Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References Nonlinear coherent states associated with a measure on the positive real half line Zouhaïr Mouayn Sultan Moulay Slimane University Morocco XXXVIII Workshop on Geometric Methods in Physics Bialowieża, June 30 - July 6, 2019 1/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References S. Twareque Ali Figure : S. Twareque Ali in Bialowieża This work is a development of project that Z. Mouayn during his visit to Concordia university on September 2014 has started with Professor S. Twareque Ali. Later, on January 2016, Professor S. Twareque Ali passed away. This work is 2/46 dedicated to his memory. Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References Plan 1 Nonlinear coherent states 2 NLCS associated with a measure 3 A polynomials realization of the basis { ϕ n } ∞ n =0 4 A class of 2D orthogonal polynomials 5 Generalized µ β -NLCS 6 The coherent states transform and its range 7 The example of the measure r β e − r dr 8 References 3/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References I- Nonlinear coherent states 4/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References The canonical coherent states CCS are written in terms of the so-called Fock basis { ϕ n } ∞ n =0 � e zz � − 1 + ∞ � z n 2 ϑ z = √ ϕ n , (1) n ! n =0 for each fixed z ∈ C where e zz is chosen so that to ensure the normalization condition � ϑ z | ϑ z � = 1 . The basis vectors { ϕ n } are orthonormal in the underlying quantum states Hilbert space H , often termed a Fock space. 5/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References The so-called deformed coherent states also known as nonlinear coherent states (NLCS) in the quantum optical literature are then defined by replacing the factorial n ! by x n ! := x 1 x 2 ...x n , x 0 = 0 , where { x n } ∞ n =0 is an infinite sequence of positive numbers and, by convention, x 0 ! = 1 . Thus, for each z ∈ D some complex domain, one defines a generalized version of CCS as + ∞ � z n ¯ z )) − 1 / 2 ϑ z = ( N ( z ¯ √ x n ! ϕ n , z ∈ D (2) n =0 where again + ∞ � ( zz ) n N ( z ¯ z ) = (3) x n ! n =0 is an appropriate normalizing constant. It is clear that the vectors ϑ z are well defined for all z for which the sum (3) converges, i.e. D = { z ∈ C , | z | < R } where R 2 = lim n → + ∞ x n , with R > 0 could be finite or infinite, but not zero. 6/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References As usual, we require that there exists a measure dη on D for which the resolution of the identity condition, � | ϑ z �� ϑ z |N ( z ¯ z ) dη ( z, ¯ z ) = 1 H (4) D holds. Here, | ϑ z �� ϑ z | ≡ T z means the rank one operator T z : H − → H defined by T z [ ψ ] = � ϑ z | ψ � ϑ z , ψ ∈ H . In order for (4) to be satisfied, the measure dη has to have the form z ) = dθ z = re iθ dη ( z, ¯ 2 π dλ ( r ) , (5) where the measure dλ is a solution of the moment problem � R r 2 n dλ ( r ) = x n ! , n = 0 , 1 , 2 , ..., (6) 0 provided that such a solution exists. In most of the cases that occur in practice, the support of the measure dη is 7/46 the whole domain D , meaning that dλ is supported on (0 , R ) . Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References II- NLCS associated with a measure 8/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References We start from a family of measures of the form dµ β ( r ) := r β dµ ( r ) (7) where dµ ( r ) is a positive measure, does not depend on β , supported by (0 , L ) , where L could be infinite. Assume that this measure have finite moments for all order and denote D L = { ξ ∈ C , | ξ | < L } . Set � L µ β := dµ β ( r ) , β ≥ 0 . (8) 0 From these moments we consider the sequence of numbers : µ n + β 1 = µ n + β n x β n − 1 . . . x β x β x β x β n ! := x β n = , , 0 ! ≡ 1 , n = 1 , 2 , 3 , ..., µ n + β − 1 µ β (9) 9/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References Definition For β ≥ 0 , the vectors ϑ z,β ≡ | z ; β � ∈ H are defined through the superposition ∞ � z n � � − 1 ϑ z,β := N β ( zz ) µ β � ϕ n , (10) 2 x β n =0 n ! 1 for each z in D β = { ξ ∈ C , | ξ | < R β = (lim n → + ∞ x β 2 } . For brevity, these states n ) will be denoted µ β − NLCS. The normalization constant is given by + ∞ � ( zz ) n N β ( z ¯ z ) = (11) x β n ! n =0 which converges for each z ∈ D β . 10/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References Proposition Let dµ β be a measure given in (7) and assuming that D L ⊆ D β , then the µ β -NLCS in (10) satisfy the resolution of the identity operator of H as � | ϑ z,β �� ϑ z,β | dη β ( z, ¯ z ) = 1 H (12) D L z ) = (2 π ) − 1 dθ N β ( zz ) dµ β ( z ¯ where dη β ( z, ¯ z ) , z ∈ D L . Proof. This can be proved by direct calculation and by using (8). 11/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References III- A polynomials realization of the basis { ϕ n } ∞ n =0 12/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
Plan Nonlinear coherent states NLCS associated with a measure A polynomials realization of the basis { ϕn }∞ n =0 A class of 2D orthogonal polynomials Generalized µβ -NLCS The coherent states transform and its range The example of the measure rβ e − rdr References On this Hilbert space H we may define the operators a β and a † β by � � x β a † x β a β ϕ n = n ϕ n − 1 , a β ϕ 0 = 0 , β ϕ n = n +1 ϕ n +1 . (13) Using these operators, it is now possible to identify the basis vectors { ϕ n } ∞ n =0 with another family of real orthogonal polynomials by following S. T. Ali & Ismail (2012). For this, we make the assumption : ∞ � 1 � = ∞ . (14) x β n =1 n 13/46 Zouhaïr Mouayn (Morocco) NLCSs associated with a measure on R +
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