Lie Algebra Contractions and Separation of Variables on Two-Dimensional Hyperboloids. Basis Functions and Interbasis Expansions George Pogosyan BLTP , Joint Institute for Nuclear Research (Dubna, Russia) Yerevan State University (Armenia) Department of Mathematics, University of Guadalajara (Mexico) NIST, Gaithersburg, September, 2016 . The work was done in collaboration with Ernest G. Kalnins and Alexander Yakhno Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 1 / 40
In this talk I would like to present investigation mainly presented in our two articles: 1. G.S.Pogosyan and A.Yakhno. Lie Algebra Contractions and Separation of Variables on Two-Dimensional Hyperboloids. Coordinate Systems. ArXiv:1510.03785. 2. E.Kalnins, G.S.Pogosyan and A.Yakhno. Separation of variables and contractions on two-dimensional hyperboloid ArXiv:1212.6123v1, SIGMA 8 , 105, 11 pages, 2012 In these articles we reconsider the problem of separation of variables of the Laplace-Beltrami (or Helmholtz) equation ∆ LB Ψ = λ Ψ , for the on two-sheeted H (+) : u 2 0 − u 2 1 − u 2 2 = R 2 , R > 0 , u 0 > 0 , and 2 one-sheeted H ( 0 ) 2 : u 2 0 − u 2 1 − u 2 2 = − R 2 . Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 2 / 40
In this talk I would like to present investigation mainly presented in our two articles: 1. G.S.Pogosyan and A.Yakhno. Lie Algebra Contractions and Separation of Variables on Two-Dimensional Hyperboloids. Coordinate Systems. ArXiv:1510.03785. 2. E.Kalnins, G.S.Pogosyan and A.Yakhno. Separation of variables and contractions on two-dimensional hyperboloid ArXiv:1212.6123v1, SIGMA 8 , 105, 11 pages, 2012 In these articles we reconsider the problem of separation of variables of the Laplace-Beltrami (or Helmholtz) equation ∆ LB Ψ = λ Ψ , for the on two-sheeted H (+) : u 2 0 − u 2 1 − u 2 2 = R 2 , R > 0 , u 0 > 0 , and 2 one-sheeted H ( 0 ) 2 : u 2 0 − u 2 1 − u 2 2 = − R 2 . Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 2 / 40
The Laplace – Beltrami operator in the curvilinear coordinates ( ξ 1 , ξ 2 ) : ∂ √ gg ik ∂ 1 ∆ LB = √ g ∂ξ k , ∂ξ i ds 2 = g ik d ξ i d ξ k , g ik g k µ = δ µ g = | det ( g ik ) | , i with the following relation between g ik ( ξ ) and the ambient space metric G µν = diag ( − 1 , 1 , 1 ) , ( µ, ν = 0 , 1 , 2) ∂ u µ ∂ u ν g ik ( ξ ) = G µν ∂ξ k . ∂ξ i Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 3 / 40
Olevskii (1950) first to show that the Laplace-Beltrami (or Helmholtz) equation allows separation of the variables in nine orthogonal coordinate systems. Thus there exist the nine sets of the wave { Ψ ( α ) } functions such that Ψ ( α ) λ 1 ,λ 2 ( ξ 1 , ξ 2 ) = N ( α ) λ 1 ,λ 2 ( R ) Ψ ( α ) 1 ( ξ 1 , λ 1 , λ 2 )Ψ ( α ) 2 ( ξ 2 , λ 1 , λ 2 ) , where λ 1 , λ 2 are the separation constants and N λ,λ 2 ( R ) is a normalization constant. Our main task is by the direct solution of Helmholtz equation in various system of coordinates to construct the corresponding Hilbert space of complete solutions satisfying the normalized condition � � √ g d ξ 1 d ξ 2 = δ ( λ 1 , λ ′ Ψ λ 1 ,λ 2 Ψ ∗ 1 ) δ ( λ 2 , λ ′ 2 ) λ ′ 1 ,λ ′ 2 We use the notation δ ( λ, λ ′ ) for Dirac delta function or Kroneker delta whichever is the constant λ discrete or takes the continuous values. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 4 / 40
The third problem have considered is the unitary transformations (interbasis expansions) relating different bases. Namely if Ψ ( I ) ρ,λ ( ξ 1 , ξ 2 ) and Ψ ( II ) ρ,µ ( χ 1 , χ 2 ) two bases corresponding separation of variables in different systems of coordinates, then � Ψ ( I ) ρ,λ Ψ ( II ) W µ ρ,λ = λ 1 ,λ 2 d µ and vise versa � � � ∗ Ψ ( I ) W µ Ψ ( II ) ρ,µ = λ 1 ,λ 2 d λ, ρ,λ Finally we also presented the contraction procedure for the separating systems of coordinate on two-dimensional hyperboloid and corresponding systems on (pseudo)euclidean spaces E 2 and E 1 , 1 , as the wave functions and interbasis coefficients. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 5 / 40
The talk is structured as follows: Some history remarks: symmetries and separation of variables, solutions and contraction Description of the general procedure "State of the art" for bi-dimensional hyperboloids New results: some new relations between coordinates systems, normalization (by inter-basis expansions), contractions of wave functions. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 6 / 40
The talk is structured as follows: Some history remarks: symmetries and separation of variables, solutions and contraction Description of the general procedure "State of the art" for bi-dimensional hyperboloids New results: some new relations between coordinates systems, normalization (by inter-basis expansions), contractions of wave functions. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 6 / 40
The talk is structured as follows: Some history remarks: symmetries and separation of variables, solutions and contraction Description of the general procedure "State of the art" for bi-dimensional hyperboloids New results: some new relations between coordinates systems, normalization (by inter-basis expansions), contractions of wave functions. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 6 / 40
The talk is structured as follows: Some history remarks: symmetries and separation of variables, solutions and contraction Description of the general procedure "State of the art" for bi-dimensional hyperboloids New results: some new relations between coordinates systems, normalization (by inter-basis expansions), contractions of wave functions. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 6 / 40
In "geometrical approach" we say that the D - dimensional Laplace-Beltrami equation ∆ LB Ψ = λ Ψ , allows the separation of variables (multiplicative) in a D -dimensional Riemannian space with an orthogonal coordinate system � ξ = ( ξ 1 , ξ 2 , . . . , ξ D ) if the substitution D � Ψ i ( ξ i , λ 1 , λ 2 , ....λ D ) Ψ = i = 1 split the Laplace-Beltrami equation to the separated equations 1 d � d ψ i � � f i + Φ ik λ k ψ i = 0 . f i d ξ i d ξ i k where f i ≡ f i ( ξ i ) , Φ ik is element of Stäckel determinant and λ 1 , λ 2 , ....λ D are the separation constants. Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 7 / 40
In 1966 Smorodinsky and Tugov proven the Theorem If the Helmholtz (or Schrödinger) equation admits simple separation of variables in the coordinate system � ξ , then there exists D linearly independent second degree operators I k , k = 1 , 2 , 3 , . . . , D (including the Laplace-Beltrami operator) commuting with with each other, and they have the form D � 1 d � d Ψ i �� � Φ − 1 � � I k = − f i , [ I k , I l ] = 0 . d ξ i d ξ i f i ik i = 1 The separation constants λ 1 , λ 2 · · · λ D are the eigenvalues of these operators: I k Ψ = λ k Ψ . Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 8 / 40
In the algebraic approach [E. Kalnins, W. Miller, Ya. Smorodinsky, P . Winternitz, etc. from 1965 till today] every orthogonal separable coordinate system is characterized by the set of second order commuting operators S α , ( α = 1 , 2 , ... D ) (including the Laplace-Beltrami operator) of enveloping algebra of the Lie algebra of the isometry group. Namely, for our case of two-dimensional hyperboloids the isometry group is SO ( 2 , 1 ) . Then we get S 1 = ∆ LB = K 2 1 + K 2 2 − L 2 , aK 2 1 + b { K 1 , K 2 } + cK 2 2 + d { K 1 , L } + e { K 2 , L } + f L 2 � � S 2 ∈ where operators K 1 , K 2 , L forms the basis of so ( 2 , 1 ) algebra K 1 = u 0 ∂ u 2 + u 2 ∂ u 0 , K 2 = u 0 ∂ u 1 + u 1 ∂ u 0 , L = u 1 ∂ u 2 − u 2 ∂ u 1 and commutation relation are [ K 1 , K 2 ] = L , [ K 2 , L ] = − K 1 , [ L , K 1 ] = − K 2 . The irreducible representations are labeled by the eigenvalue of Casimir operator ∆ LB Ψ = ℓ ( ℓ + 1 )Ψ , ℓ = − 1 / 2 + i ρ, ρ > 0 . Pogosyan (BLTP , YSU, UdeG) 2-dimensional hyperboloids: Basis Functions and Interbasis Expansions. 2016 9 / 40
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