Separation of Variables and Contractions on Two-Dimensional Hyperboloid

In this paper analytic contractions have been established in the R→∞ contraction limit for exactly solvable basis functions of the Helmholtz equation on the two-dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.

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Дата:	2012
Автори:	Kalnins, E., Pogosyan, G.S., Yakhno, A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2012
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149190
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Separation of Variables and Contractions on Two-Dimensional Hyperboloid / E. Kalnins, G.S. Pogosyan, A. Yakhno // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1491902019-02-20T01:24:45Z Separation of Variables and Contractions on Two-Dimensional Hyperboloid Kalnins, E. Pogosyan, G.S. Yakhno, A. In this paper analytic contractions have been established in the R→∞ contraction limit for exactly solvable basis functions of the Helmholtz equation on the two-dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae. 2012 Article Separation of Variables and Contractions on Two-Dimensional Hyperboloid / E. Kalnins, G.S. Pogosyan, A. Yakhno // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70H06; 35J05 DOI: http://dx.doi.org/10.3842/SIGMA.2012.105 http://dspace.nbuv.gov.ua/handle/123456789/149190 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper analytic contractions have been established in the R→∞ contraction limit for exactly solvable basis functions of the Helmholtz equation on the two-dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.
format	Article
author	Kalnins, E. Pogosyan, G.S. Yakhno, A.
spellingShingle	Kalnins, E. Pogosyan, G.S. Yakhno, A. Separation of Variables and Contractions on Two-Dimensional Hyperboloid Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kalnins, E. Pogosyan, G.S. Yakhno, A.
author_sort	Kalnins, E.
title	Separation of Variables and Contractions on Two-Dimensional Hyperboloid
title_short	Separation of Variables and Contractions on Two-Dimensional Hyperboloid
title_full	Separation of Variables and Contractions on Two-Dimensional Hyperboloid
title_fullStr	Separation of Variables and Contractions on Two-Dimensional Hyperboloid
title_full_unstemmed	Separation of Variables and Contractions on Two-Dimensional Hyperboloid
title_sort	separation of variables and contractions on two-dimensional hyperboloid
publisher	Інститут математики НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/149190
citation_txt	Separation of Variables and Contractions on Two-Dimensional Hyperboloid / E. Kalnins, G.S. Pogosyan, A. Yakhno // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 18 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kalninse separationofvariablesandcontractionsontwodimensionalhyperboloid AT pogosyangs separationofvariablesandcontractionsontwodimensionalhyperboloid AT yakhnoa separationofvariablesandcontractionsontwodimensionalhyperboloid
first_indexed	2025-07-12T21:03:37Z
last_indexed	2025-07-12T21:03:37Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 105, 11 pages Separation of Variables and Contractions on Two-Dimensional Hyperboloid? Ernie KALNINS †, George S. POGOSYAN ‡§ and Alexander YAKHNO § † Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: math0236@math.waikato.ac.nz ‡ International Center for Advanced Studies, Yerevan State University, A. Manougian 1, Yerevan, 0025, Armenia E-mail: pogosyan@theor.jinr.ru § Departamento de Matemáticas, CUCEI, Universidad de Guadalajara, Jalisco, Mexico E-mail: alexander.yakhno@cucei.udg.mx Received August 09, 2012, in final form December 19, 2012; Published online December 26, 2012 http://dx.doi.org/10.3842/SIGMA.2012.105 Abstract. In this paper analytic contractions have been established in the R → ∞ con- traction limit for exactly solvable basis functions of the Helmholtz equation on the two- dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae. Key words: analytic contraction; separation of variables; Lie group; Helmholtz equation 2010 Mathematics Subject Classification: 70H06; 35J05 1 Introduction In the recent series of papers [5, 6, 7, 9, 12, 13, 14, 15, 16] a special class of Inönu–Wigner contractions was introduced, namely “analytic contractions”. For an analytic contraction the contraction parameter R, which is the radius of an n-dimensional sphere or pseudosphere u2 0 + ε~u2 = R2, (ε = ±1) figures in the separated coordinate systems, and in the eigenfunctions and the eigenvalues for the Laplace–Beltrami operator (or Helmholtz equation). With the help of analytic contractions we have established the connection between the procedure of separation of variables for homogeneous spaces with constant (positive or negative) curvature and flat spaces. For instance it has been indicated how the systems of coordinates on the sphere S2 and hyperboloid H2 transform to four systems of coordinates on Euclidean space E2. The goal of this note is to establish the contraction limit R → ∞ for eigenfunctions (or basis functions) of the two-dimensional Helmholtz equation on the two-sheeted hyperboloid H2 : u2 0 − u2 1 − u2 2 = R2, ∆LBΨ = −σ(σ + 1) R2 Ψ, σ = −1/2 + iρ, (1) where the Laplace–Beltrami operator in the curvilinear coordinates (ξ1, ξ2) has the form ∆LB = 1 √ g ∂ ∂ξi √ ggik ∂ ∂ξk , g = \|det(gik)\|, gikg kµ = δµi , ?This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html mailto:math0236@math.waikato.ac.nz mailto:pogosyan@theor.jinr.ru mailto:alexander.yakhno@cucei.udg.mx http://dx.doi.org/10.3842/SIGMA.2012.105 http://www.emis.de/journals/SIGMA/SESSF2012.html 2 E. Kalnins, G.S. Pogosyan and A. Yakhno with the following relation between local metric tensor gik(ξ), (i, k = 1, 2) and the ambient space metric Gµν = diag(1,−1,−1), gik(ξ) = Gµν ∂uµ ∂ξi ∂uν ∂ξk , i, k = 1, 2, µ, ν = 0, 1, 2. It is well known that the Helmholtz equation (1) on the two-sheeted hyperboloid admits sep- aration of variables in nine orthogonal systems of coordinates [10, 18]. These nine systems of coordinates can be separated into three classes. The first class includes three systems of coordi- nates which are of subgroup type: viz. pseudo-spherical, equidistant and horicyclic coordinates, the second class includes three non subgroup type of coordinates: semi circular parabolic, elliptic parabolic and hyperbolic parabolic. The last two coordinate systems in the general case contain the dimensionless parameter γ and in the limiting cases when γ → 0 or γ → ∞ transform to horicyclic or equidistant systems of coordinates. These systems of coordinates have an important property; the Helmholtz equation (1) admits the exact solution (which we call exactly solvable) in terms of “classical” special functions, namely Bessel and Legendre functions or hypergeo- metric one. The third class of coordinates consists of elliptic, hyperbolic and semi hyperbolic coordinates each of which belong to the class of non subgroup type. Two of them, elliptic and hyperbolic coordinates, also contain a dimensionless parameter which is included in the metric tensor and Laplace–Beltrami operator. The Helmholtz equation (1) after the separation of vari- ables in these coordinates leads to the two Heun type differential equations with a four singular points (which we call non-exactly solvable equations) and whose solutions appear as Lame or Lame–Wangerin functions [3, 4, 8]. In contrast to the classical special functions which can be appear in terms of hypergeometric functions, the Lame or Lame–Wangerin functions are defined in terms of infinite series where the expansion coefficients can not be written in explicit form because they are the subject the third or higher order recurrence relations. The contraction limit of a basis function is not a trivial task in the case of exactly or non- exactly solvable equations. Some calculations come from the papers [5, 6, 7], but many of them are not known till now. In this paper we restrict ourselves to the contraction limit R→∞ for four kinds of orthogonal basis functions: horocyclic, semi circular parabolic, elliptic parabolic and hyperbolic parabolic, which is new. We shall present normalizable eigenfunctions but do not give their normalization constants explicitly, except in the case of horocyclic coordinates which is particularly simple. Here we have also included for completeness the contraction for pseudo spherical and equidistant wave functions (see [7, 12]). We hope that our results beside possible application to the theory of special functions will be useful also in the investigation of super integrable systems which admit separation of variables on the two-dimensional hyperboloid. We think that they can be generalized for the three and higher dimensional hyperbolic space when these six systems of coordinates are the sub systems of more complicated systems of coordinates and where the procedure of variable separation leads to similar differential equations. 2 Contractions 2.1 Pseudo-spherical basis to polar The first coordinate system is the pseudo-spherical system (τ > 0, ϕ ∈ [0, 2π)) u0 = R cosh τ, u1 = R sinh τ cosϕ, u2 = R sinh τ sinϕ, Separation of Variables and Contractions on Two-Dimensional Hyperboloid 3 and the orthogonal basis functions of equation (1) are [7, 8, 12, 13] ΨS ρm(τ, ϕ) = P \|m\| iρ−1/2(cosh τ)eimϕ, where Pµν (z) is the Legendre function. In the contraction limit R→∞ we have sinh τ ∼ τ ∼ r R , ϕ→ ϕ, ρ ∼ kR, where (r, ϕ) are the polar coordinates in Euclidean plane. There is now the matter of how this contraction affects the basic eigenfunctions that can be computed on the hyperboloid. Using the well-known representations of the Legendre function in terms of the hypergeometric function [1], P \|m\| iρ−1/2(cosh τ) = Γ(1/2 + iρ+ \|m\|) Γ(1/2 + iρ− \|m\|) ( sinh τ 2 )\|m\| ( cosh τ 2 )\|m\| 1 \|m\|! × 2F1 ( 1/2 + iρ+ \|m\|, 1/2− iρ+ \|m\|; 1 + \|m\|;− sinh2 τ 2 ) , asymptotic formulae for gamma-functions at large z [1], Γ(z + α) Γ(z + β) ≈ zα−β, and taking into account that at R→∞ F ( 1 2 + \|m\|+ iρ, 1 2 + \|m\| − iρ; 1 + \|m\|;− sinh2 τ 2 ) ≈ 0F1 ( 1 + \|m\|;−k 2r2 4 ) = ( 2 kr )\|m\| \|m\|!J\|m\|(kr), where Jν(z) is the Bessel function [2], we get P \|m\| iρ−1/2(cosh τ) ≈ P \|m\|ikR−1/2 ( 1 + r2 2R2 ) ≈ (−kR)\|m\|J\|m\|(kr). Finally, the pseudo-spherical functions in the contraction limit R→∞ take the form ΨS ρm(τ, ϕ) ≈ (−kR)\|m\|J\|m\|(kr)e imϕ, i.e., the pseudo-spherical basis up to the constant factor contracts into polar one. The correct correspondence to give limiting orthogonality relations in the polar coordinates r and ϕ can be obtained using the contraction limit of the normalization constant (see [7]) and the results sinh τdτdϕ→ 1 R2 rdrdϕ, δ(ρ− ρ′)→ 1 R δ(k − k′), and ∫ ∞ 0 J\|m\|(kr)J\|m\|(k ′r)rdr = 1 k δ(k − k′). 4 E. Kalnins, G.S. Pogosyan and A. Yakhno 2.2 Equidistant basis to Cartesian The coordinate system is the following one (τ1, τ2 ∈ R) u0 = R cosh τ1 cosh τ2, u1 = R cosh τ1 sinh τ2, u2 = R sinh τ1, From the above definition we have that sinh τ1 = u2 R , tanh τ2 = u1 u0 and in the limit of R→∞ we get sinh τ1 ∼ τ1 ∼ y R , sinh τ2 ∼ τ2 ∼ x R , (2) where x and y are the Cartesian coordinates in the Euclidean plane arising from contraction R→∞. The orthogonal wave function takes the form [7, 8, 12, 13] ΨEQ ρν (τ1, τ2) = (cosh τ1)−1/2P iρ−1/2+iν(−ε tanh τ1) exp(iντ2), where ε = ±1. To perform the contraction we use the formula [1] Pµν (z) = 2µ √ π ( 1− z2 )−µ/2[2F1 ( −1 2(µ+ ν), 1 2(1− µ+ ν); 1 2 ; z2 ) Γ ( 1 2(1− (µ+ ν) ) Γ ( 1 + 1 2(ν − µ) ) − 2z 2F1 ( 1 2(1− µ− ν), 1 + 1 2(ν − µ); 3 2 ; z2 ) Γ ( 1 2(1 + ν − µ) ) Γ ( −1 2(ν + µ) ) ] , (3) and rewrite the Legendre function in terms of the hypergeometric function P iρiν−1/2(−ε tanh τ1) = √ π2iρ(cosh τ1)iρ Γ ( 3 4 − a ) Γ ( 3 4 − b ){2F1 ( 1 4 − a, 1 4 − b; 1 2 ; tanh2 τ1 ) + 2ε tanh τ1 Γ ( 3 4 − a ) Γ ( 3 4 − b ) Γ ( 1 4 − a ) Γ ( 1 4 − b )2F1 ( 3 4 − a, 3 4 − b; 3 2 ; tanh2 τ1 )} , where a = i(ρ+ ν)/2, b = i(ρ− ν)/2. In the contraction limit R→∞ we put ρ ∼ kR, ν ∼ k1R. Then using also (2) we have the asymptotic formulae lim R→∞ 2F1 ( 1 4 − a, 1 4 − b; 1 2 ; tanh2 τ1 ) = 0F1 ( 1 2 ;−y 2k2 2 4 ) = cos k2y, (4) lim R→∞ 2F1 ( 3 4 − a, 3 4 − b; 3 2 ; tanh2 τ1 ) = 0F1 ( 3 2 ;−y 2k2 2 4 ) = 1 k2y sin k2y, (5) where k2 1 + k2 2 = k2 give us the contraction limit for P iρiν−1/2(−ε tanh τ1), P iρiν−1/2(−ε tanh τ1) ≈ P ikRik1R−1/2 ( −ε y R ) ≈ √ π2ikR exp(iεk2y) Γ ( 3 4 − iR(k+k1) 2 ) Γ ( 3 4 − iR(k−k1) 2 ) , and finally for ΨEQ ρν ΨEQ ρν (τ1, τ2) ≈ √ π2ikR Γ ( 3 4 − iR(k+k1) 2 ) Γ ( 3 4 − iR(k−k1) 2 ) exp(ik1x+ iεk2y). The last result up to the constant factor coincides with the contraction limit of equidistant wave function to Cartesian one in Euclidean space. Separation of Variables and Contractions on Two-Dimensional Hyperboloid 5 2.3 Horocyclic basis to Cartesian in E2 In the notation of the article mentioned horocyclic coordinates on the hyperboloid can be written (x̄ ∈ R, ȳ > 0) u0 = R x̄2 + ȳ2 + 1 2ȳ , u1 = R x̄2 + ȳ2 − 1 2ȳ , u2 = R x̄ ȳ . From these relations we see that x̄ = u2 u0 − u1 , ȳ = R u0 − u1 . In the limit as R→∞ we obtain x̄→ y R , ȳ → 1 + x R , where x and y are the Cartesian coordinates in the Euclidean plane. The horicyclic basis functions satisfying the orthonormality condition R2 ∫ ∞ −∞ dx̃ ∫ ∞ 0 Ψ∗HO ρ′s′ (ỹ, x̃)ΨHO ρs (ỹ, x̃) dỹ ỹ2 = δ(ρ− ρ′)δ(s− s′), have the form ΨHO ρs (x̄, ȳ) = √ ρ sinhπρ 2R2π3 √ ȳKiρ(\|s\|ȳ)eisx̄, where Kν(x) is the Macdonald function [2]. To effect the correct contraction we further require that ρ → kR, and s → k2R where k = √ k2 1 + k2 2. Consequently we need the asymptotic formula for KikR(k2(R+ x)) as R→∞. For this we use the asymptotic formula [2] Kiν(x) ∼ √ 2π (ν2 − x2)1/4 exp ( −πν 2 ) sin (π 4 − √ ν2 − x2 + ν cosh−1 ν x ) , (6) valid if ν > x � 1 and both ν and x are large and positive. If we use this formula for the Macdonald functions and take into account that √ ρ sinhπρ ∼ √ kR 2 e kπR 2 , then we finally get ΨHO ρs (ȳ, x̄) ∼ 1 Rπ √ k 2k1 sin(k1x−M) exp(ik2y), M = π 4 +R [ k cosh−1 k k2 − k1 ] . The correct correspondence to give the limiting result in the Cartesian coordinates x and y can be obtained using the delta-function contractions δ(ρ− ρ′)δ(s− s′) ∼ k k1R2 δ(k1 − k′1)δ(k2 − k′2). 6 E. Kalnins, G.S. Pogosyan and A. Yakhno 2.4 Semi circular parabolic to Cartesian coordinates These coordinates are given by the formulae (η, ξ > 0) u0 = R ( ξ2 + η2 )2 + 4 8ξη , u1 = R ( ξ2 + η2 )2 − 4 8ξη , u2 = R ( η2 − ξ2 ) 2ξη . In terms of these coordinates we see that η2 = √ R2 + u2 2 + u2 u0 − u1 , ξ2 = √ R2 + u2 2 − u2 u0 − u1 . In the limit as R→∞ we have η2 = 1 + x+ y R , ξ2 = 1 + x− y R . We see that in the contraction limit the Cartesian variables x and y are mixed up. To have a correct limit we can introduce the equivalent semi-circular parabolic system of coordinate connected with previous one by the rotation about axis u0 through the angle π/4, u0 = R ( η2 + ξ2 )2 + 4 8ξη , u1 = R √ 2 2 ( η2 − ξ2 2ξη + ( η2 + ξ2 )2 − 4 8ξη ) , u2 = R √ 2 2 ( η2 − ξ2 2ξη − ( η2 + ξ2 )2 − 4 8ξη ) . In terms of a new coordinates we have that η2 = √ 2R2 + (u2 − u1)2 − (u2 − u1)√ 2u0 − (u2 + u1) , ξ2 = √ 2R2 + (u2 − u1)2 + (u2 − u1)√ 2u0 − (u2 + u1) , and in the contraction limit R→∞ we obtain η2 → 1 + √ 2 x R , ξ2 → 1 + √ 2 y R . The suitable set of basis functions are [4, 8] ΨSCP ρs (ξ, η) = √ ξηJiρ (√ sξ ) Kiρ (√ sη ) , for s > 0 and ΨSCP ρs (ξ, η) = √ ξηKiρ (√ −sξ ) Jiρ (√ −sη ) , for s < 0. The correct limit is then obtained by choosing s = R2(k2 2 − k2 1) and ρ = kR, where k2 1 + k2 2 = k2. To find the contraction limit of the basis function let us use the asymptotic relation for the Bessel function of pure imaginary index [17] 2πJip(z) ∼ √ 2π (p2 + z2)1/4 exp ( i √ p2 + z2 − ip sinh−1 p z − iπ 4 ) exp (pπ 2 ) , and for the MacDonald function the formula (6). Then Jiρ (√ sξ ) ∼ JikR ( R √ k2 2 − k2 1 √ 1 + √ 2 y R ) Separation of Variables and Contractions on Two-Dimensional Hyperboloid 7 ∼ e π 2 kR 2 1 4 √ 2πk2R exp ( ik2y + iR [ √ 2k2 − k sinh−1 k√ k2 2 − k2 1 ] − iπ 4 ) and Kiρ( √ sη) ∼ KikR ( R √ k2 2 − k2 1 √ 1 + √ 2 x R ) ∼ 2 1 4 √ πe− π 2 kR √ Rk1 sin ( π 4 − xk1 +R [ √ 2k cosh−1 k√ k2 2 − k2 1 − k1 ]) . Using the last asymptotic formulae it is easily to get for a large R ΨSCP ρs (ξ, η) ∼ −1 R √ 2k1k2 exp ( ik2y + iδ1 − i π 4 ) sin ( k1x− π 4 + δ2 ) , where δ1 + δ2 = √ 2R(k1 + k2)−Rk sinh−1 ( k2 + k1 k2 − k1 ) , δ1 − δ2 = √ 2R(k2 − k1)−Rk sinh−1 ( k2 − k1 k2 + k1 ) . These expressions are arrived at under the assumption that k2 2 > k2 1. In case of k2 2 < k2 1 we can make the interchanges x with y, and k1 with k2. 2.5 Elliptic parabolic basis to parabolic Elliptic parabolic basis contracts into a parabolic one on E2. In these coordinates the points on the hyperbola are given by [θ ∈ (−π/2, π/2), a ≥ 0], u0 = R cosh2 a+ cos2 θ 2 cosh a cos θ , u1 = R sinh2 a− sin2 θ 2 cosh a cos θ , u2 = R tanh a tan θ. From these relations we see that cos2 θ = u0 − √ u2 0 −R2 u0 − u1 , cosh2 a = u0 + √ u2 0 −R2 u0 − u1 . In the limit as R→∞ we obtain cos2 θ → 1− η2 R , cosh2 a→ 1 + ξ2 R , where the parabolic coordinates (ξ, η) are given by x = 1 2 ( ξ2 − η2 ) , y = ξη, ξ, η > 0. The elliptic parabolic wave functions on the hyperboloid have the form [4] ΨEP ρs (a, θ) = √ cos θP iρis−1/2(sin θ)P isiρ−1/2(tanh a). To effect the contraction we take s→ κR and ρ→ kR and tanh a ∼ ξ√ R , sin θ ∼ η√ R . 8 E. Kalnins, G.S. Pogosyan and A. Yakhno The separation equation( d2 da2 + s2 − ρ2 + 1/4 cosh2 a ) F (a) = 0, becomes( d2 dξ2 + λ+ k2ξ2 ) F (ξ) = 0, where λ is now the parabolic separation constant and we impose the condition λ = R ( κ2 − k2 ) , or κ = k + λ 2kR +O ( R−2 ) . This requires taking the limit of a and θ the dependent part of the eigenfunctions P isiρ−1/2(tanh a) and P iρis−1/2(sin θ). The limit can be established from the known representation of the Legendre function (3). This results in the asymptotic formula P ikR+ iλ 2k ikR− 1 2 ( ξ√ R ) → 2ikR+ 1 4 + iλ 4k Γ ( 3 4 − iλ 4k − ikR )D 1 2(iλk−1) (√ −2ikξ ) , where Dν(z) is a parabolic cylinder function [1] Dν(z) = 2ν/2 √ πe− z2 4 [ 1 Γ ( 1−ν 2 )1F1 ( −ν 2 ; 1 2 ; z2 2 ) − z √ 2 Γ ( −ν 2 )1F1 ( 1− ν 2 ; 3 2 ; z2 2 )] . If we look at the theta-dependent part of the eigenfunctions and the corresponding limit taking sin θ = η√ R we then obtain the limit P ikR ikR+ iλ 2k − 1 2 ( η√ R ) → 2ikR+ 1 4 + iλ 4k Γ ( 3 4 − iλ 4k − ikR )D− 1 2(iλk+1) (√ −2ikη ) , and finally ΨEP ρs (a, θ) ∼ 22ikR+ 1 2 + iλ 2k[ Γ ( 3 4 − iλ 4k − ikR )]2D− 1 2 ( iλ k +1 )(√−2ikη ) D 1 2 ( iλ k −1 )(√−2ikξ ) . From these expressions we see that we do indeed obtain the correct asymptotic limit. 2.6 Hyperbolic parabolic to a Cartesian basis Hyperbolic parabolic basis contracts into a Cartesian one on E2. In these coordinates the points on the hyperbola are given by [θ ∈ (0, π), b > 0], u0 = R cosh2 b+ cos2 θ 2 sinh a sin θ , u1 = R sinh2 b− sin2 θ 2 sinh a sin θ , u3 = R cot θ coth b. From these relations we see that cos2 θ = u0 − √ u2 1 +R2 u0 − u1 , cosh2 b = u0 + √ u2 1 +R2 u0 − u1 . Separation of Variables and Contractions on Two-Dimensional Hyperboloid 9 In the limit as R→∞ we can choose cos2 θ → y2 2R2 , cosh2 b→ 2 ( 1 + x R ) . The hyperbolic parabolic basis function on hyperbolid can be chosen in the form [4] ΨHP(b, θ) = (sinh b sin θ)1/2P iρis−1/2(cosh b)P iρis−1/2(cos θ). (7) To proceed further with this limit we take ρ2 ∼ k2R2 and s2 ∼ ( k2 1 − k2 2 ) R2 (the case of s2 < 0, or k2 1 < k2 2, corresponds to the discrete spectrum of constant s, and we do not consider this case here) where k2 1 +k2 2 = k2, then using the relation between Legendre function and hypergeometric functions (3) and formulae (4), and (5), we obtain for θ depending part of basis function √ sin θP iρis−1/2(cos θ) ∼ P ikR i √ k21−k22R−1/2 ( y√ 2R ) ∼ 2ikR √ π exp(ik2y) Γ ( 3 4 − iR 2 ( k + √ k2 1 − k2 2 )) Γ ( 3 4 − iR 2 ( k − √ k2 1 − k2 2 )) . For the limit of the b dependent part of the eigenfunctions we must proceed differently. In fact we need to calculate the limit of P ikR i √ k21−k22R−1/2 (√ 2 ( 1 + x R )) as R→∞. We know that the leading terms of this expansion have the form A exp(ik1x) +B exp(−ik1x), and we now make use of this fact. By this we mean that lim R→∞ P ikR i √ k21−k22R−1/2 (√ 2 ( 1 + x R )) = A exp(ik1x) +B exp(−ik1x), where the constants A and B depend on R. It remains to determine A and B. To do this let us consider x = 0. We then need to determine the following limit lim R→∞ P ikR − 1 2 +iR √ k21−k22 (√ 2 ) = A+B. (8) From the integral representation formula [1] Γ(−ν − µ)Γ(1 + ν − µ) Γ(1/2− µ) √ π 2 Pµν (z) = ( z2 − 1 )−µ/2 ∫ ∞ 0 (z + cosh t)µ−1/2 cosh ([ν + 1/2] t) dt, (9) the above limit requires us to calculate as R→∞∫ ∞ 0 (√ 2 + cosh t )ikR−1/2 cos ([ R √ k2 1 − k2 2 ] t ) dt. It can be done using the method of stationary phase [11]. We obtain P ikR − 1 2 +iR √ k21−k22 (√ 2 ) ∼ 2 − 5 4 + iR 2 (√ k21−k22−k ) Γ ( 1 2 − ikR ) Γ [ 1 2 − iR (√ k2 1 − k2 2 + k )] Γ [ 1 2 + iR (√ k2 1 − k2 2 − k )] 10 E. Kalnins, G.S. Pogosyan and A. Yakhno × ( i Rk1 )1/2 ( k1 − √ k2 1 − k2 2 k + √ k2 1 − k2 2 )iR√k21−k22 ( k k − k1 )ikR . (10) By considering the expression for the derivatives of the Legendre function (9) at x = 0, we derive the expression d dx Pµν (z) ∣∣∣∣ x=0 ∼ −ik1P ikR − 1 2 +iR √ k21−k22 (√ 2 ) ∼ ik1(A−B), then P ikR − 1 2 +iR √ k21−k22 (√ 2 ) ∼ −A+B. Comparing the above relation with (8), we obtain that A = 0 and B is equal to (10), that is P iρ is− 1 2 (cosh b)→ Be−ixk1 . Finally, solution (7) contracts as follows Ψρs(b, θ) ∼ 2ikR √ πB Γ ( 3 4 − iR k+ √ k21−k22 2 ) Γ ( 3 4 − iR k− √ k21−k22 2 ) exp(ik2y − ik1x). 3 Conclusion In this note we have constructed the contraction limit R→∞ for the unnormalized wave func- tions which are the solution of the Helmholtz equation on the two dimensional two sheeted hyperboloid in four coordinates systems, namely, horocyclic, semi circular parabolic, elliptic parabolic and hyperbolic parabolic. Of course the complete analysis of the contraction prob- lem must include also the solutions of Helmholtz equation in the three additional systems of coordinates as elliptic, hyperbolic and semi hyperbolic. We will study this in the near future. We have not presented limits associated with nonsubgroup coordinates. The extension of these ideas to problems in higher dimensions is natural and will be presented in subsequent work. Acknowledgment The work of G.S.P. was partially supported under the Armenian-Belorussian grant 11RB-010. G.S.P. and A.Ya. are thankful to the PRO-SNI (UdeG, Mexico). References [1] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. I, McGraw- Hill Book Company, New York, 1953. [2] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. II, McGraw- Hill Book Company, New York, 1953. [3] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. III, McGraw- Hill Book Company, New York, 1955. [4] Grosche C., Path integrals, hyperbolic spaces, and Selberg trace formulae, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. [5] Izmest’ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables, J. Phys. A: Math. Gen. 29 (1996), 5949–5962. http://dx.doi.org/10.1088/0305-4470/29/18/024 Separation of Variables and Contractions on Two-Dimensional Hyperboloid 11 [6] Izmest’ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. The n-dimensional sphere, J. Math. Phys. 40 (1999), 1549–1573. [7] Izmest’ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. Two-dimensional hyperboloid, Internat. J. Modern Phys. A 12 (1997), 53–61. [8] Kalnins E.G., Miller Jr. W., Lie theory and separation of variables. IV. The groups SO(2, 1) and SO(3), J. Math. Phys. 15 (1974), 1263–1274. [9] Kalnins E.G., Miller Jr. W., Pogosyan G.S., Contractions of Lie algebras: applications to special functions and separation of variables, J. Phys. A: Math. Gen. 32 (1999), 4709–4732. 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Separation of variables and contractions, Phys. Atomic Nuclei 74 (2011), 1062–1072. [17] Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944. [18] Winternitz P., Lukac I., Smorodinskii Y.A., Quantum numbers in the little group of the Poincaré group, Sov. J. Nuclear Phys. 7 (1968), 139–145. http://dx.doi.org/10.1063/1.532820 http://dx.doi.org/10.1142/S0217751X97000074 http://dx.doi.org/10.1063/1.1666805 http://dx.doi.org/10.1088/0305-4470/32/25/312 http://arxiv.org/abs/math-ph/0310011 http://dx.doi.org/10.1063/1.1467709 http://dx.doi.org/10.1134/S1063778809050123 http://dx.doi.org/10.1134/S1063778810030130 http://dx.doi.org/10.1134/S106377881106024X 1 Introduction 2 Contractions 2.1 Pseudo-spherical basis to polar 2.2 Equidistant basis to Cartesian 2.3 Horocyclic basis to Cartesian in E2 2.4 Semi circular parabolic to Cartesian coordinates 2.5 Elliptic parabolic basis to parabolic 2.6 Hyperbolic parabolic to a Cartesian basis 3 Conclusion References

Separation of Variables and Contractions on Two-Dimensional Hyperboloid

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