Laurence Barker
Citation: Journal of Mathematical Physics 42, 4653 (2001); doi: 10.1063/1.1398582 View online: http://dx.doi.org/10.1063/1.1398582
View Table of Contents: http://aip.scitation.org/toc/jmp/42/10 Published by the American Institute of Physics
Continuum quantum systems as limits of discrete
quantum systems. III. Operators
Laurence Barkera)
Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey
共Received 29 September 2000; accepted for publication 12 July 2001兲
Convergence of a ‘‘discrete’’ operator to a ‘‘continuum’’ operator is defined. As examples, the circular rotor, the one-dimensional box, the harmonic oscillator, and the fractional Fourier transform are realized as limits of finite-dimensional quantum systems. Limits, thus defined, preserve algebraic structure. The results prepare for a sequel in which some affine canonical transforms will be ‘‘discretized.’’ © 2001
American Institute of Physics. 关DOI: 10.1063/1.1398582兴
I. INTRODUCTION
The continuum fractional Fourier transform of Namias1is the limit of two discrete fractional Fourier transforms, namely, the Kravchuk function FRFT and the Harper function FRFT共see Refs. 2 and 3兲. Some very straightforward continuum quantum systems, such as the circular rotor, the one-dimensional box and the harmonic oscillator, can easily be realized as limits of equally straightforward finite-dimensional systems whose Hamiltonians are difference operators. For many purposes, the above assertions are clear enough without ‘‘limit’’ being understood to have any abstract meaning; nevertheless, the goal of this article is to assign an appropriate general meaning to ‘‘limit,’’ to state the above assertions precisely, and to prove them. It is not that we object to the usual common sense techniques—on the contrary, we shall validate them—but subsequently, in a sequel,4 some ideas pioneered by Atakishiyev–Chumakov–Wolf5 will be de-veloped: continuum affine canonical transforms and continuum complex-order Fourier transforms will be realized as limits of analogous finite-dimensional transforms. In that application, common sense would not suffice.
Consider a Hilbert space L⬁, and Hilbert spaces Ln, where the index n runs over some infinite set of positive integers. In Sec. II, we shall interfaceL⬁with the spacesLn, and we shall assign a meaning to equations of the form
Kˆ⬁⫽lim
n
Kˆn,
where Kˆ⬁is a bounded operator onL⬁, and each Kˆnis a bounded operator onLn. In Sec. III, we shall assign a meaning to equations of the form
K⬁⫽lim
n Kn,
whereK⬁ andKn are quantum systems onL⬁andLn, respectively. Convergence of vectors has already been discussed in two prequels to the present article. The first prequel6explains howL⬁ is to be interfaced with the spaces Ln, and gives meaning to equations of the form
⬁⫽lim
n n,
a兲Electronic mail: barker@fen.bilkent.edu.tr.
4653
where ⬁苸L⬁ and n苸Ln. Some of the main definitions and results from Ref. 6 are briefly recalled later in Sec. II. The second prequel7 shows that widely used limiting techniques are in accordance with the definition of convergence.
With a view to applications, we might think of L⬁ as a ‘‘continuum’’ space, perhaps the Hilbert space formed from the space of square-integrable functions on a differentiable manifold. We might think of each spaceLnas a ‘‘discrete’’ space, perhaps a Hilbert space with a coordinate system such that the coefficients of a vector may be interpreted as sample-point values of a function on the manifold. In the case where the manifold isR, Digernes–Varadara´jan–Varadhan8 established a continuum-discrete correspondence—characterized in terms of limits—by embed-ding each Ln inL⬁. Our approach is more concerned with preservation of algebraic structure 共linearity, inner products, composition, tensor products兲. We interface L⬁ with the spacesLn by realizing the sequence (Ln)n as an inductive resolution of L⬁. The definition of an inductive resolution共recalled in Sec. II兲 is entirely algebraic, and, by this virtue, it relieves us of any need to assign any abstract meaning to the jargon ‘‘continuum’’ and ‘‘discrete.’’共As every physicist knows, these two terms often refer to different sides of the same coin.兲
The preservation of algebraic structure will be crucial in Ref. 4, where we shall be considering some Lie groups with several degrees of freedom. In subsequent work, we shall present a more systematic study of a way in which ‘‘continuum’’共usually infinite-dimensional兲 representations of Lie groups may be realized as limits of ‘‘discrete’’共usually finite-dimensional兲 representations. 共Part of the motive for this is to seek criteria for a system of numerically calculated transforms to respect ‘‘continuum’’ composition laws.兲 The results we give later, in Sec. III, and the applications we note in Sec. IV, all concern the special case of one-parameter groups. This special case is helpful as a stepping-stone because some of the concerns that arise in the general case reduce to trivialities here.
However, one-parameter systems are of interest in their own right, and can naturally be regarded as quantum dynamical systems, or, to use the language of Parthasarathy,9 quantum stochastic processes.共Let us not quibble about the flexible definitions of these terms.兲 Thus, we are led back to a question addressed by Digernes–Varadara´jan–Varadhan.8To what extent are spectra in the ‘‘continuum’’ scenario related to spectra in the ‘‘discrete’’ scenario? This question is ex-plored in Sec. V. The author would like to thank the referee for some useful suggestions concern-ing Sec. V. Although the material there is still only an initial foray into the matter, it was absent from the previous version of this article.
General motives for a continuum-discrete correspondence—characterized in terms of limits, and preserving algebraic structure—are noted in the prequels, Refs. 6 and 7. Some more extensive references for applications may be found in those two papers. The Gedankenexperiment in Ref. 7, Sec. 2, gives a heuristic introduction to our line of approach.
II. INDUCTION OF BOUNDED OPERATORS
By an operator on a Hilbert spaceL, we mean a linear map D→L, where the domain D is a dense subspace ofL. Every bounded operator on L extends uniquely to a bounded operator on L with domain L. Henceforth, all our bounded operators on a Hilbert space L shall be deemed to have domainL. We write U(L) for the group of unitary operators on L.
We must briefly review some of the definitions and results of Ref. 6. Consider a Hilbert space L⬁, a dense subspaceS of L⬁, an infinite set of positive integersN, Hilbert spaces Ln for each
n苸N, and linear maps resn:S→Ln.共The results below may easily be extended to the case where N is any directed set, as in Ref. 6.兲
The linear maps resn, called the restriction maps, are required to satisfy the reciprocity condition
具
兩典
⫽ lim n苸Nfor all,苸S. The sequence (Ln)n, equipped with the sequence (resn)n, is called an inductive
resolution ofL⬁.
Given a vector苸L⬁, and vectorsn苸Lnfor sufficiently large n苸N 共not necessarily for all
n苸N兲, we say that the sequence (n)n converges to ⬁ provided the norms 储n储 are bounded and
具
兩⬁典
⫽ limn苸N
具
resn共兲兩n典
for all 苸S. The Riesz representation theorem guarantees that (n)n converges to at most one vector in L⬁. When (n)n converges to ⬁, we call ⬁ the limit of (n)n, and we write ⬁ ⫽limn苸Nn. Note that ⫽limn苸Nresn() for all苸S.
Let us recall some results that we shall need from Ref. 6.
Theorem 2.1:共Ref. 6, Theorem 2.4兲 Any vector⬁苸L⬁is the limit of some sequence (n)n,
and, furthermore, the vectorsn苸Ln may be chosen such that储⬁储⫽储n储 for all n.
LetB⬁⫽兵j,⬁: j苸J⬁其 be any enumerated orthonormal basis forL⬁. Here, J⬁⫽N if L⬁ is infinite-dimensional, while J⬁⫽兵0,1, . . . ,d⫺1其ifL⬁has finite dimension d. By Ref. 6, Theorem 3.1, there exist Bn, indexed by n苸N, where each Bn is an enumerated orthonormal set Bn ⫽兵j,n: j苸Jn其 inLn, and
j,⬁⫽ lim n苸N
j,n
for all j苸J⬁. Note that, for each basis vectorj,⬁inL⬁, a corresponding basis vectorj,ninLn need not exist for all n, but thej,nmust exist for sufficiently large n.
As explained in Ref. 6, Sec. 3, theBn cannot always be chosen such that eachBn is a basis. 共In all our applications in Sec. IV, each of our chosen Bn is a basis. We also mention that, in all these applications,L⬁is infinite-dimensional,N is a set of positive integers, and each Lnhas finite dimension n.兲 We let L⬜n denote the subspace ofLn orthogonally complementary to the span of Bn. Given a vector⬁苸L⬁, we write
⬁⫽
兺
j⫽0
⬁
cj,⬁j,⬁
with the understanding that cj,n⫽0 for all j苸N⫺J⬁. Givenn苸Ln, we write
n⫽⬜n⫹
兺
j⫽0⬁
cj,nj,n
where⬜n苸L⬜n, and cj,n⫽0 for all j苸N⫺Jn. 共Of course, if Bnis a basis, then⬜n⫽0.兲 For later convenience, we define j,⬁ª0 when j苸N⫺J⬁, and j,nª0 when j苸N⫺Jn. Thus cj,⬁ ⫽
具
j,⬁兩⬁典
and cj,n⫽具
j,n兩n典
for all j苸N.Theorem 2.2:共Ref. 6, Theorem 3.4兲 Using the notation above,⬁⫽limn苸Nnif and only if
the norms储n储 are bounded, and cj,⬁⫽limn苸Ncj,n for all j苸J⬁.
We can now turn to convergence of operators. Let Kˆ⬁be a bounded operator onL⬁, and for sufficiently large n苸N, let Kˆn be a bounded operator Ln. We say that the sequence (Kˆn)n
converges to Kˆ⬁ provided the norms储Kˆn储 are bounded, and for all⬁苸L⬁, and all sequences (n)n withn苸Ln and⬁⫽limn苸N(n), we have
Kˆ⬁⬁⫽ lim
n苸N
共Kˆnn兲.
Theorem 2.1 ensures that the sequence (Kˆn)n converges to at most one bounded operator onL⬁. When (Kˆn)n converges to Kˆ⬁, we call Kˆ⬁ the limit of (Kˆn)n, and we write Kˆ⬁⫽limn苸NKˆn.
Remark 2.3: Given bounded operators Kˆ⬁⫽limn苸NKˆn and Kˆ⬁
⬘
⫽limn苸NKˆn⬘
, and given ,⬘
苸C, then Kˆ⬁⫹⬘
Kˆ⬁⬘
⫽limn苸N(Kˆn⫹⬘
Kˆn⬘
) and Kˆ⬁Kˆ⬁⬘
⫽limn苸NKˆnKˆn⬘
.Proof: This is obvious. 䊐
Theorem 2.4: Given any bounded Kˆ⬁onL⬁, then there exist bounded operators Kˆnon each
Ln such that Kˆ⬁⫽limn苸NKˆn and 储Kˆn储⫽储Kˆ⬁储 for all n苸N.
Proof: LetB⬁ andBn be as above. We define
Kj,k⫽
具
j,⬁兩Kˆ⬁k,⬁典
for all j ,k苸N. 共Note that Kj,k⫽0 unless j and k both belong to J⬁.兲 On each space Ln, we define an operator Kˆn
⬘
annihilatingL⬜n and such thatKj,k⫽
具
j,n兩Kˆn⬘
k,n典
for all j ,k苸Jn. Consider vectors⬁苸L⬁andn苸Lnsuch that⬁⫽limnn. Let the coefficients
cj,⬁ and cj,nbe as above. Then
储Kˆn
⬘
n储2⫽兺
j⫽0 ⬁冏
k兺
⫽0 ⬁ Kj,kck,n冏
2 ⭐储Kˆ⬁储2储n储2.So the norms储Kˆn
⬘
储 are bounded by 储Kˆ⬁储. Given⑀⬎0, then there exists a positive integer N and complex numbers c0, . . . ,cN⫺1 such that兺
j⫽0 N⫺1冏
k兺
⫽0 N⫺1 Kj,kck冏
2 ⭓共储Kˆ⬁储⫺⑀兲2兺
j⫽0 N⫺1 兩cj兩2.For sufficiently large n苸N, we have 兵0, . . . ,N⫺1其艚J⬁債Jn, whereupon 储Kˆn
⬘
储⭓储Kˆ⬁储⫺⑀. Therefore,储Kˆ⬁储⫽limn苸N储Kˆn⬘
储.We claim that Kˆ⬁⫽limnKˆn
⬘
. Let苸S. For each n, letnªresn(). To prove the claim, it suffices to show that具
兩Kˆ⬁⬁典
⫽limn
具
n兩Kˆn⬘
n典
. For each j苸N, let aj,⬁ª具
j,⬁兩典
and aj,nª具
j,n兩n典
. Thus⫽
兺
j⫽0 ⬁ aj,⬁j,⬁ and n⫽⬜n⫹兺
j⫽0 ⬁ aj,nj,n, where⬜n苸L⬜n . We have具
n兩Kˆn⬘
n典
⫽兺
j,k⫽0 ⬁ a ¯j,nKj,kck,nand a similar equation holds for
具
兩Kˆ⬁n典
.共By absolute convergence properties, all the sums we consider can be rearranged.兲 We have兩
具
兩Kˆ⬁⬁典
⫺具
n兩Kˆn⬘
n典
兩⭐冏
兺
j,k⫽0 ⬁ a ¯j,⬁Kj,k共ck,⬁⫺ck,n兲冏
⫹冏
兺
j,k⫽0 ⬁ 共a¯j,⬁⫺a¯j,n兲Kj,kck,n冏
. 共Using the boundedness of Kˆ⬁, it is easy to check that these sums are absolutely convergent.兲兺
j⫽0 ⬁冏
k兺
⫽0 ⬁ Kj,kck,n冏
2 ⭐C2储Kˆ ⬁储2for sufficiently large n. Part of Ref. 6, Lemma 3.3, says that兺j⬁⫽0兩aj,⬁⫺aj,n兩2⭐⑀2for sufficiently large n. Hence
冏
j,k兺
⫽0⬁
共a¯j,⬁⫺a¯j,n兲Kˆj,kck,n
冏
⭐⑀C储Kˆ⬁储. We may insist that C⭓储⬁储. Thereupon,兺
j⫽0 ⬁冏
k兺
⫽0 ⬁ Kj,k共ck,⬁⫺ck,n兲冏
2 ⭐4C2储Kˆ ⬁储2for sufficiently large n. The series 兺⬁j⫽0兩aj,⬁兩2 converges 共to 储储2), so there exists a positive integer M such that兺⬁j⫽M兩aj,⬁兩2⭐⑀2. We have
冏
j兺
⫽M ⬁兺
k⫽0 ⬁ a ¯j,⬁Kj,k共ck,⬁⫺ck,n兲冏
⭐2⑀C储Kˆ⬁储for large n. To prove the claim, it now suffices to show that
冏
兺
j⫽0 M⫺1兺
k⫽0 ⬁ a ¯j,⬁Kj,k共ck,⬁⫺ck,n兲冏
⫽O共⑀兲.Let j苸N. Suppose there exists some␦⬎0 such that, for every positive integer L, there exist complex numbers cL,cL⫹1, . . . satisfying
兺
k⫽L ⬁ 兩ck兩2⭐1 and冏
兺
k⫽L ⬁ Kj,kck冏
⬎␦.Then there exist complex numbers c0,c1, . . . and integers 0⫽L0⬍L1⬍¯ such that each Kj,kck is a non-negative real, and
兺
k⫽Lr⫺1 Lr⫺1 兩ck兩2⭐ 1 n2 and k⫽Lr兺
⫺1 Lr⫺1 Kj,kck⬎ ␦ 2nfor all positive integers r. The series 兺k⬁⫽0兩ck兩2 converges while the series 兺k⬁⫽0Kj,kck diverges. This contradicts the boundedness of Kˆ⬁. We deduce that, for any positive real B, there exists a positive integer L such that, for all complex numbers cL,cL⫹1, . . . satisfying兺k⬁⫽L兩ck兩2⭐B, we have兩兺k⬁⫽LKj,kck兩⭐⑀/ M . For large n, we have兺⬁k⫽0兩ck,⬁⫺ck,n兩2⭐4C2. So there exists a posi-tive integer L such that, for large n, and for all j⬍M, we have
冏
k兺
⫽L ⬁ Kj,k共ck,⬁⫺ck,n兲冏
⭐⑀/ M . Each兩aj,⬁兩⭐储储, so冏
j兺
⫽0 M⫺1兺
k⫽L ⬁ a ¯j,⬁Kj,k共ck,⬁⫺ck,n兲冏
⭐⑀储储冏
兺
j⫽0 M⫺1兺
k⫽0 L⫺1 a ¯j,⬁Kj,k共ck,⬁⫺ck,n兲冏
⫽O共⑀兲for large n. By Theorem 2.2, ck,⬁⫽limn苸Nck,n. The claim is established.
To finish the argument, we must replace the operators Kˆn
⬘
with operators Kˆn onLn such that 储Kˆn储⫽储Kˆ⬁储 for all n苸N. We may assume that 储Kˆ⬁储⫽1. From the first paragraph of the argu-ment,储Kˆn⬘
储 converges to 1. So Kˆn⫽0 for large n. When Kˆn⬘
⫽0, we put Kˆn⫽Kˆn⬘
/储Kˆn⬘
储, otherwise we put Kˆn⫽1ˆ. Then each 储Kˆn储⫽1, and 储Kˆn⫺Kˆn⬘
储→0. Since the norms 储n储 are bounded, 储Kˆnn⫺Kˆn⬘
n储→0. It was shown in Ref. 6, Remark 2.3, that, for⬁苸L⬁ andn,n苸Ln satis-fying ⬁⫽limn苸Nn and limn苸N储n⫺n储⫽0, we have ⬁⫽limn苸Nn. Therefore, Kˆ⬁⬁⫽limn苸NKˆnn. 䊐
Corollary 2.5: Given any bounded Hermitian operator Hˆ⬁ onL⬁, then there exist bounded
Hermitian operators Hˆn on each Ln such that Hˆ⬁⫽limn苸NHˆ n and 储Hˆn储⫽储Hˆ⬁储 for each n 苸N.
Proof: In the proof of Theorem 2.4, if Kˆ⬁ is Hermitian, then so is each Kˆn. 䊐 In order to accommodate the possibility of working with a compound of several quantum stochastic processes 共for example, a quantum system with several particles兲, we must discuss tensor products of inductive resolutions, and we must show how the limits of vectors and operators are compatible with the tensor product. LetL⬁
⬘
be a Hilbert space, and letS⬘
be a dense subspace ofL⬁⬘
. For each n苸N, let Ln⬘
be a Hilbert space, and let resn⬘
:S⬘
→Lnbe restriction maps. Then L⬁丢L⬁⬘
has an inductive resolution with restriction maps resn丢resn⬘
:S丢S⬘
→Ln丢Ln⬘
. Given limits of vectors⬁⫽limnnand⬁⬘
⫽limnn⬘
inL⬁andL⬁⬘
, respectively, it is clear that we have a limit of vectors ⬁丢⬁⬘
⫽limnn丢n⬘
. By considering orthonormal coordinates and applying Ref. 6, Theorem 3.4, it is easy to check that limits of bounded operators preserve tensor products in the same way. 共Warning: we are not invoking Ref. 6, Theorem 3.4, gratuitously. Not every sequence inLn丢Ln⬘
converging to⬁丢⬁⬘
has terms of the formn丢n⬘
.兲 These 共rather trivial兲 remarks show that the limits behave well in the 共rather banal兲 case of a fixed finite number of noninteracting processes. Presumably, they also behave well with respect to symmetric and anti-symmetric tensor products, and with respect to the construction of free, anti-symmetric, and antisym-metric Fock spaces共see Ref. 9, Chap. II兲. We leave that matter for further research.III. CONVERGENCE OF QUANTUM SYSTEMS
Recall that a family兵Kˆ (t):t苸R其 of operators on a Hilbert space L is said to be strongly
continuous provided each Kˆ (t) has domain L and, for all苸L, the function R→L given by
t哫Kˆ(t) is continuous. If, furthermore, Kˆ (0)⫽1ˆ and each Kˆ(t) is bounded, then we call
兵Kˆ (t):t苸R其 a quantum system on L. In that case, we sometimes consider a family of vectors
兵(t):t苸R其 such that
共t兲⫽Kˆ共t兲共0兲.
A quantum system U⫽兵Uˆ (t):t苸R其 on L is said to be unitary provided each operator Uˆ(t) is unitary. If, furthermore,
Uˆ共t兲Uˆ共t
⬘
兲⫽Uˆ共t⫹t⬘
兲for all t,t
⬘
苸R, then we say that U is conservative.The boundedness condition in our general definition of a quantum system is somewhat arti-ficial, but convenient for our purposes. Our main concern is with conservative systems, and these have been thoroughly studied in various contexts and from various perspectives. For a detailed
introduction to conservative systems as quantum stochastic systems, see Ref. 9, Chap. 1. Let us recall some well-known properties of conservative systems共introducing some notation that will be convenient in the proof of Theorem 3.5兲.
Suppose thatU is conservative. Stone’s theorem asserts that there exists a unique Hermitian operator Hˆ onL such that
U共t兲⫽exp共⫺iHˆt兲.
We call Hˆ the Hamiltonian forU. Conversely, every Hermitian operator on L is the Hamiltonian of a conservative quantum system. The bijective correspondence Hˆ↔U allows us to characterize conservative quantum systems by the Schro¨dinger equation
i d
dt共t兲⫽Hˆ共t兲.
For the sake of rigor, we must mention that, as a definition,
exp共⫺iHˆt兲ª
冕
⫺⬁ ⬁
e⫺itsdE共s兲,
where E is the spectral family for Hˆ . The notation on the right-hand side is as in Ref. 10, Chap. 7. It may be worth explaining what this equation tells us. Introducing some notation that will be of use in the proof of Theorem 3.5, let us consider an integer m, and write Eˆm for the orthogonal projection on L associated with E and the half-open interval 关m,m⫹1). 关Intuitively, we might think of Eˆm as the projection to the subspace EˆmL of L spanned by those ‘‘eigenvectors’’ whose ‘‘eigenvalues’’ are at least m and less than m⫹1. The operator Hˆ restricts to an operator on each subspace EˆmL. Vaguely, we might think of Eˆm as a kind of ‘‘eigenspace,’’ whose associated ‘‘eigenvalue’’ is spread across the interval 关m,m⫹1).兴 Any vector in L is a sum of vectors belonging to the spaces EˆmL, so the unitary operator exp(⫺iHˆt) is determined by the condition that it restricts to an operator on EˆmL given by
exp共⫺iHˆt兲⫽
兺
l⫽0⬁ 共⫺iHˆt兲l
l!
for all 苸EˆmL. 共The series converges because Hˆ restricts to a bounded operator on EˆmL.兲 Stone’s theorem may be found in Ref. 10, Theorem 7.38. The bijectivity of the correspon-dence Hˆ↔U is given in Ref. 15, Theorem 7.37. See also Ref. 9, Theorem 13.1.
Given a quantum system K⬁⫽兵Kˆ⬁(t):t苸R其 on L⬁, and quantum systems Kn⫽兵Kˆn(t):t 苸R其 onLn for sufficiently large n苸N, we say that (Kn)n converges toK⬁ provided
Kˆ⬁共t兲⫽ lim
n苸N
Kˆn共t兲
for all t苸R. Obviously, (Kn)n converges to at most one quantum system on L⬁. When (Kn)n converges toK⬁, we callK⬁ the limit of (Kn)n, and we writeK⬁⫽limn苸NKn.
Remark 3.1: LetK⬁⫽兵Kˆ⬁(t):t苸R其 andKn⫽兵Kˆn(t):t苸R其, respectively, be quantum systems onL⬁and on eachLn. Write⬁(t)⫽Kˆ⬁(t)⬁(0) andn(t)⫽Kˆn(t)n(0). Then we have a limit
of quantum systemsK⬁⫽limn苸NKnif and only if, given any initial state vectors⬁(0) inL⬁and n(0) in each Ln with ⬁(0)⫽limn苸N(n(0)), and writing ⬁(t)⫽Kˆ⬁(t)⬁(0) and n(t) ⫽Kˆn(t)n(0), we have⬁(t)⫽limn苸Nn(t) for all t苸R.
In particular, Remark 3.1 tells us that if the limit holds for the quantum systems and for the initial vectors, then the limit holds for all the time-evolved vectors. In case this seems counter-intuitive, we point out that, ifn(t) is to be a ‘‘good approximation’’ to⬁(t), one should first fix
t, and then choose n.
Theorem 3.2: Any quantum system onL⬁is the limit of a sequence of quantum systems on the spacesLn.
Proof: LetK⬁⫽兵Kˆ⬁(t):t苸R其 be a quantum system onL⬁. For each t苸R, and j,k苸N, we define
Kj,k共t兲ª
具
j,⬁兩Kˆ⬁共t兲k,⬁典
.Let Kˆn(t) be the operator in Ln constructed from the matrix entries Kj,k(t) as in the proof of Theorem 2.4. LetKn⫽兵Kˆn(t):t苸R其. Using the condition thatK⬁is strongly continuous, it is easy
to check that each Kn is strongly continuous. 䊐
Proposition 3.3: Let Hˆ⬁ and each Hˆn be bounded Hermitian operators on L⬁ and Ln,
respectively, and suppose that the norms储Hˆn储 are bounded. LetU⬁ and each Un be the
conser-vative systems with Hamiltonians Hˆ⬁ and Hˆn, respectively. Then U⬁⫽limn苸NUn if and only if Hˆ⬁⫽limn苸NHˆ n.
Proof: WriteU⬁⫽兵Uˆ⬁(t):t苸R其 andUn⫽兵Uˆn(t):t苸R其. For m苸N, let
Kˆm,⬁共t兲ª
兺
k⫽0 m 共⫺iHˆ⬁t兲k k! and Kˆm,n共t兲ªk兺
⫽0 m 共⫺iHˆnt兲k k! .Then Uˆ⬁(t)⫽limm→⬁Kˆ m,⬁(t) and Uˆn(t)⫽limm→⬁Kˆm,n(t).
Let ⑀⬎0. Consider vectors 苸S and ⬁苸L⬁ and n苸Ln such that ⬁⫽limnn. Write n⫽resn(). Let A be an upper bound for储储 and 储n储. Let B be an upper bound for 储Hˆ⬁储 and 储Hˆn储. Let C be an upper bound for 储⬁储 and 储n储. Choose m such that
2AC
兺
k⫽m⬁
兩Bt兩k/k!⭐⑀.
Then储Uˆ⬁(t)⫺Km,⬁(t)储⭐⑀/2AC⭓储Uˆn(t)⫺Km,n(t)储 for sufficiently large n. Hence 兩
具
兩Uˆ⬁共t兲⫺Kˆm,⬁共t兲兩⬁典
⫺具
n兩Uˆn⫺Kˆm,n共t兲兩n典
兩⭐⑀.If Hˆ⬁⫽limnHˆn, then, by Remark 2.3, Kˆm,⬁⫽limnKˆm,n, hence Uˆ⬁(t)⬁⫽limn苸NUˆ n(t)n. Conversely, suppose that Uˆ⬁(t)⬁⫽limn苸NUˆ n(t)n. Given t, we can put⑀⫽t2/2共and then choose m兲, where
兩
具
兩Kˆm,⬁共t兲⬁典
⫺具
n兩Kˆm,n共t兲n典
兩⫽O共t2兲for sufficiently large n. Equating coefficients of t共the sums 兺km⫽0兩Hˆnt兩k/k! and the similar sum for
Hˆ⬁ are bounded by eB兩t兩兲, we obtain Hˆ⬁⬁⫽limnHˆ nn. 䊐
Corollary 3.4: LetU⬁ be a conservative system onL⬁with bounded Hamiltonian Hˆ⬁. Then
there exist conservative systems Un on Ln with bounded Hamiltonians Hˆn such that U⬁ ⫽limn苸NUn and Hˆ⬁⫽limn苸NHˆn.
Proof: This is immediate from Corollary 2.5 and Proposition 3.3. 䊐
Theorem 3.5: Any conservative system on L⬁ is the limit of a sequence of conservative systems on the spacesLn.
Proof: LetU⬁⫽兵Uˆ⬁(t):t苸R其be a conservative system onL⬁, let Hˆ⬁be the Hamiltonian for U⬁, and let E be the spectral family for Hˆ⬁. For each m苸Z, let Eˆmbe the orthogonal projection as above, and letLm,⬁⫽EˆmL⬁. The Hermitian operator Hˆ⬁restricts to a Hermitian operator Hˆm,⬁ onLm,⬁. LetUm,⬁⫽兵Uˆm,⬁(t):t苸R其be the conservative system onLm,⬁with Hamiltonian Hˆm,⬁. Any vector⬁苸L⬁ has a unique decomposition as a sum
⬁⫽
兺
m苸Z m,⬁ where eachm,⬁苸Lm,⬁. We have Hˆ⬁m,⬁⫽Hˆm,⬁m,⬁ and
Uˆ⬁共t兲⬁⫽
兺
m苸Z
Uˆm,⬁共t兲m,⬁.
It is easy to see that there exists an enumerated orthonormal basisB⬁⫽兵j,⬁: j苸J⬁其such that eachj,⬁belongs to one of the subspacesLm,⬁. The enumerated orthonormal setsBn, as in Sec. II, may be chosen such that each Jn債J⬁. For each m苸Z, let
J⬁共m兲ª兵j苸J⬁:j,⬁苸Lm,⬁其 and Jn共m兲ªJn艚J⬁共m兲.
LetLm,n be the subspace ofLn spanned by the vectorsj,n such that j苸Jn(m). Any vectorn 苸Ln has a unique decomposition as a sum
n⫽⬜n⫹
兺
m苸Z m,n,
where⬜n苸L⬜n , and eachm,n苸Lm,n. For j ,k苸J⬁, let
Hj,k⫽
具
j,⬁兩Hˆ⬁k,⬁典
.Note that Hj,k⫽Hk, j, and Hj,k⫽0 unless j,k苸J⬁(m) for some m苸Z. Let Hˆm,nbe the Hermitian operator onLm,n such that
Hj,k⫽
具
j,n兩Hˆm,nk,n典
for j,k苸Jn(m). LetUm,n⫽兵Uˆm,n(t):t苸R其 be the conservative system onLm,nwith Hamiltonian
Hˆm,n. Let Hˆn be the Hermitian operator onLn such that Hˆn⬜n⫽0 and Hˆnm,n⫽Hˆm,nm,n. Let Un⫽兵Uˆn(t):t苸R其 be the conservative system onLn with Hamiltonian Hˆn. Then
Uˆn共t兲n⫽⬜n⫹
兺
m苸ZUˆm,n共t兲m,n.
We are to show that Uˆ⬁(t)⫽limn苸NUˆ n(t) for all t苸R.
For each n苸N, letn苸Ln, and suppose that⬁⫽limn苸Nn. Write
⬁⫽
兺
j⫽0 ⬁ cj,⬁j,⬁ and n⫽⬜n⫹兺
j⫽0 ⬁ cj,nj,nas in Sec. II. Fix t苸R, and let ⬁⫽Uˆ⬁(t)⬁ and n⫽Uˆnn. We are to show that ⬁ ⫽limnn. Write
⬁⫽
兺
j⫽0 ⬁ dj,⬁j,⬁ and n⫽⬜n⫹兺
j⫽0 ⬁ dj,nj,nas we did for ⬁ andn. The norms储n储⫽储n储 are bounded. So, by Theorem 2.2, we are to show that dj,⬁⫽limndj,n for all j苸J⬁. Fix j苸J⬁, and let m be such that j苸J⬁(m). We have
dj,⬁⫽
兺
k苸J⬁(m)
具
j,⬁兩Uˆm,⬁共t兲k,⬁典
ck,⬁.
The equation still holds with the symbol n instead of the symbol ⬁. Replacing Hˆ⬁ with the Hermitian operator EˆmHˆ⬁⫽Hˆ⬁Eˆmdoes not change Hˆm,⬁or Hˆm,n, so it does not change Uˆm,⬁or
Uˆn,m. So it does not change dj,⬁or dj,n. Therefore, we may assume that Hˆm⬘,⬁⫽0 for all integers
m
⬘
⫽m. Hence Hˆm⬘,n⫽0 for all m⬘
⫽m and all n苸N. But now Hˆ⬁ is bounded, indeed 储Hˆ⬁储 ⭐兩m兩⫹1. Furthermore, the operators Hˆn are constructed from Hˆ⬁ just as the operators Kˆn⬘
were constructed from Kˆ⬁in the proof of Theorem 2.4. So Hˆ⬁⫽limn苸NHˆ n. Thanks to Proposition 3.3,the argument is now complete. 䊐
Corollary 3.6: Any unitary operator onL⬁ is the limit of a sequence of unitary operators on the spacesLn.
Proof: Given a unitary operator Uˆ⬁ on L⬁, then by Ref. 10 Exercise 7.50, there exists a conservative system兵Uˆ⬁(t):t苸R其such that Uˆ⬁⫽Uˆ⬁(1). Theorem 3.5 now gives the assertion.䊐 A more direct way to demonstrate Corollary 3.6 is to adapt the proof of Theorem 2.4, using the Gram–Schmidt process to modify the columns of the matrices (Kj,k)j,k苸Jn. The argument is fairly routine, although it is complicated by the need to make some arbitrary choices when the Gram–Schmidt process terminates prematurely.
The existence results above can be interpreted as saying that, in principle, any ‘‘continuum’’ system共of a particular kind兲 is the limit of a sequence of ‘‘discrete’’ systems 共of the same kind兲. The next result provides one way of actually recognizing that a given ‘‘continuum’’ system is the limit of a given sequence of ‘‘discrete’’ systems.
Proposition 3.7: Let U⬁⫽兵Uˆ⬁(t):t苸R其 be a conservative system on L⬁, and for each n 苸N, let Un⫽兵Uˆn(t):t苸R其 be a conservative system onLn. Let Hˆ⬁and Hˆn, respectively, be the
Hamiltonians. LetB⬁ and Bn be as in Sec. II. Suppose that, for each j苸J⬁, there exists a real j,⬁such that
Hˆ⬁j,⬁⫽j,⬁j,⬁.
Suppose also that, for sufficiently large n, there exist realsj,nsuch that
Hˆnj,n⫽j,nj,n.
ThenU⬁⫽limn苸NUn if and only ifj,⬁⫽limn苸Nj,nfor all j苸J⬁.
Proof: This follows quickly from Theorem 2.2. 䊐
Proposition 3.7 yields an alternative共and very easy兲 proof of Theorem 3.5 in the special case of a conservative system onL⬁ with a diagonalizable Hamiltonian.
IV. SOME EXAMPLES OF CONTINUUM LIMITS OF DISCRETE SYSTEMS
In all the examples to follow, we shall apply Proposition 3.7 to show that the given ‘‘con-tinuum’’ system is the limit of the given sequence of ‘‘discrete’’ systems. Each of the inductive resolutions is a sample-point inductive resolution, as in Ref. 6, Examples 2.A–2.F. Sample-point inductive resolutions are examined also in Ref. 7.
Example 4.A: The circular rotor. The rotor, in one dimension, is a model for a particle moving
Let S be the space of smooth functions :R→C such that has period unity and is square-integrable on a bounded domain. The inner product onS is given by integration over an interval of length unity. Making a suitable choice of units, the Hamiltonian Hˆ⬁of the rotor has domainS and satisfies
Hˆ⬁共x兲⫽⫺d2共x兲/dx2
for苸S and x苸R. The completion L⬁ofS has an orthonormal basis B⬁⫽兵j,⬁: j苸N其 given by j,⬁共x兲⫽
再
& cos共jx兲 if j is even,
& sin共共 j⫹1兲x兲 if j is odd.
It is easy to check thatB⬁diagonalizesH⬁, indeed, Hˆ⬁j,⬁⫽j,⬁j,⬁, where j,⬁⫽
再
2j2 if j is even,
2共 j⫹1兲2 if j is odd.
LetN be the set of positive odd integers. For each n苸N, let Ln be the n-dimensional inner product space consisting of the functionsZ→C with period n. The inner product on Lnis given by summation over n consecutive integers. We replace the differential operator ⫺d2/dx2 with a difference operator Hˆn where
Hˆn共X兲⫽n2共⫺共X⫺1兲⫹2共X兲⫺共X⫹1兲兲 for 苸Ln and X苸Z. Given an integer j with 0⭐ j⭐n⫺1, we put
j,n共X兲⫽
再
冑
2/n cos共jX/n兲 if j is even,冑
2/n sin共共 j⫹1兲X/n兲 if j is odd.It is easy to check that 兵j,n:0⭐ j⭐n⫺1其 is an orthonormal basis for Ln diagonalizing Hˆn. Writing Hˆnj,n⫽j,nj,n, then
j,n⫽
再
2n2共1⫺cos共2j X/n兲兲 if j is even,
2n2共1⫺cos共2共 j⫹1兲X/n兲兲 if j is odd.
LetU⬁be the conservative system onL⬁with Hamiltonian Hˆ⬁. For each n苸N, let Unbe the conservative system onLn with Hamiltonian Hˆn. Of course, it is heuristically ‘‘obvious’’ thatU⬁ is some kind of ‘‘limit’’ ofUn, but in order to formulate this observation mathematically, we must realize (Ln)n as an inductive resolution ofL⬁. We define resn:S→Ln such that
resn共兲共X兲⫽共X/n兲/
冑
nfor 苸S and X苸Z with ⫺n/2⬍X⬍n/2. It is easy to check that the sequence (Ln)n, equipped with the sequence (resn)n, is indeed an inductive resolution ofL⬁. 共In fact, this is the precisely the one-dimensional case of Ref. 6, Example 2.F.兲 Given j苸N, then, for all n⬎ j, we have j,n ⫽resn(j,⬁). Therefore, j,⬁⫽limn苸Nj,n. Since n,⬁⫽limn苸Nj,n, Proposition 3.7 tells us that
U⬁⫽ lim n苸N
Un.
Example 4.B: The one-dimensional box. For each j苸N and x苸关⫺12, 1
j,⬁共x兲⫽
再
& cos共共 j⫹1兲x兲 if j is even, & sin共共 j⫹1兲x兲 if j is odd.
LetL⬁be the Hilbert space with orthonormal basis兵j,⬁: j苸N其. LetS be the dense subspace of L⬁ consisting of the smooth functions关⫺12,
1
2兴→C. The box, in one dimension, is the conservative
systemU⬁ whose Hamiltonian Hˆ⬁ has domainS and is given by
Hˆ⬁共x兲⫽⫺d2共x兲/dx2 for 苸S. Evidently Hˆ⬁j,⬁⫽j,⬁j,⬁ wherej,⬁⫽2( j⫹1)2.
Again, letN be the set of positive odd integers. Let Ln be the n-dimensional inner product space consisting of the complex-valued functions on the integers X lying in the interval ⫺n/2 ⬍X⬍n/2. As in the previous example, we replace the differential operator ⫺d2/dx2 with a difference operator Hˆn, but this time the sample-points indexed by (1⫺n)/2 and (n⫺1)/2 are to be interpreted as end-points 共they are no longer interpreted as being adjacent兲. Writing n⫽2l ⫹1, we put
Hˆn共X兲⫽
再
n2共2共⫺l兲⫺共1⫺l兲兲 if X⫽⫺l,
n2共⫺共X⫺1兲⫹2共X兲⫺共X⫹1兲兲 if ⫺l⬍X⬍l,
n2共⫺共l⫺1兲⫹2共l兲兲 if X⫽l.
The operator Hˆn is diagonalized by the orthonormal basisBn⫽兵j,n:0⭐ j⭐n⫺1其 ofLn, where
j,n共X兲⫽
冦
冑
2 n⫺1cos冉
共 j⫹1兲X n⫹1冊
if j is even,冑
2 n⫹1sin冉
共 j⫹1兲X n⫹1冊
if j is odd.In fact, Hˆnj,n⫽j,nj,n wherej,n⫽2(1⫺cos((j⫹1)/(n⫹1))).
We realize (Ln)n as an inductive resolution of L⬁ by defining resn:S→Ln by the same formula as in Example 4.A. A straightforward calculation yields, for all j苸N, all x苸关⫺12,
1 2兴 and
all sequences (Xn)nof integers such that x⫽limn苸NXn/
冑
n, the point-wise convergence condition j,⬁共x兲⫽ limn苸N
冑
nj,n共Xn/n兲.The norms储j,n储 are all unity, and, in particular, they are bounded. In Ref. 7, Theorem 3.1, it was proved that point-wise convergence of vectors with bounded norms implies convergence; in par-ticular,
j,⬁⫽ lim n苸N
j,n.
Observing that eachj,⬁⫽limn苸Nj,n, we again conclude from Proposition 3.7 that U⬁⫽ lim
n苸N Un.
Example 4.C: The Harper function harmonic oscillator. In this example and the next, we
review some results from Refs. 11 and 12, and we show how that material can be streamlined using Proposition 3.7. Recall that L2(R) has an orthonormal basis兵hj,⬁: j苸N其 consisting of the functions hj,⬁:R→C, called the Hermite–Gaussians, which are given by
hj,⬁共x兲⫽Cje⫺x
2/2
Hj共x兲,
where Hjis the Hermite polynomial of degree j , and Cjis a positive real normalization constant. The continuum harmonic oscillator is defined to be the conservative system U⬁⫽兵Uˆ⬁(t):t 苸R其 whose Hamiltonian Hˆ⬁ is given by Hˆ⬁(x)⫽(⫺d2/dx2⫹x2)(x), or equivalently,
Hˆ⬁hj,⬁⫽(2k⫹1)hj,⬁. Thus
Uˆ⬁共t兲hj,⬁⫽e⫺(2 j⫹1)ithj,⬁.
Let N be an infinite set of positive integers such that
冑
n2/n1苸Z for all n1,n2苸N with n1 ⭐n2.共At one point in the discussion, we shall make use of this peculiar hypothesis on N, but the assertions probably hold for any infinite setN of positive integers.兲 Given an element n苸N, let Ln be the n-dimensional inner product space consisting of the functionsZ→C with period n. We realize (Ln)n as an inductive resolution of L2(R) by defining restriction maps resn:S(R)→Ln such thatresn共兲共X兲⫽共n/2兲⫺1/4共共n/2兲⫺1/2X兲
for 苸S(R) and X苸Z. After Harper,13 Namias,1 Pei–Yeh3 and others, we define the Harper
function harmonic oscillator to be the conservative systemUn onLnwith Hamiltonian Hˆnsuch that
Hˆn共X兲⫽
n
2共⫺共X⫺1兲⫹共4⫺2 cos共2iX/n兲兲共X兲⫺共X⫹1兲兲
for苸Ln and X苸Z. The definition and enumeration of the Harper functions0,n,1,n, . . . may be found in Ref. 3; see also Refs. 11 and 12. The Harper functions comprise an orthonormal basis for Ln, they are eigenvectors of Hˆn, and by Ref. 12, Theorem 2.5,
hj,⬁⫽ lim n苸N
j,n
for all j苸N. 共It is here that the peculiar hypothesis on N is used.兲 Combining this result with Ref. 12, Lemmas 3.1 and 3.9, it is easy to show that the eigenvalue j,n of Hˆn associated with j,n satisfies
2 j⫹1⫽ lim n苸N
j,n. Proposition 3.7 now yields
U⬁⫽ lim n苸N
Un.
As suggested in Ref. 7, Sec. 3, the peculiar hypothesis onN can perhaps be relaxed using results that were not available when Ref. 12 was written.
Example 4.D: The Harper function fractional Fourier transform. We continue to use the
notation from Example 4.C. After Namias, the continuum FRFT is defined to be the conservative systemF⬁⫽兵Fˆ⬁t :t苸R其 such that
Fˆ⬁thj,⬁⫽e2i jthj,⬁.
As Namias observed, the continuum FRFT and the continuum harmonic oscillator are related by the equality
Uˆ⬁共t兲⫽e⫺itFˆ⬁⫺t/.
Note that Fˆ1/4is the usual Fourier transform. The Harper function FRFT comes in two versions, the import versionIn⫽兵Iˆnt :t苸R其and the domestic versionDn⫽兵Dˆnt:t苸R其. The import version, defined by
Iˆntj,n⫽e2i jtj,n,
is perhaps rather artificial共its eigenvalues being ‘‘imported’’ from the continuum FRFT兲, but it has the virtue that Fˆn1/4is the usual discrete Fourier transform. The domestic version, defined by
Uˆn共t兲⫽e⫺itDˆn⫺t/,
has the virtue that it has an explicit Hamiltonian, namely (Hˆn⫺1)/2. By Proposition 3.7, F⬁⫽ lim
n苸N
In⫽ lim n苸N
Dn.
Example 4.E: The Kravchuk function harmonic oscillator. We retain the notation from
Ex-amples 4.C and 4.D, except that we now letN be any set of positive integers. Given n苸N, let us write n⫽2l⫹1, and let Xn be the set consisting of the X such that l⫹X and l⫺X are both natural numbers. We write L(Xn) to denote the n-dimensional inner product space consisting of the complex-valued functions onXn. As in Ref. 7, Secs. 4 and 5, we realize (L(Xn))nas an inductive resolution of L2(R) by defining res
n:S(R)→L(Xn) such that resn共兲共X兲⫽l⫺1/4共l⫺1/2X兲
for苸S(R) and X苸Xn. Recall共or see Ref. 7, Sec. 5兲 that the Kravchuk functions hj,ncomprise an orthonormal basis兵hj,n:0⭐ j⭐n⫺1其 for L(Xn). The Kravchuk functions are discrete analog of the Hermite–Gaussians, and arise from a binomial weight function in place of a Gaussian weight function. By Ref. 7, Theorem 5.1,
hj,⬁⫽ lim n苸N
hj,n
for all j苸N. After Ref. 14, the Kravchuk function harmonic oscillator is defined to be the conservative system Kˆn⫽兵Kˆn(t):t苸R其 onLn such that
Hˆn共t兲hj,n⫽e⫺(2 j⫹1)ithj,n. By Proposition 3.7,
U⬁⫽ lim
n苸N Kn.
Example 4.F: The Kravchuk function fractional Fourier transform. We retain the notation
from the previous three examples, N being any infinite set of positive integers. After Ref. 2, the
Kravchuk function FRFT is defined to be the conservative systemFn⫽兵Fˆn t
:t苸R其 such that
Fˆnthj,n⫽e2i jthj,n. Equivalently,Fn may be defined by
By Proposition 3.7,
F⬁⫽ lim
n苸N Fn.
Comment: Advantages of the Kravchuk function FRFT over the Harper function FRFT. In
applications of the Harper function FRFT, one must select either the import version, whose eigenvalues are integer powers of e2it, but whose Hamiltonian is not known explicitly, or else one must select the domestic version, whose Hamiltonian is (Hˆn⫺1)/2, but whose eigenvalues are not known explicitly. Either way, the eigenvectors—the Harper functions—lack a known explicit formula, and have to be calculated numerically. The eigenvectors of the Kravchuk function FRFT Fnare integer powers of e2it. As can be gleaned from Refs. 2 and 5, the Hamiltonian forFnhas a very simple description in terms of the n-dimensional irreducible representation of the Lie algebra su(2)共see also Ref. 4兲. The eigenvectors of Fn—the Kravchuk functions—are given by a complicated but explicit formula.
V. SOME QUESTIONS AND REMARKS ON CONVERGENCE OF SPECTRA
An alternative description of a conservative system is provided by the spectral measure asso-ciated with the Hamiltonian. Throughout this section, we consider conservative systems U⬁ ⫽兵Uˆ⬁(t):t苸R其 onL⬁ andUn⫽兵Uˆn(t):t苸R其 on eachLn. Let Hˆ⬁ and Hˆn be the Hamiltonians for U⬁ and Uˆn, respectively. If U⬁⫽limn苸NUn, how is the spectral measure for the Hermitian operator H⬁ related to the spectral measure for the operators Hn? Or, more simply, how is the spectrum(Hˆ⬁) 共or the essential or residual spectrum兲 related to the spectra(Hˆn)?
On the one hand, it would be desirable to have techniques for investigating the spectrum共or spectral measure兲 of an infinite-dimensional system by examining limiting properties of the spec-tra of finite-dimensional approximations. On the other hand, finite-dimensional systems are them-selves of interest.共As a vague principle, any closed system of finite extent in space can have only finitely many independent nondecaying states.兲 Finite-dimensional systems are not always more amenable than infinite-dimensional systems共difference equations often have richer solutions than their analogous differential equations.兲 In connection with example 4.E, it is worth remembering that De Moivre, having established the correspondence between the Gaussian distribution and the binomial distribution, then employed the Gaussian as an approximation to the binomial. Con-tinuum approximation to discrete phenomena has pervaded statistical techniques ever since. It is to be expected that results relating(Hˆ⬁) and(Hˆn) could be usefully applied in either direction. As regards practical methods for relating the spectra of discrete and continuum systems, the results in this article are simply not in competition with those in Ref. 8. We do not know whether or not their results can be extended to our more general context.共It should be mentioned that the examples considered in Sec. 4 are all, essentially, in the situation they considered.兲 The following result indictates that the questions above do have answers, and that our approach can be developed to yield alternative and more general methods.
Proposition 5.1: Suppose thatU⬁⫽limnUn. Suppose also that Hˆ⬁and each Hˆn are bounded,
and that the norms 储Hˆn储 are bounded. Then every point 苸(Hˆn) is the limit ⫽limnn of pointsn苸(Hˆn).
Proof: The condition苸(Hˆ⬁) is equivalent to the condition that there exists a sequence (m)m of vectors inL⬁ such that储m储⫽1 and 储(Hˆ⬁⫺)m储→0 as m→⬁ 共see, for instance, Ref. 15, Theorem 5.10兲. Since S is dense in L⬁, we may insist that eachm苸S. Let⑀⬎0, and fix
m such that储(Hˆ⬁⫺)m储⭐⑀/2. By Proposition 3.3, the convergence hypothesis onU⬁is equiva-lent to the condition that Hˆ⬁⫽limnHˆ n. Noting that limn储resn(m)储⫽1, and putting n ⫽resn(m)/储resn(m)储, we have 储(Hˆn⫺)n储⭐⑀ for sufficiently large n. By a well-known criterion for existence of spectral points in an interval 共see Ref. 12, Theorem 5.9兲, (Hˆn)艚关
Corollary 5.2: In the situation of Proposition 5.1, suppose that the limits limnn of points n苸(Hˆn) comprise a discrete subset ofR. Then Hˆn is diagonalizable. 䊐 It seems probable that the boundedness condition in Proposition 5.1 can be removed by using a refinement of the argument 共and the rider to Stone’s theorem as recorded in Ref. 9, Theorem 13.1兲. A more systematic option would be to wait for that to become a corollary of a result expressing the condition U⬁⫽limnUn in terms of the spectral measures. We end with a few comments in this direction. Consider an interval I in R. Write I¯ and I° for the closure and the interior. Let EI,⬁ and EI,nbe the corresponding projections toL⬁ andLn associated with Hˆ⬁and Hn. To see that convergence of the sequence (Hˆn)ndoes not imply convergence of the sequence (EI,n)n, let a be an end-point of I, and let Hˆn⫽(a⫹(⫺2)n)1ˆ.
Question 5.A: Are the following conditions equivalent?
共1兲 U⬁⫽limnUn.
共2兲 If⬁⫽limnn withn苸EI,nLn, then⬁苸E¯,nI L⬁.
共3兲 If⬁⫽limnn with⬁苸EI°,nL⬁ and 储⬁储⫽limn储n储, then limn储(1ˆ⫺EˆI,n)n储⫽0.
1V. Namias, ‘‘The fractional order Fourier transform and its applications to quantum mechanics,’’ J. Inst. Math. Appl. 25,
214 –265共1980兲.
2N. M. Atakishiyev and K. B. Wolf, ‘‘Fractional Fourier-Kravchuk transform,’’ J. Opt. Soc. Am. 14, 1467–1477共1997兲. 3S-C. Pei and M-H. Yeh, ‘‘Improved discrete fractional Fourier transform,’’ Opt. Lett. 22, 1047–1049共1997兲. 4L. Barker, ‘‘Continuum quantum systems as limits of discrete quantum systems, IV: affine canonical transforms’’
共preprint兲.
5
N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, ‘‘Wigner distribution for finite systems,’’ J. Math. Phys. 39, 6247– 6269共1998兲.
6L. Barker, ‘‘Continuum quantum systems as limits of discrete quantum systems, I: state vectors,’’ J. Funct. Anal.共to
appear兲.
7
L. Barker, ‘‘Continuum quantum systems as limits of discrete quantum systems, II: state functions,’’ J. Phys. A 22, 4673– 4682共2001兲.
8T. Digernes, V. S. Varadara´jan, and S. R. S. Varadhan, ‘‘Finite approximations to quantum systems,’’ Rev. Mod. Phys. 6,
621– 648共1994兲.
9K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus共Birkha¨user, Basel, 1992兲. 10
J. Weidmann, Linear Operators in Hilbert Spaces共Teubner, Stuttgart, 1976; English edition by Springer, Berlin, 1980兲.
11L. Barker, C¸ . Candan, T. Hakiog˘lu, A. Kutay, and H. M. Ozaktas, ‘‘The discrete harmonic oscillator, Harper’s equation,
and the discrete fractional Fourier transform,’’ J. Phys. A 33, 2209–2222共2000兲.
12L. Barker, ‘‘The discrete fractional Fourier transform and Harper’s equation,’’ Mathematika共to appear兲. 13
P. G. Harper, ‘‘Single band motion of conduction electrons in a uniform magnetic field,’’ Proc. Phys. Soc. 68, 874 – 878
共1955兲.
14N. M. Atakishiev and S. K. Suslov, ‘‘Difference analogues of the harmonic oscillator共English translation兲,’’ Teor. Mat.
Fiz. 85, 64 –73共1990兲.
15P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, with Applications to Schro¨dinger Operators共Springer,
