + I)Ug,f 0) . Therefore, U _ 1 f 0 belongs to R(K-I) 1~, i f and only i f f Q i s i n RCK + I)" 1". As U i s unitary, the r e s u l t i s proved. T H E O R E M 9.7: Assume, that H C is densely defined and that any o{ the following conditions are true {on. the neat tempered distribution h . Then H lias a skew-adjoint extension. c J 2 a) h iA odd (that is, h(x 1,x 2) [f ( x ^ x ^ ] = - M x ^ x ^ [f (-x^-x^ ] , f eS(R ) ) b) M.h = -h where M . AJ, a rotation in the plane through an angle 6 . c) h ts re{lexive in any line through the origin, for example, h is re{lexive in the x^-axls i{ h ( x 1 , x 2 ) = h(x 1,-x 0) . P R O O F : a) This follows from section b) with the angle of r o t a t i o n TT radians. b) By Lemma 9.6, i t s u f f i c e s to f i n d a unitary operator that e f f e c t s the equivalence of H to - H . To t h i s end, l e t Uf(y) = f ( M . y ) . Then, f o r C C 0 a l l f eP(H cU) , we have by (6.4) U "H Uf (X) = H Uf (M Qx) c c -6 = 2 ^ { V C T ( M _ e x ) h [ F T ( M _ e x ) U f ] - V _ C T ( M _ e x ) h [ F T ( M _ Q x ) U f ] } . To write t h i s as an evolution operator, use the r e l a t i o n s below that permute the operators i n the l a s t expression and are r e a d i l y checked. i ) T ( M „x)Uf = Ur(x)f i i ) UFf = FUf i i i ) h[Uf] = if-Sitf] i v ) U"V h = V U - 1 h ±c ±c v) U " " 1 T ( M Q x ) h = T ( x ) U _ 1 h . Thus, U - 1H Uf(x) = ^ { V T ( x ) U _ 1 h [ F T ( x ) f ] - V x(x)U _ 1h[Fx(x)f]} . C ZTT C —C Since U - 1 h = -h , we have if "Si Uf = -H f . c c c) The same method as above i s used but with Uf(y) = f(y) where y i s the r e f l e c t i o n of y i n the given l i n e through the o r i g i n . Again the operators are permuted'after c a l c u l a t i n g the various r e l a t i o n s . These r e l a t i o n s are b a s i c a l l y the same as i ) to v) except that UV ch(x) = V^UhC-x) = V_ cUh(x) and UV h = V TJh . Combining these, we obtain -c c U _ 1H Uf(x) = ,f {V T(x)Uh[Fx(x)f] - V x(x)Uh[Fx(x)f]} C ZTT - C C = -H f since Uh = h . c Remark 1: The o r i g i n appears to play a c e n t r a l r o l e i n the theorem. This i s rather misleading because the symmetries about the o r i g i n are considered due to computational convenience as opposed to any i n t r i n s i c property associated to t h i s point. Any other point would do equally as w e l l since the evolution operators corresponding to h and x(x^)h are u n i t a r i l y equivalent under Uf (x) = f ( x + x n ) . 65 Remark 2: The purpose of the theorem i s to suggest cases i n which the two operators H and iA are not equivalent. Of course, they w i l l not be c c equivalent when one has a skew-adjoint extension but the other does not. From section c) of the proof, we are struck by the fact that both terms h * f and h * f i n the evolution operator are needed to insure a skew-c c adjoint extension. However, the Weyl operator i s defined through only the f i r s t term and so there i s no immediate reason why iA should have c a skew-adjoint extension. Therefore, i t i s p l a u s i b l e that the phase space formulation of quantum mechanics i s not equivalent to the usual theory on configuration space. At the very l e a s t , the symmetries of phase space contribute to the study of the evolution equation and, u l t i m a t e l y , of the Weyl equation. BIBLIOGRAPHY Anderson, R.F.V., The Weyl Functional Calculus, Journal of Functional Analysis, 4 (1969), 240-269. Anderson, R.F.V., The Multiplicative Weyl functional Calculus, Journal of Functional Analysis, 9 (1972), 423-440. Berezanskii, Ju.M., "Expansions i n Eigenfunctions of S e l f a d j o i n t Operators", Naukova Dumka, Kiev 1965; English t r a n s l a t i o n , Translations of Mathematical Monographs, volume 17, American Mathematical Society, Providence, R.I., 1968. Dixmier, J . , L'Application Exponentielle dans les Gn.oupes.de Lie ReSOlubleS, B u l l e t i n Societe Mathematique de France, 85 (1957), 113-121. Helgason, S., " D i f f e r e n t i a l Geometry and Symmetric Spaces", Academic Press, New York 1962. Hewitt, E». , and Ross, K.A., "Abstract Harmonic Analysis I I " , Springer-Verlag, B e r l i n 1970. Ichinose, T., OpehatohS on Tens Oh. VnoductS o{ Banach Spaces, Transactions of the American Mathematical Society, 170 (1972), 197-219. Jordan, J.F., and Sudarshan, E.C.G., Lie Gh.oup Dynamical fohmalism, Reviews of Modern Physics, 33 (1961), 515-524. Kobayashi, S., and Nomizu, K., "Foundations of D i f f e r e n t i a l Geometry, Volumes I and I I " , Wiley (Interscience), New York 1963. Loupias, G., and Miracle-Sole, S., C*-Algebnes des Systemes CanonlqueS, II, Annals I n s t i t u t e Henri Poincarg, Section A 6 (1967), 39-58. Moyal, J.E., Quantum Mechanics as a Statistical Theony, Proceedings of the Cambridge Philosophy Society, 45 (1949), 99-124. 67 12. Murnaghan, F.D., "The Theory of Group Representations", Dover Publi c a t i o n s , New York 1963. 13. Pukansky, L . , On the Unitary Representations o{ Exponential. Groups, Journal of Functional Analysis, 2 (1968), 73-113. JL.4.—Reed, M., and Simon, B., Tensor Products o{ Closed Operators on Banach Spaces, Journal of Functional Analysis, 13 (1973), 107-124. 15. R i e f f e l , M . , Multipliers and Tensor Products o{ L -spaces o{ Locally Compact Groups, Studia Mathematica, 23 (1969), 71-82. 16. Rubin, H., Proceedings of the International Symposium on the Axiomatic Method at Berkeley, North-Holland Publishing Company, Amsterdam 1959. 17. Treves, F., "Topological Vector Spaces, D i s t r i b u t i o n s and Kernels", Academic Press, New York 1967. 18. Weyl, H. ,• "The Theory of Groups and Quantum Mechanics", Metheuen and Company, London 1931. 19. Yosida, K., "Functional Analysis, Third E d i t i o n " , Springer-Verlag, New York 1971. 

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