d ORTHOGONAL POLYNOMIALS AND THE WEYL ALGEBRA

Transcription

1 d ORTHOGONAL POLYNOMIALS AND THE WEYL ALGEBRA Luc VINET Université de Montréal Alexei ZHEDANOV Donetsk for Physics and Technology

2 PLAN 1. Charlier polynomials 2. Weyl algebra 3. Charlier and Weyl 4. d orthogonal polynomials 5. A non linear automorphism of the Weyl algebra 6. d Charlier polynomials

3 7. difference ladder 8. Explicit expression in special case 9. Dual and orthogonality 10. Conclusion

4 1.CHARLIER POLYNOMIALS p n (k) Polynomials of a discrete variable k (used normalized version) 3 term recurrence rela@on p n +1 (k) = (k n a) p n (k) na p n 1 (k) Orthogonality rela@on k= 0 p m (k) p n (k) e a a k k! = a n n!δ m,n

5 2.WEYL ALGEBRA 3 generators: Rela@ons: a, a *, I [ a, a * ] =1, [ a, I] = [ a *, I] = 0 Canonical irreducible representa@on: a n = n n 1 a * n = n +1 n +1 I n = n a * = a + N = a + a N n Vacuum vector a 0 = 0

6 3.CHARLIER POLYNOMIALS AND WEYL ALGEBRA Oscillator a = 1 2 ( d dx + x) a+ = 1 2 ( d dx + x) Shib N = 1 2 x x λ d 2 dx x2 1 2 N(λ) = N λx + λ2 2 a(λ) = a λ 2 a + (λ) = a + λ 2 x = 1 2 (a + a+ ) = 1 2 (a + a+ ) + λ N = N + λ 2 (a + a+ ) + λ2 2

7 Let N n = n n and set k = n k n n N k = n k N + λ 2 (a + a+ ) + λ2 n 2 n k n k = (n + λ2 n k = 2 λ 2 ) n k + λ 2 n +1 n +1 k + λ 2 n n 1 k n 1 n! ν(k) p n(k) p n+1 (k) = (k n λ2 2 ) p n (k) λ2 2 n p n 1 (k) Charlier polynomials

8 obviously amount to linear automorphisms of the Weyl algebra Algebraic context allows to derive various of the Charlier polynomials In its difference and to show that they span a representa@on of the Weyl algebra

9 4. d ORTHOGONAL POLYNOMIALS P n (x) monic, degree n defined by recurrence rela@on of order d+1 Ini@al condi@ons: P n+1 (x) = xp n (x) d s=0 a n, n s P n s (x) P k (x) = 0, k < 0 ; P o (x) =1 d=1 : 3 term P n+1 (x) = xp n (x) a n, n P n (x) a n, n 1 P n 1 (x)

10 Provided u n a n, n 1 0 for d=1, orthogonal P n σ,p n (x)p m (x) = h n δ n, m h n = u 1 u 2 u n func@onal σ defined by moments c n = σ,x n

11 When d > 1 : vector orthogonality, d linear σ k k = 0,1,..., d 1 func@onals such that σ k,p n P m = 0 m > dn + k σ k,p n P dn+ k 0 n 0 E.g. d = 2 σ 0,P n P m = 0 m > 2n σ 1,P n P m = 0 m > 2n +1 σ 0,P n P 2n 0 σ 1,P n P 2n+1 0 n 0

12 Padé problems Lanczos type method for solving linear systems Random matrices

13 5. A NONLINEAR AUTOMORPHISM OF THE WEYL ALGEBRA S = e βa* e Q (a) S 1 = e Q(a) e βa* β: parameter, Q polynomial of degree N 1 SaS 1 = e βa* a e βa* = a β S 1 as = a + β Sa * S 1 = e βa* e Q(a) a * e Q(a) e βa* = e βa* ( a * + [ Q(a),a *]+ 1 [ * Q(a), [ Q(a),a ] 2 ]+ )e βa*

14 = e βa* ( a * + Q (a))e βa* = a * + Q (a β) S 1 a * S = a * Q (a) Note for deg (Q) =1, Q = const S generates transla@ons (linear automorphisms)

15 6. d-charlier POLYNOMIALS Calculate ψ nk = k S n through recurrence relation k a * a S n = k k S n = kψ nk k a * as n = k SS 1 a * as n = k S( a * Q (a))(a + β) n Q (x)(x + β) polynomial of degree N if deg Q=N Write Q (x)(x + β) = N ξ i x i i= 0 ξ i coefficients

16 Comparing ( a n = n n 1, a * n = n +1 n +1 ) k k S n = k S a * a + βa * Gives n=0: N i= 0 ξ i a i n = (n ξ 0 ) k S n + β n +1 k S n +1 ξ i n(n 1) (n i +1) k S n i kψ n,k = (n ξ 0 )ψ n,k + β n +1ψ n +1,k ξ i n(n 1) (n i +1)ψ n i,k i=1 recursively from N N i=1 ψ n,k ψ 0,k βψ 1,k = (k + ξ 0 )ψ o,k

17 N kψ n, k = (n ξ 0 )ψ n, k + β n +1ψ n+1, k ξ i i =1 n(n 1) (n i +1)ψ n i, k n=1 β 2ψ 2, k = (k = 1+ ξ 0 )ψ 1, k + ξ 1 ψ 0, k = [(k 1+ ξ 0 )(k + ξ 0 )+ ξ 1 ] ψ 0, k Hence P n (k) ψ n, k = P n (k) ψ 0, k polynomial of degree n in the discrete variable k sa@sfying the recurrence rela@ons

18 β n +1 P n+1 (k) = (k n + ξ 0 )P n (k)+ ξ i N i =1 n(n 1) (n i +1) P n i (k) P 0 (k) = 1, P n (k) = 0 n < 0 Recurrence rela@ons of order N+1 ψ 0,k = k S 0 = k e βa* e Q(a) 0 e Q(a) 0 = e Q(0) 0 k e βa* 0 = k β k (a * ) k k! 0 = β k k!

19 ψ 0, k = e Q (0) β k k! For N=1 recover ordinary case of 3 term recurrence rela@on β n +1 P n+1 (k) = (k n + ξ 0 )+ ξ 1 P n 1 (k) To make polynomials monic, set ˆ P n (k) = β n n! P n (k)

20 β n +1 β n 1 (n +1)! ˆ P n+1 = (k n + ξ 0 ) β n n! ˆ P n + N ξ i n(n 1) (n i +1) i=1 n +i β (n i)! ˆ P n i P ˆ n+1 (K ) = (k n + ξ 0 ) P ˆ n (k)+ N i =1 β i ξ i d orthogonal Charlier polynomials n(n 1) (n i +1) ˆ P n i (k) For N=1, ordinary orthogonal Charlier polynomials P ˆ n+1 (k) = (k n + ξ 0 ) P ˆ n (k)+ βξ 1 n P ˆ n 1 (k) up to shift k k + βξ 1 ξ 0

21 7. PROPERTIES Difference Start again from ψ n, k = k S n = k e βa* e Q (a) n = β n now need a recurrence relation in variable k nψ n, k = k S a * a n k S a * a n = k S a * a S 1 S n n! ˆ P n (k)ψ 0, k = k ( a * + Q (a β) )(a β)s n using the dual rela@ons

22 (x β) Q (x β) = N x i i =0 η i nψ n, k = k a * a β a * + N i =0 η i a i S n = kψ n, k β k ψ n, k 1 + η i N i =0 (k +1) (k + i)ψ n, k+ i using ψ n, k = β n n! ˆ P n (k)ψ 0, k ψ 0, k = e Q (0) β k k! we find (n k) ˆ P n (k) + k ˆ P n (k 1) = N η i β i i= 0 ˆ P n (k + i)

23 Use the standard difference operators Δf (k) = f (k +1) f (k) f(k) = f (k) f (k 1) and note that F(x + m) = m i =0 m Δ i F(x) i (n k) P ˆ n (k)+ k P ˆ n (k 1) = N β i i =0 η i ˆ P n (k + i) becomes n P ˆ n (k) = k P ˆ n (k)+ N i i η i β i Δ l P ˆ n (k) l i =0 l =0

24 n P ˆ n (k) = k P ˆ n (k)+ N i i η i β i Δ l P ˆ l n (k) i =0 l =0 Recall (x β) Q (x β) = N x i i =0 η i y Q (y) = N η i (y + β) i = i= 0 N i η i i= 0 l= 0 i N y l β i l = η l i β i i= 0 i l= 0 i l y β l n P ˆ n (k) = k P ˆ n (k)+ N µ i Δ i P ˆ n (k) i =0 N j =0 µ j y β j Eigenvalue equation: H ˆ P n = n ˆ P n

25 Ladder Operators Relation n ψ n 1, k = k S a n = k (a β)s n = k +1ψ n, k+1 βψ n, k Δ P ˆ n (k) P ˆ n (k +1) P ˆ n (k) = n P ˆ n 1 (k) Difference Appell property Similarly from n +1 ψ n+1, k = k S a * n = k ( a * + Q (a β) ) S n

26 we get P ˆ n+1 (k) = k P ˆ n (k 1)+ Q (x β) = With N 1 ν i x i i =0 N 1 ν i β i+1 P ˆ n (k + i) i =0 let N 1 T ± F(k) = F(k ±1) R = k T + ν i β i+1 i T + i= 0 Rodrigues formula : Factorization : Realization : H = RΔ [ Δ,R] = 1 ˆ P n (k) = R n 1 { }

27 Generating function Consider the generating function F(z;k) = e Q (0) β k k! n=0 ψ n, k z n n! recalling that ψ n, k = e Q (0) β k n n!k! ˆ P n (k) it yields F(z,k) = n=0 z β n! n ˆ P n (k)

28 Coherent states : z = e za* 0 n z = zn n! a z = z z hence F(a) z = F(z) z Coming back n=0 F(z,k) = e Q (0) β k k! = e Q (0) β k k! k S n n z = e Q (0) β k k! k S z n=0 ψ n, k z n n! k S z = k e βa* e Q(a) z

29 is calculated using e Q(a) z = e Q(z) z e βa* z = e βa*. e za* 0 = e (β + z) a* 0 = z + β to find We thus have (z + β) k k S z = e Q(z) k z + β = e Q(z) k! F(z;k) = e Q(z) Q (0) 1+ z β k n=0 = z β n ˆ P n (k) n! as genera@ng func@on for the d Charlier polynomials.

30 Characterization Theorem: N+1 order recurrence relation Matrix elements of S (non linear automorphisms) Appell difference property eigenvectors of H All equivalent characterization of d-orthogonal Charlier polynomials

31 8. EXPLICIT EXPRESSION IN A SPECIAL CASE Take Expanding F and some transformations yield a m = (q k + m) N m = 0,,N 1 b m = (q + m +1) N m = 0,,N 1 excluding (q m+1) N ( k) q = ( k)( k +1) ( k + q 1)

32 For N=1, we retrieve the standard expression for the Charlier polynomials:

33 9. DUAL FUNCTIONS AND ORTHOGONALITY Much more can be obtained from framework, in particular orthogonality properties Assume S invertible and define ψ n, k = k S n φ n,k = n S 1 k k =0 m S 1 k k S n = δ m, n implies the biorthogonality property φ m, k k =0 ψ n, k = δ m, n

34 φ m, k - also expressible in terms of d-charlier polynomials - Generating function for can be obtained Noteworthy: Can relate orthogonality of and to vector orthogonality of d-orthogonal polynomials φ φ m, k ψ In fact, from the biorthogonality relations and the knowledge of φ m, k can construct explicit realization of linear functionals expressing vector orthogonality.

35 10. CONCLUSIONS Thorough of d orthogonal Charlier polynomials from matrix elements of nonlinear automorphism of Weyl algebra Can establish rela@on between Meixner polynomials and linear automorphism of SU(1,1) much like the one between Charlier and Weyl. (Floreanini, LeTourneux, Vinet) One uses dila@on instead of transla@ons Can expect other d orthogonal polynomials to appear as matrix elements of non unitary automorphisms of algebra. We have worked out the case of Krawtchouk d orthogonal polynomials