Scattering theory without large-distance asymptotics in arbitrary dimensions
SScattering theory without large-distanceasymptotics in arbitrary dimensions
Wen-Du Li a and Wu-Sheng Dai a,b, a Department of Physics, Tianjin University, Tianjin 300072, P.R. China b LiuHui Center for Applied Mathematics, Nankai University & Tianjin University, Tianjin 300072,P.R. China
Abstract:
In conventional scattering theory, by large-distance asymptotics, at the costof losing the information of the distance between target and observer, one imposes a large-distance asymptotics to achieve a scattering wave function which can be represented ex-plicitly by a scattering phase shift. In this paper, without large-distance asymptotics,we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scatteringwave functions, scattering boundary conditions, cross sections, and phase shifts are givenwithout large-distance asymptotics. The importance of an arbitrary-dimensional scatteringtheory is that the dimensional renormalization procedure in quantum field theory needs anarbitrary-dimensional result. Moreover, we give a discussion of one- and two-dimensionalscatterings.
Keywords: scattering wave function, scattering phase shift, large-distance asymptotics,arbitrary dimensions [email protected]. a r X i v : . [ m a t h - ph ] O c t ontents n -dimensional scattering wave function 44 n -dimensional scattering boundary condition 4 a l ( θ ) Y ν ( z ) in odd and even dimensions 7 n -dimensional differential scattering cross section 88 One-, two-, and three-dimensional scatterings 9 n -dimensional scattering with large-distance asymptotics 1110 Conclusions 12 In conventional scattering theory, the information of the distance between target and ob-server is lost due to large-distance asymptotics. By large-distance asymptotics, in conven-tional three-dimensional scattering theory, the solution of the radial Schrödinger equation inthe asymptotic region and the scattering boundary condition are approximately representedas [2] R l ( r ) r →∞ ∼ A l sin ( kr − lπ/ δ l ) kr , (1.1) ψ ( r, θ ) r →∞ ∼ ∞ (cid:88) l =0 (2 l + 1) i l sin ( kr − lπ/ kr P l (cos θ ) + f ( θ ) e ikr r , (1.2)– 1 –espectively, where δ l is the scattering phase shift and f ( θ ) is the scattering amplitude.Without large-distance asymptotics, in ref. [1], the asymptotic solution (1.1) and theasymptotic scattering boundary condition (1.2) are replaced by the following exact solu-tions: R l ( r ) = M l (cid:18) − ikr (cid:19) A l kr sin (cid:20) kr − lπ δ l + ∆ l (cid:18) − ikr (cid:19)(cid:21) , (1.3) ψ ( r, θ ) = ∞ (cid:88) l =0 (2 l + 1) i l M l (cid:18) − ikr (cid:19) kr sin (cid:20) kr − lπ l (cid:18) − ikr (cid:19)(cid:21) P l (cos θ ) + f ( r, θ ) e ikr r , (1.4)where M l ( x ) = | y l ( x ) | and ∆ l ( x ) = arg y l ( x ) are the modulus and argument of the Besselpolynomial y l ( x ) [1], respectively, and P l ( x ) is the Legendre polynomial.In this paper, we establish a rigorous scattering theory without large-distance asymp-totics in arbitrary dimensions.In the following, we first rewrite the three-dimensional scattering theory established inref. [1] in a new form which is convenient to be generalized to arbitrary dimensions, andfrom which one can directly see what happens after a scattering. We will show that, likethat in three-dimensional cases, the scattering phase shift is the only effect in arbitrary-dimensional elastic scatterings.Moreover, it will be shown that the scattering theory is different in odd and evendimensions.An arbitrary-dimensional scattering theory is important in quantum field theory. Forexample, in scattering spectral method, to perform a dimensional regularization procedurerequires us to be able to carry out scattering theory calculations in arbitrary dimensions[3–5]. Moreover, two special cases, one- and two-dimensional scatterings, are importantboth in theories and experiments.In section 2, we rewrite the three-dimensional scattering theory established in ref. [1] ina new form. In section 3, we give an exact n -dimensional scattering wave function withoutlarge-distance asymptotics. In section 4, we construct a n -dimensional scattering boundarycondition without large-distance asymptotics. In section 5, we rewrite the results given insections 3 and 4 by sine functions, which is the form in conventional scattering theory. Insection 6, we demonstrate how the scattering phase shift appears. In section 7, we givethe n -dimensional scattering cross section. In section 8, as examples, we discuss one-, two-,and three-dimensional scatterings. In section 9, we demonstrate how to take large-distanceasymptotics of a n -dimensional scattering theory. The conclusion is given in section 10. In order to establish an arbitrary-dimensional scattering theory without large-distanceasymptotics, in this section, we rewrite the three-dimensional scattering theory withoutlarge-distance asymptotics given in ref. [1] in a new form which is convenient to be gener-alized to arbitrary dimensions. – 2 – or a three-dimensional scattering, without large-distance asymptotics, the incidentplane wave is ψ in ( r, θ ) = ∞ (cid:88) l =0 (2 l + 1) i l (cid:104) h (2) l ( kr ) + h (1) l ( kr ) (cid:105) P l (cos θ ) ; (2.1) after an elastic scattering, the wave function becomes ψ ( r, θ ) = ∞ (cid:88) l =0 (2 l + 1) i l (cid:104) h (2) l ( kr ) + e iδ l h (1) l ( kr ) (cid:105) P l (cos θ ) , (2.2) where h (1) ν ( z ) and h (2) ν ( z ) are the first and second kind spherical Hankel functions.The scattering boundary condition is ψ ( r, θ ) = e ikr cos θ + ∞ (cid:88) l =0 a l ( θ ) h (1) l ( kr ) , (2.3) where a l ( θ ) = (2 l + 1) i l (cid:16) e iδ l − (cid:17) P l (cos θ ) . (2.4) Proof.
The incident plane wave is ψ in ( r, θ ) = e ikr cos θ . (2.5)Substituting the plane wave expansion e ikr cos θ = (cid:80) ∞ l =0 (2 l + 1) i l j l ( kr ) P l (cos θ ) and j l ( z ) = (cid:104) h (1) l ( z ) + h (2) l ( z ) (cid:105) [1] into eq. (2.5) proves eq. (2.1) directly, where j l ( z ) is the sphericalBessel function [6].Next we prove eqs. (2.2) and (2.3).Without large-distance asymptotics, in ref. [1], we show that the scattering boundarycondition can be expressed as ψ ( r, θ ) = e ikr cos θ + f ( r, θ ) e ikr r (2.6)with f ( r, θ ) = 12 ik ∞ (cid:88) l =0 (2 l + 1) (cid:16) e iδ l − (cid:17) P l (cos θ ) y l (cid:18) − ikr (cid:19) . (2.7)By eq. (2.4), we can rewrite f ( r, θ ) as f ( r, θ ) = 1 k ∞ (cid:88) l =0 a l ( θ ) ( − i ) l +1 y l (cid:18) − ikr (cid:19) . (2.8)Substituting eq. (2.8) into eq. (2.6) and using h (1) l ( z ) = ( − i ) l +1 (cid:0) e iz /z (cid:1) y l ( i/z ) [1], wearrive at eq. (2.3).Finally, substituting eq. (2.1) into (2.3) and using (2.4) prove eq. (2.2).– 3 –y large-distance asymptotics, in conventional scattering theory, it is proved that thephase shift is the only effect after an elastic scattering and all information of an elasticscattering is embedded in a scattering phase shift [2].Without large-distance asymptotics, it is proved by comparing eqs. (2.1) and (2.2) that,the only effect after an elastic scattering is still a phase shift on the outgoing wave function:the incoming part, represented by h (2) l ( kr ) , does not change anymore; the outgoing part,represented by h (1) l ( kr ) , changes a phase factor e iδ l .Naturally, when taking large-distance asymptotics, the above result will reduce to con-ventional scattering theory: (cid:80) ∞ l =0 a l ( θ ) h (1) l ( kr ) r →∞ ∼ f ( θ ) e ikr /r with the scattering am-plitude f ( θ ) = (cid:80) ∞ l =0 a l ( θ ) / (cid:0) i l +1 k (cid:1) . n -dimensional scattering wave function For a n -dimensional scattering, the radial wave equation with a spherical potential reads[3] (cid:20) d dr + n − r ddr + k − l ( l + n − r − V ( r ) (cid:21) R l ( r ) = 0 . (3.1)The solution of the asymptotic equation of the radial equation (3.1), i.e., eq. (3.1) with V ( r ) = 0 , can be solved exactly: R l ( r ) = C l h (2) l +( n − / ( kr ) r ( n − / + D l h (1) l +( n − / ( kr ) r ( n − / . (3.2)It should be noted that in conventional scattering theory the exact solution of the asymp-totic equation (3.1) is approximated by an asymptotic solution of the asymptotic equation,like eq. (1.1).The n -dimensional wave function can be expressed as ψ ( r, θ ) = ∞ (cid:88) l =0 R l ( r ) C n/ − l (cos θ ) , where C λl ( z ) is the Gegenbauer polynomial, a generalization of the Legendre polynomial[6]. Then, by eq. (3.2), we arrive at ψ ( r, θ ) = ∞ (cid:88) l =0 C l h (2) l +( n − / ( kr ) r ( n − / + e iδ l h (1) l +( n − / ( kr ) r ( n − / C n/ − l (cos θ ) , (3.3)where e iδ l = D l /C l defines the phase shift [1]. n -dimensional scattering boundary condition A scattering is determined by the Schrödinger equation with a scattering boundary condi-tion. In conventional scattering theory, the scattering boundary condition is the Sommerfeldradiation condition which is constructed under large-distance asymptotics. In our precedingwork [1], without large-distance asymptotics, instead of the Sommerfeld radiation condi-tion, we construct a scattering boundary condition, eq. (1.4) or, equivalently, eq. (2.3),which preserves the information of the distance between the target and observer.– 4 – .1 Scattering boundary condition
In the following, without large-distance asymptotics, we construct the n -dimensional scat-tering boundary condition.Generally speaking, a scattering boundary condition is a wave function at an asymptoticdistance, consisting of two parts: the incident wave ψ in and the scattering wave ψ sc , i.e., ψ = ψ in + ψ sc . In three-dimensional conventional scattering theory, ψ sc is chosen as being inproportion to e ikr /r , since the asymptotics of the scattering wave function is R l r →∞ ∼ e ± ikr /r and only the outgoing wave R l r →∞ ∼ e ikr /r remains in the scattering wave function when r → ∞ [7]. Without the asymptotic approximation, as shown in eq. (3.2), the solutionis h (1 , l +( n − / ( kr ) /r ( n − / and only the outgoing wave h (1) l +( n − / ( kr ) /r ( n − / remains inthe scattering wave function. To retrieve the information of the distance, we constructthe scattering boundary condition by h (1) l +( n − / ( kr ) /r ( n − / rather than its asymptotics e ikr /r .To generalize the three-dimensional scattering boundary condition to n dimensions,we replace the three-dimensional outgoing wave h (1) l ( kr ) in eq. (2.3) with n -dimensionaloutgoing wave h (1) l +( n − / ( kr ) /r ( n − / : ψ ( r, θ ) = e ikr cos θ + ∞ (cid:88) l =0 a l ( θ ) h (1) l +( n − / ( kr ) r ( n − / . (4.1)The expression of a l ( θ ) will be given in the following. a l ( θ ) In a scattering theory without large-distance asymptotics, a l ( θ ) plays the role of the partialwave scattering amplitude in conventional scattering theory, and the information of thescattering is embedded in a l ( θ ) . In this section, we calculate a l ( θ ) in n dimensions.By using the n -dimensional plane wave expansion [6] e ikr cos θ = Γ ( n/ − √ π ( k/ ( n − / ∞ (cid:88) l =0 (2 l + n − i l j l +( n − / ( kr ) r ( n − / C n/ − l (cos θ ) (4.2)and j l ( z ) = (cid:104) h (1) l ( z ) + h (2) l ( z ) (cid:105) , we can rewrite the scattering boundary condition (4.1)as ψ ( r, θ ) = ∞ (cid:88) l =0 (cid:34) Γ ( n/ − √ π ( k/ ( n − / (2 l + n − i l C n/ − l (cos θ ) + a l ( θ ) (cid:35) h (1) l +( n − / ( kr ) r ( n − / + Γ ( n/ − √ π ( k/ ( n − / ∞ (cid:88) l =0 (2 l + n − i l h (2) l +( n − / ( kr ) r ( n − / C n/ − l (cos θ ) . (4.3)– 5 –hen a l ( θ ) can be achieved immediately by equating the coefficients in eqs. (4.3) and(3.3): Γ ( n/ − √ π ( k/ ( n − / (2 l + n − i l C n/ − l (cos θ ) + a l ( θ ) = C l e iδ l C n/ − l (cos θ ) , (4.4) Γ ( n/ − √ π ( k/ ( n − / (2 l + n − i l
12 = C l . (4.5)Substituting eq. (4.5) into eq. (4.4) gives a l ( θ ) = Γ ( n/ − √ π ( k/ ( n − / (2 l + n − i l (cid:16) e iδ l − (cid:17) C n/ − l (cos θ ) . (4.6)It should be emphasized that n = 2 is a removable singularity of a l ( θ ) , which will bediscussed in Sec. 8.2. In conventional scattering theory, the scattering wave function is approximately expressedby a sine function. In Ref. [1], we show that the scattering wave function, in fact, canbe exactly expressed by a sine function. In this section, we represent the n -dimensionalscattering wave function by a sine function exactly. In order to represent the scattering wave function by a sine function, we first rewrite thespherical Hankel function as h (1) ν ( z ) = e i ( z − νπ/ iz Y ν (cid:18) − iz (cid:19) , (5.1) h (2) ν ( z ) = − e − i ( z − νπ/ iz Y ν (cid:18) iz (cid:19) . (5.2)Here we introduce Y ν ( z ) = (cid:18) z (cid:19) ν +1 U (cid:18) ν + 1 , ν + 1) , z (cid:19) , (5.3)where U ( a, b ; z ) is the Tricomi confluent hypergeometric function [6]. Notice that Y ν ( z ) recovers the Bessel polynomial in odd dimensions.The radial wave function (3.2) then can be expressed as R l ( r ) = C l (cid:40) − e − i { kr − [ l +( n − / π/ } ikr r ( n − / Y l +( n − / (cid:18) ikr (cid:19) + e iδ l e i { kr − [ l +( n − / π/ } ikr r ( n − / Y l +( n − / (cid:18) − ikr (cid:19)(cid:41) ; (5.4)– 6 –otice that here e iδ l = D l /C l . Then the radial wave function (5.4) can be represented bya sine function: R l ( r ) = M l (cid:18) − ikr (cid:19) A l kr r ( n − / sin (cid:20) kr − (cid:18) l + n − (cid:19) π δ l + ∆ l (cid:18) − ikr (cid:19)(cid:21) , (5.5)where A l = 2 √ C l D l and M l = (cid:12)(cid:12) Y l +( n − / (cid:0) − ikr (cid:1)(cid:12)(cid:12) and ∆ l = arg Y l +( n − / (cid:0) − ikr (cid:1) are themodulus and argument of Y l +( n − / (cid:0) − ikr (cid:1) , respectively.When employing large-distance asymptotics, the radial wave function becomes R l ( r ) r →∞ ∼ A l kr r ( n − / sin (cid:20) kr − (cid:18) l + n − (cid:19) π δ l (cid:21) , (5.6)where asymptotics U ( a, b ; z ) r →∞ ∼ /z a [6] and eq. (9.4) which will be proved in section 9are used. Y ν ( z ) in odd and even dimensions As will be shown in the following, the function Y ν ( z ) defined by eq. (5.3) is different inodd and even dimensions: in odd dimensions Y ν ( z ) is a polynomial and in even dimensions Y ν ( z ) is an infinite series.The Tricomi confluent hypergeometric function U ( a, b ; z ) , when b = m + 1 ( m =0 , , , ... ), can be expanded as [6] U ( a, m + 1 , z ) = ( − m +1 m !Γ ( a − m ) ∞ (cid:88) k =0 ( a ) k ( m + 1) k k ! z k [ln z + ψ ( a + k ) − ψ (1 + k ) − ψ ( m + k − a ) m (cid:88) k =1 ( k − − a + k ) m − k ( m − k )! 1 z k , (5.7)where ( α ) n = α ( α + 1) . . . ( α + n −
1) = Γ ( α + n ) / Γ ( α ) is the Pochhammer symbol and ψ ( z ) = Γ (cid:48) ( z ) / Γ ( z ) is the digamma function [6].Then by eq. (5.3), for n ≥ , we have Y l +( n − / ( z ) = (cid:18) z (cid:19) l +( n − / l + n − (cid:88) j =0 Γ ( j + 1)Γ ( − l − n/ / j ) Γ (2 l + n − − j ) (cid:16) z (cid:17) j +1 + (cid:18) z (cid:19) l +( n − / ( − l + n − (2 l + n − − l − ( n − / ∞ (cid:88) j =0 ( l + ( n − / j (2 l + n − j j ! (cid:18) z (cid:19) j × (cid:20) ln (cid:18) z (cid:19) + ψ (cid:18) l + n −
12 + j (cid:19) − ψ (1 + j ) − ψ (2 l + n − j ) (cid:21) . (5.8)For odd-dimensional cases ( n (cid:54) = 1 ), i.e., n = 3 , , ... , the second term in eq. (5.8) equalszero because / Γ ( − l − ( n − /
2) = 0 . Moreover, because / Γ ( − l − n/ / j ) = 0 when − l − n/ / j = 0 , − , − , ... , the summation of j in fact begins with j = l + ( n − / rather than j = 0 . Therefore, in odd dimensions ( n (cid:54) = 1 ), Y l +( n − / ( z ) is in– 7 –act a polynomial, Y l +( n − / ( z ) = (cid:18) z (cid:19) l +( n − / l + n − (cid:88) j = l +( n − / Γ ( j + 1)Γ ( − l − n/ / j ) Γ (2 l + n − − j ) (cid:16) z (cid:17) j +1 . (5.9)This result can be rewritten as Y ν ( z ) = ν (cid:88) j =0 ( ν + j )! j ! ( ν − j )! (cid:16) z (cid:17) j = y ν ( z ) , (5.10)where y ν ( z ) is just the Bessel polynomial. That is to say, Y ν ( z ) recovers the Bessel poly-nomial y l ( z ) in odd dimensions.On the contrary, for even-dimensional cases, n is an even number and ν = l + ( n − / is a half-integer; as a result, Y ν ( z ) is an infinite series rather than a polynomial. In this section, similar to eqs. (2.1) and (2.2), we write out the n -dimensional wave functionsbefore and after a scattering. The phase shift is the only effect in an elastic scatteringprocess, i.e., all information of an elastic scattering process is embedded in a scatteringphase shift [2]. The result given below will show how a scattering phase shift appears.The n -dimensional incident plane wave, by eq. (4.2), can be expressed as ψ in ( r, θ ) = Γ ( n/ − √ π ( k/ ( n − / ∞ (cid:88) l =0 (2 l + n − i l h (2) l +( n − / ( kr ) r ( n − / + h (1) l +( n − / ( kr ) r ( n − / C n/ − l (cos θ ) . (6.1)After an elastic scattering, the wave function, by eqs. (4.1) and (4.6), becomes ψ ( r, θ ) = Γ ( n/ − √ π ( k/ ( n − / ∞ (cid:88) l =0 (2 l + n − i l h (2) l +( n − / ( kr ) r ( n − / + e iδ l h (1) l +( n − / ( kr ) r ( n − / C n/ − l (cos θ ) . (6.2)Comparing the wave functions before and after a scattering process, eqs. (6.1) and (6.2),we can see that after a scattering, the incoming part which is represented by h (2) l +( n − / ( kr ) /r ( n − / ,does not change anymore, while a phase factor e iδ l appears in the outgoing part which isrepresented by h (1) l +( n − / ( kr ) /r ( n − / . This reveals that the only effect after an elasticscattering is a phase shift on the outgoing wave function. n -dimensional differential scattering cross section Without large-distance asymptotics, the n -dimensional differential scattering cross sectionis dσd Ω = j sc · d S j in = | j sc | j in cos γ r n − = (cid:113) ( j scr ) + (cid:0) j scθ (cid:1) j in cos γ r n − , (7.1)– 8 –here d S = ˆn dS = ˆn r n − d Ω with the n -dimensional solid angle d Ω = sin n − θ sin n − θ ... sin θ n − dθ dθ ...dθ n − dφ and γ is the angle between j sc and ˆr , tan γ = j scθ j scr . (7.2)The n -dimensional differential scattering cross section (7.1) can be rewritten as dσd Ω = j scr j in (cid:34) (cid:18) j scθ j scr (cid:19) (cid:35) r n − . (7.3)The leading contribution of the n -dimensional differential scattering cross section is dσd Ω = j scr j in r n − = 1 k Im (cid:18) ψ sc ∗ ∂∂r ψ sc (cid:19) r n − . (7.4)Substituting ψ sc = ψ − ψ in = (cid:80) ∞ l =0 a l ( θ ) h (1) l +( n − / ( kr ) /r ( n − / into eq. (7.4), we have dσd Ω = r n − ik ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 a ∗ l ( θ ) a l (cid:48) ( θ ) W r h (2) l +( n − / ( kr ) r ( n − / , h (1) l (cid:48) +( n − / ( kr ) r ( n − / , (7.5)where W [ f ( r ) , g ( r )] = f ( r ) ddr g ( r ) − g ( r ) ddr f ( r ) is the Wronskian determinant. In this section, we discuss one-, two-, and three-dimensional scatterings, respectively. Thesethree kinds of scattering can occur in real physical systems.
For a one-dimensional scattering, n = 1 , by eq. (4.6), we have a l ( θ ) = − k (2 l − i l (cid:16) e iδ l − (cid:17) C − / l (cos θ ) . (8.1)In the one-dimensional case, θ can only take two possible values, and π . Therefore, C − / (1) = C − / ( −
1) = C − / ( −
1) = − C − / (1) = 1 ,C − / l (cos θ ) = 0 , l (cid:54) = 0 , . (8.2)Thus we have a (0) = a ( π ) = − k (cid:16) e iδ − (cid:17) ,a (0) = − a ( π ) = − ik (cid:16) e iδ − (cid:17) ,a l (0) = a l ( π ) = 0 , l (cid:54) = 0 , . (8.3)– 9 –rom eq. (7.5), we obtain the differential scattering cross section at θ = 0 and θ = π : σ (0) = sin δ + sin δ + 2 cos ( δ − δ ) sin δ sin δ , (8.4) σ ( π ) = sin δ + sin δ − δ − δ ) sin δ sin δ . (8.5)In a one-dimensional scattering, we are interested in transmissivity T and reflectivity R : T = σ (0) σ (0) + σ ( π ) = 12 + cos ( δ − δ ) sin δ sin δ sin δ + sin δ , (8.6) R = σ ( π ) σ (0) + σ ( π ) = 12 − cos ( δ − δ ) sin δ sin δ sin δ + sin δ . (8.7)It can be seen that in one dimension the scattering result is independent of the distance r . In a two-dimensional scattering, we encounter a singularity in a l ( θ ) given by eq. (4.6). Wewill show that, however, n = 2 is a removable singularity.We can see from eq. (4.6) that n = 2 is a singularity of the gamma function Γ ( n/ − ,but, meanwhile, n = 2 is also a zero of the Gegenbauer polynomial C n/ − l (cos θ ) . Thismakes n = 2 a removable singularity.When n = 2 , a l ( θ ) given by eq. (4.6) reduces to a l ( θ ) = Deg ( l ) 12 i l (cid:114) kπ (cid:16) e iδ l − (cid:17) cos ( lθ ) , (8.8)where Deg ( l ) is the degeneracy, Deg ( l ) = 1 , l = 0 , (8.9) Deg ( l ) = 2 , l (cid:54) = 0 . (8.10)The differential scattering cross section can be obtained by eq. (7.5) with n = 2 : dσd Ω = r ik ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 a ∗ l ( θ ) a l (cid:48) ( θ ) W r h (2) l − / ( kr ) r − / , h (1) l (cid:48) − / ( kr ) r − / = kr π ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 Deg ( l ) Deg (cid:0) l (cid:48) (cid:1) ( − i ) l i l (cid:48) (cid:16) e − iδ l − (cid:17) (cid:16) e iδ l (cid:48) − (cid:17) cos ( lθ ) cos (cid:0) l (cid:48) θ (cid:1) × ik W r (cid:104) √ rh (2) l − / ( kr ) , √ rh (1) l (cid:48) − / ( kr ) (cid:105) . (8.11)Performing large-distance asymptotics gives dσd Ω ∼ πk ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 Deg ( l ) Deg (cid:0) l (cid:48) (cid:1) (cid:16) e − iδ l − (cid:17) (cid:16) e iδ l (cid:48) − (cid:17) cos ( lθ ) cos (cid:0) l (cid:48) θ (cid:1) (8.12) = | f ( θ ) | . (8.13)– 10 – .3 Three-dimensional scattering The three-dimensional result can be obtained directly by setting the dimension n = 3 inthe above result.Eq. (4.6) with n = 3 , by the relation C / l (cos θ ) = P l (cos θ ) [6], gives eq. (2.4). Thedifferential scattering cross section can then be obtained by eq. (7.5): dσd Ω = r ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 a ∗ l ( θ ) a l (cid:48) ( θ ) W r (cid:104) h (2) l ( kr ) , h (1) l (cid:48) ( kr ) (cid:105) ik . (8.14)This result agrees with the result given by Ref. [1]. n -dimensional scattering with large-distance asymptotics In the above, we obtain a n -dimensional scattering theory without large-distance asymp-totics, which contains the information of the distance between target and observer. In thissection, we demonstrate how this result reduces to the conventional scattering result whentaking r → ∞ asymptotics.Using [6] h (1 , ν ( z ) = (cid:114) π z H (1 , ν ( z ) (9.1)and H (1 , ν ( z ) = ∓ √ π ie ∓ iνπ (2 z ) ν e ± iz U (cid:18) ν + 12 , ν + 1 , ∓ iz (cid:19) , (9.2)where H (1) ν ( z ) and H (2) ν ( z ) are the Hankel functions of the first kind and the second kind,we have h (1) l +( n − / ( kr ) = e ikr kr ( − i ) l +( n − / ( − ikr ) l +( n − / U (cid:18) l + n − , l + n − , − ikr (cid:19) . (9.3)Using large-distance asymptotics of the Tricomi confluent hypergeometric function, U ( a, b, z ) r →∞ ∼ z − a , we arrive at h (1) l +( n − / ( kr ) r →∞ ∼ e ikr kr ( − i ) l +( n − / . (9.4)The n -dimensional scattering condition (4.1) becomes ψ ( r, θ ) r →∞ ∼ e ikr cos θ + f ( θ ) e ikr r ( n − / , (9.5)where the n -dimensional scattering amplitude f ( θ ) = ∞ (cid:88) l =0 a l ( θ ) ( − i ) l +( n − / k . (9.6)– 11 –y eq. (4.6), the large-distance asymptotic n -dimensional scattering amplitude can beexpressed as f ( θ ) = 12 ik ( − i ) ( n − / Γ ( n/ − √ π ( k/ ( n − / ∞ (cid:88) l =0 (2 l + n − (cid:16) e iδ l − (cid:17) C n/ − l (cos θ ) . (9.7)The large-distance asymptotic n -dimensional differential scattering cross section, by eqs.(9.4) and (7.5), reads dσd Ω r →∞ ∼ r n − ik ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 a ∗ l ( θ ) a l (cid:48) ( θ ) W r (cid:20) r ( n − / e − ikr kr i l +( n − / , r ( n − / e ikr kr ( − i ) l (cid:48) +( n − / (cid:21) = ∞ (cid:88) l =0 ∞ (cid:88) l (cid:48) =0 i l +( n − / a ∗ l ( θ ) k ( − i ) l (cid:48) +( n − / a l (cid:48) ( θ ) k = | f ( θ ) | . (9.8)Integrating the differential scattering cross section gives the large-distance asymptotic n -dimensional total cross section: σ = 4 π ( n − / Γ (( n − /
2) 1 k n − ∞ (cid:88) l =0 (2 l + n −
2) ( l + 1) n − sin δ l . (9.9)
10 Conclusions
In this paper, without large-distance asymptotics, we establish a n -dimensional scatteringtheory.According to Euler’s scheme, functions can be classified by their asymptotics [8]. There-fore, for scatterings of short-range potentials, the scattering wave function is classified bythe solution of the asymptotic equation of the Schrödinger equation, i.e., the Schrödingerequation with V ( r ) = 0 . That is to say, the solution of the asymptotic equation is theasymptotics of the scattering wave function. In conventional scattering theory, however,the solution of the asymptotic equation is replaced by an approximate asymptotic solutionof the asymptotic equation. What we do in the preceding work [1] (three dimensions) andin the present paper (arbitrary dimensions) is to provide an accurate scattering theory inwhich the scattering wave function is restricted by the exact solution of the asymptoticequation rather than by an approximate solution of the asymptotic equation.An arbitrary-dimensional scattering theory has special importance in renormalizationof the scattering spectral method [3, 4]. In further research, we will discuss the applica-tion of the arbitrary-dimensional without-large-distance-asymptotics scattering theory torenormalization in the scattering spectral method.In particular, it is often useful to consider a physical problem in arbitrary dimen-sions. Arbitrary-dimensional scattering problems have been considered in many aspects:scatterings of the Fermi pseudopotential [9], scatterings of long-range potentials [10], andscatterings of black holes [11]. Besides scattering problems, there are also many arbitrary-dimensional theories. In quantum mechanics and statistic physics, one considers the n + 4 -dimensional scalar spheroidal harmonics [12], the Landau problem in even-dimensional– 12 –pace CP k [13], and Fermi gases in arbitrary dimensions [14]; in field theory, one consid-ers Yang-Mills and gravity theories in arbitrary dimensions [15]; in gravity and cosmologytheories, one considers the thermodynamic curvature of the Kerr and Reissner-Nordström(RN) black holes in arbitrary dimensions [16], an arbitrary-dimensional gravitational the-ory with a negative cosmological constant, arbitrary-dimensional AdS
Black Holes [17], andarbitrary-dimensional
AdS black branes [18].Starting from the result given by the present paper and the preceding work [1], onecan reconsider many scattering related problems. The analytic property of scatteringamplitudes, which used to be treated by conventional scattering theory [19–23], can benow discussed without large-distance asymptotics in arbitrary dimensions. An arbitrary-dimensional Lippmann-Schwinger equation without large-distance asymptotics can be con-structed in the frame of the scattering theory given in the present paper. An arbitrary-dimensional vector and tensor scattering theories without large-distance asymptotics canbe established, e.g., electromagnetic scatterings and gravitational wave scatterings, whichare usually studied in the frame of conventional scattering theory [24–27]. The acousticscattering is a scalar scattering. Two- and three- dimensional acoustic scattering theo-ries without large-distance asymptotics can also be established. In conventional acousticscattering theory, large-distance asymptotics is imposed [28–30] and, therefore, the infor-mation of the distance between target and observer is lost. A very important applicationis to consider inverse scattering problems without large-distance asymptotics in arbitrarydimensions; this is a fundamental problem and is studied under large-distance asymptotics[31, 32]. Based on the relation of two important quantum field theory methods [4, 33],the scattering spectrum method [34–36] and the heat-kernel method [37–40], we establisha heat-kernel method for calculating phase shifts [5]. This result can be generalized toarbitrary-dimensional cases. The result given by ref. [1] and the present paper can be alsoapplied to scatterings of long-range potentials, which is usually studied under large-distanceasymptotics [41]. In particular, we will discuss the scattering of a wave on a black hole in ar-bitrary dimensions, while research in literature is usually under large-distance asymptotics[42–46]. String theory related scatterings [47] can be also considered without large-distanceasymptotics.
Acknowledgments
We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supportedin part by NSF of China under Grant No. 11575125 and No. 11375128.
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