Conductance Fluctuations in Disordered Wires with Perfectly Conducting Channels
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Conductance Fluctuations in Disordered Wires with Perfectly Conducting Channels
Yositake
Takane and Katsunori Wakabayashi , Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University,Higashi-Hiroshima 739-8530, Japan PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan (Received )
We study conductance fluctuations in disordered quantum wires with unitary symmetryfocusing on the case in which the number of conducting channels in one propagating directionis not equal to that in the opposite direction. We consider disordered wires with N + m left-moving channels and N right-moving channels. In this case, m left-moving channels becomeperfectly conducting, and the dimensionless conductance g for the left-moving channels behavesas g → m in the long-wire limit. We obtain the variance of g in the diffusive regime by usingthe Dorokhov-Mello-Pereyra-Kumar equation for transmission eigenvalues. It is shown that theuniversality of conductance fluctuations breaks down for m = 0 unless N is very large. KEYWORDS: perfectly conducting channel, universal conductance fluctuations, unitary class, DMPK equa-tion
1. Introduction
The discovery of a perfectly conducting channel in dis-ordered wires provides a counterexample to the con-jecture that an ordinary quasi-one-dimensional quantumsystem with disorder exhibits Anderson localization (i.e.,conductance decays exponentially with increasing systemlength L and eventually vanishes in the limit of L → ∞ ).We have shown that perfectly conducting channels canbe stabilized in two standard universality classes. Oneis the symplectic universality class with an odd numberof conducting channels, and the other is the unitaryuniversality class with the imbalance between the num-bers of conducting channels in two propagating direc-tions. The symplectic class consists of systems hav-ing time-reversal symmetry without spin-rotation invari-ance, while the unitary class is characterized by the ab-sence of time-reversal symmetry. Much attention has recently been paid to electrontransport in the above-mentioned two universality classeswhich do not exhibit Anderson localization. For the sym-plectic class, one perfectly conducting channel is stabi-lized in the odd-channel case, while such a special channeldoes not exist in the ordinary even-channel case. We havestudied in details how the dimensionless conductance g behaves as a function of L .
7, 8
It is shown that the be-havior of g in the odd-channel case is very different fromthat in the even-channel case when L is much longer thanthe conductance decay length ξ . The dimensionless con-ductance in the even-channel case decays as g → L , while g → g with increasing L is much fasterin the odd-channel case than in the even-channel case.However, such a notable even-odd difference does notappear in the diffusive regime of L ≪ ξ .
6, 7
For the uni-tary class, the number of perfectly conducting channelsdepends on the channel-number imbalance between twopropagating directions. If m perfectly conducting chan-nels are present, the dimensionless conductance behaves as g → m with increasing L . The present authors haveshown that the notable m -dependence of g appears in thelong- L regime, by using a scaling approach and a super-symmetry approach.
13, 14
However, the behavior of g inthe diffusive regime has not been clarified. It is interest-ing to study whether the perfectly conducting channelsaffect the behavior of g in the diffusive regime.In this paper, we focus on the unitary universalityclass with the channel-number imbalance and considerthe conductance in the diffusive regime. We here presentthe basic framework to describe the electron transportin systems with the channel-number imbalance on thebasis of the scaling approach. Let us consider disor-dered wires with N right-moving channels and N + m left-moving channels. In this case, m left-moving chan-nels become perfectly conducting, and the dimensionlessconductances g and g ′ for the left-moving and right-moving channels, respectively, differ from each other.Each dimensionless conductance is determined by a cor-responding set of transmission eigenvalues. If the set ofthe transmission eigenvalues for the right-moving chan-nels is { T , T , . . . , T N } , that for the left-moving channelsis expressed as { T , T , . . . , T N , , . . . , } , where we haveidentified the N + 1 to N + m th channels as the per-fectly conducting ones. The dimensionless conductance g is given by g = P N + ma =1 T a = m + P Na =1 T a , while g ′ = P Na =1 T a . We observe that g = g ′ + m . It shouldbe noted that the mean free paths l and l ′ for the left-moving and right-moving channels, respectively, are notequal due to the channel-number imbalance. Indeed, theysatisfy l ′ = ( N/ ( N + m )) l . The statistical behavior of g ,as well as g ′ , is described by the probability distribu-tion for { T , T , . . . , T N } . We define λ a ≡ (1 − T a ) /T a and introduce the probability distribution P ( { λ a } ; s ) for { λ , λ , . . . , λ N } , where s is the normalized system lengthdefined by s ≡ L/l . The Fokker-Planck equation for P ( { λ a } ; s ), which is called the Dorokhov-Mello-Pereyra- Full Paper
Yositake
Takane and Katsunori
Wakabayashi
Kumar (DMPK) equation,
16, 17 is expressed as ∂P ( { λ a } ; s ) ∂s = 1 N N X a =1 ∂∂λ a (cid:18) λ a (1 + λ a ) J ∂∂λ a (cid:18) P ( { λ a } ; s ) J (cid:19)(cid:19) (1)with J = N Y c =1 λ mc × N − Y b =1 N Y a = b +1 | λ a − λ b | . (2)The factor Q Nc =1 λ mc in J represents the repulsion aris-ing from the m -fold degenerate perfectly conductingeigenvalue. This eigenvalue repulsion suppresses thenon-perfectly conducting eigenvalues { T , T , . . . , T N } . Itshould be emphasized that all the influences of the per-fectly conducting channels are described by this factor.The purpose of this paper is to study how the per-fectly conducting channels affect the dimensionless con-ductance g in the diffusive regime. Particularly, we focuson the variance, var { g } ≡ h g i − h g i , which character-izes the magnitude of conductance fluctuations. We knowthat in the ordinary case of m = 0, the variance takesthe universal value 1 /
15 irrespective of the normalizedsystem length s ≡ L/l . This is called universal conduc-tance fluctuations. Does this universality hold even in thepresence of the perfectly conducting channels? To answerthis question, we obtain var { g } by using two approachesbased on the DMPK equation. First, we analytically cal-culate the variance as a function of s by using an N − ex-pansion approach. This is applicable to the case in which N ≫ max { m, } . We show that the variance is given byvar { g } = 1 /
15 irrespective of m in the diffusive regime of N ≫ s ≫
1. This indicates that in the large- N limit, theuniversality of conductance fluctuations hold even in thecase of m = 0. To study the case in which the ratio m/N is not very small, we numerically calculate var { g } by us-ing a classical Monte Carlo approach based on an approx-imate probability distribution for transmission eigenval-ues. This is applicable to the case of an arbitrary m aslong as s/ N ≪
1. We treat the cases of N = 6 and 20with m = 0 , , ,
3. We show that in the case of m = 0,the variance approximately takes a constant value nearlyequal to 1 /
15 for N & s &
1. However, deviation fromthis universal behavior arises when m = 0. We show thatfor m = 0, the variance does not take a constant value,but decreases with increasing s , and the correspondingdeviation becomes more pronounced with increasing m .We also show that the deviation in the case of N = 6 ismore noteworthy than that in the case of N = 20. Theseresults indicate that the universality of conductance fluc-tuations breaks down for m = 0 unless N is very large.In the next section, we analytically obtain var { g } byusing the N − expansion approach. In §
3, we numeri-cally calculate var { g } by using the classical Monte Carloapproach. The Monte Carlo results are compared withthose obtained in §
2. Section 4 is devoted to summary. N − Expansion Approach
We consider var { g } ≡ h g i−h g i in the diffusive regimeof N ≫ s ≫
1. The ensemble average of an arbitrary function F ( { λ a } ) is defined by h F i = Z ∞ d λ · · · d λ N F ( { λ a } ) P ( { λ a } ; s ) . (3)Let Γ be Γ ≡ N X a =1 T a = N X a =1
11 + λ a . (4)We can express var { g } in terms of h Γ p i with p = 1 and2. To obtain h Γ p i , we derive the evolution equation forit on the basis of the DMPK equation. Combining theDMPK equation with eq. (3), we obtain N ∂ h F i ∂s = * N X a =1 J ∂∂λ a (cid:26) λ a (1 + λ a ) J ∂F∂λ a (cid:27)+ = * N X a =1 (cid:26) λ a (1 + λ a ) ∂ F∂λ a + (1 + 2 λ a ) ∂F∂λ a + λ a (1 + λ a ) N X b =1( b = a ) λ a − λ b + mλ a ! ∂F∂λ a (cid:27)+ . (5)Replacing F by Γ p , we obtain N ∂ h Γ p i ∂s = − mp h Γ p i + p ( p − h Γ p − (Γ − Γ ) i − p h Γ p +1 i , (6)where Γ q = N X a =1 λ a ) q . (7)Note that eq. (6) is not closed since its right-hand sidecontains h Γ p − Γ i and h Γ p − Γ i . Even if eq. (6) is com-bined with differential equations for h Γ p Γ i and h Γ p Γ i ,they do not form a closed set of equations and we can-not obtain h Γ p i in a simple manner. To overcome thisdifficulty, we adapt the N − expansion approach pre-sented by Mello and Stone. This approach is adaptableif N ≫ max { m, } . We expand h Γ p i in a power series of N − , and obtain it up to order of N − p . To do so, wemust supplement eq. (6) by the following equations, N ∂ h Γ p Γ i ∂s = − m ( p + 2) h Γ p Γ i + 4 p h Γ p − (Γ − Γ ) i + p ( p − h Γ p − (cid:0) Γ − Γ Γ (cid:1) i + 2 h Γ p +2 i − ( p + 4) h Γ p +1 Γ i , (8) N ∂ h Γ p Γ i ∂s = − m ( p + 3) h Γ p Γ i + 6 p h Γ p − (Γ − Γ ) i + p ( p − h Γ p − (cid:0) Γ Γ − Γ (cid:1) i + 6 h Γ p +1 Γ i − ( p + 6) h Γ p +1 Γ i− h Γ p Γ i , (9) N ∂ h Γ p Γ i ∂s = − m ( p + 4) h Γ p Γ i + 8 h Γ p (Γ − Γ ) i . Phys. Soc. Jpn. Full Paper
Yositake
Takane and Katsunori
Wakabayashi + 8 p h Γ p − (Γ Γ − Γ Γ ) i + p ( p − h Γ p − (cid:0) Γ − Γ Γ (cid:1) i + 4 h Γ p +2 Γ i − ( p + 8) h Γ p +1 Γ i . (10)These equations can be derived from eq. (5). We employthe following expansions, h Γ p i = N p f p, ( s ) + N p − f p, ( s ) + N p − f p, ( s ) + · · · , (11) h Γ p Γ i = N p +1 j p +1 , ( s ) + · · · , (12) h Γ p Γ i = N p +1 k p +1 , ( s ) + · · · , (13) h Γ p Γ i = N p +2 l p +2 , ( s ) + · · · (14)with f p,n (0) = j p,n (0) = k p,n ( s ) = l p,n (0) = δ n, . (15)Substituting these expansions into eqs. (6) and (8)-(10),and equating the coefficients of the various powers of N ,we obtain a set of closed differential equations for f p, ( s ), f p, ( s ), f p, ( s ), j p, ( s ), k p, ( s ) and l p, ( s ). We solve theresulting equations under the initial conditions given ineq. (15). These procedures are outlined in Appendix. Wefinally obtain h Γ p i = N p (1 + s ) p − mpN p − s ) p +1 ( s + 2 s )+ m pN p − s ) p +2 (cid:0) (3 p − s + (12 p − s + 12 ps (cid:1) + pN p − s ) p +4 (cid:0) (3 p − s + (18 p − s + (45 p − s + (60 p − s + (45 p − s (cid:1) . (16)The variance of Γ, which is equal to var { g } , can be de-rived from this expression. Up to order of N , we obtainvar { Γ } = 115 s + 6 s + 15 s + 15 s (1 + s ) . (17)This results in var { g } = 115 (18)irrespective of m in the diffusive regime of N ≫ s ≫ { g } in the large- N limit.
3. Monte Carlo Approach
To obtain var { g } without the restriction of N ≫ m ,we employ a Monte Carlo approach based on an analyticexpression of the probability distribution for transmis-sion eigenvalues. Let us introduce a set of variables x a ,related to λ a by λ a = sinh x a . We analytically obtain asimple approximate expression of the probability distri-bution P ( { x a } ; s ) from the exact solution of the DMPKequation. The DMPK equation has been solved exactlyfor the ordinary case of m = 0. The exact solutionfor an arbitrary m has been obtained in ref. 20. In the notation which is convenient for our purpose, the exactprobability distribution is given by P ( { x a } ; s ) = const . N Y a,b =1( a>b ) (cid:0) sinh x a − sinh x b (cid:1) × N Y a =1 sinh 2 x a sinh m x a × det (cid:8) I b ( − sinh x a ) (cid:9) a,b =1 , ,...,N , (19)where det { A ab } a,b =1 , ,...,N denotes the determinant ofthe N × N matrix ˆ A and I b ( − sinh x a ) = Z ∞ d kk b − c m ( k )e − k N s × F m ( k, − sinh x a ) (20)with c m ( k ) = 14 π (cid:12)(cid:12) Γ (cid:0) m +1+i k (cid:1)(cid:12)(cid:12) Γ ( m + 1) | Γ(i k ) | , (21) F m ( k, − sinh x )= F (cid:18) m + 1 − i k , m + 1 + i k , m + 1; − sinh x (cid:19) . (22)When s/ N ≪
1, the dominant contribution to the in-tegration over k comes from the large- k region of k & p N/s ≫
1. In this region, we can approximate that c m ( k ) = 12 m +1 ( m !) k m +1 , (23) F m ( k, − sinh x )= F (cid:18) − i k , k , m + 1; − sinh x (cid:19) . (24)We express the asymptotic form of F m ( k, − sinh x ) interms of a Bessel function. We start with the followingexpression F (cid:18) − i k , k , m + 1; − sinh x (cid:19) = m ! (cid:18) cosh x sinh x (cid:19) m P − m i k − (cosh 2 x ) . (25)We replace the Legendre function by its asymptotic form P − m i k − (cosh 2 x ) ∼ (cid:18) k (cid:19) m + (cid:18) π sinh 2 x (cid:19) × cos (cid:16) kx − mπ − π (cid:17) (26)for large k . This expression can be derived fromeq. (8.723) of ref. 21. Combining eqs. (24)-(26) andthe asymptotic form of the Bessel function J m ( y ) ∼ (2 /πy ) / cos( y − mπ/ − π/
4) for large y , we arrive at F m ( k, − sinh x ) ∼ m ! (cid:18) k (cid:19) m (cid:18) cosh x sinh x (cid:19) m (cid:18) x sinh 2 x (cid:19) × J m ( kx ) . (27) J. Phys. Soc. Jpn.
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Yositake
Takane and Katsunori
Wakabayashi
Substituting eqs. (23) and (27) into eq. (20) and carryingout the k -integration, we obtain I b ( − sinh x a ) = ( b − m +2 m ! (cid:0) s N (cid:1) b + m (cid:18) x a cosh x a sinh x a (cid:19) m × (cid:18) x a sinh 2 x a (cid:19) e − Nx as L mb − (cid:18) N x a s (cid:19) , (28)where L mb − is the Laguerre polynomial. Substitutingeq. (28) into eq. (19) and using the following relationdet (cid:26) L mb − (cid:18) N x a s (cid:19)(cid:27) a,b =1 , ,...,N = const . N Y a,b =1( a>b ) (cid:0) x a − x b (cid:1) , (29)we obtain P ( { x a } ; s )= const . N Y a,b =1( a>b ) (cid:8)(cid:0) sinh x a − sinh x b (cid:1) (cid:0) x a − x b (cid:1)(cid:9) × N Y a =1 (cid:26) ( x a sinh 2 x a ) m + e − Nx as (cid:27) . (30)For the case of m = 0, the identical probability distri-bution has been obtained in ref. 19. It is convenient torewrite eq. (30) as P ( { x a } ; s ) = const . e − H ( { x a } ) , (31)where H ( { x a } ) = N X a =1 (cid:18) γx a − (cid:18) m + 12 (cid:19) ln | x a sinh 2 x a | (cid:19) − N X a,b =1( a>b ) (cid:16) ln (cid:12)(cid:12) sinh x a − sinh x b (cid:12)(cid:12) + ln (cid:12)(cid:12) x a − x b (cid:12)(cid:12)(cid:17) (32)with γ ≡ N/s . From eq. (31), we find that the averageof Γ p = ( P Na =1 / cosh x a ) p is expressed as h Γ p i = Z − Z ∞ d x · · · d x N Γ p e − H ( { x a } ) (33)with Z = Z ∞ d x · · · d x N e − H ( { x a } ) . (34)Strictly speaking, our approach is justified only when γ − is sufficiently small, since we have assumed s/ N ≪ γ − is notsmall. To see this, let us consider the large- γ − limit.In this limit, the variables { x a } become much greaterthan unity and are widely separated with each other, sothat we can assume 1 ≪ x ≪ x ≪ · · · ≪ x N . Underthis assumption, H ( { x a } ) can be approximated as H ( { x a } ) = N X a =1 (cid:0) γx a − (2 a − ηm ) x a (cid:1) (35) with η = 2. Equation (35) is identical to the correctexpression in the large- γ − limit if we substitute η =1. This implies that the probability distribution givenin eq. (31) is qualitatively reliable even for not small γ − although it overestimates the influence of perfectlyconducting channels.Note that we can interpret H ( { x a } ) as the Hamil-tonian function of N classical particles in one dimen-sion. This analogy allows us to adapt a Monte Carlo ap-proach to numerical calculations of h Γ p i . Using a simpleMetropolis algorithm,
22, 23 we compute var { g } for N = 6and 20 with m = 0 , , , γ − = s/N .The results are shown in Fig. 1, where γ − is restrictedto 1 ≥ γ − ≥ × Monte Carlo steps. Figure 1 shows that var{g} γ −1 (a) var{g} γ −1 (b) Fig. 1. The variance of g for (a) N = 6 and (b) N = 20 as afunction of γ − ≡ s/N . Circles, squares, triangles and diamondscorrespond to m = 0 , , var { g } depends on m . In the ordinary case of m = 0, weobserve from this figure that var { g } increases with in-creasing γ − for N − & γ − ≥ / γ − & N − . However, deviation from this universalbehavior arises when m = 0. For m = 0, we observe thatvar { g } does not take a constant value, but decreases withincreasing γ − for γ − & N − . The decrease of var { g } becomes more pronounced with increasing m . We also . Phys. Soc. Jpn. Full Paper
Yositake
Takane and Katsunori
Wakabayashi observe that the deviation in the case of N = 6 is morenoteworthy than that in the case of N = 20. We concludethat the universality of conductance fluctuations breaksdown for m = 0 unless N is very large. This conclusion isnot inconsistent with the result of §
2, because the N − expansion approach is justified only in the large- N limit.
4. Summary
We have studied the variance of the dimensionless con-ductance g in disordered wires with unitary symmetryin the diffusive regime. We have focused on the case inwhich the number of left-moving conducting channels is N + m , while that of right-moving ones is N . In thiscase, m left-moving channels become perfectly conduct-ing. First, we have analytically obtained the variance asa function of s = L/l by using an N − expansion ap-proach. We have shown that the variance is given byvar { g } = 1 /
15 irrespective of m when N ≫ max { m, } .This indicates that we cannot detect influences of theperfectly conducting channels in the large- N limit. Sec-ond, we have numerically calculated var { g } by usinga classical Monte Carlo approach to study the case inwhich the ratio m/N is not very small. We have shownthat var { g } approximately takes a constant value nearlyequal to 1 /
15 only in the ordinary case of m = 0, butdeviation from it appears when m ≥
1. We have alsoshown that the deviation becomes more pronounced withincreasing m , particularly for small N . We conclude thatthe universality of conductance fluctuations breaks downfor m = 0 unless N is very large. Acknowledgment
K. W. acknowledges the financial support by a Grant-in-Aid for Young Scientists (B) (No. 19710082) fromthe Ministry of Education, Culture, Sports, Science andTechnology, also by a Grand-in-Aid for Scientific Re-search (B) from the Japan Society for the Promotionof Science (No. 19310094).
Appendix: Derivation of h Γ p i We substitute eqs.(11)-(13) into eq. (6) and equate thecoefficient of N p + q ( q = 1 , , −
1) in the left-hand sidewith that in the right-hand side. We obtain f ′ p, ( s ) = − pf p +1 , ( s ) , (A · f ′ p, ( s ) = − mpf p, ( s ) − pf p +1 , ( s ) , (A · f ′ p, ( s ) = − mpf p, ( s ) + p ( p −
1) ( j p − , ( s ) − k p − , ( s )) − pf p +1 , ( s ) . (A · j ′ p, ( s ) = 2 f p +1 , ( s ) − ( p + 3) j p +1 , ( s ) , (A · k ′ p, ( s ) = 6 j p +1 , ( s ) − ( p + 5) k p +1 , ( s ) − l p +1 , ( s ) , (A · l ′ p, ( s ) = 4 j p +1 , ( s ) − ( p + 6) l p +1 , ( s ) . (A · h Γ p i by solving eq. (A · · · f p, ( s ) = 1(1 + s ) p . (A · · f ′ p, ( s ) + pf p +1 , ( s ) = − mp (1 + s ) p . (A · f p, ( s ) = pχ p ( s ) with χ p +1 ( s ) = (1 + s ) − χ p ( s ), the above equation is reduced to χ ′ p ( s ) + p + 11 + s χ p ( s ) = − m (1 + s ) p . (A · f p, (0) = pχ p (0) = 0, we ob-tain χ p ( s ) = − ( m/ s + 2 s ) / (1 + s ) p +1 . This resultsin f p, ( s ) = − mp s + 2 s (1 + s ) p +1 . (A · j p − , ( s ) and k p − , ( s ) to obtain f p, ( s ). Sub-stituting eq. (A ·
7) into eq. (A · j ′ p, ( s ) + ( p + 3) j p +1 , ( s ) = 2(1 + s ) p +1 . (A · j p, ( s ) satisfies j p +1 , ( s ) = (1 + s ) − j p, ( s ), eq. (A ·
11) is reduced to j ′ p, ( s ) + p + 31 + s j p, ( s ) = 2(1 + s ) p +1 . (A · ·
12) with j p, (0) = 1 is given by j p, ( s ) = 2 s + 6 s + 6 s + 33(1 + s ) p +3 . (A · l p, ( s ) which is necessary to obtain k p, ( s ).Substituting eq. (A ·
13) into eq. (A ·
6) and assuming that l p, ( s ) satisfies l p +1 , ( s ) = (1 + s ) − l p, ( s ), we obtain l ′ p, ( s ) + p + 61 + s l p, ( s )= 43(1 + s ) p +4 (cid:0) s + 6 s + 6 s + 3 (cid:1) . (A · ·
14) with l p, (0) = 1 is given by l p, ( s ) = 19(1 + s ) p +6 (cid:0) s + 24 s + 60 s + 84 s + 72 s + 36 s + 9 (cid:1) . (A · ·
13) and (A ·
15) into eq. (A ·
5) andassume that k p, ( s ) satisfies k p +1 , ( s ) = (1 + s ) − k p, ( s ).We then obtain the following differential equation k ′ p, ( s ) + p + 51 + s k p, ( s )= 13(1 + s ) p +7 (cid:0) s + 48 s + 120 s + 162 s + 126 s + 54 s + 9 (cid:1) . (A · ·
16) with k p, (0) = 1 is given by k p, ( s ) = 115(1 + s ) p +6 (cid:0) s + 48 s + 120 s + 165 s + 135 s + 60 s + 15 (cid:1) . (A · J. Phys. Soc. Jpn.
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Wakabayashi
We now turn to the evaluation of f p, ( s ). Substitutingeqs. (A · ·
13) and (A ·
17) into eq. (A · f ′ p, ( s ) + pf p +1 , ( s ) = m p s ) p +1 (cid:0) s + 2 s (cid:1) + p ( p − s ) p +5 (cid:0) s + 12 s + 30 s + 40 s + 30 s + 15 s (cid:1) . (A · f p, ( s ) as f p, ( s ) = m u p ( s ) + v p ( s ), weobtain the differential equations for u p ( s ) and v p ( s ) as u ′ p ( s ) + pu p +1 ( s ) = p s ) p +1 (cid:0) s + 2 s (cid:1) , (A · v ′ p ( s ) + pv p +1 ( s ) = p ( p − s ) p +5 (cid:0) s + 12 s + 30 s + 40 s + 30 s + 15 s (cid:1) . (A · ·
19) by assuming u p ( s ) = p µ p ( s ) + pν p ( s ) with µ p +1 ( s ) = (1 + s ) − µ p ( s ) and ν p +1 ( s ) =(1 + s ) − ν p ( s ). Equation (A ·
20) can also be solved byassuming that v p ( s ) = p ξ p ( s ) + pτ p ( s ) with ξ p +1 ( s ) =(1 + s ) − ξ p ( s ) and τ p +1 ( s ) = (1 + s ) − τ p ( s ). Combiningthe resulting u p ( s ) and v p ( s ), we obtain f p, ( s ) = m p s ) p +2 (cid:0) (3 p − s + (12 p − s + 12 ps (cid:1) + p s ) p +4 (cid:0) (3 p − s + (18 p − s + (45 p − s + (60 p − s + (45 p − s (cid:1) . (A ·
21) Substituting eqs. (A · ·
10) and (A ·
21) into eq. (11),we finally obtain eq. (16).
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