AN ABSTRACT OF THE THESIS OF
BERNARD W. BANKSfor the degree DOCTOR OF PHILOSOPHY (Name) (Degree) in MATHEMATICS presented on (Major Department) (Date) Title: TIME DERIVATIVES OF OBSERVABLES AND
APPLICATIONS Redacted for Privacy Abstract approved: Stuart Newberger
LetA andH be self -adjoint operators on a Hilbert space. Conditions for the differentiability with respect totof -itH -itH
by Bernard W. Banks
A THESIS submitted to Oregon State University
in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1975 APPROVED:
Redacted for Privacy Associate Professor of Mathematics in charge of major
Redacted for Privacy
Chai an of Department of Mathematics
Redacted for Privacy
Dean of Graduate School
Date thesis is presented March 4, 1975 Typed by Clover Redfern for Bernard W. Banks ACKNOWLEDGMENTS
I would like to take this opportunity to thank those people who have, in one way or another, contributed to these pages. My special thanks go to Dr. Stuart Newberger who, as my advisor, provided me with an inexhaustible supply of wise counsel. I am most greatful for the manner he brought to our many conversa- tions making them into a mutual exchange between two enthusiasta I must also thank my parents for their support during the earlier years of my education.Their contributions to these pages are not easily descerned, but they are there never the less. Finally, I want to thank Don Solomon, Greg Fredricks and Gary Greenfield for the special atmosphere of enthusiasm for mathematics that has existed between us during our years as graduate students.I learned much from them. TABLE OF CONTENTS Chapter Page
INTRODUCTION 1 The Born Interpretation 1 Ehrenfe st cs Theo rem 2 Bound and Unbound States 3 Stone's Theorem for Real Hilbert Spaces 4
TIME DERIVATIVES OF UNBOUNDED OBSERVABLES 5 Potentials and States 32 The Polynomials Po /ax) 34
APPLICATIONS 37 Ehrenfest's Theorem 37 2 3 The Coulomb Potential on L(R) 38 The Case of Smooth Potentials 40 The Behavior of the Mean 43 The Absolutely Continuous States 47 The Wave Operators 48
BOUNDED OBSERVABLES 55 Multiplication Operators 59 Application 61 Integral Operators 62
A REAL STONE'S THEOREM 65 The Complexification of H 65 Operators on Hc 66 Complexification of a Real Operator 70 A Spectral Theorem for Real Operators 71 Further Properties of El and E2 78 The Uniqueness of E1 and E2 82 A Function Calculus for A 84 Stone's Theorem--Real Case 85
BIBLIOGRAPHY 92 TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS
INTRODUCTION
The Born Interpretation
Let 9be a normalized state function defining the state of a quantum mechanical system.For a long time in the development of quantum mechanics it was not clear how to interpret 9 physically.
2 In the 1920's Max Born proposed that I 9 I should be regarded as the probability distribution of the position of the particle in state cp.
Thus the probabilitTthat the particle is in regionR is I912dx. Suppose thatH = Ho + Vis the Hamiltonian of a particle in a potential fieldV, then in the Schrodinger picture the fundamental dynamical law of quantum mechanics is that if i(x, t)is the state
t, then4(x, t)- iFhp(x, t). at time at That is,Ox, t)satisfies the Schrodinger wave equation.Well known considerations connected with Stone's theorem show that the solution inL2(R1)is itH Ox, t)-= e where may be thought of as the initial state at t = 0.Thus,eitHco is the state at any timet.
For eacht,eitHis unitary, and so if 11911= 1 then
II itHii 2 itHitH 1.The Born interpretation now says that some self -adjoint operators can be interpreted as observable quanti- ties.For such an operatorA the quantity
Indeed, ifAis multiplication byx., which has a natural inter-
) E Rn), pretation as the j-coordinate of position(x =(x1" xn itH itH e yo> x. eitHyo I 2dx, which is exactly the then If we setA(-o)(t) = Ehrenfest's Theorem dx Consider a particle of mass 1.In classical mechanics dt d2x is the momentum and is the force ifx(t) is the position of dt2 the particle. In the quantum mechanics the expected value of momentum in coordinate direction is given by.The thejth a x. question arises whether the derivative of the mean ofx,(?.(49)(t)), 3 3 is the expected value of momentum. Moreover, is the second deriva- tive of the mean ofx.,(x'((p)(t)), equal to the mean or expected a V < eitHitHccp,e > ?Paul value of the force which is given by Ox. 3 3 Ehrenfest [9, 455] asserted that the answer is yes!Note that in bibliographic citations the first number locates the reference in the bibliography and the second is the page number. Ehrenfest's justifi- cation of his assertion was not rigorous, nor could it be, for he gave no hypotheses. Here we shall give sufficient conditions under which Ehrenfest s Theorem is true. This will involve us in the question of the differentiability of means to which the bulk of Chapter I is devoted.It is in Chapter II that Ehrenfest's Theorem is taken up. Bound and Unbound States The theory of self -adjoint operators on a Hilbert space recognizes both eigenstates and absolutely continuous states. We can ask what the behavior ofx.(99)(t) is when cp is an eigenstate or an absolutely continuous state.For an eigenstate, isx.(P)(t), bounded? an absolutely continuous state, does x.- (cpFor)(t)converge to astconverges to co? We address ourselves to these questions in Chapter II. In Chapter III we shall give some results on the differentiability of the functionA((p)(t) where Ais assumed bounded. Under this assumption we no longer require Ato be self-adjoint. 4 Stone's Theorem for Real Hilbert Spaces LetU(t)be a strongly continuous group of unitary operators on a complex Hilbert space.Then Stone's theorem says that itH U(t) = e , for some unique self-adjoint operator H, and Ut(t) = iHeit. In Chapter IV we shall formulate and prove Stone's theorem for real Hilbert spaces. To some extent Chapter IV is independent of the preceding chapters. On the other hand,it is quite reasonable to ask about the differentiability of means on real Hilbert spaces.But then a Stone's theorem in the real case becomes indispensable, and so the material of Chapter IV is not as independent of the main theme of this paper as it might at first appear. 5 I.TIME DERIVATIVES OF UNBOUNDED OBSERVABLES LetA and H be self-adjoint operators on a Hilbert space A((p)(t) A--,((p)(t,) exists and is equal to 9such thatdA(9)(t)dt 1 -itH -itH a , a a , Ais a polynomial in In these two cases ax ax ° ax we shall assume thatH = + Vwhere I-I is the self-adjoint H0 0 realization of the Laplacian =- L2and V is a suitable ax. potential. We shall begin by considering any self -ad joint operatorsA and Hon a Hilbert spacefi The first theorem gives a general criterion for the differentiability of A(co)(t).IfAis any operator we shall useD(A)to denote the domain ofA. We shall assume all operators are densely defined, but not not necessarily bounded unless it is so stated. Theorem 1.1.Let A be a symmetric operator andH a self- ad joint operator on a Hilbert space Suppose E -it 0(pH - e D(HA-AH),e-itHcp, D(A), andhAe II is bounded itH -itH for alltin a neighborhood ofto' andA((P)(t) = Then -itoH -itoH A1(9)(to) = Proof.Without loss of generality setto = 0,for otherwise we can -itoH replace 9 bye p in the following argument. itH -itH (9)0) = lim -itHco-e-itHA9 -itH = lim< ,e 9> t 0 e-itHco-Aco-fe - itHA9 -A9] - itH = lim< , e > tø.0 e-itH49-A9 -itH e-itHAco-A9 -itH lim< ,e (p> - lim < ,e 9> t 0 t 0 Ae-itH49-A49 -itH tourn< , e co> + We next show that -itH Ae 9 - A9 - urn < ,eitH(p> = . t 0 We have 7 Ae-itHyo-Acp ,e-itH(p> <-iA1-1(p, co> I -itH =I < e cp,Ae (p> --itH-Ae-itH(p>+ AeitH(p> - < A(p>I sinceAis symmetric, I-1(pE D(A), and by adding and subtracting <-iHcp, Ae-itH(p>. Thus, Ae-itHgo - Acp - itH < ,e (r>- (p>1 -itH + AHT e- itH49-(P < II e t iffirdiII Ae-itH II But the right side of this inequality tends to zero astconverges to zero since II Ae-itHcp II is bounded in a neighborhood of zero. q. e. d. Corollary 1.2.IfA and H are bounded self-adjoint operators, then 7511(ep)(t) exists for allt, and - itHp> :711 ((p) (t) = X(v)(t)we must establish the following three things: e-itH(pE D(A)for alltin a neighborhood ofto II Ae-itHcoll is bounded fortin a neighborhood of -itr,H and 3) e u E D(HA-AH). 8 To this end we shall have need of the following general concepts and lemmas. Definition 1.3.LetX be a Banach space andB: (a, b] Xbe a function on the interval[a, b]with values inX. Then we define B(t)dt = lim B(t,')At. a Ati II 0 i=1 if the limit exists in norm. This is the usual Riemann integral of a function with values in a Banach space.This integral exists if B is continuous. Lemma 1.4.Let A be a closed operator onX, and suppose that b bB(t)dtand AB(t)dtexist.Then B(t)dt E D(A), and a a a bB(t)dt= bAB(t)dt. ASa a Proof. B(t)At, = AB(t)t. 1 1 I. 1 n=1 n=1 Now B(t)dt, n2--1 9 and B(t. )Lt. AB(e)a. AB(t)dt . 1 1 a 1= 1 i=1 But Ais closed hence B(t)dt E D(A), and ab B(t)dt = AB(t)dt . q.e.d. a a Corollary 1. 5. Let A be a closed operator onX, and suppose L70 that the improper integrals B(t)dtand AB(t)dtexist.Then a a 00 00 B(t)dt E D(A), and AsB(t)dt = AB(t)dt . a a a Proof. We have lirn bB(t)dt B(t)dt =boo a and Sc°AB(t)dt = lim bAB(t)dt. a boo a But, Bet)dt AB(t)dt a a by Lemma 1.4.Thus, 10 lim A B(t)dt = AB(t)dt. Again, Ais closed, so oo B(t)dt E D(A), and B(t)dt = AB(t)dt. a a q. e. d. In the sequel we shall often be concerned with integrals over paths in the complex plane of the following type. Definition 1.6.Letrcbe the path defined byz(t) = c + itand z (t) = -c + itwhere t E(-°0,()°). IfT is an operator and X a complex number we shall - (X-T)1 writeR(X, T) = for the resolvent ofT. We sayf().)is of order g(X) at X cc provided that f(X) is. bounded for large values of X . g(X) Lemma 1.7.LetT(t)be a strongly continuous group of unitary operators on the Hilbert space , and letT be the infinitesimal generator.Let c be a positive real number, and a a complex number with0 < c < I R(0)1, R(0-) the real part ofa.Then, for E D(Tn), withnan integer, n > 2, 11 T (t )cp =--1- eXt R(X, T)(c-LI-T)n(pdX . 2Tri (a_mn co -X Proof.First we note thatR(X, T) = e tT(t)dt,(see [1, 622]), forX E p(T).Moreover, R(X, T) II< K e((c/4)-R(X))tdtfor R(X) >-4 , so R(X, T )11 is uniformly bounded in the half-plane R(X) >. Similarly II R(X, T)11 is uniformly bounded in the half- -c planeR(X) < . Thus, the integrand is of order!XI-nas onr and so the integral t)(0-I-T)ncpdX j1-SeXtR(X,(a_mn is well-defined. 1 eXtR(X, T)(aI-T)119 dX . Let cp ED(Tn)and let 13(t)(P = 2Tri (a_mn Fc Now choose p. so thatR(1j) > c, and assumet > 0,then 1 extR(X,T)(cLI-T)119 B(t)codt c°e(X-p.)tdtdX 2Tri ( So°° a-x) 0 Fc 1 R(X,T)(aI-T)ncodX 2Tri (N.-X)(a-Mn Because of the order of the integrand this last integral can be replaced by integrals over two small negatively oriented circles about 12 a andi.These integrals can then be evaluated by the residues at a and L. The Residue at Immediately we see that the residue at p, is -R(11, T)(aI-T)nyo The Residue at a:Consider f(X) -It(X' T) (-1)n+1 R(X, (11-)(a (h- con (X -p,) R(X. , T) We must compute the -1)-thderivative of - (X-la)-1R(X,T ). 0-[-L) n-1 dr1-1(X-p.)R(X,T) = )Dn-l-m(X-p.)-1DmR(X, T) dxn-1 m=0 n-1 n-1)(_1)(n-l-m) (n-l-m)! DmR(x,T) rn n-m m=0 whereD = ButDmR(X, T) =-1)mm ![R(X, T )1m+1, SO dX 00 dn-1 -1 [R(X., T )]m+1 (X-P4 R(X,T) - 1(n-1) !(x_on-m The residue atais 13 00 -1-m (aI-T) 9 (aI-T)cp (aI-T)n-1c9 (a-P-) (a-02 rn (al.on-m -cp (aI-T)R(p., T)(P But we have the identityR(I, T)9 - + so we have in general: -1 ( I-T)kk-1 (aI-T)R(p.,T)(ca-T)k-lcp (a-p4(a-P-)k-1 (aOk Making substitutions and telescoping we get R(11, T )( a I-T )n9 , R(11,T)(aI-T)n9 e-tB(tdt)cp - + R (11, T )49- 0 (a-.L)n (a-On = T)cp = e-iltT(t)cpdt . 0 Now apply linear functionals to both sides to see thatB(t)cp = T(t)q' for allt >0and all E D(T). NowT(-t)is a unitary group with generator-T.So we have X 1 et R(X, T(t)_cp -T)(aI+T)ncpdX ZITI 'c (a-X)n fort > 0Substituting -X. for X and recalling that 14 R(-), -T = -R(X,T)we see that 1 e-xtR(X,T)(aI+T)ricp T(4)9 - (a-mn dX. 2Tri rc If we put-a fora and-tfort,we get eXtR(X,T)(aI-T)ncp 1 dX , fort <0. T(t)- S (a-mn Thus, B(t)cp = T(t)9for allt. q. e. d. Lemma 1.8. Let Abe a closed operator and Ha self-adjoint operator on a Hilbert space1-1 . Let cp and suppose that for an integern > 2and all X Erc,R(X,iH)(aI-iH)ncpE D(A), and ix.1n-(1+() AR(X,iH)(aI-ill)n911 is of order as 1X1-"- on for someE > 0, andAR()',iH)(aI-iH)ncpis continuous in X on F. Then for each realt,eitH9E D(A), and 11 AeitH911is bounded on any finite interval as a function oft.Here ais as in Lemma 1.7. Proof.Note that iHis the infinitesimal generator of the unitary itH groupe . From Lemma 1.7 we have itH 1 extR(X,iH)(aI-iH)11cp e cp 27ri dX (a- Mil 15 eAR(h, iH)( But is continuous on and its norm is (a,-)on rc, Thus, of order I k I -(1+e) as I XI onrc. 1 eXtAR(k, iH)(0,I-iH)n9 dX 2Tri jrc (a-x) exists.Consequently by Corollary 1.5eitH9E D(A) and 1 itH S eXtAR(X, iH)(aI-iH)119dX. Ae (P-2_, (a-m "1- rcn But then 1 r Aeit91 I e II AR(X, iH)(aI-iH)n9 III dXI and sinceeXt=e(±c+ib)t,where b E -co , 00) , we have bounded on any finite interval of the real line. q. e. d. Remark 1.9.Let Vbe a closed operator and H self -adjoint. Suppose D(H) C D(V). ThenVR(X,iH)is a bounded operator, since it is closed and everywhere defined. andH be self-adjoint operators and V Lemma 1.10. LetH0 ) C D(V). Moreover, let H = H0 +V. a symmetric operator withD(H0 LetX E p(iHo) cm p(iH),the intersection of the resolvent sets of iH andiH,thenR(X., iH) - R(X, iH0 )= R(X, iH0 )iVR(X, iH). 16 Proof.Forv E D(H0) D(H)we have(X-iH0)(P - (X-iH)go = iVy9. But yo = for some 4i, and so - =iV(X.-iH)-111.) for all LIJ E Ran(X =S4 Ran(X-iH0)is the range of(X-iF10). But then we have -(X-iH0)-14,= (X-iH0)1iV(X-iH)-14) for all E Here I-1 is the Hilbert space in question.q.e.d. Remark 1.11.LetA, B and Hbe self-adjoint operators on andaandbany complex numbers. Suppose - itH9> X(49)(t) = (aA+bB)(V)(t) is differentiable and(aA+bB)I(9)(t) =L7iqco)(t)+bBli(9)(t). Proof. We have (aA+bB)(c)(t) =<(aA+bB)e-itHv,e-itHv> =a = aA(9)(t) +b73(9)(t) But each term on the right is differentiable. q. e. d. 17 Remark 1.12. We emphasize the fact shown in the proof of Lemma 1.7 that ifH is self -adjo int then II RR, iH) II is uniformly bounded onrc. We generalize this fact as follows. Lemma 1.13. Let Hbe a self-adjoint operator and Va closed operator on withD(H) C D(V).Let qE D(H), then vR(X, is bounded onrc. Proof.ForX E r we recall that X = c + ibwithc > 0and . We assume first that X c + ib.But we have R(X,iH)(p =Swe-Xteitlicpdt, (see Dunford and Schwartz [1, 622]). 0 oo Now Vis closed, so VR (X, iH)co =Se-":tVeitHcpdtby 0 Corollary 1.5 if the integral exists.But, -Xt itH e Ve co dt = e-XtV(c-iH)-leitH(c-iH)ca dt since cpE D(H).ButV(c-iH)-1is bounded, and so the integrand is continuous int. Moreover, -ct < Me-ct 18 00 But this is integrable. Thus 13 For X -c + ibwe note thatR(-X, -iH) = -R(X, iH).But then 11VR(X, ii-1)q)11 II -iH)II, and so the above argument now applie s. q. e. d. Polynomials P(x) In this section we shall concern ourselves with the case of A being the maximal multiplication operator determined by the poly- nomialP(x)in the variablex =(x ,... , xn).Therefore we shall , assume throughout that the Hilbert space1-1 isLZ(Rn).More - over, will be the self -adjoint closure of the Laplacian-6 0 defined on A the space of rapidly decreasing functions onRn. Throughout, the potentialV will be a symmetric operator whose domain containsA such that(-A + V)I has a self -adjoint closure H=H +VI withD(H ) = D(H).In consonance with the 0 A 0 physical considerations in the introduction we shall refer toH0as the free Hamiltonian, and H as the Hamiltonian. We shall make repeated use of the Fourier transform of func- tions (1,E L2(Rn). We shall denote the transform by1(0or 0 We shall take as a definition of the transform restricted toA 1(9)(k) = e- k>,p(x)dx Rn 19 Ifx = (x , ) anda = (al, ... ,an ,where is a al a non-negative integer, then we definexa,-- x1 ...xnn .Moreover, we set I = i=1 In view of the linearity described in Remark 1.11, weneed only study multiplication operators of the formxa. We setA = x and interpretAas the self-adjoint closure ofxa definedonA We now set about establishing for Athe three properties listed on page 9.We begin with properties 1) and 2). Leta = (a , , a) where eacha. is a non- Remark 1.14. 1 n negative integer.LetSa be thespace of functionsf EL2 (R) p < a(i.e.,p. < a.),f is in the domain of such that for each 1 aP aPis the closure ofaPdefined onA Here, ax 'whereax ax 8Pf 8'PIf a means. Then S is a Banach space in the ax P1 Pn ax'1 ..axn P 2. The proof is Sobolev norm given by 111.11 2a= 11 112 s p < a L standard and can be found with slight modification in[2,323]. LIJ 13 < a. Remark 1.15.If LIJ E D (Xa ) then E D(x)for all Note that we shall, from time to time, confuse the operators a act with their closures.It will be clear from the context and ax which is meant. Of course in this case D(xa)= E L2(Rn) I xafELZ(R)},n andxa 20 Proof.Let LjiE D(Xa), then + I X I 2dx lxal2kPl2dx X 1)CHI. 1 lxl >1 But for >1, <1 xa1 so 4J12dx I xPI2,IqJI2dx+ xal21ip2dx< Ix1 <1. >1 So, E L2(Rn). q. e. d. If E thenR()., E D(XCL) Lemma 1.16. D(xa) iHo)tp ;1131al31 Proof. Note that the operator - ax is essentially self- adjoint, and its closure has for its Fourier transform (27r)1131k13. Now let qi E D(xa),then there is a sequence {qin}c A , the space of rapidly decreasing functions,such that and a a, x x LIJ inL2 (Rn ), since xais closed.But, 12 II xallin-x% 2= dx and 2dx ixa-i2Hin_Lidzdx 21 2 n Thus, xPtiin--" x LP inL (R ) for each p<_ a.But then for all R a Ad -1 I P1(xP" ButSais a Banach space, so 8k 9' n 2Tri) a in S . So ifLP E D(xa)then/L ES. With this fact we have 1 4 [xaR(x., iHo)LPii]- )1 al ar ] 2Tri ak Lk-472ilki2 n a-13 (a) a 1 2Tri 13 ak 2 ak p< a x_-472iikl 8a-I3 1 X. is a bounded multiplication But for fixed each ak X-4Tr2iiki2 aP.inL2(Rn), so operator. Moreoverak ak 4 r.aR(x., iH0)4in]converges.ButR(X, iHo)4inconverges to a R(X, Since x is closed, R(X,iHo)tp ED(xa).Indeed we have the useful formula a-13 a 1xaR(X,iH0 )4J] - (a) a p ak1X-4Tr2ilk2 1 8ki p < a and -a 13 -1 )1 al (-1 Ipi(a)s-ira xaR =( Scx ). iH0 27ri 27i ak 2,, 2 p < a q. e. d. 22 1 is a sum of terms of the form Remark 1.17.ak X-4Tr2iik12 a al f(k) = C k . k n (X.-4TrZi Ik2)mwhere al < m and a,m 1 n I al 5.H. Proof.We proceed by induction on yj.If = 0 then the assertion is clearly true. Now suppose the assertion is true for IYI= n, and a a k1 kn f(k) = C am (X-4TrZiI k) 2)m is one term in the sum. Then a a;-1 a a a+1 a i n a.k1 ..k.'...kn k1 af(k) i 1 n 1 ..ki...kn c +C' ak.-am 2 , 12 m am 2 1 12 m+1 (X.-4Tr ilkl ) (X-4Tr ilkl ) Hence the assertion is true for M= n+1. q. e.d. a We regard )- where al < 2m,as a (X-4Tr2i Ik I2)m multiplication operator. Since X. will be onFc,f (k)is con- tinuous ink, and so the norm of this operator is the supremum of Ifx(k)I We setg(X) = Sup Ifk(k)l .The next lemmas show how g(X) behaves as a function of X. onrc. a Letg(X) = Sup with X E and Lemma 1.18. 2 Fc , k (X-4Trijk 23 I al < 2m. Then there is a constantK > 0such that g(k) < KI XII al /2on F. and so Proof.If Ercthen X = c + ib, I f(k)- 2 2 , 2 2 m/2 (c +(b-4Tr lki ) ) To find the supremum we may as well usethe square of this expression, 2 1k2aI I - fX(k) 21 12 2 m (c2+(b-47r ) ) 2 Considerkon a sphere of radiusr,thenr2 k. , so i=1 Therefore, al (k)j 2 [c+(b-4Tr2r2)21m We shall find the supremum of h(r) = rzl [c2+(b-4tr2r2)21m Differentiating we get 24 [al al r 2I -1(c2+(b-4Tr2r2)2)m +16n-mr(c2+(b-4Tr2r2)2)rn-1(b-41T2r2)r2lal ht(r) = [c2+(b-4-72r2)212m Settinghi (r) we get 4 16n-4()a-2mr +82(m-7 I al )br+ la!(c2+b2) 1 Now ifI al = 2mandb < 0thenh (r) < and is asymptotic X 161T4 2(c2+b 2) to this value.Ifb > 0thenr- . For thisrthen 4n-2b 22 1 1 ' u ' a'for some constantK.Ifb > Ic)then h (r) z+b2) I al h (r) 4Trb [c2+(b...(c2+b2) 2]m < K(c2+b2)2m < K2 for some positive constants shows that for all K1 andK2, 0 < b Now if < 2m,andhi(r) = 0,and we sett =r2,then -82(m-I a )b± 64Tr4(m- I al)2b2-64Tr4(1al -2m)1 (c2+b2) t= 32Tr4(lal -2m) Now withlal < 2m,h(r) ---' 0asr and hk(0) = 0, so one of these values oftdetermines a value ofrat whichh(r) is maximum. Letr0be one of these values, then l al ro rOal g(X) < c+(b-4Tr2r2)m/2 cI /2 [1x1m-a -4(m- a)2X -a a -2m) - 47211a1-2m1 I al /2 Lemma 1.19.Forlal < 2rnthe function g(X) is continuous onrc. Ikla I where Proof. We have fX(k)I /2 (c2+(b-47r2Ik I 2)2)m X = I converges to uni- c + ib. We shall show that IfX formily ink E Rn.Indeed, 26 2 21 Ifk(k) - Ifx (k) 20.1 1 1 c2+(b-427 2)2)m (c+(b0-4721 Id2)2)m m (c2+(b -4Tr21k12)2)m-(c2+(b-427 Ikl2 )2 ) Ik2ai 0 2 m (c2+(b -4271- )2) (c2+(b-4721k 0 2Ial k Now sinceI al < 2 the quantity 2 is bounded 1 (c +(b-4Tr I kl2) ) by sayC1 for allk E Rnand allb such thatlb-b0 I< 1. 2 1 Moreover(c2+(b0-4 I k I 2)2)m-(c2+(b-4Tr2Ik I 2)2)mis a poly- n.omial of degree at most4m-1with coefficientsa (b)such that 2 a (b) 0 as bo .So for allkwith I k I ro,for some rO, c2+ Ik12)2)m-(c2+(b-4Tr2Ikl 2)2)m a (b)I (c+(b0-4.721k I 2)2)m But this is as small as we please for b close enought tolac Now I f I is continuous on Bn-1,where observe that x(k) [b0-1, bo+I] x This is a compact set, so is B'B is ther0-ball inRn. I fx(k) I uniformly continuous there.But the square root function is uniformly 2 continuous on the non-negative real axis, SO 1 fk(k) I= X(k)I converges uniformly inkto I fx (k) I ash X. onrc. 0 II 1100 Nowg(X) = Sup I fx(k) I= III fxIlloo But is continuous, k 27 so is continuoussince g is a composition of the continuous X I I , 11 11 oo:F supI F(k) . q. e. d. maps fx(k) and H + Vwhere is the The reader is reminded that H0 H0 closure of -AonAthe space of rapidly decreasing functions, and V is a symmetric operator such that H is the self -adjoint -A + VonA We assumeD(H) = ), or closure of D(H0 equivalentlyD(H ) C D(V), a a a Theorem 1.20.Letx be a monomial on Rn,and let x11...xn I al m - > 1 E where E > 0.Let m be an integer such that 2 9E D(Hm+1)and(aI-iH)mco E D(xa),whereais fixed and O.Let V (3 < a, C D(xPv). Re(a) be such that for all D(Ho) a Then, eitHcoE D(x )for each realt, and 11xaeitHco II is bounded for alltin any finite interval. Proof. We shall employ Lemma 1.8.As a maximal multiplication a operatorx is closed. We must show that if =(ai-iFI)mcp,then a x R(X, is defined, and is continuous in X on rc.Moreover, -(1+E ) we must show that II xaR(X, is of order Xim as Ix' We haveR(X,iH)4J R(X, iElo)tP R(X,iHo)iVR(X,iH)4-' But Lp 13 andVR(X,iH)L1) D(xa), and hence both are inD(x ) for 13 Then by Lemma 1.16, R(X, E D(xa), and 28 a x R(X., =xaR(X, + xa R(X, iHo)iVR(X, iH)) 0 a -1 ad -01 ;-1[a-P 1 ]4. p)(-2Tri akx.-4Tr2ilki2 p < a 01 1 But eachB (X) -= is continuous in ak\4k2 X. on and11B ()11 < M 1\1 /2by Remark 1.17 and rc, N Lemmas 1.18 and 1.19.Here so each of the terms Ba-p(X)qis continuous onrc, and 1 /2 11B _p()xPLP11 < M ala_p 11xPt-P11. In addition each termx VR(X, iH)L1J is continuous in on rc,since by the resolvent equation for X., p.E xPVR(X, - VR(, iH)4. ([-L-))xPVR(X, iH)1(N-,iH)tli NowR(p., iH)4, E D(H)CD(xPV), so by Lemma 1.13 x vR(x., a-14is uniformly bounded for X.E r and p. fixed.So xPVR(X,iH)11i is continuous. Moreoversince E D(H) xPVR(X,iH)1.1) is uniformly bounded onrcby Lemma 1.13 and the B (X)xPiVR(X, iH)L1) fact thatxPV is closable.Thus, the terms a- P lx.11 al /2 are continuous and bounded by some a-p asiXi on F. B ut thenxa R(X, iF)LPis continuous, and 11xaR(X., iH)14)11 29 ILL1 I al /2.However, > 1 E SO is of orderlxi m 2 xR(X,iH)t.iII is at most of order IX1111-(1+E). Thus the theorem follows from Lemma 1.8. q. e. d. In order to apply Theorem 1.1 we must show that CL a eco E D[Hx H]. But if Hco E D(Hm+1) and (aI-iH)m(Hco) E D(ca)then from Theorem 1.20 we have a D(xa e H2 E D(x).Thus e-itHcpE H).Therefore, it remains to find conditions such that e-itHyoED(Hxa ). I 2 + E Theorem 1.21.Let m be an integer such that m - 2 > for some f > 0 . Let co E D(Hm+1) arid (aI-iH)mv E D(Xa ). Suppose thatD(Ho) CD(x13V) for allp < a.Then xaeitHE D(H) for allt E R. eitH(p D(xa Proof. By Theorem 1.20 we have ) We have the general assumption thatD(H) =D(H0 ), so we need only show that xaitH e cp E D(H0 ).Therefore, we need only show that 4 (xa eitHco) 2 , 12 E D(K2), whereK = k I co, this being a constant times the Fourier transform ofHo' We have seen that x a itH 1 X-t'R(x, iH)(aI-iH)mco x e dX (a-x)1n rc 30 by the hypotheses and the proof of Lemma 1.8. NowS is unitary, so -P 1 e "3"ixa R(X,iH)(aI-iH)mco] a itH clX 9) = 2177 (x e F (a-X) C As in the proof of Theorem 1.20 we see that if =(aI-iH)mcpthen al &a-13 1 a)(--7--1)1 [ 4 (xaR(), 2Tri ak 2 k < a A PPcvR(x, A X {ak +ak 1 ). We observe that sinceVR(X, iH)4i E D(xP),(VR(X,iH)i)AEakD(a But eacha'P is a sum of terms of the form ak 2 X-4Tr2ilki kr1 with < pby Remark 1.17. So if we multiply (X-472ijk I 2)P by 1k12we get a -1I al I (xaR(X, iH)) = p)( 2Tri a13 A X [ + ivR(x, , --ak where < p +1 < 2P . But the operators 31 B ()'.)- \I P (X-47r2i1k1-) are bounded and continuous in norm onFc.Moreover, II(X)II <1\Alx1IN112, whereIN1 < lad + 2, (see Remark 1.17 and /2)+1 Lemma 1.18). Thus 111k12 4 (xaR(X, iH)J11< c 1X1(1a1 But, 12±. m - 2 - 1 > 1+ E) so the integral eXt2 -P kaR(X, iH)(ai-iH)m(P1clX rc (a-)'.)na exists.Consequently Corollary 1.5 implies that -P7(xaeitHcp) E D(K2), a itH and sox e goE D(H). q. e. d. Theorem 1.22.LetP(x)be a polynomial of degreer.Let m E > 2 +E Let be an integer such that for some >0, m -2 H = + V, and co be a state such that coE D(Hm+1), and H0 (aI-iH)m+lcp E D(xa)for all1 al _< r.Let the potential Vbe such that for all1a1 < r.Then, D(H0) C D(xaV) itH itH P(x)(cP)(t) = PI(x)((p)(t) = Proof. By Theorems 1.1, 1.20 and 1.21 the result is true for a 32 with al < r.So by Remark 1.11 the result follows forP(x). q. e d. Potentials and States In the above theorem it is clear that the hypotheses can be satisfied for a, large class of potentials and states.Moreover, the choice ofV restricts the choice of co to some extent, and vice versa..However, from a physical point of view it is more likely that the potential would be determined a-priori, and so we take the point of view that the states are to be determined after a potential is given. The time is ripe to give sufficient conditions under which our + Vhold.Recall that we have blanket hypotheses on H H0 D(H) = ). assumed thatHis self -adjoint and D(H0 Definition 1.23.LetT and V be operators on some Hilbert space V is relatively T-bounded if an only ifD(T) C D(V) and there are non-negative numbersaandb such that for every E D(T) 1114'11 al19911 b II TC°11 The relative T -bound of V is the infimum of all suchb. Theorem 1.24.LetTbe essentially self-adjoint.IfV is symmetric and relatively T -bounded with bound less than1,then T + Vis essentially self-adjoint and the closure (T+V) T + V. 33 The proof may be found in [3,288]. H = + Vwhere onR3and V In the case Ho Ho is a multiplication operator it is enough ifV = V1 + V2with oo 3 V1 EL2(R3)and V2 EL (R ) [3,302].In this case the fact that ) C D(V)depends on the fact that in dimension three the ele- D(H0 ) are bounded and continuous.Again this may be ments ofD(H0 found in [3,302]. Corollary 1.25. LetP(x)be a polynomial of degreerin x E R Let m be the least integer strictly greater than2 + H = H + Vwhere -A I a and V is a real C func- Let H0 a tion onRnwithx Vbounded for all asuch that al < r. Moreover, suppose that < C (1+1x12)a for k 2m, c9 E the space of whereCaand kaare constants.Let rapidly decreasing functions.Then P(x)1(co)(t)exists, and _p(x)Hie -itHgo, e-itHco> --P(x)1(42)(t);_- < i[Hp(x) Proof. Vis a bounded operator so Theorem 1.24 applies. Now if cp EAthen cp ED(Hm+1)and(at-iH)mcoE . But A cD(xa) for all al< r.But ) C D(xaV),and m - 27- > 2+E for D(H0 - 2 someE >0. Hence the result follows by Theorem 1.22. q. e.d. 34 The Polynomials P(a lax) In this section we study the constant coefficient partial a differential operatorP( ) wherePis a polynomial. The investigation follows similar lines as the foregoing. We therefore il on and their self -adjoint consider the operators 8x A closures A. We setH H + V where H= -A I and V is a closed 0 0 A symmetric operator such thatH is self -ad joint andD(H) D(H0). Theorem 1.26.Suppose D(H0) C D(AaV).Let(aI-iH)299 E D(Aa ) a and9 D(3).H TheneitHE D(A ) for each realtand a itH 11 hAe diis bounded for all realtin a finite interval. Proof.Let qi =(aI-iH)2cpthen qiE D(H).Observe thatAa R(X, ).But then we have permutes with iH0 a AaR(k, iFIN) = R(k, iH0 )A + R(k , iHo)iAa VR(X., iH)iP, since E D(Aa)andR(k,iH)kii E D(H). Now sinceAaVis closable and qi D(H),AaVR(X., iH)ti) is continuous and bounded in norm on by Lemma 1.13.ButR(X, is bounded in norm and con- rc iHo) tinuous onrc.Thus, we can apply Lemma 1.8. q. e. d. Recall again thatH permutes witheitH, so if Hyo ED(H3) or dE D(H4)and(aI-iH)2I-1(pE D(Aa)then itH itH e Hcp = He 9 E D(A).a So in order to show that 35 a a eitHE D[HA -A H]it remains to show whenAaeitHyoE D(H). a Theorem 1.27.LetH = + V, and suppose )C D(A V). H0 D(H0 Let cpE D(H3)and(aI-iH)2coE D(Aa). Then, AaeitHc0 E D(H). Proof. We set IP =(aI-iH)z(p, and as before AaR(X, iH)4)= R(X, iHo)Aa+ R(k, iHo)iAaVR(X, iH)LP. ButHois a closed operator and it is easy to see thatH0R(X,iH0)is bounded for all X onrc.Indeed, 4n-2iki2 R(X, n - 02X-4Tr2ilki and so by Lemma 1.19HoR(X, iH0)is also continuous in X on AaR(X, is defined and is a continuous and I. HenceH0 X onF But then bounded function of C. a R(X,iH)(aI-iH)2(p xtHoA dX (a-X)2 aitH exists, and so e E D(H0 )= D(H) by Corollary 1.5.q. e. d. Theorem 1.28.LetPbe a polynomial of degreeronRn. Let co be a state such that (pE D(H4)and(aI-iH)zyoE D(Aa) for allasuch that a< r.For all al letD(Ho)c D(Aav). a Then for every realt Pt ax)(cow is differentiable and 36 a a -itH -itH - P( )Hje ,e >. P(axr(9)(t) = a a P(ax ) = CA.a al < r I Proof.This result is an immediate corollary of Theorems 1. 1, 1.26 and 1.27. q. e. d. It should be noted that the hypotheses here have little chance of being satisfied if Ial > 2and V is a multiplication operator since we require )C D(Aa V).So even ifV is smooth we are ask- D(H() ing for twice differentiable functions to be differentiable to a higher order.But if V is say an integral operator whose effect is to smoothL2functions then the hypotheses can be satisfied. We shall Aa- a in Chapter II. treat the case where ax. 37 II.APPLICATIONS In this chapter we shall give sufficient conditions under which Ehrenfect's theorem holds.In addition we shall examine the of the mean of positionx.((Pbehavior )(t)as it relates to the choice of initial state p. Ehrenfest's Theorem We have seen in the introduction that the classical formulas, dx. d2x. m - momentumand m - force, have, according to dt dt2 Ehrenfest, an analog in the quantum setting.Indeed, Ehrenfest's theorem in physics [9, 455] asserts that the same formulas hold if is replaced by its mean value as a function of time, and the momentum and force are replaced by their mean values. H + V where Just as in Chapter I we shall assume that 1-I0 is the free Hamiltonian onL2(R)and V is a potential such H0 thatH is self-adjoint and D(I-I ) = D(H). 0 Theorem 2.-1.Letx.be thej-thcoordinate operator on Let H + V, with V a multiplication operator, and let co H0 4 be such that E ) and (aI-iH)4E D(x.).LetD(x.V) D D(H ) Then x!(,(P)(t) exists and itH x!(co)(t) = Proof.In Theorem 1.22 setr = 1and m = 3.Observe that x.VVx.on the intersection of their domains. q. e. d. 3 The Coulomb Potential onL2(R3) 1 In the Coulomb potential V is given by V(x) = Observe that V E L2(R3), and so V satisfies our blanket require- H = + Vbe self -adjoint withD(H) = ) so that ments that H0 D(H0 Ix. D(H ) C D(V). Moreover, Ix V I= < 1, sox.Vis bounded and D(H0 ) C D(x.V). Let be the subspace ofA consisting in all those A0 functions co E that vanish in a neighborhood of the origin. Definition 2.2.Let A be an operator on a Hilbert space14 . Then cpE is said to be an analytic vector for A if and only if there is at > 0such that 00 II An9 II tn<00. n! n= Theorem 2.3 (Nelson [4, 572]).IfA is a symmetric operator on D(A), and if D(A) contains a dense set of analytic vectors then Ais essentially self-adjoint. Lemma 2.4.Consider . ax.a as the self -adjoint realization of 39 . a ax. onA . Theni(Hx.-x.H)C 2i J 3 ax. Proof.One easily calculates thati(Hx.-x.H)is symmetric on 3 J D[Hx.-x.HJ.Also, for9 E-9) ,i(Hx.-x.H)co = i(H x.-x.H )9 = 2i-. J J 0 J J 0 8x. 33 J a Now let = i(lix.-x.H)andB = i(Hx.-x.14)1 = 21ax then both 3 3 3 3 A are symmetric operators withBC AC A. Now B(henceA) a contains a dense set of analytic vectors.Indeed, consider8x.in oo the Fourier transform representation.Let 9 E Ccwith support in a ball of radiusR. Then 1/2 lik71/011= Ikr.n121°M2dk < Rrn c/n L2(Rn) 3 L2(Rn),and for0 < t <1 and But -1[eRn)]is dense in R -1(Coos, ) so we have 00 II Any, II tn Rnilatn n. n! .11C911 nO n= Thus, by Nelson's theorem Ais essentially self-adjoint.So, since BCA, BCA,so AC AC B,by taking adjoints. q. e. d. Corollary 2. 5.Let Vbe the Coulomb potential then for all - , <2iae-itHcp,e-itH9> E xj""," 8x. 40 Proof.Since coE 0,Hyo = Hoy° + VcoEA0.So by repetition Hm EA c D(x.)for m < 4.So by Theorem, 2.1 and the fact that .x V is bounded we have xil(49)(t) Li J by Lemma 2.4 we have a - it H -itH co,e (p> . q. e. d. x!((p)(t) = This corollary says that the velocity of the mean is the mean of momentum. This is the first of the two assertions of Ehrenfest's theorem.It is unfortunate that we cannot continue with the Coulomb is not potential beyond this point, because the composition va xJ ).We shall therefore turn our attention to defined on all ofD(H0 smooth potentials. The Case of Smooth Potentials In this section we shall assume that V E Cx(Rn) and is a real a and are bounded. valued ,function on Rnsuch thatv.,)c.V ax. Li Therefore, H = Ho .+ Vis self -adjoint andD(Ho) D(H), by Theorem 1.24. a Remark 2.6. (-a-;---c.) ° V is defined on D(H0 ). Li a a v a(p Proof.Let q2EAthen (V(p) = -67 cp v a . Now if 41 a (I)E then coE 1)( ) and so there is a sequence{'p} C D(H0) ax. 3 a n a co co But, Vyan Vcp and andax. ax. aVcpn8V a so coE D( 0V) since is closed. ax. ax. ax' ax. ax. 3 3 3 3 Indeed for cpE D(H) we have the formula 0 (v_) av q. e. d. ax. ax. ax. Now observe that since V is smooth the requirements that 4 E D(}{) and(aI-iH)49 E D(x)are satisfied ifHmy)E D(x.) 3 for0 < m < 4. Corollary 2.7.Let 'pbe a state such thatFlmcp E D(x.)for 3 -a- - itH -itH x!(v)(t) = <2i e r ca> 0 Corollary 2.8.LetV be a realCxpotential onRnwith av 'Orand bounded.Let be a state such that Hm E D(x) ax. 3 3 m < 2.Then for all0 < m < 4,and Hm D()ax. for 3 -itH ---a-Hje ,e-itH(p>. x:1(40(0 =-2 <[Hax ax. 42 Lemma 2. 9. H8x.-ax.H -ax. 3 _ .... 00 n a a ) H - H agrees with Proof.OnCc(R the operator ax. ax. 8V 3 3 - by the product rule for differentiation. Moreover, these 8x. _ _ 3 a a Mr operators are symmetric.SetA ,-- H 7;7. - ax. HandB =--ax.--..- I 00. 3 C , J J c We claim thatCc°(R)is a dense set of analytic ThenB C A . c 00 n (p E (R ) then vectors for B (hence A).Indeed, if Cc Amy, E C°° (R n)and has support inside that of (p. So c II Amc911 C II (-°1-7-1: -:1,": m (P 00 11m,plImA t <00 m! m.0 for somet >0. So by Nelson's theorem Aand B are essentially self -ad joint.HenceB C A, soB C A ,and so by taking ad joints we haveA C A C B. q. e. d. _g2x2x2.z. 1,p_LEInnleAtl.Let V be a C00 potential withx,V 3 8V Let (p be a state such that HmcpE D(x.)for andax. bounded. E then m <4 andHm(p D()ax, for m < 2, 43 T -itH-itHcp, r p>,and 1) x 1.(cp)(t) = < ax.e av 2)x'.1(cP)(t)= e-itHe-itH(p> . Proof.Use Corollaries 2.7, 2.8 and Lemma 2.9. e. d. Remark 2.11.Corollary 2.10 is true for any state co E pro- vided the derivatives of V up to order six are of polynomial growth. The Behavior of the Mean The Bohr model of the hydrogen atom suggests that the electron can be in two different types of state, either orbitting the nucleus or free.If we assume that eigenstates are orbital states and absolutely continuous states are free states, then we can conjecture that the mean of an eigenstate is constant, and that the mean of an absolutely continuous state goes to infinity as time increases.In this application we shall study these possibilities. Definition 2.12. Let Ebe the spectral measure of the self- adjoint operator H on some Hilbert spacen . Then we define three subspaces of SA as follows: 1)1/-1 e this is called the point space of pt(H) = E{X}Si ER H. 44 -1-is(H) = {v/i 1-lac(H)= {yo E All of these spaces are closed, and in particular lApt C e and ac We setcrpt(H) =fx.E R I E{X} 01, this is called the point spectrum ofH.It is well known that for separable Hilbert Cr (H) is countable. Moreover is the spaces pt Tpt(H) set of eigenvalues ofH.It is also not difficult to see that (H) Efcr(H)1. pt pt Remark 2.13.Let be any Hilbert space andA and H self-adjoint operators on 11 . Letvbe an eigenvector ofH, E D(A)then ii.(go)(t)= Proof. Let X. be the eigenvalue ofvthen it is well known that 0 -itH -itHe e Consequently e D(A). Thus -itX0 -itXo A(cp)(t) We can say a little more than this. Theorem 2. 14.Let1-1 be a separable Hilbert space withH and 45 Aself-adjoint operators onIA . LetAbe defined on every eigenvector ofH.Let be an orthonormal basis of eigen- { vectors forIA Let =is be such that I ak I and pt(H). akyok k=1 k=1 converge.Then, I akI II 49kII k=1 itH e E D(A)for allt, and x(cp)(t) Proof.Let Lt? eitH =eitH = akyok ake kcpk k=1 k= 1 then 00 itk AeitH itXk = kA e kAco ake k=1 k=1 since 00 itX k ake Acok k=1 itH converges.ButAis closed, so e cpE D(A), and itH = kAca . Ae ake k=1 Now 46 00 it itH itH co> = e arie 00 it(X ) kn . akane oo 00 q. e. d. 00ak IIAq'k < n=1 k=1 Example 2.15.In order to show that Theorem 2.14 is not vacuous we again study the free Hamiltonian perturbed bythe Coulomb poten- tial.In order to get a point spectrum we insist that the potential be attractive.Thus we consider the self-adjoint closure H of -A-V where V - 1 on R.3 lx1 It is well known that the point spectrum consists in the eigen- C value s X. = forn = 1,2,3... . Moreover, it is also well n n2 known that, in spherical coordinates, a basis of eigenstates for the point spacel(H)is given by the functions (n+1/ )! 11/2 (r)Yrn(0,4)) ni m(r,0,4)) = C [n!(n_i-1)!(2.e+l)! ' 47 I-C2,/nr is a Here, Rni(r)has the formr e P(r) where P(r) polynomial of degreen'such that1+1-nn'.The functions Yi? 0,0 are the usualspherical harmonics. We also have the condi- tions thatn 1, 2, 3, ..., 1= 0, 1, 2, ,n-1,and m = -1,..., 0, For a discussion of these facts see [5, 1643. Now is positive so the functions decrease rapidly. C2 tljni IfAis the multiplication operator Acp = r co , we see that each E D(A).Thus the hypotheses of Theorem 2.14 are easily satisfied in a practical situation. The Absolutely Continuous States We have conjectured that is the subspace of states --S-Aac(H) for which the mean of position is unbounded.In this section we shall show that under certain conditions this is the case. Our approach will be to compare the mean of position in the + V with the mean in the case H ast CC case H H0 H0 itHo It is natural in this context to compareeitHco ande co as t co as these quantities play a central role in the definition ofthe respective means. We are therefore led in a reasonable way to a consideration of the wave operators. 48 The Wave Operators The behavior ofeitH9for largetmay be discovered by itHo comparing it with the behavior of the well known quantity e 4i.We would say the two quantities are close if they are asymptotic. Definition 2.16.We shall say that qi E D(W)and that Wq ç2 if -itH -itH0 and only iflie 9-e 0 ast 00 ort -00.But itH-itH4) this is equivalent to lico-e e II Again, equivalently tH-itHo W '47S -limeie whereS - linndenotes the strong limit. For heuristic reasons it is clear that the wave operators as LIJ E many of defined above are very unlikely to exist if pt(H ), these states corresponding to bound states physically. So we shall define W only for states ac(H0)such that -itH lirneitHe 9 exists.It is easy to show that the domains W±9 t'±00 of W are closed linear subspaces of hc(H0),and thatW± are bounded operators. We must therefore providesufficient condi- tions thatD(W )=c(H0). D be a subset of (H ) such that Lemma 2. 17 ([3, 533]). Let ac 0 (H ).Suppose that for each the closure of the span of is ac 9E D e-itHocpE D(H ) n D(H)for there is a real s such that 0 -itHo co t, and s similar result holds forW_ with the obvious modification. d itH-itH0 itH If cc, (e e (p) ie e-itH0 , Proof. E D we havedt (H-Ho) and this derivative is continuous by hypothesis.Thus, if itH-itHo W(t) e e then t" 4_ u. -itH0c9 W(t")co W(tI)co= 0)e dt t so rt" -itHO IIW(t")cp-W(e)(P11 < II(H-H0)e dt. But by the integrability conditionW(t) is seen to be Cauchy, so But then since is continuous on Dand W+cpexists. W+ D(W+) the result holds for all is a closed linear subspace of1-1ac(H0) cp Eac(H0). q. e. d. Corollary 2. 18 ([3, 535]). Let0 be the free Hamiltonian on 3 (R ) V is real valued andV V L2 and H = H0V where 1 bounded. Suppose that withV1 EL2(R3)andV2 0-1+E I V(x)12dx < cofor some E >0. 3 Then W exist. We remark that the wave operators do not exist for the Coulomb potential. 50 We introduce at this point a few facts from Richard Lavine 's paper [6, 368]. Definition 2. 19.We setA(t) =eitHAe-itHwhere His the + VonL2(R)andA is a suitable Hamiltonian H =Ho (H).Let operator.LetQ be the projection ofL2(R)on ac a P. = coordinate momentum operator.Let f ax, be thei-th be a bounded complex valued continuous function on R. We define A = f(P1' ,Pn)by considering its Fourier transform to be the multiplication operatorf(ki, ..,k). We say thatH satisfies the weak scattering axiom two (w.s.2) if and only if for every A = f(P1' ,Pn)S - limA(t)Qexist. - is compact.If A Lemma 2.20 (Lavine). Assume thatV(H0+i)1 is a possibly unbounded operator whose domain containsD(H0), -l andA(H +i) is bounded, (which is true ifA is closed), and 0 S [A(140+i)-1](t)Q B±exist, then for all qiE D(H), t *09 lirn A(t)Qqi = B t ±00 Proof. We remark first that ifK is a compact operator then lirn K(t)Q 0.Indeed, K(t)Q = KeitHQco II. Since (H), fiac -iH where p.() = Now let E D(H) and cp. = (H+i)Iii.Then flA(t)Q() -B±(H+i)ip= 11A(t)(H+i)-1Q(P-B±911 < A(t)[{(H+i)-1-(Ho+i)-11(t)}Q((P)II +11A(t)(H0+i)-1(t)Q((P)-B±(PII Now the second term goes to zero by hypothesis.But the first term is the same as 11[A(t)(Ho+i)-1(t)][V(t)(H+i)-1Qh9II But this last term has the form JIK(t)QcoII for a compact operatorK.So the term goes to zero. q. e. d. Corollary 2.21 (Lavine).LetP. = iax.. Suppose that H 4, E D(H) satisfies w. $.2 andV(Ho+i)-1is compact. Then for all urn P.(t)(211) exist. t-~-±00 Proof.We apply the above lemma withA = P. Observe that - D(H) is compact. Moreover, D(Ho), sinceV(H0+i)1 D(H ) C D(A),and 1- is bounded.But, A(Ho+i) 52 -1 S - lim 1A(H +i)](t)Qexists, since t ±cc itHA(}{+0-1e-itH [A(H+i)--11(t) andA(H i) 1 is a continuous 0 0 0 function ofPi, ., Pn.But w. s. 2 is satisfied so S -lim[A(Fl0+i)-1(t)Qexist.So the result follows from Lemma t ± 00 2.20. q.e.d. We remark thatW±are complete if and only if Ran W± = ac(H). Remark 2.22.If the wave operators W±ofH = Ho + Vexist and are complete then Hsatisfies w. s.2. Proof.Let A = f(P1,... ,Pn)where f is a bounded, complex itHO continuous function onRn. Then A commutes with e . But. -itH itH itH 0 O A(t)Q = e Ae-itHitHQ = ee Ae e-itHQ, and so S -lim A(t)Qexist. q. e. d. R3 H = + Vthe free Hamiltonian We note that in with H0 perturbed by a potentialV cLin L2the wave operatorsW± are complete [3, 5461. - to be compact it is Remark 2.23.We note that forV(H0+i)' enough thatV be a real function with V locally square integrable and V(x) --1" 0 as lx1 co, forn < 3.See [10,109]. We are now ready for our results on the unboundedness of the 53 mean of position for absolutely continuous states. Theorem 2.24.Let B be as in Lemma 2.20 where A -a--x.. LetH = + v H satisfies w. s. 2 and +i)-1is Ho be such that V(H0 compact.Letx.be the usualj-thcoordinate operator on L2(Rn).Let )C D(x.V).Let co be a state such that D(H0 3 cp D(H4) E D(x.)for and yoe (H). and Hm m <4, ac Suppose that 0.Thenx.((p)(t) CO ast +00. +i) is compact we have D(H) ).By Proof.SinceV(H0 D(H0 Corollary 2.21 and Lemma 2.20 itH. -itH (p,(p> c Co ast Co by the mean value theorem. xJ.(0(t) if c < 0thenx.( (PSimilarly)(t) -°° q. e. d. Corollary 2.25.Let B be as in Lemma 2.20.LetV(H0+i)-1 ) C V).Let cp be a state such as in be compact, and D(H0 -D(xj Theorem 2.24 above. Suppose also that the wave operatorW+ exists, g, 0. is complete, and cp, = for some tp. Suppose axa. 54 Thenx.(v)(t) ±00 ast oo. 3 Proof. We must show that i -1 -1 * +i) W (H+i)cp, W > = a -1 this last since (H +i) is bounded. Now (H+i)yoE Ran W ax. 0 sinceRanW+ reducesH the operator W+ is intertwining. (see Theorem 3.2 [3, 529] ).So Thus W (H+i)ca (H0+i)W+(p, a * a Now Theorem 2.24 applies. q. e. do Entirely similar results may be given fort -00using B and W. 55 III.BOUNDED OBSERVABLES In this chapter we present a few results on the differentiability of bounded observables.This added condition on the observables will enable us to weaken the hypotheses of some of the theorems as our first theorem shows. Theorem 3. 1.LetAbe a bounded operator and H a self-adjoint operator on a Hilbert space 14 Let cp EA and -it H 0 E D[HA-AH], and A--((P)(t) Proof.Without loss of generality we may assume thatt0 -it 0H 0 for otherwise we replace co bye (p in the following argument. -itH -itH Ae-itH(p-Aca But Ais bounded so AM(cp)(0) = + = Note that we have no need of any symmetry condition onA. -it H For differentiability we require only that e ° E D[HA-AH]. We now give a criterion of differentiability which is of greater utility. H = + V, with V Theorem 3.2.Suppose Ais bounded, Ho symmetric, Hoself-adjoint, and V relatively Ho-bounded with C ).Then for all bound less than one. SupposeA(D(Ho)) D(H0 E D(H)and allt, A-(ç9)(t) is differentiable and A-1((p)(t) = Proof. he hypotheses imply that His self -adjoint and D(H) D(H0).But thenA(D(H)) C D(H).If cp E D(H) then e-itHcE D(H)for allt, and soAe-itHE D(H).Then, e-itH ED[HA-AH], and so the theorem follows by Theorem 3.1. q. e. d. the Example 3.3.Consider again the spaceL(R3)andHo -A. LetH = + Vwhere V is any self-adjoint closure of Ho operator onL2 (R)3 satisfying the hypotheses of Theorem 3. 2.We 57 remind the reader that the Coulomb potential is included.Let i forj 1,2 and 3be the momentum operators on P.j ax. L2(R3). iLetfbe any bounded, complex, measurable function on R3, and defineA = f(P1, P2, P3)by (AT) (k1,k2,k3) = f(k1,k2,k3)49 Then Ais bounded. Moreover, A[D(110)] C D(H ). Hence for every q E D(H), .A.-1(c9)(t) exists at allt. Definition 3.4.We shall say that a bounded operator Ais H-differentiable if and only ifA(9)(t)is differentiable for each D(H)and for eacht,and -itHcp> Alko)(t) It is worth noting that Theorem 3.2 says that ifAis H0 -differentiablethen for "small" perturbationsV, Ais + V -differentiable. Ho The above example suggests a Banach algebra of bounded operators that are H-differentiable. and H be as in Theorem 3. 2.Let Theorem 3. 5.LetH0 = {A EB(P )1 AH0 C then P is a 13*-algebra of H-differentiable operators. 58 Proof.IfAE fi then Theorem 3.2 ensures thatA is H-differentiable.Certainly p is an algebra. We must show it is closed under the taking of limits and adjoints. and A in the norm topology. Suppose{An} C P An Suppose (pc D(H0), then ED(H0)and Act.But An`p Act' cp)= H cp) and is closed.Thus n(H0 0 n A(Ho9), Ho AH cp, and so p is closed. H°Act) Suppose AE 3.Letqi ED(F10), SO kp,A cp> = co> = 9> <4.1, A H cp> 0 So A yoE D(140), andH0A = A Hoc).So A E P q. e. d. The above example and theorem suggests the following algebra of bounded operators is an algebra of H-differentiable operators. Theorem 3. 6. Let andH + Vbe self -adjoint operators H0 H0 with V symmetric, and Vrelatively Ho-bounded with bound less than one.Letfbe any bounded Borel measurable function on R. Thenf(H) is H-differentiable. Proof. Byf(H0)we meanSf(X)dEwhereE is the spectral R ), but measure ofHO, We need only show that f(H0)H0CH0 f(H0 this is well known. So the result follows by Theorem 3.2. q. e.d. 59 Multiplication Operators We now turn to the question of H-differentiability of multiplica- tion operators.Throughout this section-Si L2(Rn), andH0 will be the self -adjoint closure of the Laplacian as usual. In view of Theorem 3.2 it will be enough to find conditions on a bounded multiplication operator Asuch thatA[D(H0)] C D(H0). To do this we shall need to make use of a product rule for distribu- tions given in the following lemma. Lemma 3.7.Suppose 1 < p , q < 00 ,1 +!= 1,and P q f,D.f ELP(R), andg, D.g ELq(Rm) Then 3 3 D.(fg) = D.f)g + f(D.g) ae.Note that the derivatives are in the dis- J J J sense. Moreover, S)tributionf(D.g dx = -(D.f)g dx . J J co m Proof. Choose a molifier p, that isp E Cc (R ) withSpdx =1 and0 < p < 1.Let f * p where p np(nx)then fn = 1 /n 1/n(x) = f E CX(RM)rmLP(Rm), andf fin LP(Rrn).Moreover, n D.f= (D f) * p and soD.f E LP(Rrn) andD.f D.fin in j 1/n in in i LP(Rn).Butfn E CCXD(RM)) so Leibnitz formula for distributions gives D,(f g) = (D.f )g + f (D.g) . n n n So D.(f g E li(Rn). Sincefng ET,l(Rn)we have n 60 (D.f )g dx = (D.g)dx.Thus, Holder's inequality shows that n n dx = -Sf(D.g )dS(D.f)gx. J Now let cpE D(Rm) (C°°(Rm) with compact support).Then =-SfgD.dx<13.(fg),9> . But by Leibnitz' formula again J - = -gD.9D.(cog) +D.g, SO D cp.( g) E Lq(Rm), and so cp> = -SfD.(cpg)dx+Sf(D.g ) But we have seen that -5fD.(9g)dx=S(D.f)gyodx . But p was arbitrary so fD.( g) = (D.f)g + f(Dg)ae. q. e. d. 3 is exactly the Sobolev space of all We remark thatD(Ho) elements ofL2(Rn)whose derivatives up to order two are also in L2(Rn). Corollary 3.8.LetA E L2(R)nL00(R)and suppose that the distribution derivatives ofA up to order two are in L°°(Rn).Then C L2(R)n A[D(H0 D(Ho) . 61 Proof.Ifcp E D(H0)we must show thatAlp E D(H0 ). AcoE L2(Rn) a and so it is a distribution.Let D be a derivative withI a< 2. By an immediate extension of Lemma 3.7, Da(A)= a Da-pA But D139E L2(Rn)andDa-pAis essentially bounded, so Da(A9) EL2(Rn).Hence Act°E D(H0) q. e. d. Theorem 3.9.Let Vbe a symmetric operator withV relatively H -bounded with bound less than one.Let 2 n in AE L (R) r L (R ),and suppose the distribution derivatives up to order two are inL2(Rn) Lg°(Rn). Then Ais H + V -differentiable. Proof. Immediate from Corollary 3.8 and Theorem 3.2.q. e.d. Application SupposeV E C (Rn)is a smooth potential satisfying the hypotheses of Theorem 3. 9.Then the expected value of potential energy at timetis V(cp)(t) = - itH(p,-itH(p> V'((,0)(t) = In this section we give a result in which Ais an integral operator with kernelG(x, y).We shall find conditions on G so thatA[D(1-10)] C D(H0). a bounded Lemma 3. 10.LetHo be a closed operator, and A operator on a Hilbert space14 . Let C be a dense subspace of andA(C) C Suppose is bounded. Then D(Ho), D(Ho). H0 AlC A[D(Ho] C D(I-10). Proof.Letyo E D(H0 ), then there is a sequence {4911} C C such that p cp. But then --ATsinceA is bounded.But A9n E D(H0), and kii for some qi, since Ais H0Acpn-` H0 bounded onC.But is closed, so Aco E ) and H0 D(H0 H Acp = LP. q. e. d. 0 is the Laplacian on Rn, and the In the next TheoremLx subscript denotes the variables with respect to which the derivatives are taken. Theorem 3. 11.Let be the free Hamiltonian on L2(Rn).Let H0 H + V, where V is a symmetric operator relatively H0 -bounded with bound less than one.LetA be an integral H0 2 operator with kernelG(x, y)such thatG(x, y) E Co(R2n ).Then 63 A is H-differentiable. Proof. By Theorem 3.2 we need only show that A[D(Ho)] C D(H). We apply Lemma 3.10.Let q'ECC°0 (R n) (Cc° -functions with compact support), then A G(x, y)cp(y)dy.ButG(x, y)has compact Rn Acp EC2(R11), so so Acp has compact support. Moreover, 0 is bounded on Ac' E D(H0).Next we show that HOA 0 n SinceAc' E Co(R z AAc' =ASG(x, y)co(y)dy = (A G(x, y))49(y)dy. Rn Rx Thus we have, nA x G(x, y)(p(y)dy I2dx A xG(x, y) (p(y)dy)2dx. But for eachx,A xG(x, y)and are inL2(Rn), so by Holders inequality we have: G(x, y) II (19(y) I dY )2 < .51Rr A G(x, y) I2dylk112. Thus, 64 2 HAq,I12 00 n Hence Ais bounded on and so A[D(H0)]C D(Ho) H0 Co(R ), by Lemma 3.10. Thus Ais H-differentiable. q. e.d. 65 IV. A REAL STONE'S THEOREM In this chapter we shall prove a spectral theorem for normal operators on a real Hilbert space.Such a theorem has been given for bounded normal operators by R. K. Goodrich [7, 123].Here we shall treat the bounded and unbounded cases at one stroke, and we shall use a method that makes the results completely natural.In addition we shall provide a functional calculus for unbounded or bounded normal operators, and thence obtain Stone's theorem for real Hilbert spaces. Thus the treatment of the unbounded case and the provision of Stone's theorem extend Goodrich's note. It is natural to complexity the real Hilbert space lsolve our problems in the complexification, and then lower the results to the real Hilbert space. The Complexification of H Let H be a real Hilbert space with inner product ( ).We ofH in the following way.Let define the complexificationHc [(p, 4,]EHxH anda+iloc4then we set (a+ib)[cp, 4J] = [ aqi-Fb].This yields a complex vector spaceHc. We define an inner product onHc by <['][i]> ((P,) i(go, 1-1) + i(u,) + 66 It is easy to verify thatHc , with inner product <, >,is a complex Hilbert space.Note that [(1 ), 4)1 =[p ,O] 0] + 0].Consider the map9 [9, This map is an isometric imbedding ofH in H with imageH x {0}. Indeed, 2 11[9, °]ll 2 = <[9 01, Ec 0,01> =((f(P) = Thus, we shall identifyH with the real subspace H x {0}ofHc, so that yo + itp , t Then as expected(491+iq + (T2+4)2) = (T1+T2) + i(4)1+14)2), and (a+ib)((p+i0 = (aco-b) + i(a41+139). We shall use the language of complex numbers in this context where the meaning is clear.For example given cp + i4J we shall refer to9as the real part and i as the imaginary part. Operators on H Suppose Aand B are real operators onH then we shall A + iBon by define an operator Hc (A+iB)(y7+i4, ) = (Ay9-13Lp) + i(A4J+Bco). Remark 4.1. A + iBis a complex linear operator on H. A + iBis bounded if and only ifA and B are bounded onH. IfA + iB + A = and B Al iBIthen Al B1. 67 Proof. (A+iB)[(p+iip+s+it] = (A+iB)[(p+s+i(Lii+t)] Acp + As - Bi - Bt + i(Aili+At+Bcp+Bs) = (A+iB)(40+iLP) + (A+iB )(s +it).A similar calculation can be made for scalar multiples. Suppose A + iBis bounded then SO (A+iB)co II 2 = A(Pil +II BC° 112 II A+iB211911 2 II A II II A+iB and B < A+iB Conversely, ifA andB are bounded thenA + iBclearly maps convergent sequences to convergent sequences. IfA + iB = A + iB then(A+iB)(r = Ago + i139, and 1 1 = A + q. e. d. (A1+iB 1) iB19. A Counter Example.It is very tempting to conjecture that A+iB II2=II All 2 It B II 2, but this is not so as the following example shows. Let H =0*2=R2+iR2andH =R2.Let A(cp1, k 2) = ((p1' 0)and B (cp1, co2) = (O, q2).Then II A II= 1 and JIBJI 1.Now, A+iB = Sup 11 (A+iB)(go+i4J) = SupNiAco -Bip2+11 A4J+Bcpil 2 2 2 2 2 = Sup +cp2+ =1 68 Remark 4. 2.Let A and Bbe bounded operators on H then 11 A+iB 5_ 2[11 All2+11 BII 2] and 11 All II A-FiB 11 and II B II<11 A+iB 11 A+1B 11 II A II+II B Proof. 1) 11 A+iB II Sup II (A+iB)(cp+i4) 119+411=1 = Sup (A9-13qi)+i(ALIJ-1-Bcp) 2 SupNI Ay9-BLp 112+ IIA4J+Bco II But A9-13LPII 2 411 A911 2 + BLP II and Att)+B(P II2 2{11 ALP II 2+1113911 21 by the parallelogram law.Thus 11 A-FiB II __Sup 2[11A11211(P+4112+11B11211(P+i4)1121 < 2[IIA112+11B112] Parts 2) and 3) are obtained by direct computation with the definition of the norm of an operator. q. e. d. The above considerations raise the question that ifEis a bounded operator onEic are there bounded real operatorsEland 69 such thatE = E + iE2. The answer is yes! E2 1 Lemma 4.3.LetEbe a bounded operator on Hcthen there such that exist unique bounded operatorsEl andE2on H E=E +iE2* Proof.Let9E H thenEyo = Eicp + iE29. We claim that E =El+iE2.Indeed, E((p+i) = E+ iE= E +iE29+ i E, qyfiE =(E 1ç-E2) + i(E14J+E29) = (E1+iE2)(9+i4J). The linearity ofElandE2is established by direct calculation. q. e. d. We can get the same sort of results for unbounded operators but a little care is required with domains.IfA and B are any real operators on Hthen we can defineA + iBon D(A) m D(B) + i(D(A)(mD(B)) We note that ifA and B have the common dense domain Din H then A + iBis defined on the dense domain D + iDinH Conversely ifEis densely D thenD mH + C D, so defined onHc with domain + E. we may defineElandE2as before and getEl iE2 70 Complexification of a Real Operator LetAbe a normal operator on H, that is letAbe - closed and densely defined andAA = AA. We defineA = A + 0i, so thatA(cp+iLli) = Acp + iAi. We note that in the bounded case 11 A 1 1 = 11 A 1 1 Lemma 4.4.IfAis normal then so is A. Proof. A is densely defined.Next Ais closed for if +iiand + LALIJn p + iqthen Acp = pand n Ay)n = q. Now we show that A(A)= (A) A .We may equivalently show thatD(i)=D[(A-)*] and i(cP+i4)) II II (171)*((P+it-P) for all * + iE D(A). Observe thatp + iqE D[(A) if and only if (A) (p+iq) = s + it.But, = (p ,$) s) - i( ,s) + i(co, t) + t) Setting4, = 0we get (Ay', p) (p, s)and (AE9,q) = (6.0,t), SO p, q ED(A- )and A p = sandA qt.But A is normal so p, qE D(A).Similarly ifp + iq E D(A)thenp + iq E DE(A)1. 7-* Indeed we see that (A)= (A). But then 71 112 11 (X)*(co+i) 11=11 (A*)((p+it) 11=Ni11 A*co 11 2+ 11 A*LP Ac9 2+11 A 2 X j)ll. q. e d. Consequently A has a spectral representation zdE Where Eis a projection valued spectral measure on the Borel sets and such of0.But then we get two operatorsE1 E2on H thatE( ) = E () + iE2( ).If we calculate purely formally we get 1 x+iy)d(E1+iE2) =SxdE ydE2 + i[xdE2+.5AydEl] Therefore, it is completely natural to conjecture that A SxdEi SydE2. A Spectral Theorem for Real Operators We begin by discussing some of the properties ofEland E2. are bounded operator valued functions.Indeed FirstElandE2 they are measures. We haveE(13) = 0 = Ei(0 + iE2(4)), isE1(0 = E2(0 =0. 00 Let M be a countable disjoint union of Borel sets in0,then n=1 n 72 oo oo 00 oo E( M) 2j E(M)40 = E iE(M) 1(1\4n)co+ n=1 n=1 n=1 n=1 co oo ( M )(.,9 + iE2( M E1 n=1 n=1 for anyco EH. So oo oo Ek( M)n = Ek(Mn)cp n=1 n=1 Moreover, the sum is independent of order of summation. Conse- quently, for co , E H, (E k( )cp, = ( ) is a real measure k, co , fork 1, 2. We remark thatE1(4) = IandE2(c1) = 0since E(4) = I = I + i0. Remark 4.5.LetEbe the spectral measure ofAand EE1 +iE2. Then, 1) E is self -adjoint and is skew-adjoint. 1 E2 2) For Borel setsM1 andM2, E1(M1(---NM2) = E1 (M1 )E1 (M2) E2(M1)E2(M2) E2(M1r M2) = E2(M1)E1 (M2)+ E1(M1)E2(M2) 3) For any Borel setM, 73 E ( E2(M)- Ez(M), and 1 2 (M) . E2(M) = E1(M)E(M) +E(M)E1 Proof.1) follows from the fact thatE is self-adjoint, and the fact that a little computation shows thatE= El - iE2 .The rest follows from the fact thatE(M1r-)M2) = E(M1 )E(M2). q. e. d. co E H, ()9, (r) Remark 4.6.For (E1 is a positive measure and E,( )(p, 9) = 0. Proof,Note thatE2is skew-adjoint so(E2( )9, (P) = 0.Now 2 E1( )9,9) = (Ez(1 )9-E2( )9,(P) = (E1 ()(P E1)9) (E2 ()9, E ( )9) >0 q. e. d. Integrals.We shall need to define the operators f(z)dEkwhere f is a measurable complex valued function on Since p. ( ) = (E k( ) o , tli) is a bounded real valued k, , LI) measure we may define =Sf(z)h(z )d Sf(z)dp.k, y where Ik,I is the total variation of p. andh is the k H. Radon-Nikodym derivative of with respect to I k, LI) ilk, 74 We observe that for any complex measures p. and X, iff is I and X integrable then fdp. +SfdX Sfd(p.+X) We shall also set p. () Theorem 4.7. Consider p. and fork = 1, 2where (P, 4) .n., ( p , 9,1P E H.Then, p- , = II, , + ii-L . Moreover, if f is any co, 14, ...,(p,y 2, (p, LP complex Borel measurable function on .* thenf is p. -integrable if and only iff is p, -integrable fork = 1 (Poli k, cp , 4, and2.In this case = + ,cf(z)dp Sf(z)dp.1, Sf(z)dp. Proof.First = = (E1( )co, 4J) i(E2( )(P -1, , + Now letfbe p. -integrable.But for any Borel measurable set (Pqi M, Iii. (M)1 < IP, (M)I < (M).But by the minimum (P4) bounding property of total variation we must have 75 k = 1 and2.But thenfis I (N) co4i1 for Ii -integrable. k, , 1.1 k = 1and2, Conversely iffis k, -integrable for 4j1, fis [-I, -integrable. then, since II-Lcoopl <.....11 I + Ip. 11,co , iP (P , q. e. d. We are now in a position to define theoperatorskcxdEk andYk =,S1ydEk fork = 1,2.Recall thatE = Ei + iE2is the spectral measure ofA = A + i0where Ais a normal operator. and the We shall take as the domain of these operatorsXk Yk spaceD(A). We observe that if cp E D(A)then < co 4iE H. SIxldl Slz Idlc(p I for all The same can be said for soxand yare both 1-1. integrable for all k, LP H by Theorem 4.7. We are going to use the Riesz representation theorem to define andY .We would like to define linear functionals by Xk ) = xd(Ek( )(P,),P) and R ..cyd(E ( )(P k,cp() = onH. and R are linear and real valued, and Clearly Lk, k, the next theorem establishes the continuity. 76 Theorem 4.8.For each cpE D(A)andk = 1or2,L and k, co Rk, are continuous. Proof.Observe that ,)II xd(Ek()49,01 91x1d111"41 But from arguments in the complex case we know that I 5._Sizi2dIN-covilliP11 Hence, is continuous, and the same argument may be applied Lk, (p Rk, . q. e d. Thus we may writeL (1J) = (Xkc 0 , , and R (4i) 44, by the Riesz representation theorem.Clearly k, (P (Ykg), X and D(A). Ykare linear on Theorem 4. 9 (The Spectral Theorem). LetA be a densely defined normal operator onH, and letE = El + iE2be the spectral measure ofA, thenA =SxdEl-SydEz= X1 - Y2. 77 Proof.For p,4iE D(A)we have S(x+iy)d(111, ) L P t-P vz, X lydllLi + i (x+iy)chi2 = S-ydp. 2,9,LP because (A(p, 4J) is real.So (Acp, LP) =(SixdE19,LP) - ( ydEzp, LP) =([9xdEi-SydE2](p,Lp). q. e. d. We emphasize at this point that the above theorem is a special case of the more general problem of defining operators of the form where u is any real valued Borel measurable function.If SuclEk is a bounded real valued function all domain difficulties disappear = on all ofH.Indeed, we set and we may defineu udEk (SudE ,4J) =Sud Further Properties ofElandE2 In order to establish a uniqueness theorem for the spectral representation ofA we need to studyElandE2 moreclosely. Let us suppose thatA is a bounded normal operator onH, thenA = A + i0is bounded and normal, and the spectral measure Eof A is concentrated in a disk D centered at the origin in Then from the spectral theorem in the complex case we get n * m (A) ((A) ) -ZnZmdE. Butwe observe that (A)n((;:)*) = A(A)m + i0.Now a straightforward computation using polar coordinates for convenience, that isx = r cos 0and y = r sin 0, shows that n+m ("A-)«A-)*)m= r cos(n-m)OdE1 rn+msin(n-m)OdEz ID + i[ rn+msin(n-m)OdEi+,r rn+mcos(n-m)OdE Thus, n+nn . A (A*)rn = rcos(n-m)OdE1n+Tn - r sin(n-m)OdE2 and rn+msin(n-m)0dE1 = rn+mcos(n-m)OdE2 . 79 Lemma 4.10 [7,123].IfA is a bounded normal operator on the real Hilbert spaceH,andS CR2is a Borel set, and is the reflection of S across the x-axis, that is S= {(x, -y) I (x,y) ES}, thenE (S) = E /(S) and E2(S ) -E2(S). If cpE )rp,y9) =0, since is skew-adjoint. Proof. H then(E2( E2 Therefore0 = rn+mcos(n-m)0d(E2()cp, cp) = - rn+msin(n-m)0dia where p.(S) (Ei(S)cp,cp), cp E H.Letfbe continuous on D and f(r, -0) = -f(r, 0).Given E >0the Stone-Weierstrass theorem provides a trigonometric polynomialP(r, 0)such that If(r,0)-P(r, 0)1 < -2-onD, where P(r, 0) = anmrn+m sin(n-m)0 + bnmrn+mcos(n-m)0. n+m , E I coskn-m)01 < But then we must have and so bnmr n+m Consequently f(r, 0)4. = 0. 1f(r,0) - anmr sin(n-m)01< E If S is a compact set in the upper half plane then there exists a sequence of continuous functions {fn},withIf} bounded converging poitwise to and vanishing off the upper XS half plane. is the characteristic function ofS.Letgn Here XS equalfnin the upper half plane and equal-fnin the lower half plane, thenSgndp.= 0.But g pointwise, so by the S-S* 80 dominated convergence theorem .11Xs- Xs *11-1 - 0, orp(S) =i-L(S*). LetB be the collection of Borel sets lying in the upper half plane and such that 11(S) = p.(S) . This is easily seen to be a cr-algebra which as we already know contains the compact sets of the upper half plane. Hence B contains all the Borel sets in the upper half plane.But it is then immediate that 11(S) = P.(S*) for every set.Thus, (E1(S)9 , (p) = (E1(S*)cp, (p) for allyo E H. Now E1(S) is self -adjoint and so the polarization principle in the complexifica- (S) = (S*). tion gives(E (S)9,0 (Ei(S*)(P, 0, soE1 E1 Now rn+mcos(n-m)OdE2 = - rn+msin(n-m)OdE 1 andElis x-axis symmetric, so rn+mcos(n-m)0d(E2( )4 0 = 0. Thus similar arguments giveEz(S) = q. e.d. Suppose now that A is an unbounded normal operator onH. We shall employ F. Riesz reduction to the bounded case to show that the above theorem is still true.In what follows we refer the reader 18,307 fa 81 Lemma 4.11.LetA be an unbounded normal operator onH, andE = + the spectral measure ofA.IfS CR2is a El iE2 S (S) ), and Borel set and its x-axis reflection, thenE1 E1(S E(S)-E2(S*). Proof.Let A= A + i0as before, and let B (I+(A)(A---)*)-1. B is bounded and self-adjoint, and its corresponding spectral measure Fis concentrated in[0,1]. Now if B C + iDthen (C-fiD)(I+A(X)*)=(C+iD)[(I+AA*)+i0)= I + i0.ThusC(I+AA ) I andD(I+,AA ) = 0.But(I+AA ) is one to one and has a dense range. SoD = 0.ThereforeB = C + Oi.But then if F = F1 + ir2 F = 0by Lemma 4.10.Set 1 I as in the reduction 2 Pn = n+1 ),n to the bounded case, then Pn:H H; that is, His invariant under the projectionPn.By reduction to the bounded case we have 00 CO = H +iH , Hc = PHn c Hcn n n) n=1 n=1 = PH and Moreover we have where Hcn n c Hn = PnH. or) oo where is the spectral n=1n= 1 En En measure ofA:H H isAl . Moreover if n cn cn.HereAn Hcn A = A1 then =2 =A+ Oi.But each is bounded H .71,1H n n A-n n cn and normal, so by Lemma 4.10, ifEn =Eln+ iE2nthenEin 82 satisfyEin(S) and ), and E2n =Eln*(S) E2n(S) -E2n(S S CR2.Let H,S CR2, and E = E1+ iE2,then E (S)(p = Re E(S)'(the real part) 1 oo = Re = Re [E /4331. En(S )Pn E01. ln(S)Pn -E2n(S)Pnco] n= 1 00 ln(S)Pncc. n=1 On the other hand 00 E1(S)co = ReE(S)c9 E (S)P (/', in n n=1 soE1(S) = E1(S). Similarly,E2(S) = -E2(S ). q. e. d. The Uniqueness of Eand E2 Definition 4. 12 [7, 125].LetElandE2be bounded operator valued measures on the Borel sets ofR2in the real Hilbert space H. Then(Er E2)is a spectral pair if and only if forS, S1 and R2, and S S we S2Borel sets in the x-axis reflection of have: E1(S)is self-adjoint and E2(S)is skew-adjoint, E1(S) E1(S ), andE2(S) -E2(S) 83 E (S nS2) = E1(SI)E1(S2)E2(S1)E2(S2), and (S E(S + E(S)E and E2ln S2) 1 1)E2(S2) 2 1 l(S 2,) , (R2)= I, and E1 E2(R2) = 0. Remark 4.13. IfAis normal andE El + iE2is the spectral measure ofA then(E1E2)is a spectral pair. Theorem 4. 14.Let Abe a normal operator on a real Hilbert spaceH, and let be a spectral pair such that (El E) 2) A=SxdEl-irydE2. ThenEl = ElandE2 = E2where E = E + iE is the spectral measure of Proof.Since Es ) is a spectral pair the integralsSydEli (E''l 2 andSxdE2are also defined onD(A), the domain ofA.Indeed they are both zero. Thus A + i0 =S3cdE - + i[,S4ydE l+SxdE2] S(x+iy)dEl where Indeed, E = El + iE'2 .E'is a spectral measure. (S))2- (E'(S))2,+ i[E'(S)E'(S)+E' (S)E' (S)] (E'(S))21 El 2 2 1 1 2 = E(S). El (S) + iEl(S)2 84 ClearlyE'(S)is self-adjoint andE'(R2)= I.The strong add itivity comes from the strong additivity of the components. Thus, the uniqueness theorem in the complex case gives E q. e. d. A Function Calculus for A Letf(z) = u(z) + iv(z)be a Borel measurable function onR2. LetE =El+ iE2be the spectral measure ofA + i0 = A, where Ais a normal operator on the real Hilbert spaceH. Assume that f is bounded. Then f(A) Sf(z)dE= (u+iv)d(E +iE ) 1 2 = SudE1 - SvdE2 + i[ SudE2+SydE1] . Now if we wantf(A)to be a real operator we must have vdE+SudE2= 0.But we can achieve this ifvis odd with respect to the x-axis and u is even.This is because E1(S) = E (S) andE2(S) = -E2(S). Remark 4.15.The set of allf = u + ivwhere f is a bounded Borel measurable function onR2, and u is even with respect to the x-axis andvis odd is a real algebra. We shall denote this algebra by EL . 85 Definition 4. 16. ForfEet. we define f(A) =SudES.vdE =S u+iv)dE I . 1 2 Theorem 4.17.The map F: L(H), whereL(H)is the set of bounded operators onH, defined byF(f) = f(A)is an algebra homomorphism. Moreover, (A) = (f(A))*,wheref = u - iv. Proof. We havef(A) =Sf(z)dEIH,so the additivity and real scalar multiplication are clear.But from the same result in the complex case f1.f(A) =1(z)f2(z)dE1(z)dE1HSf2(z)dEl = f1 (A). f2(A) Also from the complex case we havef (A) = (f(A)). q. e. d. Stone's TheoremReal Case We begin this section with the construction of a particular unitary group on Hassociated with a given normal operator A onH. For each realtconsider the function ft(z) cos(ty) + i sin(ty).Observe thatcos(ty)is even with respect 86 to the x-axis andsin(ty)is odd.Thus for eacht,ftEet. Let (E1 ,E2) be the spectral pair ofA, and set ft(A) =Scosty dE1 sin(ty)dE2 . Theorem 4.18. ft(A)is a strongly continuous one parameter unitary group. Proof.Clearlyf0(A) = I. Now f = cos(t1-f-t2)y + i sin(t1+t2)y t 1+t2 = cos tly cos t2y - sin tiy sin t2y + i[sin tor cos t2y + cos tor sin t2y] =f f . t t 1 2 f (A) ,--- f(A)f(A).So Thus, by Theorem 4. 17, t1+t2 t 1 t2 * f(A) = (f (A))-1. Butf = T so (f (A))-1 = f (A) = (f (A)) -t t -t t' t t t H. We must establish Therefore, ft(A) is a unitary operator on t tothen thatft(A)is strongly continuous.Observe that if cos(t-t0)y 1 andsin(t-t0 )y 0, where the convergence is pointwise.But then 87 !Ift(A)T-ft(A)(,9112= 0 (ft(A)(P-fto(A)co,ft(A)cc-ft0(A)(P) z1 = 2-2(ft(A)(p, ft(A)co) 0 =211(1'112 2(ft 49) -to(A)(P, = 211 4911 24cos(t-t0)Yd -Ssin(t-to)yd So by dominated convergence II ft(A)49-ft (A)q1! 0 q. e. d. 0 We are going to show that the unitary group described above is the only one. Letu(t)be a strongly continuous one parameter group of unitary operators on a real Hilbert spaceH. We define u(h)co -ct) u(0) cp = 0 if the limit exists. We define D(ul(0)) = {coE HI u1(0)(pexists}. Certainlyu1(0) is an operator on this domain. We shall use the method of complexification again to show thatu'(0)is densely defined and skew-adjoint. Theorem 4.19.Letu(t)be a strongly continuous unitary group 88 (in brief, a unitary group) on the real Hilbert spaceH.Let u(t) = u(t) + i0be the complexification ofu(t).Then u(t)is a unitary group onHc, D(u1(0)) = D(u1(0)) + iD(u1(0)), andut(0) = ul(0), and Ifuandvare unitary groups andu'(0) = v'(0)then U = V. Proof. 1)Ifu(t)(cp+iLli) = 0thenu(t)9 u(t)4J = 0, and so = = 0. T, + ir E 9 and Lp E H such Let Hcthen there exist thatu(t)9 = andu(t)4i 11 SO u(t)(9+i4J) = + ill.Thus u(t)is a bijection onHcMoreover, (u + + i0 = u(t1)u(t2) + i0 tt1 u(t1+t2) = (u(t1)+i0)(u(t2)+i0) = u(t1)u(t2). Finally, II 71(t)(9+iLP)112 =II u(t)(P112 11u(t)4J112=11(P+i112 Henceu(t)is a unitary group. u(h)(9+i0) -(9 tli) 2) u.'(0)(9+iLli) lim 0 u(h)92-9 u(h)4J-0 h0lim + ilim 89 E D(U1(0)) if and only if + i4i E D(u'(0)) + iD(u'(0)) . 3) Supposeuandvare unitary groups on Hwith u1(0) = v1(0), thenu(0) = ul(0) + iur(0) = v'(0) + iv1(0) = v'(0). So by the complex Stone theoremu(t) = v(t),sou(t) v(t). q.e.d. Corollary 4.20.Ifu(t)is a real unitary group then u'(0)is densely defined. Proof. '(0)is densely defined, soD(u'(0)) = D(u'(0)) + iD(u'(0)) is dense.ThusD(u'(0))is dense inH. q. e. d. Corollary 4.21.u'(0) is skew-adjoint. Proof. We setG =-; us(0)for brevity.By the complex Stone theorem CT= G + i0is skew-adjoint. We show that (G) G Suppose cp + i4i E D(G ) then cp, E D(G ).Thus for every .* + ifl E D(6"), cp) = G*(p) and(GT1,0 = (11, G 4)).But then (P) i(G 4J) + i(Gri, (P) + (G11, Lp) -= G (p)- G LIJ) + i(n, G (p) + (n, G ) (y9+iip> Hence, G* C(C) * Conversely, ifco + ii E (G) then for every + in E D(E) we have, 90 = -G'-iG4J> sinceG is skew-adjoint.Thus = R, -Gm) - + + Set1 = 0to get =(, -G9)and(G,Lp) = -GLIJ).So -G(p. = G and7Gt.li.= G. So q2 + E D(G ), and ;1/4 (G) (cp+i0 G (cp+ikP).So G = (G) , and thus G is easily seen to be skew-adjoint q. e. d. Recall the particular unitary group corresponding to the normal operator A on Hgiven by ft(A) =Scosty dEi -9sinty dEz. Remark 4.22.Letu(t)be a unitary group on Hwith infinitesimal generatoru'(0).Then the above arguments have shown that the infinitesimal generator ofu(t) isu'(0). Theorem 4.23.LetA be a skew-adjoint operator onH, then A. ft(A)has infinitesimal generator 91 Proof.Observe that ft (A) = ft (A) + i0 =scos ty dEl-Ysinty dE2 + icos tydE2+Ssinty dEl] =SeitYd(Ei+iE2). Now Ais skew-adjoint, soA = A + i0is skew-adjoint with spectral measureEl + iE2concentrated on the imaginary axis. But because of thisX= S (x+iy)d(El+iE2) = iy d(E1+iE2) Therefore-iXis self-adjoint, and -iA = yd(E1FiE2)where we regardEl + iE2as a measure onR.But then if we set B = -iXtheneith=5eitYd(E1+iE2)-= ft(A).But this implies that iB = 71 So by the preceding remark is the generator offt(A). A = (ft (A)'(0) =ft(A)'(0) =A + i0.So Ais the infinitesimal gen- erator offt(A). q. e. d. Theorem 4.24.Letu(t)be a unitary group on the real Hilbert spaceH. Then there exists a unique skew-adjoint operator A such thatu(t) = ft(A). Proof. By Corollary 4.21A = u'(0)is skew-adjoint.Set v(t) thenv1(0) = A by Theorem 4.23. So by Theorem 4.19 ft(A) u = v.Suppose now thatu(t) = ft(B)where B is skew-adjoint. Thenu'(0) = B. q. e. d. 92 BIBLIOGRAPHY Dunford, N. and Schwartz, J. T.Linear Operators, Part I: General Theory.Interscience, New York, (1958). Treves, Fransois.Topological Vector Spaces Distributions and Kernels. Academic Press, New York, (1967). Kato, Tosio.Perturbation Theory for Linear Operators. Springer-Verlag, New York, (1966). Nelson, E. Annals of Mathematics, 67,(1959). Prugovecki, E.Quantum Mechanics in Hilbert Space. Academic Press New York and London, (1971). Lavine, R.Journal of Functional Analysis, Vol. 5,No. 3 June, (1970). Goodrich, R. K.Acta Sci. Math. (Szeged),33,(1972). Riesz, F. and Sz-Nagy, B.Functional Analysis.Ungar, New York, (1971). Ehrenfest, P.Zeitschrift der Physik, 45,(1927). Rejto, P.Pacific Journal of Math.,19,(1966).