CCommutators of spectral projections of spinoperators
Ood ShabtaiAugust 4, 2020
Abstract
We present a proof that the operator norm of the commutator of cer-tain spectral projections associated with spin operators converges to inthe semiclassical limit. The ranges of the projections are spanned by alleigenvectors corresponding to positive eigenvalues. The proof involves thetheory of Hankel operators on the Hardy space. A discussion of severalanalogous results is also included, with an emphasis on the case of finiteHeisenberg groups. Contents P x,j,m (cid:48) ,m . . . . . . . . . . . . . . . . . . . . 114.3 Matrix elements of ( a ( j + ) , ∞ )( J x ) . . . . . . . . . . . . . . . . 13 P x,j,m (cid:48) ,m . . . . . . . . . . . . . . . . . . . . . . . 175.4 Limits of central elements of (( j + ) a, ∞ )( J x ) . . . . . . . . . . . 171 a r X i v : . [ m a t h - ph ] A ug Miscellaneous proofs 18
Let J x , J y , J z denote the generators of an irreducible, unitary, n -dimensionalrepresentation of SU (2), satisfying the commutation relations[ J x , J y ] = iJ z , [ J y , J z ] = iJ x , [ J z , J x ] = iJ y . Consider the commutator C n = (cid:2) (0 , ∞ ) ( J x ) , (0 , ∞ ) ( J z ) (cid:3) , where (0 , ∞ ) denotes the indicator function of (0 , ∞ ) ⊂ R . The main results ofthe present work are that Theorem 1.1 (L. Polterovich) . (cid:107) C n +2 (cid:107) op = for every n ∈ N , and Theorem 1.2. lim n →∞ (cid:107) C n (cid:107) op = . The sequence ( (cid:107) C n (cid:107) op ) ∞ n =2 is bounded from above by due to a general factabout commutators of orthogonal projections ([10]). Nonetheless, it is perhapsnot evident a priori that the sequence should converge at all, let alone to thelargest possible value.As it turns out, however, analogues of Theorem 1.2 hold for several otherfamilies of pairs of spectral projections arising from non-commuting observables.A few such examples are formulated in Section 1.3, and a modest extension ofTheorem 1.2 is included in Section 3.1. Ultimately, we suspect that the variousresults presented here are instances of a rather general phenomenon. We referthe reader to Section 7 for details and remarks along these lines. The numerical simulations (originally by Y. Le Floch) of ( (cid:107) C n (cid:107) op ) ∞ n =2 implyfurther intriguing properties. Here, [
A, B ] = AB − BA denotes the commutator of a pair of linear operators A, B . (cid:107) C n (cid:107) op as a function of n .Figure 2: ln (cid:0) − (cid:107) C k (cid:107) op (cid:1) as a function of ln(4 k ).Notably, (cid:107) C n (cid:107) op appears to depend on the dimension of the representationmodulo 4. More precisely, Conjecture 1.3. (cid:107) C n +3 (cid:107) op −(cid:107) C n +1 (cid:107) op (cid:107) C n +3 (cid:107) op −(cid:107) C n (cid:107) op = o (1) and (cid:107) C n +3 (cid:107) op −(cid:107) C n +1 (cid:107) op12 −(cid:107) C n + p (cid:107) op = o (1) for p = 0 , , . However, the convergence rate of the various sequences is presently un-known . Finally, it holds that (cid:107) C n (cid:107) op ≥ for every n ≥
2, as depicted above. The application of linear regression suggests a rate not faster than O (cid:16) n − (cid:17) , but otherwisedoes not seem to provide a clear answer. C n ) ∞ n =2 , joint with Y. Le Floch and L. Polterovich.The proof that (cid:107) C n +2 (cid:107) op = for every n ∈ N is quite straightforward andrequires little but symmetries. Otherwise, the lower bound is derived throughspecific elements of the matrix representing C n in some orthonormal eigenba-sis of J z . The arguments involved are relatively simple when n is odd, thanksto symmetries again. When n ≡ C n is complicated toestimate directly as n → ∞ (and involves sums of products of certain specialfunctions). The proof of Theorem 1.2 also revolves around the matrix elements of C n andtheir limits as n → ∞ . The proof appears in Section 3, relying on two mainingredients which are presented subsequently. Namely, in Section 4 we derivea concise integral formula for the elements of C n in terms of the correspondingmatrix elements of the one-parameter subgroup (cid:8) e − iθJ y (cid:12)(cid:12) − π ≤ θ < π (cid:9) . Thus, we avoid the initial, complicated expressions for the elements of C n al-together. In Section 5, we use a classical ([26], 8.21.12) asymptotic estimatefor the elements (up to normalization ) of e − iθJ y to study the aforementionedintegral formula. The estimate is valid in a restricted range of indices, whichnonetheless suffices for our purpose, i.e., to calculate the limits of sufficientlymany of the elements of C n .Ultimately, for every positive integer N and every n > N + 1, we invert therows of some sub-matrix of − C n to obtain a collection C n,N ∈ M N ( C ), suchthat C n,N is a sub-matrix of C n,N +1 whenever n > N + 3, and furthermore,lim n →∞ C n,N exists and has constant elements along the anti-diagonals.The latter, we recall, is the defining property of (possibly infinite) Hankelmatrices ([13]). More generally, any operator on some Hilbert space whosematrix relative to some orthonormal basis is a Hankel matrix may be consideredas a Hankel operator. Thus, we show that (cid:16) lim n →∞ C n,N (cid:17) N ≥ is the sequence of truncated matrices of some fixed Hankel operator whosenorm, it turns out, equals . This suffices to conclude the proof.We note (again) that a slightly extended version of Theorem 1.2 is alsoincluded in Section 3.1. The proof is pretty much identical, and involves slightlymodified Hankel operators. The asymptotic estimate is formulated in terms of Jacobi polynomias, rather than Wignerd-functions, and applies in significantly more general settings. See (14) for the original result,and Conclusion 5.1 for the adaptation to our present settings. Notably, the same Hankel operator appears in all of the cases addressed in Section 7. .3 Analogues of Theorem 1.2 The results of this part are proven in Section 6. Our first example involves thestandard quantum model for a particle in a line (i.e., in R ). The next two exam-ples are similar to the first, but involve the configuration spaces T , Z n = Z /n Z rather than R . The last example is formulated in terms of the representationtheory of the group of orientation preserving Euclidean plane isometries.Let X, Ξ denote the position and momentum operators on L ( R ), acting ona smooth f ∈ L ( R ) by Xf ( x ) = xf ( x ) , Ξ f ( x ) = − i (cid:126) f (cid:48) ( x ) . Theorem 1.4.
Consider the commutator C (1) (cid:126) = [Π X , Π Ξ ] , where Π X = (0 , ∞ ) ( X ) , Π Ξ = (0 , ∞ ) (Ξ) . Then C (1) (cid:126) ≡ C (1) is independent of (cid:126) , and (cid:107) C (1) (cid:107) op = . Similarly, define the operators Θ , Z on L ( S ) (cid:39) L (cid:0) [0 , π ) , π dθ (cid:1) byΘ u ( θ ) = θu ( θ ) , Zu ( θ ) = − i πn u (cid:48) ( θ ) , where u ∈ C ∞ ( S ) and n ∈ N . The operators Θ , Z may be used to constructan analogue of Weyl quantization for the cylinder T ∗ S ([11], [20], [19]). Theorem 1.5.
Let C (2) n = [Π Θ , Π Z ] , where Π Θ = (0 , ∞ ) (cos Θ) , Π Z = (0 , ∞ ) (cos Z ) . Then lim n →∞ (cid:107) C (2) n (cid:107) op = . The same result holds if we replace the Heavisidefunction with ( a, ∞ ) , where a ∈ [0 , . The previous two examples may be formulated in terms of the representa-tion theory of the Heisenberg groups H ( R ) and H ( T ) (i.e., the group of unitaryoperators on L ( T ) generated by translation operators and by operators of mul-tiplication by characters). A similar result holds for representations of finiteHeisenberg groups associated to Z n as n → ∞ ([24, 18, 30, 27]). This problemwas suggested to us by D. Kazhdan, and its solution provided the model of theproof for SU (2) (as well as for the rest of the examples).Let g , g define an irreducible unitary representation of the finite Heisenberggroup H ( Z n ) on l ( Z n ) by g ( f )( x ) = f ( x + 1) , g ( f )( x ) = e πxin f ( x ) . Let Π , Π denote the orthogonal projections on the subspaces of l ( Z n ) spannedby eigenvectors of g , g corresponding to eigenvalues with positive real part.Consider the commutator C (3) n = [Π , Π ] . The parallel of Theorem 1.2 is the following.5 heorem 1.6. lim n →∞ (cid:107) C (3) n (cid:107) op = , and the same holds if we replace theHeaviside function with ( a, ∞ ) , where a ∈ [0 , . Additionally, as in the case of SU (2), Theorem 1.7 (Y. Le Floch) . (cid:107) C (3)4 n +2 (cid:107) op ≡ for every n ∈ N . The numerical simulations of the sequence (cid:16) (cid:107) C (3) n (cid:107) op (cid:17) ∞ n =2 feature some strik-ing similarities with the equivalent simulations (1) for SU (2), including theconjectured dependence on n mod 4.Figure 3: The norm of C (3) n as a function of n for the Heisenberg groups H ( Z n ).Note the similarity to the graph for SU (2). In particular, the graph also appearsto depend on n mod 4.Our final example may be derived as a consequence of Theorem 1.2. Let SE (2) denote the group of orientation preserving Euclidean plane isometries.The asymptotic formula 5.1, which underlies the proof of the Theorem 1.2,provides a non-trivial relation ([7, 22, 25]) between the representations of SU (2)and of SE (2). This fact led us to study the analogue of C n for the irreduciblerepresentations of SE (2).Consider L ( S ) (cid:39) L (cid:0) [0 , π ) , π dφ (cid:1) as before, and fix R >
0. Let X , X denote the multiplication operators M R cos φ , M R sin φ respectively, and let Φdenote the differentiation operator f (cid:55)→ − if (cid:48) . Then X , X , Φ satisfy the com-mutation relations[ X , X ] = 0 , [ X , Φ] = iX , [Φ , X ] = iX , hence they generate a unitary representation of SE (2) on L ( S ), which is ir-reducible since X ± iX = M Re ± iφ act as raising and lowering operators on6he standard basis of L ( S ). In fact, every non-trivial irreducible unitary rep-resentation of SE (2) is equivalent to the representation generated by X , X , Φfor some
R >
Theorem 1.8.
Let C (4) R = [Π X , Π Φ ] , where Π X = (0 , ∞ ) ( X ) , Π Φ = (0 , ∞ ) (Φ) . Then C (4) R ≡ C (4) is independent of R , and (cid:107) C (4) (cid:107) op = . Theorem 1.2 is essentially reduced to the problem of the calculation of (cid:107) H E (cid:107) op for some Hankel operator H E which we now specify. The operator H E appearsand plays roughly the same role in all of the results of Section 7 as well.Let T ⊂ C denote the unit circle, and declare the functions z (cid:55)→ z p , p ∈ Z to be an orthonormal basis of L ( T ). Let Π T : L ( T ) → L ( T ) denote theCauchy-Szeg¨o projection on the Hardy space H ( T ) = { f ∈ L ( T ) | ˆ f ( p ) = 0 for every p < } . Here, ˆ f ( p ) = (cid:104) f, z p (cid:105) L ( T ) denotes the p -th Fourier coefficient of f . Finally, let M φ denote the multiplication operator defined by a function φ ∈ L ∞ ( T ). Definition 2.1.
The Hankel operator corresponding to the symbol φ ∈ L ∞ ( T ) is defined as H φ = (Id − Π T ) M φ Π T : H ( T ) → (cid:0) H ( T ) (cid:1) ⊥ . Let [ H φ ] = ( h k,l ) k,l ≥ denote the matrix representing H φ in the standardbases B = { z p − | p > } , C = { z − p | p > } of H ( T ) and H ( T ) ⊥ , respectively. Then h k,l = (cid:104) φz l − , z − k (cid:105) = ˆ φ (1 − k − l ) . The truncated matrices associated with a symbol φ ∈ L ∞ ( T ) are relevant to usas well. For an infinite matrix A = ( a k,l ) k,l ≥ denote A N = ( a k,l ) ≤ k,l ≤ N . Wewill require the following basic fact. Lemma 2.2. lim N →∞ (cid:107) [ H φ ] N (cid:107) op = (cid:107) H φ (cid:107) op . Finally, let E = { z ∈ T | (cid:60) z > } denote the right half of the unit circle.Denote H E = H E , where E is the indicator function of E . The Fouriercoefficients of E are specified byˆ E ( p ) = (cid:26) if p = 0 , sin (cid:0) πp (cid:1) πp if p (cid:54) = 0 . (1)Perhaps somewhat surprisingly, the operator H E is closely related to the com-mutators C n . In particular, the proof of Theorem 1.2 relies on the following.7 emma 2.3. (cid:107) H E (cid:107) op = . Hence by Lemma 2.2, lim N →∞ (cid:107) [ H E ] N (cid:107) op = . We present the (simple) proof of the lemma in the following subsection. Theinequality (cid:107) H E (cid:107) op ≤ follows from the contents of Section 3, though we includea separate proof using Nehari’s Theorem on Hankel operators. The complemen-tary inequality (cid:107) H E (cid:107) op ≥ is a direct consequence of Power’s Theorem onHankel operators with piecewise continuous symbols. In this part, we apply two fundamental theorems on Hankel operators to obtainthe proof of Lemma 2.3. For a complex sequence a = ( a k ) k ∈ N , define the Hankelmatrix S a = ( a k + l ) k,l ∈ N : l ( N ) → l ( N ). Theorem 2.4 ([12]) . S a is bounded on l ( N ) if and only if there exists φ ∈ L ∞ ( T ) such that a k = ˆ φ ( k ) for every k ≥ . In this case, (cid:107) S a (cid:107) op = inf {(cid:107) φ (cid:107) ∞ | ˆ φ ( k ) = a k for every k ≥ } . We recall that [ H E ] = ( h k,l ) k,l ≥ = (cid:0) ˆ E (1 − k − l ) (cid:1) k,l ≥ , so the sequence associated with [ H E ] is (cid:0) ˆ E ( − − k ) (cid:1) k ∈ N = (cid:16) (cid:100) ¯ z E ( k ) (cid:17) k ∈ N . If we define φ ( z ) = ¯ z (cid:18) E ( z ) − (cid:19) , and choose k ≥
0, thenˆ φ ( k ) = (cid:104) E − , z k +1 (cid:105) = ˆ E ( k + 1) = ˆ E ( − − k ) , therefore Conclusion 2.5. (cid:107) H E (cid:107) op ≤ (cid:107) φ (cid:107) ∞ = . To obtain the complementary inequality, assume that φ ∈ L ∞ ( T ) has welldefined one-sided limits at every point of T . For α ∈ T , define the jump of φ at α as φ α = 12 lim t → + (cid:0) φ (cid:0) αe it (cid:1) − φ (cid:0) αe − it (cid:1)(cid:1) . Then
Theorem 2.6 ([17]) . The essential spectrum of the Hankel operator H φ is givenby σ ess ( H φ ) = [0 , iφ ] ∪ [0 , iφ − ] ∪ (cid:16) ∪ α ∈ T \{± } (cid:104) − (cid:112) − φ α φ ¯ α , (cid:112) − φ α φ ¯ α (cid:105)(cid:17) . The inequality (cid:107) H E (cid:107) op ≥ now follows, since Conclusion 2.7.
The essential spectrum of H E equals (cid:2) − , (cid:3) . Note that σ ess is a (closed) subset of the spectrum of H E , hence (cid:107) H E (cid:107) op ≥ . The proof of Theorem 1.2
We begin with a few preliminary notations and definitions ([3], [28]). Recall thatthe spectrum of J x , J y , J z equals the set { j, j − , ..., − j } , where j = ( n − E z,j = { e m | m = j, j − , ..., − j } denote an orthonormal eigenbasis of J z , such that J z e m = me m . The matricesrepresenting (0 , ∞ ) ( J x ) , (0 , ∞ ) ( J z ) in E z,j may be written as P x,j = ( P x,j,m (cid:48) ,m ) | m (cid:48) | , | m |≤ j = (cid:18) P ,x,j P ,x,j P ∗ ,x,j P ,x,j (cid:19) , P z,j = (cid:18) ˜ I j
00 0 (cid:19) . Here, ˜ I j is the identity matrix of size (cid:4) j + (cid:5) . The matrix of C n is given by[ P x,j , P z,j ] = (cid:18) − P ,x,j P ∗ ,x,j (cid:19) , hence (cid:107) C n (cid:107) op = (cid:107) P ,x,j (cid:107) op . In our notations, P ,x,j = ( P x,j,m (cid:48) ,m ) j ≥ m (cid:48) > ≥ m ≥− j . We turn our attention to the ”central elements” of P x,j , that is, to the se-quences ( P x,j + k,m (cid:48) ,m ) k ∈ N with j, m (cid:48) , m fixed such that m (cid:48) , m ∈ { j, j − , ..., − j } .According to Conclusion 5.5,lim k →∞ P x,j + k,m (cid:48) ,m = ˆ E ( m − m (cid:48) ) , where we recall that E is the indicator function of the right half of the unitcircle in C , as well as the symbol of the Hankel operator H E of Lemma 2.3.Thus, evidently, for fixed N ∈ N , the bottom left N × N corner of P ,x,j converges in M N ( C ) as n → ∞ . More precisely, Conclusion 3.1.
Let N ∈ N , and assume that j > N . Let C n,N = ( c n,k,l ) k,l =1 ,...,N ,where c n,k,l = (cid:26) P x,j,k, − l if n ∈ N + 1 P x,j, − + k, − l if n ∈ N Then lim n →∞ C n,N = [ H E ] N . The proof of Theorem 1.2 easily follows now, since lim N →∞ (cid:107) [ H E ] N (cid:107) op = by Lemma 2.3, and clearly12 ≥ lim inf n (cid:107) C n (cid:107) op ≥ lim inf n (cid:107) C n,N (cid:107) op = (cid:107) [ H E ] N (cid:107) op . (2)Letting N → ∞ , we obtain the desired result.9 .1 An extension of Theorem 1.2 In this part, we outline the proof of the following extension of Theorem 1.2.
Theorem 3.2.
Fix a ∈ [0 , and b ∈ (0 , , and let C n,a,b = (cid:104) ( a ( j + ) , ∞ )( J x ) , ( ,b ( j + )]( J z ) (cid:105) . Then lim n →∞ (cid:107) C n,a,b (cid:107) op = . This can be extended further by replacing ( a ( j + ) , ∞ )( J x ) with spectralprojections corresponding to (not necessarily open) intervals whose end-pointsare a (cid:0) j + (cid:1) , a (cid:0) j + (cid:1) , where 0 < a < a ≤ ∞ . However, once 0 < a ischosen, the projection arising from J z must correspond to an interval of the formabove (or (0 , ∞ )), due to the limitations of the asymptotic estimate underlyingthe results of Section 5.The proof of Theorem 3.2 is rather identical to that of Theorem 1.2, exceptthat we apply the more general Conclusion 5.6, instead of Conclusion 5.5. How-ever, we note that the conjectured modulo 4 dependence on the dimension n seems to be (more or less) a unique feature of the case a = b = 0 (see Figures6, 7, for example).Let P x,a,j , Q z,b,j denote the matrices representing the spectral projectionsin E z,j . Then P x,a,j = ( P x,a,j,m (cid:48) ,m ) | m (cid:48) | , | m |≤ j = (cid:18) P ,x,a,j P ,x,a,j P ∗ ,x,a,j P ,x,a,j (cid:19) , Q z,b,j = (cid:18) ˜ I b j
00 0 (cid:19) , where ˜ I b j = (cid:18) I b j (cid:19) , and b j = (cid:4) b (cid:0) j + (cid:1)(cid:5) , so that lim j →∞ b j = ∞ . Hence, C n,a,b is represented bythe matrix [ C n,a,b ] = (cid:32) (cid:104) P ,x,a,j , ˜ I b j (cid:105) − ˜ I b j P ,x,a,j P ∗ ,x,a,j ˜ I b j (cid:33) . We may proceed to define a sequence of sub-matrices of [ C n,a,b ] as before, since b j j →∞ −−−→ ∞ . According to Conclusion 5.6,lim k →∞ P a,x,j + k,m (cid:48) ,m = ˆ E a ( m − m (cid:48) ) , where E a = { z ∈ T | (cid:60) z > a } . It follows that the bottom left N × N corner of˜ I b j P ,x,a,j converges in M N ( C ) to the truncated matrix of the Hankel operator H Ea , whose norm equals by the same arguments that were applied to H E .10 Matrix elements of spectral projections
In this section, we establish a concise integral formula for the coefficients of P x,j = ( P x,j,m (cid:48) ,m ) | m (cid:48) | , | m |≤ j , which is the matrix representing the spectral pro-jection (0 , ∞ ) ( J x ) in the basis E z,j . The latter, we recall, is an eigenbasis of J z .The eventual formula that we obtain for P x,j,m (cid:48) ,m treats the cases m − m (cid:48) ∈ Z and m − m (cid:48) ∈ Z + 1 separately. The Wigner small d-matrix d j ( θ ) = (cid:16) d jm (cid:48) ,m ( θ ) (cid:17) | m (cid:48) | , | m |≤ j is the matrix of e − iθJ y in the basis E z,j . It is fundamental in the representation theory of SU (2). TheWigner d-functions d jm (cid:48) ,m ( θ ) are real valued 4 π -periodic trigonometric polyno-mials, commonly specified by the formula d jm (cid:48) ,m ( θ ) = (cid:115) ( j + m )!( j − m )!( j + m (cid:48) )!( j − m (cid:48) )! · (cid:88) s ( − m (cid:48) − m + s (cid:18) j + m (cid:48) j + m − s (cid:19)(cid:18) j − m (cid:48) s (cid:19) (cid:18) cos θ (cid:19) j + m − m (cid:48) − s (cid:18) sin θ (cid:19) m (cid:48) − m +2 s . The parity of d jm (cid:48) ,m with respect to m (cid:48) , m and θ is specified by ([28], 4.4) d jm (cid:48) ,m ( − θ ) = ( − m (cid:48) − m d jm (cid:48) ,m ( θ ) = d jm,m (cid:48) ( θ ) = d j − m (cid:48) , − m ( θ ) . (3)Another useful relation is d jm (cid:48) ,m ( θ + π ) = ( − j − m d jm (cid:48) , − m ( θ ) . (4)Finally, we will rely on the Fourier expansion of d jm (cid:48) ,m , specified by ([3], 3.78,[6]) d jm (cid:48) ,m ( θ ) = e i π ( m − m (cid:48) ) j (cid:88) µ = − j d jm,µ (cid:16) π (cid:17) d jm (cid:48) ,µ (cid:16) π (cid:17) e − iµθ . (5) P x,j,m (cid:48) ,m We may rotate one spin operator to another, and in particular, J x and J z arerelated by the formula e i π J y J x e − i π J y = J z . This means that the vectors f m = e − i π J y e m , m = j, j − , ..., − j form an orthonormal eigenbasis of J x , with J x f m = mf m . Note that P x,j,m (cid:48) ,m = (cid:104) (0 , ∞ ) ( J x ) e m , e m (cid:48) (cid:105) = (cid:88) µ> (cid:104) e m , f µ (cid:105)(cid:104) f µ , e m (cid:48) (cid:105) . P x,j,m (cid:48) ,m = (cid:88) µ> d jm,µ (cid:16) π (cid:17) d jm (cid:48) ,µ (cid:16) π (cid:17) . (6)Comparing the last expression with (5), we deduce that P x,j,m (cid:48) ,m equals thesum of negative Fourier coefficients of d jm (cid:48) ,m , up to multiplication by e i π ( m (cid:48) − m ) .Equivalently (see Section 2), P x,j,m (cid:48) ,m = e i π ( m (cid:48) − m ) (Id − Π T ) ( d jm (cid:48) ,m )(0) . Π T acts on L ( T ) by z p (cid:55)→ [0 , ∞ ) ( p ) z p . A closely related operator is theperiodic Hilbert transform H T , which acts by z p (cid:55)→ − i sgn( p ) z p . Thus, anotherequivalent formula is P x,j,m (cid:48) ,m = 12 e i π ( m (cid:48) − m ) (cid:18) d jm (cid:48) ,m (0) − (cid:68) d jm (cid:48) ,m , (cid:69) L ( T ) − i H T (cid:16) d jm (cid:48) ,m (cid:17) (0) (cid:19) . (7)Furthermore, by (5), the zeroth Fourier coefficient of d jm (cid:48) ,m is specified by (cid:104) d jm (cid:48) ,m , (cid:105) L ( T ) = (cid:26) e i π ( m − m (cid:48) ) d jm, (cid:0) π (cid:1) d jm (cid:48) , (cid:0) π (cid:1) if j ∈ N j ∈ N \ N . (8) H T maps even functions to odd functions and vice versa. In light of the parityproperties of (3), we finally obtain the following. Conclusion 4.1.
The matrix elements of (0 , ∞ ) ( J x ) in the eigenbasis E z,j of J z are given by P x,j,m (cid:48) ,m = e i π ( m (cid:48) − m ) (cid:16) δ m (cid:48) ,m − (cid:104) d jm (cid:48) ,m , (cid:105) L ( T ) (cid:17) if m (cid:48) − m ∈ Z − i e i π ( m (cid:48) − m ) H T ( d jm (cid:48) ,m )(0) if m (cid:48) − m ∈ Z + 1 . Here m (cid:48) , m ∈ { j, j − , ..., − j } and δ m (cid:48) ,m is Kronecker’s delta. The periodic Hilbert transform admits the representation H T f ( θ ) = 14 π lim ε → + (cid:90) ε ≤| θ |≤ π f ( θ ) cot (cid:18) θ − θ (cid:19) dθ, where f ∈ L ( T ) is a 4 π -periodic function. For f = d jm (cid:48) ,m with m − m (cid:48) ∈ Z +1,we obtain the formula H T ( d jm (cid:48) ,m )(0) = − π (cid:90) π d jm (cid:48) ,m ( θ ) cot (cid:18) θ (cid:19) dθ, (9)which will be studied in the next section. i.e., functions f ∈ L ( T ) with ˆ f ( p ) = ˆ f ( − p ) for every p ∈ Z . .3 Matrix elements of ( a ( j + ) , ∞ )( J x ) In this subsection, we extend (7) to projections of the form ( a ( j + ) , ∞ )( J x ),where 0 ≤ a <
1. More generally, the method may be used to obtain similarformulas for the elements of spectral projections corresponding to (not neces-sarily open) intervals with end-points at a (cid:0) j + (cid:1) , a (cid:0) j + (cid:1) , where a < a (hence also for projections corresponding to combinations of such intervals).The matrix elements in the present case are given by P x,a,j,m (cid:48) ,m = (cid:68) ( a ( j + ) , ∞ )( J x ) e m , e m (cid:48) (cid:69) = (cid:88) µ>a ( j + ) (cid:104) e m , f µ (cid:105)(cid:104) f µ , e m (cid:48) (cid:105) = (cid:88) µ>a ( j + ) d jm,µ (cid:16) π (cid:17) d jm (cid:48) ,µ (cid:16) π (cid:17) . This formula is the analogue of (6), and it also admits an interpretation throughthe Fourier expansions of the Wigner d-functions. Indeed, denote the Fouriercoefficients of d jm (cid:48) ,m by ˆ d jm (cid:48) ,m ( p ). Thenˆ d jm (cid:48) ,m ( p ) = (cid:40) e − i π ( m (cid:48) − m ) d jm, − p (cid:0) π (cid:1) d jm (cid:48) , − p (cid:0) π (cid:1) if p = 2 j, j − , ..., − j , therefore P x,a,j,m (cid:48) ,m = e i π ( m (cid:48) − m ) (cid:88) p< (cid:100)− a (2 j +1) (cid:101) ˆ d jm (cid:48) ,m ( p ) . Thus, P x,a,j,m (cid:48) ,m equals the sum of Fourier coefficients of d jm (cid:48) ,m correspondingto indices lesser than (cid:100)− a (2 j + 1) (cid:101) . Equivalently, we may shift the Fourierexpansion of d jm (cid:48) ,m using the relation (cid:91) z − p f ( l ) = ˆ f ( l + p ) to obtain P x,a,j,m (cid:48) ,m = e i π ( m (cid:48) − m ) (Id − Π T ) (cid:16) g jm (cid:48) ,m (cid:17) (0) , where g jm (cid:48) ,m ( θ ) = e − ia j θ d jm (cid:48) ,m ( θ ) and a j = (cid:100)− a (2 j +1) (cid:101) . As in (7), we translatethe former expression to P x,a,j,m (cid:48) ,m = 12 e i π ( m (cid:48) − m ) (cid:16) δ m (cid:48) ,m − (cid:104) g jm (cid:48) ,m , (cid:105) L ( T ) − i H T ( g jm (cid:48) ,m )(0) (cid:17) . (10)In the final part of the next section, we will study lim k →∞ P a,x,j + k,m (cid:48) ,m . In this section, we use an asymptotic approximation of Wigner d-functions byBessel functions of the first kind in order to compute lim k →∞ (cid:104) d j k m (cid:48) ,m , (cid:105) L ( T ) and lim k →∞ H T (cid:16) d j k m (cid:48) ,m (cid:17) (0), where j, m (cid:48) , m are fixed, k ∈ N and j k = j + k .Since P x,j,m (cid:48) ,m are the elements of a symmetric matrix, we further assume that m − m (cid:48) ≥
0. The values of the limits lim k →∞ P x,j k ,m (cid:48) ,m will then follow fromConclusion 4.1. 13 .1 Asymptotic approximation of Wigner d-functions The relevant asymptotic relation between Bessel functions and Wigner d-functionsfollows from a formula for the latter in terms of Jacobi polynomials.Let p ∈ N . The Bessel function of the first kind J p may be specified by ([1]) J p ( x ) = ∞ (cid:88) k =0 ( − k k !( k + p )! (cid:16) x (cid:17) k + p = 1 π (cid:90) π cos( pt − x sin t ) dt. (11)We note, for later use, that for x ∈ R , it holds that ([1], 9.1.7, 9.2.1) J p ( x ) = O ( x p ) , J p ( x ) = O (cid:16) x − (cid:17) (12)as x → x → + ∞ , respectively. The Bessel functions associated withnegative integers are specified by ([1], 9.1.5) J − p ( x ) = ( − p J p ( x ) = J p ( − x ) . (13)The Jacobi polynomials P ( α,β ) k are a class of classical orthogonal polynomialsspecified by ([26], 4.3.1) P ( α,β ) k ( x ) = ( − k k k ! (1 − x ) − α (1 + x ) − β d k dx k (cid:2) (1 − x ) α (1 + x ) β (1 − x ) k (cid:3) . They are orthogonal on the interval [ − ,
1] with respect to the weight function W ( α,β ) ( x ) = (1 − x ) α (1 + x ) β .The results of the present section are based on the following classical esti-mate. Assume that α > − , β ∈ R . Then ([26], 8.21.12) (cid:18) sin θ (cid:19) α (cid:18) cos θ (cid:19) β P ( α,β ) k (cos θ ) = ( k + α )! r α k ! (cid:114) θ sin θ J k ( rθ ) + E ( α,β ) k ( θ ) , (14)where r = k + α + β +12 and E ( α,β ) k ( θ ) = √ θ O ( k − ) in intervals of the form[0 , π − δ ]. Wigner d-functions are related to Jacobi polynomials by ([3], 3.72) d jm (cid:48) ,m ( θ ) = (cid:115) ( j + m )!( j − m )!( j − m (cid:48) )!( j + m (cid:48) )! (cid:18) sin θ (cid:19) m − m (cid:48) (cid:18) cos θ (cid:19) m + m (cid:48) P ( m − m (cid:48) ,m + m (cid:48) ) j − m (cos θ ) . Thus, choosing k = j − m, α = m − m (cid:48) , β = m + m (cid:48) , we obtain a powerfulasymptotic approximation of d jm (cid:48) ,m . Conclusion 5.1.
Fix m, m (cid:48) ∈ N or m, m (cid:48) ∈ N \ N such that m − m (cid:48) ≥ − .Then d jm (cid:48) ,m ( θ ) = C j,m (cid:48) ,m (cid:114) θ sin θ J m − m (cid:48) (cid:18) j + 12 θ (cid:19) + E ( m − m (cid:48) ,m + m (cid:48) ) j − m ( θ ) , We refer the reader to [22] for a survey of the asymptotic properties of Wigner d-functions. here E ( m − m (cid:48) ,m + m (cid:48) ) j − m = √ θ O (cid:16) j − (cid:17) in intervals of the form [0 , π − δ ] , and C j,m (cid:48) ,m = (cid:115) ( j − m (cid:48) )!( j + m )!( j − m )!( j + m (cid:48) )! 1 (cid:0) j + (cid:1) m − m (cid:48) satisfies lim j →∞ C j,m (cid:48) ,m = 1 . The asymptotic approximation may be extended to d jm (cid:48) ,m with m − m (cid:48) < Conclusion 5.1 together with the fact that lim x →∞ J m − m (cid:48) ( x ) = 0 by (12) implythat lim j →∞ d jm (cid:48) ,m ( θ ) = 0 for m − m (cid:48) ≥ θ ∈ (0 , π ) fixed. This remainstrue when m − m (cid:48) <
0, as may be shown using the parity properties of d jm (cid:48) ,m .In particular, if j ∈ N and m (cid:48) is fixed, then lim k →∞ d j + km (cid:48) , (cid:0) π (cid:1) = 0. Thus, inlight of (8), we find that lim k →∞ (cid:104) d j + km (cid:48) ,m , (cid:105) L ( T ) = 0 . (15)The latter also follows from the next lemma, which will be used in the analysisof H T ( d jm (cid:48) ,m )(0). As before, denote j k = j + k . Lemma 5.2.
Let f ∈ L ∞ ( T ) . Then lim k →∞ (cid:104) d j k m (cid:48) ,m , f (cid:105) L ( T ) = 0 .Proof. Recall that d jm (cid:48) ,m ( θ ) is an element of a unitary matrix, so | d jm (cid:48) ,m ( θ ) | ≤ j, m (cid:48) , m and θ . Moreover, lim k →∞ d j k m (cid:48) ,m ( θ ) = 0 for θ ∈ (0 , π ). By thedominated convergence theorem, it follows thatlim k →∞ (cid:90) π f ( θ ) d j k m (cid:48) ,m ( θ ) dθ = 0for every f ∈ L ∞ ( T ). The symmetries (4), (3) of d jm (cid:48) ,m ( θ ) imply, similarly, thatthe integrals over the intervals [ π, π ] and [ − π,
0] converge to 0.The lemma is not immediately applicable to H T ( d jm (cid:48) ,m )(0), since by (9), H T ( d jm (cid:48) ,m )(0) = − π (cid:90) π d jm (cid:48) ,m ( θ ) cot (cid:18) θ (cid:19) dθ, and cot (cid:0) θ (cid:1) is unbounded. However, it allows us to truncate this integral to aninterval of the form [0 , δ ], where δ > Conclusion 5.3.
Fix < δ < . Then lim k →∞ (cid:104) H T ( d j k m (cid:48) ,m )(0) − I j k ,δ (cid:105) = 0 ,where I j,δ = − π (cid:90) δ d jm (cid:48) ,m ( θ ) cot (cid:18) θ (cid:19) dθ.
15t this point, we wish to use the asymptotic formula of Conclusion 5.1. Theerror satisfies E ( m − m (cid:48) ,m + m (cid:48) ) j − m ( θ ) = √ θ O (cid:16) j − (cid:17) , and the function √ θ cot θ isintegrable, hence lim k →∞ (cid:90) δ E ( m − m (cid:48) ,m + m (cid:48) ) j k − m ( θ ) cot (cid:18) θ (cid:19) dθ = 0 . Therefore, we have obtained thatlim k →∞ H T ( d j k m (cid:48) ,m )(0) = − π lim k →∞ (cid:90) δ (cid:114) θ sin θ J m − m (cid:48) (cid:18) j k + 12 θ (cid:19) cot (cid:18) θ (cid:19) dθ. (16)Let H R : L ( R ) → L ( R ) denote the standard Hilbert transform, specified by H R f ( x ) = − π lim ε → + (cid:90) ∞ ε f ( x + t ) − f ( x − t ) t dt. As in the case of H T , if f ∈ L ( R ) is even then H R ( f ) is odd, and vice versa.Finally, we are ready to prove the main result of this subsection. Claim 5.4.
Let j k = j + k as above, with j ∈ N fixed and k ∈ N , and fix m (cid:48) , m ∈ { j, j − , ..., − j } . Then lim k →∞ H T (cid:16) d j k m (cid:48) ,m (cid:17) (0) = H R ( J m − m (cid:48) ) (0) . When m − m (cid:48) ∈ Z , this simply says that H T (cid:16) d j k m (cid:48) ,m (cid:17) (0) = 0 = H R ( J m − m (cid:48) ) (0) .Proof. Assume that m − m (cid:48) ∈ Z + 1. By the substitution x = (cid:0) j + (cid:1) θ in (16),it suffices to establish that lim k →∞ I j k = H R ( J m − m (cid:48) ) (0), where I j = − π (cid:90) ( j + ) δ (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) xj + sin (cid:16) xj + (cid:17) J m − m (cid:48) ( x ) cot (cid:32) x (cid:0) j + (cid:1) (cid:33) dxj + with j ∈ N . To this end, denote f (1) j ( x ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) xj + sin (cid:16) xj + (cid:17) , f (2) j ( x ) = 1 j + cot (cid:32) x (cid:0) j + (cid:1) (cid:33) ,f j = ( ,δ ( j + )) f (1) j f (2) j J m − m (cid:48) , Then lim j →∞ f (1) j ( x ) = 1 and lim j →∞ f (2) j ( x ) = x . Additionally,0 < f (1) j ( x ) < M δ = max θ ∈ [0 ,δ ] (cid:114) θ sin θ , < f (2) j ( x ) ≤ x j ∈ N , x ∈ (cid:2) , δ (cid:0) j + (cid:1)(cid:3) . Thus, | f j ( x ) | ≤ x | J m − m (cid:48) ( x ) | , lim j →∞ f j ( x ) = 4 x J m − m (cid:48) ( x )for every x ∈ (0 , ∞ ). Finally, (0 , ∞ ) ( x ) x J m − m (cid:48) ( x ) ∈ L ( R ), hence by thedominated convergence theorem, we deduce that − π lim j →∞ (cid:90) R f j ( x ) dx = − π (cid:90) ∞ J m − m (cid:48) ( x ) x dx = H R ( J m − m (cid:48) )(0) , where the last equality holds since J m − m (cid:48) is an odd function by (13). P x,j,m (cid:48) ,m The results of the previous subsection, together with Conclusion 4.1, imply thatlim k →∞ P x,j k ,m (cid:48) ,m = (cid:26) if m − m (cid:48) = 0 , − i e − i π ( m − m (cid:48) ) H R ( J m − m (cid:48) )(0) if m − m (cid:48) (cid:54) = 0 , where we recall that H R ( J p )(0) = 0 for p even. The Hilbert transform of theBessel function J p admits the alternative representation ([16], 15.9.1) H R ( J p )( x ) = 1 π (cid:90) π sin( x sin t − pt ) dt. Thus, when p (cid:54) = 0, we find that H R ( J p )(0) = − πp (1 − ( − p ). Additionally, ie − i π p = sin (cid:0) π p (cid:1) whenever p is odd, therefore finallylim k →∞ P x,j k ,m (cid:48) ,m = (cid:26) if m − m (cid:48) = 0 , sin (cid:0) π ( m − m (cid:48) ) (cid:1) π ( m − m (cid:48) ) if m − m (cid:48) (cid:54) = 0 . In light of (1), this establishes the relation between the elements of (0 , ∞ ) ( J x )and the Fourier coefficients of E , where we recall that E denotes the right halfof the unit circle in C . Conclusion 5.5. lim k →∞ P x,j k ,m (cid:48) ,m = ˆ E ( m − m (cid:48) ) . (( j + ) a, ∞ )( J x ) In the final part of the previous section, we obtained a formula for the matrixelements of the projection ( a ( j + ) , ∞ )( J x ) in the basis E z,j , with 0 ≤ a < g jm (cid:48) ,m ( θ ) = e − ia j θ d jm (cid:48) ,m ( θ ) , a j = (cid:100)− a (2 j + 1) (cid:101) , we saw that P x,a,j,m (cid:48) ,m = 12 e i π ( m (cid:48) − m ) (cid:16) δ m (cid:48) ,m − (cid:104) g jm (cid:48) ,m , (cid:105) L ( T ) − i H T (cid:16) g jm (cid:48) ,m (cid:17) (0) (cid:17) . | g jm (cid:48) ,m | = | d jm (cid:48) ,m | , therefore most of the arguments in the analysisof P x,j,m (cid:48) ,m = P x, ,j,m (cid:48) ,m remain valid for P x,a,j,m (cid:48) ,m with a >
0. Specifically, g jm (cid:48) ,m (0) = d jm (cid:48) ,m (0) = δ m (cid:48) ,m , lim k →∞ (cid:104) g j k m (cid:48) ,m , (cid:105) L ( T ) = 0 , and since lim j →∞ a j j +1 = − a , we can also establish thatlim k →∞ H T (cid:16) g j k m (cid:48) ,m (cid:17) (0) = H R ( f a,m − m (cid:48) )(0) , where f a,p ( x ) = e aix J p ( x ). Moreover, using the parity of cos( ax ) , sin( ax ) and J p ( x ), we see that H R ( f a,p )(0) = − iπ (cid:82) ∞ ax ) J p ( x ) x dx if p ∈ Z , − π (cid:82) ∞ ax ) J p ( x ) x dx if p ∈ Z + 1 . The Bessel function J p is part of the integral kernel of the p th order Hankeltransform, which provides a straightforward way to evaluate the integrals above.For a <
1, we have that ([2], 8.2.33, 8.7.2, 8.7.27) (cid:90) ∞ sin( ax ) J ( x ) x dx = sin − ( a )and otherwise when p ∈ Z \ { } , (cid:90) ∞ sin( ax ) J p ( x ) x dx = 1 p sin (cid:0) p sin − ( a ) (cid:1) . Similarly, if p ∈ Z + 1, (cid:90) ∞ cos( ax ) J p ( x ) x dx = 1 p cos (cid:0) p sin − ( a ) (cid:1) . Combining the above, we obtain a generalization of Conclusion 5.5.
Conclusion 5.6.
Assume that a = cos α = sin (cid:0) π − α (cid:1) , with α ∈ [0 , π ) . Then,using trigonometric identities for angle difference, we obtain that lim k →∞ P x,a,j k ,m (cid:48) ,m = (cid:26) απ if m − m (cid:48) = 0 π ( m − m (cid:48) ) sin (( m − m (cid:48) ) α ) if m − m (cid:48) (cid:54) = 0 = ˆ E a ( m − m (cid:48) ) , where E a = { z ∈ T | (cid:60) z > a } . Recall that X, Ξ act on a smooth function f ∈ L ( R ) by Xf ( x ) = xf ( x ) , Ξ f ( x ) = − i (cid:126) f (cid:48) ( x ) . σ (cid:126) denote the rescaling f ( x ) (cid:55)→ √ (cid:126) f ( (cid:126) x ). Then σ (cid:126) is unitary with respectto the inner product (cid:104) f, g (cid:105) = (cid:90) ∞−∞ f ( x )¯ g ( x ) dx, since (cid:104) σ (cid:126) f, σ (cid:126) g (cid:105) = (cid:126) (cid:90) −∞ f ( (cid:126) x )¯ g ( (cid:126) x ) dx = (cid:104) f, g (cid:105) . Let F denote the (unitary) Fourier transform on L ( R ), acting on a function f ∈ L ( R ) ∩ L ( R ) by F f ( ξ ) = 1 √ π (cid:90) ∞−∞ f ( x ) e − ixξ dx. We define the semiclassical Fourier transform F (cid:126) , using the scaling propertiesof F , as F (cid:126) = σ (cid:126) − F = F σ (cid:126) . The observables X, Ξ are conjugated by F (cid:126) , that is,Ξ = F − (cid:126) X F (cid:126) . Consequently, so are the functional calculi of Ξ , X , where the latter consists ofmultiplication operators. Recall thatΠ X = (0 , ∞ ) ( X ) = M (0 , ∞ ) , Π Ξ = (0 , ∞ ) (Ξ) , hence Conclusion 6.1.
The projections Π X , Π Ξ are related by Π Ξ = F − σ (cid:126) M (0 , ∞ ) σ (cid:126) − F = F − M (0 , ∞ ) F , where M f denotes the operator of multiplication by f . In particular, Π Ξ is independent of (cid:126) , hence C (1) (cid:126) = [Π X , Π Ξ ] = C (1) for some fixed, bounded operator C (1) on L ( R ). Next, we recall that the Hardyspace on R is given by H ( R ) = { f ∈ L ( R ) | F f ( ξ ) = 0 for every ξ < } , therefore Π Ξ = Π R is the Cauchy-Szeg¨o projection on H ( R ), and consequently C (1) = (cid:2) M (0 , ∞ ) , Π R (cid:3) . Let C ( z ) = z − iz + i denote the Cayley transform (which maps (0 , ∞ ) ⊂ R onto {(cid:61) z < } ⊂ T ). The unitary operator U C : L ( T ) → L ( R ) specified by U C f ( x ) = π − ( x + i ) − f ( C ( x ))is known ([21], p.92) to map H ( T ) onto H ( R ).19 emma 6.2. U ∗ C Π X U C = M {(cid:61) z< } and U ∗ C Π Ξ U C = Π T , where the latterdenotes the Cauchy-Szeg¨o projection on H ( T ) .Proof. Note that U ∗ C ψ ( z ) = 2 i √ π − z ψ (cid:0) C − ( z ) (cid:1) . For a bounded function ψ : R → R and f ∈ L ( T ), we obtain that U ∗ C M ψ U C f ( z ) = 2 i √ π − z ψ (cid:0) C − ( z ) (cid:1) · ( U C f ) (cid:0) C − ( z ) (cid:1) = (cid:0) ψ ◦ C − (cid:1) ( z ) f ( z ) . Then, (0 , ∞ ) ◦ C − = C ((0 , ∞ )) = {(cid:61) z< } , which means that U ∗ C Π X U C = M {(cid:61) z< } .Next, U C is unitary and maps H ( T ) onto H ( R ), hence it maps H ( T ) ⊥ onto H ( R ) ⊥ . It follows immediately that U ∗ C Π R U C = Π T .In light of the previous lemma, we conclude that (cid:13)(cid:13)(cid:13) C (1) (cid:13)(cid:13)(cid:13) op = (cid:13)(cid:13)(cid:13) U ∗ C C (1) U C (cid:13)(cid:13)(cid:13) op = (cid:13)(cid:13)(cid:2) M {(cid:61) z< } , Π T (cid:3)(cid:13)(cid:13) op . Consider the translation operator τ on L ( T ), specified by f ( z ) (cid:55)→ f (cid:0) e i π z (cid:1) .Then τ z m = e i πm z m for every m ∈ Z , therefore τ ∗ Π T τ = Π T . At the sametime, τ ∗ M f τ = M τ ∗ f , therefore τ ∗ M {(cid:61) z< } τ = M E , where we recall that E = { z ∈ T | (cid:60) z > } . Finally, if we denote Π ⊥ T = Id − Π T ,then [ M f , Π T ] = (cid:0) Π T + Π ⊥ T (cid:1) [ M f , Π T ] (cid:0) Π T + Π ⊥ T (cid:1) = Π ⊥ T M f Π T − Π T M f Π ⊥ T = H f − H ∗ f , where H f denotes the Hankel operator with symbol f . Conclusion 6.3. (cid:13)(cid:13)(cid:2) M {(cid:61) z< } , Π T (cid:3)(cid:13)(cid:13) op = (cid:107) [ M E , Π T ] (cid:107) op , where [ M E , Π T ] = H E ⊕ ( − H E ) ∗ : H ( T ) ⊕ H ( T ) ⊥ → H ( T ) ⊥ ⊕ H ( T ) . (17) It follows from Lemma 2.3 that (cid:107) [Π X , Π Ξ ] (cid:107) op = . .2 Proof of Theorem 1.5 Recall that we have defined the operators Θ , Z on L ( T ) (cid:39) L (cid:0) [0 , π ) , π dθ (cid:1) by Θ u ( θ ) = θu ( θ ) , Zu ( θ ) = − i πn u (cid:48) ( θ ) , where u ∈ C ∞ ( T ) and n ∈ N . We are interested in C (2) n = [Π Θ , Π Z ], whereΠ Θ = (0 , ∞ ) (cos Θ) = M E , Π Z = (0 , ∞ ) (cos Z ) . The proof that lim n →∞ (cid:107) C ( n ) (cid:107) op = immediately reduces to Lemma 2.3,since { z k | k ∈ Z } is an eigenbasis of Z (analogous to E z,j for SU (2)), with Z ( z k ) = 2 πkn z k . This means that Π Z ( z k ) = E ( λ k,n ) z k , where λ k,n = e πkn i . The matrix elements of Π Θ are specified by (cid:104) Π Θ z l , z k (cid:105) = ˆ E ( k − l ) . Consequently, the matrix elements of C (2) n are specified by c (2) n,k,l = (cid:104) C (2) n z l , z k (cid:105) = (cid:104) Π Θ Π Z z l , z k (cid:105) − (cid:104) Π Z Π Θ z l , z k (cid:105) = E ( λ l,n ) (cid:104) E z l , z k (cid:105) − (cid:104) E z l , Π Z z k (cid:105) = ( E ( λ l,n ) − E ( λ k,n )) ˆ E ( k − l ) , In particular, when n < l < n and 0 ≤ k < n , c (2) n,k,l = − ˆ E ( k − l ) . Conclusion 6.4.
Fix some positive N ∈ N , and assume that n > N . Define C (2) n,N = ( a k,l ) k,l =1 ,...,N = (cid:16) c (2) n, (cid:100) n (cid:101)− k, (cid:100) n (cid:101) + l − (cid:17) k,l =1 ,...,N . Then a k,l = − ˆ E (1 − k − l ) . It follows that − C (2) n,N = [ H E ] N is the truncated Hankel matrix associated with H E . By Lemma 2.3 and the sameargument as in (2), we deduce that lim n →∞ (cid:107) C (2) n (cid:107) op = . If we replace (0 , ∞ ) with ( a, ∞ ) , where a ∈ (0 , H Ea (as in Con-clusion 5.6) instead of H E . 21 .3 Proof of Theorem 1.6 The proof of Theorem 1.6 may be obtained by straightforward computations.However, we will use a geometric model as follows.We identify the standard basis of V n = l ( Z n ) with∆ n = (cid:110) δ πkn | k = 0 , , ..., n − (cid:111) = (cid:110) δ πkn | k ∈ Z (cid:111) , where δ πkn is the Dirac measure supported in πkn ∈ Z n ⊂ T (cid:39) [0 , π ). For avector v = (cid:80) n − k =0 v k δ πkn and a bounded, measurable function f : T → C we willuse the notation f v = n − (cid:88) k =0 f (cid:18) πkn (cid:19) v k δ πkn , and refer to the operator v (cid:55)→ f v as the multiplication operator M f : V n → V n .Given f : T → C , define the discretization A n ( f ) = f A n (1) = 1 √ n n − (cid:88) k =0 f (cid:18) πkn (cid:19) δ πkn . The representation of H ( Z n ) is realized on ( V n , (cid:104)· , ·(cid:105) n ), where (cid:104)· , ·(cid:105) n is specifiedby (cid:104) δ πkn , δ πln (cid:105) n = δ kl . In these settings, g is the multiplication operator M z ,and g is the operator of translation by πn .In particular, g δ πkn = δ π ( k − n , and we note that g A n ( f ) = 1 √ n n − (cid:88) k =0 f (cid:18) πkn (cid:19) δ π ( k − n = 1 √ n n − (cid:88) k =0 f (cid:18) π ( k + 1) n (cid:19) δ πkn = A n ( τ n f ) , where τ n f ( θ ) = f (cid:0) θ + πn (cid:1) . Thus, g A n ( z k ) = e π kn i A n ( z k ) = λ k,n A n ( z k ) , therefore E n = { e k,n | k = 0 , , ..., n − } = (cid:8) A n (cid:0) z k (cid:1) | k ∈ Z (cid:9) is an eigenbasis of g , orthonormal with respect to (cid:104)· , ·(cid:105) n (as may be seen by astraightforward calculation). ∆ n is clearly an orthonormal eigenbasis of g .Let F n denote the (unitary) discrete Fourier transform, specified by (cid:104)F n v, δ πkn (cid:105) n = 1 √ n n − (cid:88) l =0 v l e − πkln i . Then
Lemma 6.5. g = F − n g F n . roof. Note that (cid:104)F n δ πmn , δ πkn (cid:105) n = 1 √ n e − π kmn i , hence F n δ πmn = 1 √ n n − (cid:88) k =0 (cid:16) e πkn i (cid:17) − m δ πkn = A n ( z − m ) , which means that g F n δ πmn = 1 √ n n − (cid:88) k =0 (cid:16) e πkn (cid:17) − ( m − δ πkn = F n δ π ( m − n . We conclude that F − n g F n δ πmn = δ π ( m − n = g δ πmn , therefore g = F − n g F n .We have that Π = E ( M z ) = M E , therefore Π = F − n M E F n . Since E n is an eigenbasis of g , it holds thatΠ e k,n = E ( λ k,n ) e k,n . The matrix elements of Π in E n are given by (cid:104) Π e l,n , e k,n (cid:105) n = (cid:104) A n (cid:0) E z l (cid:1) , A n (cid:0) z k (cid:1) (cid:105) n = (cid:104) z l A n ( E ) , A n ( z k ) (cid:105) n = (cid:104) A n ( E ) , A n ( z k − l ) (cid:105) n . Here, we have used the fact that M f A n ( f ) = f A n ( f ) = A n ( f f ) and that (cid:104) f A n ( f ) , A n ( f ) (cid:105) n = (cid:104) A n ( f ) , ¯ f A n ( f ) (cid:105) n . The proof of Theorem 1.6 reduces to Lemma 2.3, as in all previous cases. Wedemonstrate this using E n (though ∆ n works just as well). The matrix elementsof the commutator C (3) n = [Π , Π ] are specified by c (3) n,k,l = (cid:104) C (3) n e l,n , e k,n (cid:105) n = ( E ( λ l,n ) − E ( λ k,n )) (cid:104) Π e l,n , e k,n (cid:105) n . In particular, when n < l < n and 0 ≤ k < n , c (3) n,k,l = (cid:104) A n ( E ) , A n ( z k − l ) (cid:105) n . If f , f ∈ L ( T ), then (cid:104) A n ( f ) , A n ( f ) (cid:105) n = 12 π n − (cid:88) k =0 (cid:20) f (cid:18) πkn (cid:19) ¯ f (cid:18) πkn (cid:19) πn (cid:21) n →∞ −−−−→ (cid:104) f , f (cid:105) L ( T ) . Thus, 23 onclusion 6.6.
Fix some positive N ∈ N , and assume that n > N . Define C (3) n,N = ( b n,k,l ) k,l =1 ,...,N = (cid:16) c (3) n, (cid:100) n (cid:101)− k, (cid:100) n (cid:101) + l − (cid:17) k,l =1 ,...,N . Then lim n →∞ b n,k,l = lim n →∞ (cid:104) A n ( E ) , A n ( z − k − l ) (cid:105) n = ˆ E (1 − k − l ) . It followsthat lim n →∞ C (3) n,N = [ H E ] N is the truncated Hankel matrix associated with H E . By Lemma 2.3 and the sameargument as in (2), we deduce that lim n →∞ (cid:107) C (3) n (cid:107) op = . As in the case of Theorem 1.5, if we replace (0 , ∞ ) with ( a, ∞ ) , where a ∈ (0 , H Ea (as in Conclusion 5.6) instead of H E . The proof is immediate. Indeed, Π Φ ( f ) = Π T f − ˆ f (0), andΠ X = M (0 , ∞ ) ( R cos φ ) = M E , therefore C (4) R = [ M E , Π T ] , and (cid:107) C (4) R (cid:107) op = , as we have already seen in (17). We begin with an informal interpretation of Theorem 1.2, based on a realizationof the representations of SU (2) through Berezin-Toeplitz quantization of theunit sphere S ⊂ R . This will lead us to formulate a conjectured, generalizedversion of Theorem 1.2, using the language of quantization. Subsequently, wewill explore the conjectured formulation in a number of concrete examples.In what follows, L ( H ) denotes the space of self-adjoint operators on a finitedimensional Hilbert space H . Let ( M, ω ) denote a closed , quantizable sym-plectic manifold. A Berezin-Toeplitz quantization ([4, 23, 9]) of M produces asequence of finite dimensional complex Hilbert spaces ( H (cid:126) ) (cid:126) ∈ Λ , where 0 is anaccumulation point of Λ ⊂ (0 , ∞ ) and lim (cid:126) → + dim H (cid:126) = + ∞ , together withsurjective linear maps T (cid:126) : C ∞ ( M ) → L ( H (cid:126) ), such that1. T (cid:126) (1) = Id H (cid:126) ,2. if f ≥
0, then T (cid:126) ( f ) ≥ (cid:107) f (cid:107) ∞ − O ( (cid:126) ) ≤ (cid:107) T (cid:126) ( f ) (cid:107) op ≤ (cid:107) f (cid:107) ∞ , i.e., compact and without boundary. i.e., ω π represents an integral de-Rham cohomology class. (cid:13)(cid:13) i (cid:126) [ T (cid:126) ( f ) , T (cid:126) ( g )] − T (cid:126) ( { f, g } ) (cid:13)(cid:13) op = O ( (cid:126) ),5. (cid:13)(cid:13) T (cid:126) (cid:0) f (cid:1) − T (cid:126) ( f ) (cid:13)(cid:13) op = O ( (cid:126) )for every f, g ∈ C ∞ ( M ). Here (cid:107) f (cid:107) ∞ = max M | f | is the uniform norm and { f, g } is the Poisson bracket of f, g . The existence of a Berezin-Toeplitz quantizationin these rather general settings is a non-trivial fact, though if ( M, ω ) is a closedK¨ahler manifold, then the construction itself is quite direct. Item 4 above isknown as the correspondence principle , and it is central to our interpretation.Let us identify S with the complex projective space C P via the stereo-graphic projection through the north pole, and let ρ denote the standard actionof SU (2) on C P , given by ρ ( U ) ([ z : z ]) = [ αz − ¯ βz : βz + ¯ αz ] , U = (cid:18) α − ¯ ββ ¯ α (cid:19) ∈ SU (2) . In addition to the properties specified above, the Berezin-Toeplitz quantizationof S (cid:39) C P is SU (2)-equivariant, meaning that H (cid:126) carries an irreducible,unitary representation ρ (cid:126) of SU (2) such that T (cid:126) (cid:0) f ◦ ρ ( U ) − (cid:1) = ρ (cid:126) ( U ) T (cid:126) ( f ) ρ (cid:126) ( U ) ∗ for every (cid:126) ∈ Λ, f ∈ C ∞ (cid:0) C P (cid:1) and U ∈ SU (2). Here, (cid:126) − = n = dim H (cid:126) , andΛ = (cid:8) n − (cid:12)(cid:12) n = 1 , , ... (cid:9) . The spin operators J x , J y , J z ∈ L ( H (cid:126) ) are then, up tonormalization, the quantum counterparts of the Cartesian coordinate functions x, y, z : C P → R . Specifically, T (cid:126) ( x ) = 1 n + 1 J x , T (cid:126) ( y ) = 1 n + 1 J y , T (cid:126) ( z ) = 1 n + 1 J z . Since (0 , ∞ ) is unaffected by positive rescalings, Theorem 1.2 means thatlim n →∞ (cid:107) C n (cid:107) op = lim (cid:126) → + (cid:13)(cid:13)(cid:2) (0 , ∞ ) ( T (cid:126) ( x )) , (0 , ∞ ) ( T (cid:126) ( z )) (cid:3)(cid:13)(cid:13) op = 12 . Finally, our loose interpretation of this result goes as follows. We considerthe spectral projections (0 , ∞ ) ( J x ) , (0 , ∞ ) ( J z )as a pair of observables that are somehow related ([31, 32, 33]) to the indicatorfunctions of the hemispheres { x > } , { z > } ⊂ S . Thus, we interpret Theo-rem 1.2 as an informal attempt to explore the correspondence principle (item 4above) in the context of discontinuous classical observables . At the moment, itis unclear whether C n corresponds to a well-defined classical object as n → ∞ .Still, the behavior of ( C n ) n ≥ appears to be related to the intersection of theboundaries of the respective hemispheres, that is, to the points ± (0 , , To the best of our knowledge, a well-defined, useful (in the context of quantization) notionof Poisson bracket which is applicable to discontinuous observables does not exist.
25o see this, note that H (cid:126) may be identified with the space of homogeneouspolynomials of degree n − ρ (cid:126) becomes thestandard irreducible unitary representation of SU (2) in the latter space. Assumethat v n ∈ H (cid:126) is a polynomial which realizes the norm of C n , i.e., assume that (cid:107) C n v n (cid:107) = (cid:107) C n (cid:107) op . Our numerical simulations suggest that v n concentratesabout the points ± (0 , ,
0) when n → ∞ , as illustrated in the following images.Figure 4: (originally by Y. Le Floch) The modulus of (unit) eigenvectors of C corresponding to extremal eigenvalues, realized as polynomials on C .Figure 5: The image above to the left, reproduced with the eigenvector realizedas a function on S using the stereographic projection.More generally, assume that T (cid:126) ( f ) , T (cid:126) ( g ) are a pair of quantum observablesarising from smooth, non-commuting observables f, g on some quantizable phasespace M , and let I, J ⊂ R denote some intervals. As before, we view theprojections Π (cid:126) ,f,I = I ( T (cid:126) ( f )) , Π (cid:126) ,g,J = J ( T (cid:126) ( g ))as a pair of observables that are related to the domains f − ( I ) , g − ( J ) ⊂ M .The numerical evidence (Figures 6, 7 in particular) and the results presented26n this work appear to support the following conjecture, which was inspired byrecent findings pertaining to quantization of domains in phase space ([14, 15, 5]). Conjecture. If M is -dimensional, and if the intersection of the boundariesof the domains f − ( I ) , g − ( J ) is non-empty and transversal, then Π I,f, (cid:126) , Π J,g, (cid:126) are maximally non-commuting in the semiclassical limit, i.e., lim (cid:126) → + (cid:107) [Π I,f, (cid:126) , Π J,g, (cid:126) ] (cid:107) op = 12 . In the context of S , Theorem 3.2 is a modest extension of our main result,and agrees with the conjecture. Similarly, consider the sequence C n,a = (cid:104) ( a ( j + ) , ∞ )( J x ) , ( a ( j + ) , ∞ )( J z ) (cid:105) , where a ∈ [0 , n →∞ (cid:107) C n,a (cid:107) op for a > (cid:107) C n,a (cid:107) op ) n ≥ as a crosses the value √ . According to our numericalsimulations, this indeed seems to be the case. The following images are theanalogues of Figure 1 above for the choices a = 0 . , .
75 and a = 0 . , . (cid:107) C n,a (cid:107) op as a function of n , where a = 0 .
25 (top), a = 0 . (cid:107) C n,a (cid:107) op as a function of n , where a = 0 . a = 0 . a = 0 . Acknowledgements
This research has been partially supported by the European ResearchCouncil Advanced Grant 338809, and by the European Research CouncilStarting Grant 757585. I wish to express my sincere gratitude to theEuropean Research Council.I wish to thank my supervisor Leonid Polterovich for his suggestionto pursue the questions addressed in this manuscript, for his interest,commitment, and insightful guidance throughout the project, and for themany useful ideas, remarks and corrections. I also wish to thank YohannLe Floch for numerous contributions to this project throughout the years,and for his patience, help and encouragement. The results presented here ould not have been obtained without the contributions and assistance ofboth.I wish to thank my co-supervisor Lev Buhovsky for his support, formany clarifications, and for his kindly dedication, during our meetings,to carefully review the details of certain key parts of this work. I wish tothank Mikhail Sodin for his involvement during key stages of the projectand for his clear, illuminating advice and input.Finally, I wish to thank David Kazhdan for invaluable conversationsand for his suggestion to consider the finite Heisenberg groups, prior towhich the problem seemed entirely intractable. References [1] Abramowitz, M., Stegun, I.A., eds.
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