aa r X i v : . [ h e p - t h ] O c t Boundary Shape and Casimir Energy
H. Ahmedov ∗ Feza G¨ursey Institute, P. K. 6 C¸ engelk¨oy, 34684 Istanbul, Turkey. Tel:+90-216-308 94 30
I.H. Duru † Izmir Institute of Technology, 35430, Izmir, Turkey (Dated: November 19, 2018)Casimir energy changes are investigated for geometries obtained by small but arbitrary deforma-tions of a given geometry for which the vacuum energy is already known for the massless scalarfield. As a specific case, deformation of a spherical shell is studied. From the deformation of thesphere we show that the Casimir energy is a decreasing function of the surface to volume ratio. Thedecreasing rate is higher for less smooth deformations.
PACS numbers: 03.65.-w, 03.70.+kKeywords: Casimir energy, boundary variations and approximations.
I. INTRODUCTION
Casimir energies are known for several cavities in several spatial dimensions for electromagnetic or massless scalarfields. Exact Casimir energy calculations are available for rectangular prisms [1], for spherical shell [2], for cylindricalregion [3], for a pyramidal cavity and for a conical cavity [4]. All these geometries have definite boundary wall shapes.For example the prisms are all with right angular wedges, the sphere is the perfect one; and, the pyramid and thecone are of very special types. The obvious reason of these restrictions is the fact that these are the regions for whichone can calculate the exact field modes with the required boundary conditions. Limited number of examples ( allwith rigid walls ) do of course not give much hint about the dependence of the Casimir energy on the shapes of theregions.There are to our knowledge two approaches in dealing with rather arbitrary geometries. First one is theproximity force approximation [5]. It is applicable to two body systems which are close to each other. It employsthe parallel plate modes in every cylindrical region of infinitesimal base between the bodies and then integratesover these regions [6]. Second approach is the one called multiple scattering expansion. It formulates the vacuumenergy for the electromagnetic field in terms of the successive scattering from the conducting boundaries [7]. In thisapproach between two successive scatterings free Green function is employed. The method enables one to investigatethe connections between the divergencies and the geometrical details of the boundaries. Employment of the freeGreen functions however from one scattering to the next one is not of much practical value in compact space regions:Specially for more curved boundaries one may need to consider large number of scatterings to approximate the exactGreen function. On the other hand it may be more reasonable to employ the exact Green function ( if it is alreadyknown ) of the compact region ( instead of the free one ) if the region under consideration is sufficiently close to theoriginal region. Variations around the exactly solvable geometries may give some hint about the dependence of thevacuum energies on the shape of the boundaries.In present work we try to investigate the shape dependence of the Casimir energy for massless scalar field bycalculating the effect of the small but arbitrary deformations of a given geometry for which we already know thevacuum energies. We use Green function to formulate the perturbation theory around the exact solution of theregion with boundary S . Suppose G Sω is the exact Green function for the massless scalar field confined to the regionin S , and β is the small deformation of S . Converting a boundary problem into an integral equation we arrive at theperturbation series G ˜ Sω = G Sω + βG Sω + β G Sω + · · · ∗ Electronic address: [email protected] † Electronic address: [email protected] where G jSω is the correction to the original Green function G Sω resulting from the j times reflections from the deformedboundary ˜ S . One reflection gives information about the size of the new boundary. To get information about theshape dependence we need to take into account at least two reflections.Having in hand the Green function we can construct the zeta function which is useful tool in Casimir energycalculation. The zeta function of the system can be expressed in terms of the heat kernel coefficients which arefunctionals of the geometrical invariants of the boundaries [8]. For arbitrary geometries these coefficients are toocomplicated and therefore are not available for practical purposes [9]. From this point of view we think that theperturbation around a known geometry may be an effective approach.For massless fields which are the only fields for which the Casimir energy is meaningful, there is no uniqueway of getting rid of the infinities if the heat kernel coefficient a of the zeta function expansion is not zero [10]. Thesituation can be improved if one considers the whole space for when one sums the zeta functions of the in and outregions a coefficients cancel each other. Of course for such cancelation the boundaries should be free of sharp corners.In the coming section we briefly review the zeta function approach to the vacuum energy calculations andthe regularization scheme which we employ in the our work.In Section III we present the general formulation of the Casimir energy contribution of small deformations ofthe boundaries.In Section IV deformation of the sphere is discussed.In Section V we analyze the dependence of the energy on the shape of the boundary.Details of the involved calculations are given in the Appendices. II. A BRIEF REVIEW OF THE ZETA FUNCTION METHOD
Formally the calculation of the Casimir energy is reduced to a treatment of a sum over all one particle energyeigenvalues E = 12 X λ ∈ Λ p E λ (1)This sum is divergent and regularization is needed. For the scalar field confined in a compact three dimensional regionwith the Dirichlet boundary condition the zeta function ζ ( z ) = ∞ X n =1 E − zn (2)is well defined by Weyl theorem for Rez > / ζ ( z ) = 1Γ( z ) Z ∞ dtt z − K ( t ) (3)with the heat-kernel K ( t ) = ∞ X n =1 e − tE n . (4)For t → ∞ the integral is well behaved. Possible poles arise from t → K ( t ) ∼ ∞ X n =0 , / , ,... a n t n − / . (5)Splitting the integral as R dt + R ∞ dt we arrive at Res ( ζ ( z )Γ( z )) | z =3 / − n = a n (6)where a n are heat kernel coefficients which depent on the geometry of boundary which confines the scalar field.When the coefficient a is nonzero the value of the zeta function at z = − / a heat-kernel coefficient depends on an odd power of extrinsic curvature. Two of them add to cancel each other when weapproach the surface both from in and out region of the ball. This does not hold only for the spherical shell butis a general property for boundaries of an arbitrary shape. This cancelation of poles occurs only for infinitely thinboundaries. Once a finite thickness is introduced the absolute value of the extrinsic curvature at the inner and outerside of the boundary is different and divergencies do not cancel each other.In the present work we try to investigate the shape dependence of the Casimir energy for the massless scalarfield by calculating the effect of the small but arbitrary smooth deformations of the boundary of given regions forwhich we already know the vacuum energies. We restrict our attention to the deformations of the spherical shell.Using the scattering theory in the spherical coordinates one arrives at the zeta functions inside ζ in ( z ) = sin πzπ ∞ X l =0 (2 l + 1) Z ∞ dωω − z ddω ln( ω − l − / I l +1 / ( ω )) (7)and outside the ball [9, 10] ζ out ( z ) = sin πzπ ∞ X l =0 (2 l + 1) Z ∞ dωω − z ddω ln( ω l +1 / K l +1 / ( ω )) (8)The zeta function in the whole space ζ ( z ) = 12 ( ζ in ( z ) + ζ out ( z )) (9)or ζ ( z ) = sin πzπ ∞ X l =0 ( l + 1 / Z ∞ dωω − z ddω ln( I l +1 / ( ω ) K l +1 / ( ω )) (10)is well defined at z = − /
2. To find this value we use the uniform asymptotic expansions for Bessel functions [11] K ν ( νx ) = r π ν e − νη (1 + x ) / (1 + ∞ X k =1 ( − ) k u k ( t ) ν k ) (11)and I ν ( νx ) = 1 √ πν e νη (1 + x ) / (1 + ∞ X k =1 u k ( t ) ν k ) (12)where t = 1 √ x , η = p x + ln x √ x (13)and coefficients u k ( t ) satisfy the recurrence relation u k +1 ( t ) = 12 t (1 − t ) u ′ k ( t ) + 18 Z t dτ (1 − τ ) u k ( τ ) (14)with the initial condition u ( t ) = 1. The uniform asymptotic expansions implyln I ν ( νx ) = ∞ X − X k ( t ) ν k (15)and ln K ν ( νx ) = ∞ X − ( − ) k X k ( t ) ν k (16)where the first four terms of X n ( t )s are X − = 1 t + ln t t ,X = 12 ln t,X = t − t ,X = t − t t . (17)The zeta function in the whole space becomes ζ ( − /
2) = 2 π ∞ X m =0 ζ (2 m − Z ∞ dxX m ( t ) (18)where ζ ( s ) = ∞ X n =0 / n ) s (19)is the Riemann zeta function which vanishes for z = 0 and z = − R is [12] E sph = 12 R ζ ( − / ≃ αR , α ≃ , . (20) III. CONTRIBUTION OF SMALL BOUNDARY DEFORMATIONS TO THE VACUUM ENERGY
In this section we use the Green function representation of the zeta function for the massless scalar field in thethree dimensional space vanishing on a surface Sζ S ( z ) = sin( πz ) π Z ∞ dωω − z +1 Z R d ~xG Sω ( ~x, ~x ) (21)where G Sω ( ~x, ~y ) = { G inSω ( ~x, ~y ) , ~x, ~y ∈ Ω in G outSω ( ~x, ~y ) , ~x, ~y ∈ Ω out (22)is the Green function in R satisfying the boundary problem( − ∆ + ω ) G Sω ( ~x, ~y ) = δ ( ~x − ~y ) , G Sω ( ~x, ~y ) = 0 , ~x ∈ S. (23)Here G inSω ( G outSω ) is the Green function in the in-region Ω in ( the out-region Ω out ). To have a well defined integralover R in (21) one considers a ball of radius L and then let it go to infinite. Divergent terms in powers of L willcorrespond to the infinite vacuum oscillations of the free Minkowski space. The variation δζ ( z ) = ζ ˜ S ( z ) − ζ S ( z ) (24)resulting from the deformation of the boundary S is then given by δζ ( z ) = sin( πz ) π Z ∞ dωω − z +1 Z R d ~xδG ω ( ~x, ~x ) (25)where δG ω ( ~x, ~y ) = G ˜ Sω ( ~x, ~y ) − G Sω ( ~x, ~y ) . (26)Due to (23) it satisfies the wave equation ( − ∆ + ω ) δG ω ( ~x, ~y ) = 0 (27)and the boundary condition δG ω ( ~x, ~y ) = − G Sω ( ~x, ~y ) , ~x ∈ ˜ S. (28)The above boundary problem is equivalent to the integral equation [14] G ˜ Sω ( ~x, ~y ) = G Sω ( ~x, ~y ) − Z ˜ S d ˜ s ∂G ˜ Sω ( ~x, ~v ) ∂m ( ~v ) G Sω ( ~v, ~y ) , (29)where ∂∂m ( ~v ) = ~m ( ~v ) ∂∂~v (30)is the derivation along the unit vector ~m ( ~v ) normal to the wall ˜ S at a point ~v . In the parametric representation ~v = ~v ( τ ), τ = ( τ , τ ) the integration measure on ˜ S is d ˜ s = p | ˜ g | d τ (31)where | ˜ g | is the determinant of the induced metric˜ g ab = ( ∂~v∂τ a , ∂~v∂τ b ) (32)with ( · , · ) being the scalar product in the three dimensional space.The solution of the integral equation (29) up to the second order ( which we can also interpret as the secondreflection ) is G ˜ Sω ( ~x, ~y ) = G Sω ( ~x, ~y ) − Z ˜ S d ˜ s ∂G Sω ( ~x, ~v ) ∂m ( ~v ) G Sω ( ~v, ~y ) + Z ˜ S d ˜ s Z ˜ S d ˜ s ∂G Sω ( ~x, ~v ) ∂m ( ~v ) ∂G Sω ( ~v, ~v ′ ) ∂m ( ~v ′ ) G Sω ( ~v ′ , ~y ) . (33)The property Z R d ~xG Sω ( ~z, ~x ) G Sω ( ~x, ~z ′ ) = − ∂∂ω G Sω ( ~z, ~z ′ ) (34)allows us to integrate explicitly the perturbation solution over the three dimensional spatial space to get G ˜ Sω = G Sω + 12 ∂∂ω Z ˜ S d ˜ s ∂G Sω ( ~v, ~v ) ∂m ( ~v ) − ∂∂ω Z ˜ S d ˜ s Z ˜ S d ˜ s ∂G Sω ( ~v ′ , ~v ) ∂m ( ~v ) ∂G Sω ( ~v, ~v ′ ) ∂m ( ~v ′ ) . (35)We consider deformations of the boundary S along the unit vector ~n ( z ) normal to the surface S at a point ~z : ~v = ~z − β~n ( ~z ) f ( ~z ) (36)where β is the dimensionless deformation parameter of the surface S . This deformation formula implies˜ g ab = g ab − β ( ∂~z∂τ a , ∂f~n∂τ b ) + ( a → b )) (37)where g ab is the metric tensor on S . Using δ p | g | = 1 / p | g | g ab δg ab we arrive at the variation of the integrationmeasure d ˜ s = ds − βg ab ( ∂~z∂τ a , ∂ ( f~n ) ∂τ b ) ds. (38)which together with the Taylor expansion G Sω ( ~x, ~v ) = − βf ( ~z ) ∂G Sω ( ~x, ~z ) ∂n ( ~z ) + β f ( ~z ) ∂ G Sω ( ~x, ~z ) ∂n ( ~z ) + · · · (39)implies δG ω = − β ∂∂ω Z S dsf ( ~z ) ∂ G Sω ( ~z, ~z ) ∂n ( ~z ) + β ∂∂ω Z S dsg ab ( ∂~z∂τ a , ∂ ( f~n ) ∂τ b ) f ( ~z ) ∂ G Sω ( ~z, ~z ) ∂n ( ~z )+ β ∂∂ω Z S dsf ( ~z ) ∂ G Sω ( ~z, ~z ) ∂n ( ~z ) − β ∂∂ω Z S ds Z S ds ′ f ( ~z ) f ( ~z ′ )( ∂ G Sω ( ~z ′ , ~z ) ∂n ( ~z ) ∂n ( ~z ′ ) ) (40)Up to the second order in β the zeta function variation (24) becomes δζ ( z ) = sin( πz ) π Z ∞ dωω − z +1 δG ω . (41) IV. DEFORMATION OF THE SPHERICAL SHELL
The in and out-Green function for the massless scalar field vanishing on the sphere of the radius R are G inω ( r, ~n ; r ′ ~n ′ ) = − π √ rr ′ ∞ X l =0 (2 l + 1) P l (( ~n, ~n ′ )) I l +1 / ( ωr )( K l +1 / ( ωR ) I l +1 / ( ωr ′ ) − I l +1 / ( ωR ) K l +1 / ( ωr ′ ) I l +1 / ( ωR ) (42)and G outω ( r, ~n ; r ′ ~n ′ ) = − π √ rr ′ ∞ X l =0 (2 l + 1) P l (( ~n, ~n ′ )) ( K l +1 / ( ωR ) I l +1 / ( ωr ) − I l +1 / ( ωR ) K l +1 / ( ωr )) K l +1 / ( ωr ′ ) K l +1 / ( ωR ) (43)where 0 ≤ r ≤ r ′ ≤ R , P l ( x ) is the Legendre polynomial and ~n is the unit vector normal to the sphere ( see AppendixB ). The derivative normal to the sphere is ∂∂n ( ~z ) = ∂∂r . We have the derivatives of the following type ∂ G S ω ( r, ~n ; r ′ ~n ′ ) ∂r∂r ′ | r,r ′ = R = − ω πR ∞ X l =0 (2 l + 1) P l (( ~n, ~n ′ )) T l +1 / ( ω ) (44)where T l + ( ω ) = 12 ddω ln( I l + ( ω ) K l + ( ω )) (45)is the spectral function in the whole space. Inserting the above type terms in (40), (41) becomes δζ ( z ) = − z sin( πz ) πR ∞ X l =0 (2 l + 1) Z ∞ dωω − z T l + ( ω ) Z d Ω( β πR f ( ~n ) + β πR f ( ~n ))+ 8 z sin( πz ) πR ∞ X l,l ′ =0 ( l + 12 )( l ′ + 12 ) D ll ′ Z ∞ dωω − z ( T l + ( ω ) − T l ′ + ( ω )) (46)where D ll ′ = β π R Z d Ω Z d Ω ′ P l (( ~n, ~n ′ )) P l ′ (( ~n, ~n ′ )) f ( ~n ) f ( ~n ′ ) , (47) d Ω = dφdθ sin θ is the integration measure on the sphere. The first term in (46) at z = − / δE = E sph Z d Ω( β πR f ( ~n ) + β πR f ( ~n )) + 2 πR ∞ X l,l ′ =0 ( l + 12 )( l ′ + 12 ) D ll ′ Z ∞ dωω ( T l + ( ω ) − T l ′ + ( ω )) (48)The expansion (B5) and the addition formula (B4) imply D ll ′ = β ∞ X J =0 J X M = − J K Jll ′ | f JM | (2 J + 1) (49)where K Jll ′ are the Clebsch Gordon coefficients (B6) and f JM are the expansion coefficients of the deformation function f in the spherical harmonics: f ( ~n ) = ∞ X J =0 J X M = − J f JM Y JM ( ~n ) . (50)Using (49) we may represent (48) as δE = E sph [ β πR Z d Ω f ( ~n ) + β πR Z d Ω( f ( ~n ) + f ( ~n ) ˆ Hf ( ~n ))] (51)where ˆ H is an energy operator ˆ HY JM ( ~n ) = H ( J ) Y JM ( ~n ) (52)with e-values H ( J ) = 1 πα ∞ X l = J ( l + 12 ) J X N = − J Λ JN G JN ( µ ) Z ∞ dωω ( T l + ( ω ) − T l + N + ( ω )) . (53)Here we used the formulae (B6) and (B12) for the Clebsch Gordon coefficients. The evaluation of H ( J ) which isquite involved, is given (together with all auxilary formulas ) in the Appendices. The energy operator H ( J ) has thefollowing expansion H ( J ) = α J + α J + α J + α − J + + α − ( J + ) + · · · . (54)Terms with coefficients α − n , n = − , − , − , . . . in the above expansion give negligible contributions to the Casimirenergy compared to the first three ones H ( J ) = − , J − , J − , J. (55)The Casimir energy in the cavity obtained by the small but arbitrary deformations of the spherical region of radius R is, therefore given by E ˜ S = αR (1 + β πR Z d Ω f + β πR Z d Ω f ( ˆ H + 1) f ) . (56) V. SHAPE DEPENDENCE OF THE CASIMIR ENERGY, DISCUSSION
It is instructive to compare the Casimir energy (56) with the energy in a spherical cavity with equal volume. Thevolume and the area of the cavity after deformation are ( up to β R order )˜ V = 4 π R (1 − β πR Z d Ω f + 3 β πR Z d Ω f ) (57)and ˜ S = 4 πR (1 − β πR Z d Ω f + β πR Z d Ω f (1 −
12 ∆) f ) (58)where ∆ = 1sin θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂φ (59)is the Laplace operator on the sphere. The ratio of the energy (56) and the Casimir energy E of the sphere withvolume (57) is E ˜ S E | eq. vol. = 1 + β πR ( Z d Ω f (2 + ˆ H ) f − π ( Z d Ω f ) ) . (60)This ratio to be examined by its dependence on the shape of the cavity after deformation. One way may be to studyits dependence on the ratio of the surfaces of the deformed and spherical cavities with equal volumes:˜ SS | eq. vol. = 1 − β πR ( Z d Ω f (1 + 12 ∆) f − π ( Z d Ω f ) ) . (61)We then express the energy ratio (60) as E ˜ S E | eq. vol. = 1 + β πR Z d Ω f ( ˆ H − ∆) f −
2( ˜ SS | eq. vol. −
1) (62)or E ˜ S E | eq. vol. = 1 − R d Ω f (1 + ˆ H ) f − π ( R d Ω f ) R d Ω f (1 + ∆) f − π ( R d Ω f ) ( ˜ SS | eq. vol. − . (63)We know from (55) that the operator ˆ H has negative e-values. Thus both of the above relations show that the Casimirenergy linearly decreases by the increase of the surface. To have a better feeling of this inverse proportionality let usconsider a simple example. Suppose the deformation function is given by f ( θ, φ ) = P l (cos θ ) (64)with P l being the Legendre polynomials. By using (52) and (53), we can write (63) for this specific example as E ˜ S E | eq. vol. = 1 − Λ( l )( ˜ SS | eq. vol. −
1) (65)where the coefficient Λ is given by Λ( l ) = 2 2 + H ( l )2 − l ( l + 1) (66)or by using (55) can be written as Λ( l ) = 2 2 − , l − , l − , l − l ( l + 1) . (67)Thus for large values of l we can write (65) as E ˜ S E | eq. vol. = 1 − l ( ˜ SS | eq. vol. − . (68)We then can conclude that for less and less smooth deformations we get smaller and smaller Casimir energies. Acknowledgments
The Authors thank the Turkish Academy of Science (TUBA) for its supports, to D.A. Demir and T.O. Turgut fordiscussions and to P. Talazan for helping in ”Mathematica”.
APPENDIX A: THE CALCULATION OF THE VACUUM ENERGY
Using the uniform expansion formula (16) for the spectral function (45) we represent the energy operator (52) (with µ = l + 1 / ω = µx and ε = Nµ ) as H ( J ) = 1 πα ∞ X n,m =0 ∞ X l = J µ − m − n J X N = − J Λ JN D l ( N, J ) Z ∞ dxx Y n ( x, ε ) Y m ( x, ε ) (A1)where Y n ( x, ε ) = ddx ( X n ( t ( ε ))(1 + ε ) n − X n ( t (0))) (A2)and t ( ε ) = 1 + ε p (1 + ε ) + x . (A3)After the change of variables 2 n = s + t , 2 m = s − t we have H ( J ) = ∞ X s =0 H s ( J ) (A4)where H s ( J ) = 1 πα ∞ X l = J µ − s J X N = − J Λ JN G JN ( µ ) Z ∞ dxx F s ( x, ε ) (A5)and F s ( x, ε ) = s X t = − s Y s + t ( x, ε ) Y s − t ( x, ε ) . (A6)By using the Taylor expansion at ε = 0 F s ( x, ε ) = ∞ X k =0 ε n n ! F ( n ) s x ) (A7)and the asymptotic expansion (B13) we arrive at H s ( J ) = ∞ X τ =0 b τ ζ (2 τ + 2 s − , J ) (A8)where ζ ( z, J ) = ∞ X k =0 J + + k ) z (A9)is the Riemann zeta function b τ = 1 πα τ X p = − τ h Z τ + p N τ − p i ( τ − p )! Z ∞ dxx F ( τ − p ) s ( x ) (A10)and h f ( N ) i = J X N = − J Λ JN f ( N ) . (A11)0For large J we have ζ (2 τ + 2 s − , J ) ≃ J − τ − s (A12)and h Z τ + p N τ − p i ≃ J τ . (A13)This asymptotic expressions imply H s ( J ) ≃ J − s , J ≫ . (A14)Thus H ( J ) and H ( J ) give the main contributions to the energy operator H ( J ). Using the summation formulae(B18) we get H ( J ) = 1128 α ( J ( J + 1) ζ (0 , J ) − J ( J + 2)( J − ζ (2 , J )) (A15)and H ( J ) = − α ( J ( J + 1)2 ζ (2 , J ) + 9916 ·
24 ( J + 12 ) ζ (4 , J )) (A16)from which by the virtue of ζ (2 , J ) ≈ J + 1 / J + 1 / + 16( J + 1 / + o ( J ) ζ (4 , J ) ≈ J + 1 / + 12( J + 1 / + o ( J ) (A17)we read α = − α, α = − α, α = − · α (A18) APPENDIX B: THE KLEBSCH GORDON COEFFICIENTS
The spherical harmonics are Y lm ( θ, φ ) = { P ml (cos θ ) cos mφ, m = 1 , , . . . lP l (cos θ ) , m = 02 P | m | l (cos θ ) sin | m | φ, m = − , − , · · · − l (B1)where P l ( x ) is the Legendre function and P ml ( x ) = s l − m )!( l + m )! (1 − x ) m d m dx m P l ( x ) (B2)is the associated Legendre function. We use the notation Y lm ( θ, φ ) = Y lm ( ~n ) where ~n = (cos θ, sin θ sin φ, sin θ cos φ ) (B3)is the unit vector on the sphere.The addition formula for the spherical harmonics is P l (( ~n, ~n ′ )) = l X m = − l Y lm (ˆ n ) Y lm (ˆ n ′ ) (B4)where the argument of the Legendre function is the scalar product of the unit vectors ~n and ~n ′ .1The Klebsch Gordon coefficients K Jll + N defined by the expansion P l ( x ) P l ′ ( x ) = l + l ′ X J = | l − l ′ | K Jll ′ P J ( x ) (B5)are [15] K Jll + N = ( J + 12 )Λ JN F l ( N, J ) (B6)where ( µ = l + 1 / F l ( N, J ) = 1 µ + N + J Γ( µ + N − J )Γ( µ + N + J + )Γ( µ + N − J + )Γ( µ + N + J ) (B7)and Λ JN = 1 π Γ( J − N +12 )Γ( J + N +12 )Γ( J − N +22 )Γ( J + N +22 ) (B8)These coefficients are nonzero if | N |≤ J ≤ l + N and N + J is even number.The formula ln Γ( z ) = z ln z − z −
12 ln z + ln √ π + n − X k =1 B k k (2 k − z k − (B9)implies ln Γ( z + Γ( z ) = 12 ln z − z + 1192 z (B10)up to the third order in z . The last formula allows as to get the asymptotic expansion of the functions F l ( N, J ). Upto the third order in µ for large values of µ we have: F l ( N, J ) = 1 q ( µ + N − J )( µ + N + J ) (1 + J µ − N J µ ) (B11)For the function G JN ( µ ) = ( µ + N ) F l ( N, J ) (B12)we get the following asymptotic expansion G JN ( µ ) = 1 + Z µ + Z µ + Z µ + · · · (B13)where Z = N ,Z = J ( J + 1) − N ,Z = N N − J ( J + 1)16 . (B14)Let J is even. Putting J = 2 j and N = 2 n we have J X N = − J Λ JN e iNθ = j X n = − j π Γ( j − n + )Γ( j + n + )( j − n )!( j + n )! e i nθ (B15)2In the new variable m = n + j we have J X N = − J Λ JN e iNθ = j X m =0 π Γ( m + Γ(2 j − m + ) m !(2 j − m )! e i (2 m − j ) θ (B16)which is exactly the series representation for zonal spherical functions Y j (cos θ ) J X N = − J Λ JN e iNθ = Y J (cos θ ) . (B17)The same formula is true for odd J . For example we have J X N = − J Λ JN = 1 , J X N = − J Λ JN N = J ( J + 1)2 , J X N = − J Λ JN N = 3 J ( J + 2) + J ( J − . (B18)(B18)