Fortschritte der Physik / Progress of Physics: Volume 32, Number 2 [Reprint 2022 ed.] 9783112656082

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 1984 Number 2 Board of Editors F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin F. Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay CONTENTS: V . G . DUBROVSKY, a n d B . G . KONOPELGHENKO

The General Form of Nonlinear Evolution Equations Integrable by Matrix Gelfand-Dikij Spectral Problem and Their Group-Theoretical and Hamiltonian Structures

25—60

S . CIECHANOWICZ, a n d Z . OZIEWICZ

Angular Correlation for Nuclear Muon Capture

61—88

AKADEMIE-VERLAG • BERLIN ISSN 0015-8208

Fortschr. Phys., Berlin 82 (1984) 2, 2 5 - 8 8

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text on and the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawing should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawing and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary uprigth typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vektors): wavy underlined twice Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): green underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upringth Greek letters (symbols of elementary particles): red and green underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o 0, p P, s S, u U, v V, w W, x X, y Y, z Z). I t will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, bj, Mil, Mij, Please differentiate between following symbols: a, a ; a, a , oo; a, d; c, C, c : ; e, I, S, e, k, K, x, x; x, X, x, X; 1; o, 0, a, 0; p, Q; U, U, U> V, V, V, 0 (h = 0,l,...,N—2) and N is arbitrary one, in the framework of AKNS-technique. This technique has been formulated in Refs. [11, 12] for the second order linear spectral problem dW/dx = XAW + P(x, t) W. Then AKNS-technique has been generalized to the case of linear spectral problem of the arbitrary matrix order [13—20], The AKNS-technique in the form which we will use in the present paper (generalized AKNS-technique) has been developed b y one of the authors (B.G.K.) [16—19], The advantage of AKNS-technique consists in the following: it allows 1) to find the general form of nonlinear equations connected with given spectral problem in simple and convenient form, 1

Fortschi-. Phys., Bd. 32, H. 2

26

V . G. D t j b b o v s k y

and B .

G. K o n o p e l c h e n k o ,

Nonlinear Evolution Equations

2) to calculate the infinitedimensional group of general Backlund transformations for these equations and 3) to investigate the Hamiltonian structure simultaneously of the whole class of the equations integrable by given spectral problem. The main result of the present paper is a construction of the infinitedimensional abelian group of transformations (BC-group) connected with the spectral problem (1.1). This group acts on the manifold of the scattering matrices {S(X±t)} for the problem (1.1) in the following simple linear manner: S{il, í) -> S'(K, t) = (B{A, í))" 1 S(X, t) G{jI, t) where B(X, t) and C(A, t) are arbitrary diag' >al matrices with elements Bk(X, t) which are arbitrary functions entire on A. The action: V —> V of this BC-group on the manifold of potentials {V(x, t)}, where V(x, t) is a column with N — 1 matrix components V0(x, I), Fj(a;, t), ..., Vjv-2(%, t), is given by the formula

*Eb¿a+,

4=0

t) ( J f k r - J¿kv) = 0

(i.2)

where Bk{A+, t) are arbitrary functions entire on A+ and A+, c£~k, Jik are certain matrix integro-differential operators which depend only on V and V. The explicit form of these operators is presented in sections 3 and 4. The infinitedimensional abelian group of transformations (1.2) plays a fundamental role in the analysis of nonlinear systems connected with spectral problem (1.1) and their group-theoretical properties. As we shall see, the nonlinear evolution equations integrable by (1.1) are the infinitesimal form of transformations (1.2) generated by time displacement. The general form of the integrable equations is ^ where ük(L+,

^ ot

- 5 W k=i

+

> t)SftV

= 0

t) are arbitrary functions meromorphic on L+,

(1.3) def and L+ = A+{ V =

V),

j?k^Ljfk(v = V) - j?k(V' = V). The infinitedimensional group of transformations (1.2) is the group of general Backlund transformations for the equations (1.3). At dBJdt = 0 (k = 0, 1, ..., JV — 1) the transformations (1.2) are auto Backlund transformations for the equations (1.3): they convert the solutions of definite equation of the form (1.3) into the solutions of the same equation. If 8Bk/dt =(= 0 then the transformations (1.2) are generalized Backlund transformations for the class of equations (1.3): such transformations convert the solutions of given equation (1.3) into the solutions of other equations of the form (1.3). BC-group of transformations (1.2) contains also the infinitedimensional abelian symmetry group of the equations (1.3) as a subgroup. In the infinitesimal form these symmetry transformations are ( 7 —> F ' = V + 6V) dV(x, t) = 5 W ) 4=1

&tV

(1-4)

where fk(L+) are arbitrary entire functions. Symmetry transformations (1.4) are connected with the infinite set of integrals of motion of the equations (1.3). Symmetry group (1.4) as well the group of auto Backlund-transformations is a universal one, i.e. any equation of the form (1.3) (with any functions Ok(L+, t), is invariant under transformations (1.4). We consider also the Hamiltonian structure of the equations (1.3). Hamiltonian character of the equations connected with problem (1.1) was proved in the first papers [2],

27

Fortschr. Phys. 82 (1984) 2

Here we show with the use of AKNS-technique that the infinite family of Hamiltonian structures is connected with the equations (1.3), namely the following infinite family of Poisson brackets {,}„ (n = 0, ± 1 , ¿ 2 , ...): + 00 d x i v

— oo

\ôW^

L + ) n J

w)

(L5)

where J is certain matrix differential operator (Gelfand-Dikij operator). Bracket {,}0 is a well-known Gelfand-Dikij bracket [2]. Bracket {,}-, corresponds to second Hamiltonian structure which was considered in Refs. [4, 21, 22, 8, 9]. The paper is organized as follows. In the second section we consider the direct scattering problem for (1.1) which is rewritten in Frobenius form and obtain some important relations. In section 3 the recursion operators which play a main role in all our constructions are calculated. The infinitedimensional group of transformations (1.2) is constructed in section 4. The general form of the integrable equations is found in section 5. In section 6 the general group-theoretical properties (namely, group of Backlund transformations and symmetry group) of the equations (1.3) are discussed. Hamiltonian structure of the equations (1.3) is considered in section 7. In section 8 the explicit forms of recursion operators, integrable equations and Backlund transformations are presented in the cases N = 2, 3,- 4. In the conclusion we briefly consider some other approaches to Gelfand-Dikij problem (1.1).

II. Some preliminary relations

First of all let us rewrite spectral problem (1.1) in the matrix form (in the well-known Frobenius form (see e.g. [23])) dW ox

P{x, t) W

(2.1)

where 1 0 0 1 W =

A = 0 UAT1

0 0

0 0 (2.2)

P =

0

0

0

0

0

0

-F 0 , — F i , . . . , and 1 is identical matrix M X M. The elements of column W and matrix elements of A and P are matrices M X M. The Frobenius form (2.1) of Gelfand-Dikij spectral problem (1.1) is more convenient for our purpose. We will assume that Vk(x, t) —> 0 at \x\ —> oo so fast that all integrals which will appear in our calculations exist.

V. G. D u b r o v s k y and B. G. K o n o p e l c h e n k o , Nonlinear Evolution Equations

28

Now we proceed to construction of transformations (1.2.). Let us introduce, according to standart procedure (see e.g. [24]), the fundamental matrices-solutions F+(x, t, X) and F~(x, t, X) of the problem (2.1) given by their asymptotic behaviour t, X )

)

A),

E ( x ,

U)

A)

(2.3)

where E ( x , A) is fundamental matrix-solution of matrix equation d E / d x = A E . This system of equations have as it is well known (see e.g. [23]). the infinite set of fundamental solutions. W e will consider the solution of this system, i.e. the asymptotic E of problem (2.1), of the form E ( x , X)

=

exp

2>(X)

where J is diagonal matrix: and

q

exp

=

(2.4)

( A x )

A

=

i k

,

Here and below

( 2 n i / N ) .

d

3>ik

=

(l/|/JV) ( V 1 ) 4 - 1

is Kronecker symbol

i k

\d

1

i k

( i , k =

=

|

1, ... {

N ) , ) .

[ 0, i =F «/ \ (i = 1, ..., N) are eigenvalues of matrix A, by definition A > 0

Let us note that and A = In standard manner we introduce the scattering matrix S(X, t): +

F

( x ,

A)

t,

=

F ' ( x ,

t , X)

(2.5)

S ( X , t)

or S ( X , t ) — ( F ~ ( x , t , A)) -1 F ( x , t , X ) . Transition from one choice of the asymptotic of the problem (1.1) to the other (E1 —> E2 = EXK where K is some nondegenerate matrix) leads only to a trivial redefinition of the scattering matrix ($! S = K ~ S K ) . ) Let us take now two arbitrary potentials P(x, t) and P'(x, t) [P(x, t) ^ ^ ^ -> 0, P'(x, t) ij^pp > 0) and two corresponding solutions 'F(x, t, X) and W(x, t. ?J) of the problem (2.1). With the use of the equation (2.1) and the equation for W^1 (dW^/dx = — W~\A + P)) one can to show that +

1

2

W ' ( x , t, X )

=

-

-

W ( x , t,

W ( x , t , X)

1

1

A)

J d y V - i ( y ,

t , X)

(.P'(y, t) -

P ( y , t))

W ' ( y , t, X ) .

(2.6)

X

Putting W = F+

in (2.6) and going to the limit x

—oo we obtain

+ 00 S ' ( X , t)

-

S ( X

t) =

- S ( X

t) f

d x F

— oo

+

- \ x ,

t , X)

( P ' ( x , t) -

P ( x , t)) F

+

' ( x , t,

A).

(2.7)

Formula (2.7) which relates a change of potential P(x, t) to a change of the scattering matrix S(K, t) plays a fundamental role in our further constructions. The mapping P(x, t) —> 8(X t) given by the spectral problem (2.1) establish a correspondence between the transformations P P' on the manifold of potentials {P(x, t), P(x, t) > 0} and the transformations S S' on the manifold of the scattering One of the simplest asymptotic of (2.1) is Ed = exp Ax. I t connected to asymptotic E by formula E j > =

E S I -

1

.

29

Fortschr. Phys. 32 (1984) 2

matrices { S'{X, t) = B~\X, t) S(X, t) C(X, t)

where B(X, t) and C(X, t) are arbitrary block-diagonal matrices (i.e. Bik = bi(X, t) k = = H «t-l-m I d{X*>) jw=0 m=0 I

0 ••• 0\ (3.16)

A 0 . . . u0

1 0 ••• oy

Using the simple properties of matrices A and R one can show that '

fk =

0

... 0

0 •• 0

0

0

... 0

0 •• 0

1

0

0 •• 0

—led

1

... 0 ... 0

—kd ... 0

0 •• 0

0

*

*

*

-kd

0 •• 0

1

(k =

1, 2, . . . , N )

(3.17)

0 •• 0

k and (s*)l 1 = ck'(s*)li = 0,

I = 1, 2 , Z = k + 2 , . .., N;

_ n jf+i-i _ (sN)M — has many zero elements and have in mind further Hamiltonian treatment

(

++

+-fc

\T

®x-in) • T i l e relation

(3.28)

++

is now of the form

=

(3.30)

Matrices & and «# are of the order N — 1 and equal {&)u = S (fmhb M

(^)ur=

oVm) -

(3.31)

-¿{Sm)u(m=0

where (R^-d)^

=

(Rx-xh,

-°

*)

where summation in ( 3 . 3 4 ) is performed over all possible dividing of integers from k to i into pairs. For example, (£);i1 = - m a ma

A h

= - m a ^(^r,

1

ma

= - m a

+ m ^ ^ m a

y^ma, ^(^n1-

Fortschr. Phys. 32 (1984) 2

35

One can verify that matrix operator i f - 1 has no nontrivial kernel. Thus, from (3.30) we have A%w{x, t, X) = XNxw{x,

t, X) '

(3.35)

where A =

(3.36)

The operator A is just a recursion operator which we are interested in. I t plays-a fundamental role in AKNS-technique. The explicit form of A can be found by formulas (3.36), (3.34), (3.31), (3.32), (3.15) and (3.16). I n our further constructions we will need the operator A+ adjoint to operator A with def N-l +oo respect to bilinear form ((x'x)) = E f dx tr (/¿'(e) Xi(x))- The operator A+ is calculated ¿=1 -co by standart rule ( « x ' A x » = « ( A + X ' ) x))) and it is A+ = # + ( # + ) " 1 . Operators

and

(3.37)

are

N )ik — — E Vm(sm+)i,k+1 m=0

r+

N — 2J li+Vm{sm+)xk> m=0

= E Vm(rm+)Lk+1 - J i f t + l 1 , ro=0

(3.38)

- (t, k = 1, 2, ..., N - 1)

where

- 4

* m=0

.

1, 2,...,N).

(3.39)

I n the operators l k + (3.39) and in all adjoint operators (denoted by symbol + ) which def

will further appear (S"1/) ix) = +

+

The operators rk and sk to rk and sk. Namely 0>+k

=

¿Nr +

f

x

dyf(y).

— CO

are block matrices of the order N and defined analogously s+

(3.40)

where /8

XN — (• VQ')

0 - 0 0

— (• V\) ( 3.41

»

1

- (• n - 2 ) 8

The explicit forms of sk+ and rk+ are calculated by formulas sk+ = 0>+k(XN = 0)

(3.42)

36

G. D u b r o v s k y and

V.

B.

G. K o n o p e l c h e n k o , Nonlinear Evolution Equations

and /0-0 d0>+k / 0 - 0 = ZJ k—1—w d(X») XN = N — m=0

rk+

\ o - - o

1 0 oy

From (3.38) it follows that is upper-triangular matrix: (&+)nt = 0, i > k and (#+)ii = N8 (i = 1, 2, ..., N — 1). Matrix elements of the inverse matrix ( # + ) " 1 are i>k, = 0, = (1 /N) 3 - S

(3.43)

(&+)Tk = L

-

+

where summation is performed as well as in (3.34) over all possible dividing of integers from i to k into pairs. Formulas (3.37)—(3.43) give us somewhat cumbersome but direct procedure for calculation of the operator A+. Explicit form of the operator A+ for some concrete examples (N = 2, 3, 4) will be given in section 8. IV. Construction of the transformations (1.2) ++

In the previous section it was shown that matrix elements of can be expressed through the quantity (formulas (3.2), (3.6), (3.25)). Let us transform therefore the equality (2.15) into the form which contains only independent quantity Using the properties of bilinear form ( ) and equalities (3.6), (3.25) we obtain (AkP'&*>

- Ak$WP)

PAk&*l)

= (P'&w&ft

-

= ((P'0>wM

- PAkM)

(4.1)

Let us single out the explicit dependence on XN in the operators 3P(k)M and AkM. virtue of (3.6), (3.13), (3.14), (3.17) and (3.27) we have = X»$ ( k ) + ¿ F w , k

N

r N k

A M = X {R ) ~

In (4.2)

k

(4.3)

+ RM

where 9

w

-=Erk-

&(ty

m = 0

m

( - oF„_m),

=£sk_m((M.)o

(4.4)

F,_m),

(4.5)

and matrix operators rk and sk are given by formulas (3.15), (3.16). Substitution of (4.2) and (4.3) into (4.1) gives ++

(AkP'0w

++

— Aki>{-*')P)

= ((XNP"$lk)

+ P'^w

- XNP{RT)N~k

- PRkM)

(4.6)

Fortsehr. Phys. 32 (1984) 2

37

If one introduce N++— 1-componentcolumn V(x, t) = = (V0{x, t), V±(x, t),..., • and proceed from to then from (4.6) one obtain ++

(AkP'0w

++

Ak0wP)

—

= where

VA--_2(x, t))T

-

F'#

w

+ XNVKW

+ VNW)

*»

(4.7)

k C#(k))iV = 27 {rk-m)il (• o Vn-m), m=0 = 2 7 (sk-M)il (• ° VN-M) + 2 7 (SA: -m

m=0

Fiv-mj

m=0

(4-8)

(K(k))il = di.l+N-k^ > (#(«)«

=

(», lc,l = 1,2

tf

Further in virtue of (3.35) for arbitrary entire function Bk(lN) t, A) = Bk(A)

Bk(XN)

-

1). we have (4.9)

t, X).

With the use of the equalities (4.7) and (4.9) one can rewrite (2.15) in the form

((

-

+ VK(k)A

+ VN(k)) Bk(A, t) ¿*\k) j J = 0. (4.10)

The equality (4.10) is the form of the equality (2.15) in which the explicit dependence on XN is eliminated. This elimination become possible due to existence of the recursion operators. At last, the equality (4.10) is equivalent to the following one ^*XX)ZBk{A\

t) ({A+0{k) + # ( + *>) V - (A+K(+k) +

7) J J = 0 (4.11)

where A+,

K{k), N{'k) are operators adjoint to operators, A, K{k), +-00 Nik) with respect to bilinear form ({%'%)) = J dx tr [%'T(x) %{x)). Operator A+ is given by formulas (3.37)—(3.43) and k {^tk))il = 27 Fy-m(*i-m)ii' m=0

m »

=

_

m=0

)il

+27 m=0

( 4 -12)

^ij-jv+t'i >

(Ntk))u = du+k1 + d

L N

J f ,

(i, k, I = 1, 2 , . . . , N -

1)

where operators rk+, sk+ and lk+ are calculated by formulas (3.42), (3.39).

38

V. G. DUBROVSKY and B. G. KONOPELCHENKO, Nonlinear Evolution Equations

The equality (441) is just the relation between V, V and y}*) under the transformations of the scattering matrix of the form (2.8). The equality (4.11) is satisfied if t) ( J f k V ' -

N£Bk(A+,

k=o

J?kV) = 0

(4.13)

where

+

^ k= def

~

, -

.

(

4 1 4

)

If quantities y}*\x, t, A)j form a complete set (as in the case N = 2) then the equation (4.13) is also necessary condition of fulfilment of (4.11). Thus, we find the transformations of potential V{x, t) —> V'(x, t) which correspond to the transformations (2.8). These transformations are given by the relation (4.13) where Bk(A+, t) are arbitrary entire functions on A+. I t is important that the relation (4.13) contains only the potential V and transformed potential V. We restricted overselves by the transformations law of scattering matrix of the form (2.8) just in order that it will be possible to convert the transformation law of scattering matrix into the explicit transformation law of potential which contains only V and V. It is remarkable that these "restricted" transformations (2.8), (4.13) are quite wide and, as we shall see, contain all the transformations typical for equations integrable by the spectral problem (2.1) and these integrable equations themselves. It easy to see that the transformations (2.8), (4.13) form a group. Indeed, letwe have two transformations o f t h e type (2.8), (4.13): S - > S 1 = B ^ S C » -> = B^SJJ^. Since matrices B1i_B2> C 1; C2 are diagonal one, then S —>S2 = B2~1S1C2 = B2~1B1~1SC1C2 — {B2B1y~L S{G2G-¡), i.e. the product of transformations of the type (2.8) is the transformation of the same type. In virtue of commutativity of diagonal matrices the group of transformations (2.8), (4.13) is abelian group, more exactly, abelian infinitedimensional group. The transformations from this group are indicated by N functions Bk(?.N, t) (k = 0, Í , N — 1) entire on Xs, which can be arbitrary one. The group of transformations (2.8), (4.13) can be considered as infinite-parametrical Lie group with parameters bkn(t) which are coefficients of the expansion of In Bk(XN, t) on F rtef

=

\

00

Z &*,( S'{)I, t) = z; exp f dsQk{XN, s)J (A-*)* k=0 \i N-1 / (' \ X S(X, t) £ exp - / dsQ^, 8) • A1 = S(X, t'). (5.2) ;=o \ t I The corresponding transformation of potential is V(x, t) formula J exp ( - f dsQk{A\ «)) {rkV(t') k=0 \ i /

- JíkV(t))

V(x, t') and is given by

= 0

(5.3)

where in the operators A+, X'k and J4k one must put V'(x, t) = V(x, s). For the first time the transformations of the type (5.3) were considered in Ref. [27], (for problem (1.1) at N = 2, M = 1). See also Refs. [28, 29, 16, 17, 19]. At fixed functions Qk(XN, t) one-parameter group of the transformations (5.3) determine a flow Yq \ V(x, t) —> V(x, t'), in other words, an evolution system. This evolution system can be also described by certain nonlinear evolution equation. Indeed, let is consider the infinites imal displacement in time: t -> í' = t + « where s 0. In this case

Bk{lN, t) = dka - eQk{XN, t),

(k =

Substituting (5.4) into'(5.3), taking into account that (/l+)° = keeping the terms of the first order on e we obtain of

- £ W

,

t)

kV

(5.4)

0,l,...,N-l).

= J í ü = *\NM) and

= 0

def def where L+ = A+\y!=v, ££k = CtiTk\y=v — ^k\v'=v- Operators L+, are calculated by formulas (3.37)-(3.43), (4.14), (4.12) at V = V.

(5.5) JÍ¿\V'=v

40

V. G. D u b r o v s k y and B. G. K o n o p e l c h e n k o , Nonlinear Evolution Equations

I t is not difficult to show that operators 2L+ "¿BfcS)

can be also represented in the form dk~m + 2 k ¿ ^ { R í n - d T ' 1

m=0

m=1

(1-V).

m=0 - 1 (.-DM '¿\(8k~m~1 F A '_ m _i) • where (^(a-d)« = ¿+1(¿, A; = 1, . . N — 1) and

if =

/0

0 - 0

1

0 ••• 0

/,+

0

0 — 0

/^„a

^0

0 ••• 1

'

In particular, for arbitrary jV J ? ! = 1 (jV _ 1)M 9. For the scattering matrix under transformation (5.4) S{l,t') and from (5.2) we obtain linear evolution equation ^

(5.6)

=

i) +

s8S(?.,t)/dt

=

(5.7)

where Y(A, t)

4=1

f) J * .

Thus the consideration of the infinitesimal transformation (5.3) leads to nonlinear evolution equation (5.5). So evolution equation (5.5) give the flow Yq : V(x, t) V(x, t') in the infinitesimal form. The relation (5.3) which does not contain the derivative dV/dt is an "integrated" form of the evolution equation (5.5). Class of the equations (5.5) is characterized by integers N, M, by recursion operator L+ and N — 1 arbitrary functions Ü1(XN, t), ..., í3JV_1(AJsr, í) entire on XN. The choice of concrete N, M and functions Ot(XN, t) give us concrete equation of the form (5.5). A few examples will be given in section 8. The nonlinear evolution partial differential equations (5.5) are just the equations integrable by the inverse scattering transform method with the help of spectral problem (1.1). Using the equations of the inverse scattering problem for (1.1) (Gelfand-LevitanMarchenko type equations) one can find, in principle, a broad class of exact solutions of the equations (5.5). At N = 2 and arbitrary M the inverse scattering problem was considered in Ref. [30], At N = 3, M = 1 it. was discussed in Refs. [31, 32]. See also Refs. [24, 45]. In the form (5.5) one can represent a broader class in the integrable equations. These are the equations ¿Mi\ m=1

h,...,

tp) w&'ti'-'tp)

-N£ük{L+, i¡ = 1

,...,

tl

y xkv

= o

(5.8)

where jm(p, ij, ..., tp) and Q^fx, tlt ..., tp) are arbitrary functions entire on a. In this case the^scattering matrix S()., tx, ..., tp) satisfies an equation ÍUX-,, ro=1

vim

4=1

h, ...,tp)

A",

i1;

At N = 2 the equations of the form (5.8) have been considered in Ref. [29].

...,tp)

41

Fortschr. Phys. 32 (1984) 2

In the case p = 1 the equations (5.8) are equivalent to the equations (5.5) with meromorphic functions Qk{fi,t) = Qk{fi, í)//i(//, t). In the present paper we will consider the equations (5.8) (or (5.5)) only with one time-type variable t. Let us turn the attention to the fact t h a t in virtue of (5.7) at any functions F ' = V + ÔV symmetry transformations are ôV(x,t)

^¿h(L+)^kV. k=0

(6.2)

The infinitedimensional abelian symmetry group can be also considered as infiniteparameter abelian Lie group. Indeed, let us expand entire functions fk{L+) in the power oo series: fk{L+) = JJ fkn(L+)n. As a result symmetry transformation (6.2) is rewritten as »=o JV-l oo dv = £ Zhn(L+r&kV 4=1 «=0

i.e. as a superposition of the infinite number of one-parameter symmetry transformations = /,.(£+)" J2?*7

(6.3)

where expansion coefficients fkn{— oo < / * „ < + oo) play a role of the transformation parameters. Let us note that symmetry transformations (6.3) are connected with integrals of motion O n (i:) . If one omit the transformation parameters fk„ from ôikt„) V then the corresponding quantities ôikjn) V(à{kiTl)V = fkrÂ(k.n)V) are related each to other by a simple formula k.n+l)V = L+ô{k,a)V,

(» = 1 , 2 , . . . ) .

(6.4)

At N = 2, M = 1 an analogous properties of the operator L+ was firstly noted by LENABT (see Ref. [35]).

Let us emphasize now that B-group of Backlund-transformations and symmetry group are universal one: B-group and symmetry group are those for any equation of the form (5.5). The universality of the symmetry group is of the same nature as the universality of the integrals of motion. Let us, lastly, consider the transformations (4.13) with time-dependent functions B(X, t) = 0(2., t). In virtue of (2.8) the evolution law of the scattering matrix (5.7) conserve its form under such transformations too, i.e. under S S' = B~\t) S(t) B(t): ^ where

=

_ Y'{X, t) = 7(A, t) - (d/dt) In B(X, t).

(6.5) (6.6)

I t follows from (6.5) that the transformations (2.8), (4.13) with time-dependent functions Bk(X, t) convert definite equation of the form (5.5) (with given Y(X, £)) into other equations (with Y'(X, t) given by (6.6)) of the form (5.5). We refer such transformations, follows Refs. [27—29], as generalized Backhand transformations. These transformations connect the solutions of different equations from the class (5.5). It follows from (6.6) that generalized Backlund transformations act in a transitive manner on the whole family of the equations (5.5) (at fixed N and M).

43

Fortschr. Phys. 32 (1984) 2

Thus, we see that BC-group of the transformation (2.8), (4.13) contains complete information on the general group-theoretical properties of the equations integrable by (1.1) and these integrable equations themselves. VII. Hamiltonian structure of the integrable equations

Let us consider the equations (5.5) with arbitrary entire functions oo t ) = E aWO n—0

where

( L

+

)

Q

k

{ L

+

, t),

i.e.

n

(7-1)

are arbitrary functions. We will use the method developed in Refs. for proof of the Hamiltonian character of these equations. First of all let us note that from the relation (2.7) we have cokn (t)

[12,

16—19]

— oo (see formula (7.10)) we obtain from (7.14) the following relation U A) =

&

+

= -oo)

(7.17)

where # = - N d). Finally, multiplying left-and right-hand sides of (7.17) by (Ak)aa/SM and summing over oc we have t, X) = Lnw

+

=

-oo)

{h = 1 , 2 , . . n -

i)

(7.18)

where 77 ( 4 ) are defined by (7.4) and

l ^

p r

The equations (7.18) are just the equations for / J ® which we are needed. Inhomogeneous terms '¡STI(K)(x = — oo) in (7.18) are easily calculated. Really, with the use of

46

V. G. Dubrovsky and B. G. Konopelchenko, Nonlinear Evolution Equations

(7.10) we have

[n(-

k )

= — oo)), =

(x

d

^

k + l f

and therefore

= -oo)), =

1

0,

l +

k




N

(7.19) where summation is performed as in formula (7.16). The shortcoming of the equations (7.18) is that they contain the operator L instead of operator L+ which define the nonlinear part of the equations (5.5). However, it is valid the following important The relation

T h e o r e m :

IL = L+I

(7.20)

holds where I is matrix (N — 1) X (N — 1) differential operator. Its matrix elements are N + l - i - k

i

I

*

=

2 7

=

i k

{ C i T U V M + ^ d * .

1=0

-

c f r ^ i - d y

( . v

l + i + k

^ ) } ,

(7.21)

0 ,

i

+

k

^

N

+

1

( i , k = i , 2 , . . . , N ~ l )

where G = {ll)/(k\(l jfe)!) This theorem is proved by direct calculations with the use of the relations (7.15), (7.16), (7.9), (3.15), (3.16), (3.21), (3.37)—(3.39), (3.42), (3.43). t k

One can show that the operator I is invertable and I + = —7. Let us note that the operators

and ( # + ) _ 1 7 are pure differential one and

)

=

—(#+)-1/.

Therefore the operator

= L + I is pure differential one too. Let us also emphasize that in virtue of (7.20) the operators L + and L can be represented as followsl + = r j - , L = 1

The relation (7.20) allow us to connect the terms Let us substitute the asymptotic expansion of

(L+)

n

on

kV

with the quantities i.e.

II{k).

oo n

\ x ,

( k

t,

x)

=

27

x - «

N

n j

\ x ,

k

t)

n=0

into the equations (7.18). As a result we obtain the following system of relations I T o

W

( x ,

n ^

k

n

W ( x ,

n

t)

\ x , t )

t)

= #/7(® = - o o ) ,

(7.22)

=

L i r

(7.23)

=

L n < »

0


.]

+

where operator L+ is calculated b y formulas (8.8) a t F 0 ' = F 0 , 7 / = F j . W e present here t h e explicit forms of m a t r i x elements of L+ in t h e scalar case M = 1 (/ x + = —1/3 X ( S 2 + V1), U+ = - 0 ) : ^

S3 +

= J

V0 + J

= -§-^ + T +

ii 2 = - |

0 4

F, 8 + I

+ T

+ |

3 F

( a F l )

(0F O ) a - i ,

0-1

o - I " F,

'

(8.11)

-

i^F,-) - |

Fj

2

+ j (S2F0) 0"1 - •§• F1(071) 0_1 - | a2((0Fx) 0-i.), ¿2+2 =

+

-J

d

(V

v

) -

± 0 ( ( 0 F , ) 0-i.) + ±

0(7o0-i.).

I n scalar case l i = 1 t h e equation (8.10) with linear f u n c t i o n s +

Q2 = TT>20 + (O21L

0F - ^ =

WNL+,

is t h e f o l l o w i n g s y s t e m of t w o e q u a t i o n s

Wlo

0Fo+Wll

(

2 1 - - 0 « F X + -0"FO

+ f 9(TV) - j

V, 0 3 F j - |

+ «a,o{s 2 F 0 - -g-0 3 F 1 -

= w1(J +

I

2 +-0(7,070)

(07.) (02Ft) -

± Fi 2 0 F x J

7 , 07 X J + a>21 j - ±

0SFO

FXF0) - J 8{ F 0 0 2 Fj) + ± 0( F 0 0 7 0 ) - J 0( F 0 F X 2 I (8.12)

^

= «wio SF X + + cu2o{207o -

J - l ^ F , + j8W

0

+ ±0(70 7 0 - 1 - 0 ( 7 ,

0»F,} + co21 { — J - 0 « F 1 -

|"0(F¿07x) -

j

Vi 2 0 7 , +

± 0 ^

10(Fo2)J.

8WJ

07,)|

52

V . G. DUBROVSKY and B. G. KONOPELCHENKO, Nonlinear Evolution Equations

The system of equations (8.12) contains some well-known equations as particular cases. At w1(j = cou = co2i = 0 system (8.12) is

(8.13) 8V

I t is easy to see that the system (8.13) is equivalent to the equation

that is Boussinesq equation at 10 = con = oj20 = 0, i.e. the system dV

/ = %

dV

1

5

5

5

— g S ' F . - g T O - g

-j^VoVS)), / 1 5 = « « ( - j S6 F, - J

F,

a2

W ^ )

FO -

5 ¥

+ J 8(V0 dv0)

5 0( F 0 a 7,) - -

(8.14) V,' d V,

Under the reduction F 0 = 1/2 8V1 and at OJ21 = —9 the system (8.14) is reduced to the equation =

35 F i

+

5 F i 33Fi +

^

e 7 i

02 Fi

+

5Fl

2gFi

which was considered earlier in Refs. [8, 31]. The system (8.14) admits also a reduction F 0 = 0. In this case the system (8.14) for a>2i = — 9 is the equation

ot

=

+

57

S3Fi +

5 S F i 32Fi +

5Fi2

0 7

I t was considered previously b y SAWADA and KOTERA [45] and DODD and GIBBON [46].

Let us note that the third order Gelfand-Dikij spectral problem (1.1) has been considered also b y FORDY and GIBBONS [S] and b y BRUSCHI and RAGNISCO

[47].

Fortschr. Phys. 82 (1984) 2

53

The case N — 4 O p e r a t o r s /¿ + a r e 4+ =

- j

4+ =

-0« -

4+ =

— 1 0 + j

For operator

8s ì

j

r2 a -

F2 + Ì

ì

v1 + j

((0-1.) F i ' -

((0- 1 -) F 2 ' -

= #+(#+)-i

Foia- 1 -)),

((0-1.) f 0 ' F1(0-1-)),

(8.15)

F2(0-I.)).

h a v e («' = 1, 2, 3)

w e

= 4 / ^ / 3 0 + 4(0.) F/2+ 0 -

4 ( 4 + 0 - ) F j _ ! + 2 / ; + 0 ( . F 2 ' ) + 24+(0.) F 2 '

+ 4(0.) 0F v„+(Z-

1, A)%

(3.1)

The theory for this process has been developed by BUKHVOSTOV and POPOV [15], [IS], OZIEWICZ and PIKULSKI [16] and [5]. Recently, the paper on the angular correlation in this reaction has been published by DEVANATHAN and SUBRAMANIAN [17], The measurement of the gamma-neutrino correlations in ¡¿-capture b y 2 8 Si has been done b y MILLER et al. [75]. The detailed kinematical analysis of the results of this experiment was performed in ref. [15] and the role of the nuclear structure was inv e s t i g a t e d i n r e f . [20] a n d b y PAKTHASABATHY a n d SRIDHAR i n r e f .

[21],

Here we describe the angular correlations in terms of the helicity amplitudes for the decay of the muonic atom as describe'd in the previous section.

Fortschr. Phys. 32 (1984) 2

65

The formula giving the angular correlations in the process (3.1) is defined as

where P^ k and v are the muon rest polarization on if-orbit, the gamma radiation linear momentum and the neutrino momentum respectively, and,

Qt, = Jdk jdv J ^

gfi.

(3.3)

Qjj is the final state density matrix, Sjj

= HyTeiT+Hy+,

(3.4)

and Qi is the initial state density matrix,

Qi = pei+ + (i-p)er-

(3-5)

The gamma-neutrino (y-v) angular correlation is defined as

W(k, v) = i - J dPpWiPp, k, v).

(3.6)

With this, we have

. W(k,v) = £ (-iy(W>+h) BsvasPs(k • v). ,« the Bsri term describes the nuclear y-radiation

(3.7)

Here, process and as the weak vertex. The explicite dependence of the correlation formula on the neutrino helicity has been extracted. This amounts to a convention that the helicity amplitudes in the correlation coefficients as must be calculated for the left-handed neutrino (for the details see ref. [5]). This remark is relevant also for all the remaining formulas. Then, as is given by

Aas = pA+as+ + (1 - p) A~as-,

(3.8)

where,

The correlation coefficients as depend on time in the same manner as the probability coefficient A. Therefore, as may be expressed as a, = of + (asstat -

a/) e~Rt,

(3.10)

where, - .ut

Ji

+

1 A +

a

+ I

~ 2Jf + 1 A

Ji

-

( 3 l l )

2J{ +1 A

Next, we describe the y-radiation term

B.1 = [B,i(L, L)

A~ a

Bsrj,

+ 2 ridR^L,

L

{3A1)

and it looks like + 1) +

Ô*RS(L

+ 1 ,L+

1)] (1 +

(3.12)

where R^L, L') is given by

RAL, L') = sUf

o

)

W{iLJfs

> JfL"> '

(3-13)

66

S.

CIECHANOWICZ

and

Z . OZIEWICZ,

Angular Correlation

The circular polarization of the nuclear deexcitation is denoted by rj = ± 1 , for the right and left polarized radiation, respectively. The mixing ratio 0

s= 1

fl

s= 2

1

1 2

2 y2

-ll-

2 y2 + -•

10 V2 i

^ + ( l - 2 > ) ^ ,

(3.16)

where A± characterize the polarization of the muon in the HF states, then WIP,,, v) = 1 - 2h• Jf with change of parities, and inversely, by substitution ML++(-l)*EL,

and

A

L

^{-\fV

L

.

(5.2)

•The quantities A±, and also as±, are related by eqs. (2.9) and (3.11), respectively, therefore we have only calculated A~ and as~, which describe [¿-capture by nuclei with positive dipole magnetic moment pt. For convenience, we add to each of the tables examples of the reactions which are described by the particular formulae. In these examples we restrict ourselves to the capture from ground state of light nuclei, however, the corresponding formulae are strictly valid for both light and heavy nuclei. To summarize, the following look-up procedure for using the tables is recommended: Rule 1. If the change of parities of the nuclear levels in the transition under consideration is opposite to that shown in the tables, the substitution (5.2) should be made. Rule 2. In the case of capture by spin nuclei with negative magnetic dipole moment ¡x, in the absence of formulae for A+ and as+, they must be calculated from eqs. (2.5) and (3.9). As it has been pointed out at the end of sect. 3 there is some degeneracy of the correlation coefficients due to which some of them may be dependent on the others. Example of such degeneracy has been also given by BERNABBTJ [10], and we show here several new casss of this degeneracy. Their origin is obvious. capture by zero spin targets. This most important case includes, e.g., the following reactions: 12C -> B, 1 6 0 -> N, 1 8 0 -> N, 20Ne F, 22Ne -> F, 24Mg -> Na, 26Mg -> Na, 28 32 34 Si —> Al, S —> P, S —> P, etc. Each such transition is fully described by two independent amplitudes. Exceptions are the simplest (in the kinematical sense) 0 0 transitions each of which is described by one amplitude only. The amplitudes are real numbers if T-invariance holds. As an example of the additional kinematical degeneracies between the correlation coefficients in this case, we mention that the circular polarization of a gamma ray is completely determined by the effects of directional angular distribution. The circular polarization measurement is not an .independent experiment and therefore, one can generally perform the summation over rj in eq. (3.27) without the loss of any information. Then, only the terms with s even will be remain. If we denote the ratio of the multipole amplitudes by Muon

(5.3) then the general formulae (2.5), (3.9), (3.19) and (3.22) look simply as follows; the probability coefficient, A =

|2V|2

+

the y — v correlation in 0

as =

4

( - 1 ) %

Fortschr. Phys., Bd. 32, H. 2

(5.4)

|2j°|2

I transition,

I +

1 +

I x f

(5.5)

74

S. CIECHANOWICZ and

Z . OZIEWICZ,

Angular Correlation

the angular distribution of recoil,

âi

1/2

75

stat

_ 2

=

V0A1

+

«- = 0, 2 M S ,

&r

=

A - .

3/2:

A«w = 2Jf x » + ¿ i « +

#22 +

= (i^ - A1 - j E2 /t+

F22,

V2J,

= j M S + - | j M i + AS +

== 3 ( j f i + J

^

-

(A -

= fi MS + ^

F2)2,

1/2 -

40

- A1V2 - Ì

¡3/2

s 3/2\

\l/2

0 1/2/

ES -

VS,

,

= s

á2+

= (M, - ¿ 0 ( I - M ! + Ax - I F 2 j + ( 3 J Í ! + -jr/:

F2)

E2¿J,

a,

5 / 2 , sr, =

+

« - = 0,

+ 2A,V2 + j

á2stat = MS - AS + 3M^E2 Ô3SUt = ^

-

' ^

-

{^E2

-

V2J.

76

S.

CIECHANOWICZ

and Z.

5+ = 2 [M2 +

a 2 3Ut = - 2 |

OZIEWICZ,

E3J - (A2 -

M

S

Angular Correlation 73)2,

+ At* - M2E3 + Ea* + F 3 2 j ,

Muon capture by spin 1 targets. There are two well known reactions for this case in the region of the very light nuclei both corresponding to positive value for magnetic dipole moment of the target, 6 Li —> He and 14 N C. The helicity amplitudes T / expressed in terms of the multipole amplitudes; / - > 0:

1

1:

1-fO:

Fortschr. P h y s . 3 2 (1984) 2

77

1 -*• 1 : ¿«ut

/I-

v0

=

=

a2stat

Vo -

- 2 v

=

F0

a2- =

2 MS

+

2

+ AS

+ — ES

|/g ( 2 J f i +

0

v

-

2

(j^« -

a s )

V3

[ 3 ^ ( 1 3 ^

-

F22,

+

A )

-jr{M1-A1) -

+

+

+

3

-

- 2F2

-

I/- ( ^ - ^ > ( 3 ^ 1

lLáx) +

1

5«+ =

1

-

(Fo -

l/6^4i)2 +

+

[ 2 ]/2 F 0 -

3(Jfx

-

+

]/3

+

( 3 ^ - 4 F „ )

E

>

2

4 y2

+

+

F22),

4 0

(3 E 2

+

2F2)2,

-

]/3 ( 3 M 1 -

-l/2 F0 I p - F0 +

J

13Ai)]

2F2(M1 -

5 y6 Ä- =

(

A,)}

5^3

-

( 3 E 2 + 2 F 2 ) + -j- ( 3 E 2 +

2F2)2,

3JS72 -

3 J M i

• E ^ M i

- A , )

-5-^1+

V2

( t * +

2F2j +

-

V ^ M ,

J f ^ -

-

Sil,)

4

2: Asm =

/I" =

2MS +.AS + j E22+ F, 2 + j MS + ¿3 2,

(Jf, -

Atf -

+

-

^

äa 4 stat

- y f

l/lfjf, -

F

2)

( M S - A S

^

(

J

i

A )

+

2F2) +

+ ^s) +

+ 1/15

A

-

^

E ¿ +

F2(#2 +

+

-

)

1

y - 1

F22

+

, M

3

)

F,)

78

S. CIECHANOWICZ and Z. OZIEWICZ, Angular Correlation = - 1 / ^ (Mi - A ) 2 + l / y ( ^ 1 -

V

V22

+ E2V2+

14

A) F2) ( ± M , +

+

^

- 2(£?2 -

-

F2

5 /£ lf, + A , 2 \3

+

8/11 5 \ 2

F2) (2£ 2 + F2)

+

F2)(-Jf3+^3

Muon capture by spin 3/2 targets. Examples of reactions with negative ¡1 are 9 Be —> Li, Ne -> F, etc. In the case of positive ¡x we have n B -> Be, 23Na -> Ne, 33S -> P, 35 C1 —> S, 39K —> Ar, etc. Muon capture by 9 Be and n B has been studied both experimentally and theoretically, see ref. [33]. The helicity amplitudes given by the multipole amplitudes; 21

1/2: _ —1/2 —

(_iy+u/2)

J71+

l/6J(J + 1) X Vj-•-(1/2) — yj-di2) +

77+ _— 1/2

_ 12,J + 3

Uj

1 \2J + 1

1/2) + */j+(l/2)

(_l)^+d/2) , / 2 J + 3

J

]/6J(J + 1 ) X

]/ 2 / + 1

w

J-(i/2) — 2A/-( 1/2)

(_l)-Ml/2) -1/2

(_l)J-d/2) ^1/2 —

J l/6 X

Note:

1/2) + 2/J-C1/2) +

J]/6 n

J2J

1

J _ /I2J 2J + 3 + 3 1 \2J + 1

1/2) + 2/J+( 1/2)

1

2J + 1

"2J + 3 2J - 1 *V-(l/2) + ^-(1/2)

= 1/2) = 0.

I2J -

V 1 \2J + 1 J+(l/2) — 2A/+(l/2)j

J

/2J + 3 I 3)

6«i) -

2/2 -

j / y 2/3 + j

l / y (32/3 +

3« 2 +

4»,)

I / - (32/3 + 3)

5/2: ^-5/2 — 1/ Ï 5 2»i -

T

J

-3'2

+

1/6 5 j 2/i + »1 +

-

-l/2 -

- g - «1 + 2/1 +

21/42 (2/2 - j

v.) +

(2/2 - f

-

g -

J 3,2 _

Vx +

- j / y

1/ y (2/3 - «3) +

5 y 3

2/!-

(»+J-«.) /io /

5

«4

(2/3 -

(2/3 + j

/10

V

»3

j

»3)

80

S. CiECHANOWicz and Z. OZIEWICZ, Angular Correlation /_2_

T.—3/2

+

^

=

^

=

VI — «i

15

I/42

3/2

V2

% 3 + y «3)

1 ^ + 4

i / 5

-

1,1

-

K i r r

i/„

2/1 -

+1/21

It ( % 2 -+

+

]¡\ [y*

+

T"

«i

#

4

+ Y"2) -

I/y

-

'

7 \y*~

2

HI

(;Y3 + 3«3)

+

(2/4

1/2:

^stat = 2 M i 11 = Y ^ i

Sjsut

+

2

=

2

A

Z¡

10 + y J M i + ^i2 -

2A1V2

+

3 ó r =-- - I M S + -

E 2 2 + j/,,2,

+

+ 3 ^ 2

2 J

R—

-

AX)

/3

S- =

+ ¿X -

&+ =

[M,

-

F2

E^,

2 J I M i + A,2 + ( ö J f j + 3Ai) (3E2 +

(3®, +

+

2F2)

2F2)2,

-

F2j2,

A1 - Ì ET +

VTJ.

3/2 ->• 3/2: /l s u t = F 0 2 + 2 J ^ 2 + A' =

+

F0 F 0 - - 3 V 6 (2^1 + 4 1/5

2

+ ^ V5

! -

-

+

^0

Fa 2 + j M

+ -

[E ( e2 , + J V F2 2)j ++ 2E2(E2+ F2) ' 1 j f , + A ^3

3

) + ( i

* +

AS,

+ A)

+

3

F2) +

j f , + ,

F,2

¥

Fortschr. Phys. 32 (1984) 2 &2stat =

_2F0F2 + j

(M?

-

AS)

-±E2M3-j(M3*

á2- =

w

F

1

0

- ± ( M

-

A0 -

j

( A M +

A,A3)

Vt) + i - J f , +

J/5 ( Ä 2 +

- A,) (Jf x - 34,) +

¿1 ( * , + y

— — A2

-

+ A3*),

{ M l

t

^-j.M.E,

F,)] -

I

y l 2 J f ! (2E 2 + A F 2 j

(if, +

(•§•*, +

.^m3

+ Y (4M3 + ¿3)

+

As)

A3J,

—

12/

3 ^ ( 2 ^ + F2) +

A11 h{ rj EE2t ++ 2V 2 F t; yj + 3E 2 * -

= V,0 ( - v0 + 21/5 A, + 6E 2 + 4 F 2 ) + -g- i i ^ ! 2 + 2 i l M i -

w + A

5«+ =

(2®, +

F0 [—5F0 -

F2) (4M? + 34,) +

6l/5^x +

+

-

rt F2) -

- -|r [4Jf3(2£2 -

F2) - ZA3(2E2 - 37,)]

V5

A,(E2

4 F 2 ) ] + 3£ 2 * -

-

F22

Á3J,

+ 4F2] + - [ d i f , ^

+ -=[2^(2®, ]/5

24x) 0 5F22

5/2: ¿.ut

+ A12 + - | ® 2 2 +

=

= (Mt -

A,) M

1

- A

+ J f i

{E>

-F'>

+ ^m32

+ ^M3A3

+

-

1

(j

m

1 Jf,«

(3E2 +

> + A>)

+ A3*,

+

+

2F2)

J

E

1^ \ J2 ^ i 2 )

' + JE*V>

+

11^

82

S. CiECHANOwicz -and Z. Oziewicz, Angular Correlation

^=«2*" =

-

A

«2

+ 3 j / y M 1 JS 2 -

+

~ m sut

2

4 F

-

=

*

2 )

^

(iM.M, +

- T / T ^a

M x i)f3) + y W

&M,)

~ ï ï

+

'

F22)

-

=

X

14

( 1 0 F , + 3 9 Et)

+ 7Jf3 +

9A3

1/2

- -L— [263^22 + 5V2(E2 + 3F2)]

21 y y j/io

21 31

23

5 ]/Ï4

27

if32

+

11 31 1/35

Eo o+— -7 V11// 52- 71 ( ^ 2 — F 2 ) ( 4 £ 2 + 1 +

7/Ï5jL

+ 31/42 &-

E,

79

—

+ 4

+

4

i / ê

25«+ = 9 7 J Í ! 2 + 6 J M 1 -

¿s(14M, +

) A , - M , -

x

-

E.

F2(15^2 +

y

21

—

+ 2 9 4 3j

Z

9¿3)],

| / Ç ( ^

2

- 2 F

2

) - 8

2 / 4 |/=.[^.Jlf3+4 3 V3

17 F 2 )

\M3 + + 3A 3 43)* ) + 2 7 2 ( i t f 3 +

23^!2 +

+ 9A3)

3F2)

M 3 + 47^3) + 2 V 2 { ^ M

^

[37Jf32 -

— - M

=

(m3

-

241/2

lM^UEz

81/2 (M1 - A , ) ( 4 3 f j + 3 4 3 ) + ¿

+ 2F2) +

[3^(73^ -

2M3(7E2 - V2) - SA3{E2 - 4F2)

2^3)

AX{E2

20 F2) -

+

6F2)]

110 F 2 2 ]

Fortschr. Phys. 32 (1984) 2

83

Muon capture by spin 512 nuclei. The formulae presented here describe the muon capture processes by targets with negative ¡i\ 1 7 0 - > N , 25 Mg-> Na, 47Ti etc., and by targets with positive ¡i as, e.g. 27A1 Mg. 5/2

1/2: Q

A

+ F,2 + -M32

A«« = -E*

+

A3\

5A- = 11E? + UE2 F2 + 5 F22 + (# 2 - F2) istat = - y Ef - f

3 / &~ = j (2E2

4 V2 + jM3

+

7«+ = —3 ( e 2 -

3

+

A3J,

- 2F 2 ^3 +

- 2E 2 F 2 + F22 - 2(7E2 + 57.)

5«r =

5/2

Jf, + A3j + 5 M

F2 + j

Jf, +

+

Jf 3 + -¿s)* >

\2 + A3 J,

Ms

-

3/2: = 2Mx* + AS + j E22 + 7 2 2 + 5/1" = 17MS + 1 4 J M ! + BAf + lli? 2 7 2 + 5F 2 2 + 16 | / A + TLM3* + ^-M3A3 5a.2stat =

_

A

+

Jf 3 2 + A 2 +

,

(JF! - A^ (3E2 + 2F 2 ) + _ F2) ( - | i l f 3 +

ZA3\

2 _ f m M 1 E 2 - 4 | / | (4Jf!Ji 3 + 3^M 3 )

+ § W

+ 2F22) - 4 | / Y £ 2 M 3 + M32 +

25A2" = 1 9 J f ! » - 1 4 ^ ! - B A J — J/J[M 1 (67I7 2 + 1 0 F2) + 4 , ( 1 7 ^ , - 1 0 7 , ) ] +

/6 -Mi ( y M, + 74.) + 7 \T

+

5FA 2

) ~ T / T

( y M3 + (LIEI +

543j

10FA) ( M 3

+

84

S. CiECHANOWicz and Z. Ozievvicz, Angular Correlation 25 -ór

= 61

+ 2 8 J I M i + Aj2 -

]/l05

Jf,( 1E2 + 4F 2 ) + 2A1(E2 + V2)

- 4 | / | (Jf x - 4 , ) (43f, + 3P3) + y (51E2* + 10E 2 V 2 - 5F 2 2 ) + f l / y J0,(58Jf, + 17 — J / 3 2 +

+

= -{M, -

3943) + 2F2(13Jf3 +

124,)

43

- Axf + I / . y ( J f , - i o (¿?2 -

1 1 / 6 (Jf, -

A0 (i-Jf, + 4,) -

- T 1/T h ( f

~

13

^3)+

2F2)

¿ ( Ç ^

2Fa

2

-

5£2F2 +

13F22)

" T^3)

K

1 (22 ,, „ 11 „, , „ , „\ + y l y ^ 2 + TM3A3 - SA A. 5/2

5/2:

^stat = 5

Vo2 + 2M2 «2stat =

+

%+

Â1

V22 + ±.M2

+

- y ï ï F0F2 +

_ 4t») -

+ 91/^-E2M3

- i Jf,«.+

/Ö"

1/42 &t«* = 16 y - ( M A - 4 , 4 , ) Jf,« _ 34.«

_

5

=

]/jE

8

+

U M l

¿

yo

-

(E2*

+

»., /7i5

+

+

Äl)

+ 4,J -

fññ I/35 [^2(4J/3 +

+ ^t432

(3^2 +

- F 2 2 ) - 10 1 j — E 2 M 3

i - (Mí,« -

+ y j / y [ ^ i ( 2 ü f 3 - 343) - 64jilf3] -

+

3^3Ö M,E2 _ g

43* +

07 T

+

5V¡)

4M141 -

- y £

5 4 !2)

2

3 90 74,) + 5F243] + yitf,2 + ^

F

2

+

M3A3

2F,2)

Fortschr. Phys. 32 (1984) 2

J

I

=

F

- ¥ l / f +

85

^

~ ^

2As)

A l { m

J ]/T

-

- ¿ ( ^ 2 -

F 2 ) ( 1 1 # 2 + 5F 2 )

_7_ E2 15 -

= - V

+ 7 f s

+ ^

+ H

msA,

-

V^A,

-

AJ

* +

p i t3^2^

3

104^)

+

+

+ J L |41Jf 3 2 - M 3 ^ 3 -

= - V

24 /

3

^2ii3 +

F 0 (3E 2 + 2F 2 ) 17

+ y ] / | [6^(3^ -

F

- i? 2 F 2 - 3F 2 2 )

™ ¿s2) + - ,

- 2 j / ^ F04x +

o /07 ~ TI/35 t 3 ^ 2 ^

5As)]

+

j / | [ 6 i l f a ( 3 # 2 + F 2 ) + A ( 3 # 2 + 8F 2 )] + y

J

*>

+ QE2 + 4F 2 )j

+ ¿ ( 1 6 ^ - 32JMt -

+

F

I V2(28M3 + 6J3)

+

j

-

- ^