Analytical solution of the non-discretized radiative transfer equation for a slab of finite optical depth

Pergamon

J. Quanr. Spectrosc. Radiat. Trattsfer Vol. 53, No. I, pp. 59-74, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-4073(94)00101-4 0022-4073/95 $9.50 + 0.00

ANALYTICAL SOLUTION OF THE NON-DISCRETIZED RADIATIVE TRANSFER EQUATION FOR A SLAB OF FINITE OPTICAL DEPTH G. V. EFIMOV,? W. VON WALDENFELS,$

and R. WEHRSE§

TJINR, Laboratory of Theoretical Physics, Dubna, Russia, SInstitut fur Angewandte Mathematik der Universitat Heidelberg, Im Neuenheimer Feld 294, D 69120 Heidelberg, and @nstitut fur Theoretische Astrophysik der Universitiit Heidelberg, Im Neuenheimer Feld 561, D 69120 Heidelberg, Germany (Received 4 February

1994)

Abstract-An analytical solution of the radiative transfer equation for a plane parallel slab of finite depth is presented without requiring discretization in space or angles. The solution, expressed in terms of the matrix hyperbolic tangent function, avoids the problem of exponentially increasing terms so that it is also suitable for numerical calculations. By using the resolvent of the transfer equation we obtain simple series expansions for the matrix elements.

INTRODUCTION

Methods for the numerical solution of the radiative transfer equation for plane-parallel static media are well developed. ‘9’Except for the Monte-Carlo method, in all of these methods the transfer equation, either in differential or an integral form for the specific intensities or the mean intensities, has to be discretized in depth and angle coordinates. The resulting system of linear equations is then usually solved by decomposition and back-substitution or by a Jacobi iteration (“accelerated n iteration”). Only the discrete-ordinate-matrix-exponential (“DOME”) method3 does not require depth discretization in a large variety of cases. The modern methods are usually quite accurate and very fast so that they can be employed in iterations toward radiative equilibrium and in the synthesis of stellar spectra. However, these numerical methods are usually not well suited for media with steep gradients and highly peaked redistribution functions. Furthermore, they can hardly be used for error or sensitivity analyses or for radiative transfer problems in stochastic media as algorithms have to be used which are essentially analytical. While such methods have been available for a long time semi-infinite configurations, the boundary value problem occurring for finite optical depths, which is simultaneously sufficiently general to deal with situations of practical importance, seems not to have been published. In this paper we present a method that meets these requirements. It is based on formalisms developed in field theory and quantum stochastic problems.4 It is in many respects the generalization of the DOME formalism3 to a continuous distribution of angles. However, it does not contain terms that increase exponentially since the relations between ingoing and outgoing intensities are expressed in terms of hyperbolic tangent functions. The resolvent is used to derive the required eigenvalues and -vectors in the spectral decomposition of the kernel.5 The eigenvectors for the continuous problem are generalized eigenvectors introduced, e.g., in Gelfand and Wilenkin6 and Efimov and von Waldenfels.4 Computations show that the code becomes more compact than the code of the DOME method since the eigenvalues and -vectors need no longer to be calculated numerically (as for example by means of the QR algorithm) and many matrix manipulations can be avoided as positive eigenvalues do not have to be eliminated explicitly. The CPU times may become very short if a fast realization of the tanh function is used. In the next section the formal solution of the transfer equation is presented in terms of the hyperbolic tangent function. The spectra1 decomposition of the kernel M and the evaluation of tanh A/2 (A = optical depth of the configuration considered) are subsequently described. In two 59

G. V. Efimov et al

60

appendices some formulae and estimates are derived in detail. In a final section we discuss the numerical performance. FORMAL

SOLUTION

We consider the transfer equation for monochromatic unpolarized radiation in a slab of finite optical depth A. Let us first assume that there are no photon sources in the medium so that the transfer equation is homogeneous

s +I

dZ(tvP) ’ ~ dt = -Z(t,p)+/3

Z(t,P) dp

(1)

-I

where ~6 [ - 1, l] is a parameter and 0 < /I 6 l/2. We introduce k = l/p and rewrite dZ(t, k) -=-kZ+pk dt

Z(t, k’) dk, s I!il2i k” =-

s

Mkk’Z(t, k’) dk’

Ik’lPI

with

A&=kh(k-k’)-flk&.

(2)

(3)

If we replace Z(t, k) by the vector Z(t) then the obvious formal solution is Z(t) = e p”‘Z(0).

(4)

This does not lead, however, to a usable solution for two reasons: Firstly, the spectrum of A4 extends from - co to + co and emM’cannot be defined easily. Secondly, the physical boundary conditions are not of the form implicitly assumed above. It is better therefore to express Z(t) as

(5) and to consider a layer defined by to < t < t, and A = t, - to. The physical problem is to calculate the outgoing intensities Z_(t,,) and Z+(t,), if the ingoing intensities I+(&,) and Z_(t,) are given. Now, YJO= Yin =

I+ (to) Z-(h)

( > (8)

Truncate M in such a way that emM’can be defined, then Z(t,) = epMAZ(to).

(9)

Define the projectors r+=

1 0 o o

( >

(10)

and 0

0

z-=o ( 1)

(11)

then

Z(t,1=

I+ (4)

(z- > (4 )

= z,

Y, +

T-

Y’o

(12)

Radiative transfer equation for a slab of finite optical depth

(to)=

61

I+ (to) =T+Yl)+T_Y, I- 00)

(13)

+ I@,) = Ye + Y,

(14)

( >

I&)

and z., Y, +

Y.

7_

e-““(7

=

+ Ye +

7_

Y,).

(15)

Hence, (7, -

eeMA7_)Y, = (emMAT+- 7_)Yy,.

(16)

Define 73=

1 ()

0 -1

( >

(17)

then 7

--

1 +73

f-

W)

2

and the equation becomes [(l - emMA)+ (1 + e-MA)73]Y, = [(eVMA- 1) + (eeMA+ l)73]Yy,.

(19)

Using 1 -e-MA

= tanh MA/2 1 +eWMA

(20)

one has (z3

After multiplication

with

z3

(21)

obtain finally

we

(1 +

tanh MA/2)Y, = (z3 - tanh MA/2)Y,.

+

73

tanh MA/2)Y,

= (1 -

z3

tanh MA/2)Y’,.

(22)

The quantity to be calculated is tanh MA/2 which is very satisfactory as tanh x stays bounded for x + + co so that the infinite spectrum of M does not cause any difficulties. If photon sources are present in the medium, the treatment is similar. The transfer equation is now inhomogeneous and reads p) + /I

WYP>dp + B(t, cc)

-I

or

Wt, k ) --=dt

s +I

W, PU) = -IQ, ’ ___ dt

Mk+,Z(t, k’) dk’ + kB(t, k). s

Calling kB(t, k) = b(t)

(25)

we have dI z = -MI+b(t)

(26)

and I(tl 1)= e -“(‘l-‘~)Z(to) +

s II

e-(‘~-“~b(t) dt.

‘0

(27)

62

G. V. Efimov et al

Equation (19) now becomes f[(l - emMA)+ (1 + e-MA)r,]Y, = i[(ePMA- 1) + (emMA+ l)r,]Y’, +

” e-(‘l-‘)“b(f) dt. s '0

(28)

Multiplying by eMAl and dividing by cosh(MA/2) = i(e““‘* + emMA’*) we

(29)

get - ?)I [r3 + tanh(MA/2)] Y, = [z, - tanh(MA/2)] Y, + s” “‘[,(’ cosh(MA/2) '0

b(t) dt

(30)

and, finally,

[l + rj tanh(MA/2)]Y,

= [l - rj tanh(MA/2)]Y,

-06(r)dr + z3 [‘exp[M(’ cosh(MA/2) '0

(31)

Here again the infinite spectrum of A4 does not present any difficulty in computations. SPECTRAL

DECOMPOSITION

OF

M

We have Mkk’= Dkk’+ u,v,

(32)

DkK=k6(k

(33)

with -k’),

uk = -fik,

vk=-.

(34)

1

(35)

k*

So M has the form treated in the Appendix M=D+lu)(vl.

The set Bis the subset B=(-oo, -l]u[l,co) The resolvent of M is (cf. 119) 1 =-+-u z-D

(36)

of R. 1 z-D

1

1

I> qijv- (I

z-D

(37)

with (38)

We transform back to ,u and obtain +’

dp -= s -, pz-1

C(z)=l+jI The function is holomorphic - 1 and from 1 to 00. As.

1+;1og,

l-z

in C - B, i.e., in the complex plane with two cuts from -co

s +I

ImC(x + iy) = -2bxy

dpp*

-I KG - l)* +

P2Y21W + I)*+ P2Y21

(39)

to

(40)

there exists no zero outside the real axis. If z = x E R then C(x)=l+~lOgl+x.

l-x

(41)

Radiative transfer equation for a slab of finite optical depth

63

In the interval - 1 < x Q 1 the function C(x) has a maximum for x = 0, one has C(0) = 1 - 28 and C(x) decreases monotonically to - 00 for x + + 1 or x+ - 1. If fl < l/2 then C(x) has two simple zeros I,, A2 = -I,, if /3 = l/2 then C(x) has a double zero at A,,= 0. If /I = (1 - ~)/2 then I, N &[l

+ O(6)]

for 6+0.

(42)

For x E B one has

Rec(x)=c,(x)=l-(ul~/u) x-l x+l

=l+$og I

I

= Cp( -x)

(43)

where P indicates the “principal value”. For rZE B one has c(n f 2.0)= C,(A) f i7rUj.Q

C(z) has only finitely many zeros outside B. We therefore apply the expressions of the appendix and obtain the following results: In the case B c l/2 the spectrum of A4 consists of the two discrete eigenvalues A,, A2 and the set B. The eigenvectors are

(46)

(47) N; = C,(l)‘+

A*U;U;

where qf~~(x)= s(A - x). The dual eigenvectors are

Using k, k’ E B as indices we have Nf=

.-/j

‘J’i(k)=

s

dk ,Jm--

-k & I

I

Bk,

I

(55)

64

G. V. Efimov et al

In the case j? = l/2 the two roots I, and A2= -rZ, converge to a double root & = 0. We have in z = 0 the expansion qz)

= -fz’

_

$4

+“-

. . .

(56)

so C(0)

=

C’(0)

=

cm(o)

=

(57)

0,

$U(O) = -: = c,.

(58)

We can again apply the formulae of the appendix. We have the vectors

IY,>= IYY,)=

”

l l lu>,

ED’

(IvA9

=IJc,

@I

(!&I=

lu),

AD

-(VI;. ’ &

(62)

Here ( !P, ) and ( !Pyz I are eigenvectors and I Y2 ) and (Y, I are principal vectors for the eigenvalue 0, explicitly: Y,(k) = - l Jc,

(63)

(-B)

Y,(k) = -&-P/k)

(64)

0

Q/k)

=L-4

(65) Jc,

q’,(k)

=

Lk-’

(66)

Jc,

CALCULATION

OF tanh

M

We want to calculate in this section t3 tanh MA/2 using the spectral decomposition resolvent. At first we use the well-known series tanh(x) =

and the

1

F “=-“$(2m

(67) + 1)+x

Hence tanh(MA/2) =

y m=-w -_i!!(2m

= --

1 + I)+!!$

imEm&M

(68)

or tanh(MA/2) = -i

E

m- m

R(iy,)

(49)

65

Radiative transfer equation for a slab of finite optical depth

with (70) Using Eq. (119) we obtain tanh(MA/2) = -i

I-& m (71)

with (72) The first term can be summed up using the consideration r3 tanh(MA/2) = tanh(KA/2) - i C &

above. Then 1

u ~ ’ iy,,,-D

t3 1u )

m

(73)

with

K=r,D

=(lkld(k

-k’)),,..

(74)

Finally, [r3 tanh(MA/2)],,P = (tanh Ik IA/2)6(k - k’) + z 1

$-&s m

&

m

m

1 (75)

This expression gives good results for small A and small p, it fails for /3 = l/2. For /I = l/2 it is better to extract the pole of R(z) in the origin.

R(z)=f~~luXvl~+~~ 0

0

(

j$u)~vl;+;lu~~v,$

)

+&

+ 4 (2)

R,(z)=&lu> ~~~l;:~-~~~l~~(~l~-~~(~l~~~~l~+~l~)~.,~). 0

This corresponds

to the decomposition

(76)

0

(153). We have

tanh(MA/2) = AA/2 + tanh(DA/2) - f c R, (iy,,,)

(77)

with A =&

Iu)(vl;

(78)

0

and ~~tanh(MA/2) = &A tj I u )(v I i $ + tanh(KA/2) - i 1 r3 R, (iy,,,) 0

(79)

and [rj tanh(MA/2)],,.

= iEkk’-‘A/2 + (tanh 1k 1A/2)6(k - k’) Ikl

k’-*

I

-----TW3-iym GY, - k iym - k’ WY,,,)

3 Y,

3

(80)

66

G. V. Efimov et al

with Ek=sgnk

= )l

(81)

and find finally [r~ tanWW2)1k,k --%

z k k’-3A/2+(tanh]k]A/2)6(k

t82j . -C(iy,)(k2+y~)( ii k1 -k’)

Iklk’-2(kk’-_v:)

3 ~ k,_3

Here C(Q)=

1 -i--arctany

= +fy2-$fY4+

**a

(83)

[tanh(A A/2)lpi, dl.

(84)

We go back to the general formula [Eq. (163)] and write tanh(MA/2) = AA/2 + s i.EB

For large A, up to an error of order e-A, this quantity can be approximated

by

tanh(MA/2)=AA/2+~~pidn-/~~~idl=AA/2+B

(85)

Recalling pi. =

we see that B can be approximated

& [R(1 -

io) - R(1 +

io)]

(86)

by

B=&

R(z)dz s Iwhere T is the contour of two curves around the cuts (see Fig. 1). After the subtraction singular part in 0 we can deform the contour to twice the contour r’ and obtain tanh(MA/2) = AA/2 + sgn D - iR :m R(Q) dy. s m

Fig. 1. Integration paths in the complex plane, cf. Eq. (87).

(87) of the

(88)

67

Radiative transfer equation for a slab of finite optical depth

As y,,,+ 1- y, = 2n/A this is exactly the limit of Eq. (79) for d-+00. We obtain [z~tanh(MA/2)]k,K = &kk’-3 A/2 + 6(k - k’)& 1 +5

+oD s _m

1 1 - t arctan y

Ik((k’)-*(kk’-y*)

3 - --ickk’-3 y

(k*+y*)(k’*+y*)

1

dr.

(89)

IMPLEMENTATION

We implemented this new algorithm for the solution of the radiative transfer equation in two ways. Firstly, we calculated the matrix tangent hyperbolic function by means of a diagonalization, and then used Eq. (22) to derive the specific intensities. The results were compared for a very wide range of parameters b and A with the corresponding intensities calculated by means of the DOME method.3*7 Applying 12 angles per half sphere with a Gaussian distribution and double precision arithmetic we consistently found agreement to better than 10A6. Since many matrix manipulations which are necessary in the DOME formalism for the elimination of the positive eigenvalues are no longer required the code becomes much shorter and more compact. However, due to the fact that the eigenvalue and eigenvector routines* take most of the CPU time there is only a modest speed up in the execution time. Secondly, we calculated the matrix elements of tanh MA/2 directly from the series expansion 75. Unfortunately, this expansion frequently does not converge well, so that we had to use up to 3000 terms to find agreement to better than 10m6 for all matrix elements and for the specific intensities which are again calculated from Eq. (22). Depending on the machine implementation of the division and the tanh function, the execution time for one parameter combination (fl, A) may be reduced by a significant factor compared to DOME calculations. More importantly, the total number of FORTRAN statements can be decreased by more than 80% since the eigenvalue and -vector routines as well as many matrix-matrix and vector-matrix operations are no longer needed. Furthermore, the memory requirements are appreciably smller since many auxiliary arrays are now not required. Up to now all our calculations have been made with non-optimized codes using the formulae given above. Further significant improvements in the numerical performance of the algorithm can certainly be achieved if, e.g., convergence acceleration methods are applied to Eq. (75) and/or pretabulated values for the tanh function are employed. On the other hand, we would like to stress again that the main advantage of the method presented here is the fact that it provides an analytical solution of the radiative transfer equation for plane parallel media of finite optical depth and therefore opens, e.g., an easy access to the transfer through slabs with stochastic density distributions or with stochastic wavelength dependencies of the absorption coefficient. Note in addition that the algorithm can easily be generalized to include differential velocities. Acknowledgemenf-This

work has been performed under the Sonderforschungsbereich

359.

REFERENCES 1. W. Kalkofen, ed., Numerical Methods in Radiative Transfer, Cambridge Univ Press (1987). 2. L. Crivellari, I. Hubeny, and D. G. Hummer, eds., Beyond Classical Models, Kluwer, Dordrecht (1991). 3. M. Schmidt and R. Wehrse, in Numerical Methods in Radiative Transfer, p. 341, W. Kalkofen, ed., Cambridge University Press (1987). 4. G. V. Efimov and W. von Waldenfels. Annals of Physics 233, 182 (1994). 5. T. Kato, Perturbation Theory@ Linear Operators, Springer, Berlin (1980). 6. I. M. Gelfand and N. J. Wilkenin, Verallgemeinerte Funktionen IV(Distributionen), VEB Deutscher Verlag der Wissenschaften, Berlin (1964). 7. P. H. Hauschildt, R. Wehrse, S. Starrfield and G. Shaviv, Astrophys. J. 393, 307 (1992). 8. G. Engeln-Miillges and F. Reutter, Formelsammlung zur Numerischen Mathematik mit Standard FORTRAN Programmen, BI-Wissenschaftsverlag, Mannheim (1988).

G. V. Efimov et al

APPENDIX

Spectral

Decomposition

1

of a Matrix

of the Form

D + lu>(vl

We recall at first the connection between a finite dimensional matrix M and its resolvent 1

R(z) = z-M

Assume for example that M has different non-degenerate exists an invertible matrix X such that

eigenvalues I,, AZ,. . . , A,. Then there

X-’

M=X

where e,is the vector (0 ,... , l,..., 0)r and the 1 on the i-th place. Calling X,. = xi and e’X-’ then the xi form the columns of Xand the ri the rows of-X_‘. As - _’

(91)

= &,

e Tei = 6,

(92)

Mxj = MXej = XAej = XAjej = AjXej = Ajxj

(93)

Mxj = Ajxj

(94)

tjM = Ajtj.

(95)

we have

hence

and equally

The xj form a basis of eigenvectors, and tj form the dual basis as &xi = e,X-‘Xej = eTej = 6,

(96)

We call the tj the dual eigenvectors. The matrices Pizxi5i

(97)

are projectors, i.e., p: =pi and they fulfil the orthogonality

conditions

PiPj = 6i,jPi.

(98)

Then the resolvent is given by -I

z -1,

I

X-'

(99) So R(z) is a matrix valued function holomorphic in z outside the eigenvalues ,I.,, . . . , I,. The eigenvalues are poles of R(z) and the pi are the residues of R(z) in J., , . . . , I,. So when R(z) is given it is easy to calculate the eigenvalues and the eigenvectors of M. The situation changes if M has degenerate eigenvalues. Assume, e.g., that M is a 2 x 2 matrix with double eigenvalues 0, that M#O

but

M’=O,

e.g.,

M=

(100)

69

Radiative transfer equation for a slab of finite optical depth

Then - 111 =-Z-M

=i+M 2 z2’

M

zl-_

(101)

Z

A little bit more general assume that M has one double eigenvalue but all other eigenvalues are simple and that it4 has the Jordan form

Then M =X/IX-’

(103)

we have for xi = Xe, and ti = eTX_’ A4xi = Aixi i = 3, . . . n, tiM

=

Mx,

=

Mx2

Aox,

=

Ati,

(105)

+

X2)

(106)

(107)

&x2,

(log)

r,M = &
=

5,

(104)

(109)

+&t,.

so x2,x3,..., x, are eigenvectors, 5,) &, . . ., 5. are dual eigenvectors, xl is a principal vector and t2 is a dual principal vector, i.e., (M - &)‘x, = 0 and <,(M - &)2 = 0. The resolvent has the form

R(z)

A

=hM =*+

(z -Jo)2

++iE3Z 2,

(110)

So R(z) has in Is, a double pole. The residue is PO= XI(1+ x2r2

(111)

the projector into the principal space belonging to the eigenvalue Iz, and the coefficient of (z - &-’ is A

=x25,

(112)

which is nilpotent, A2 = 0. We have to consider a matrix of the form M =D +lu>(vl

(113)

i.e., B t R is a suitable subset (we think of an interval or a union of intervals) forming the index set of our matrices D,,.=kd(k (I u xv

-k’)

I)/# = %cfi/c

(114) (115)

where uk and vk are two suitable functions of k E B. The resolvent of M is very easy to calculate using the formula 1 z-D-V

=-+

1

z-D

1 1 ------V------z-D

z-D

+

(116)

G. V. Efimov et al

70

One obtains

(117) Hence

with C(z)=

(uI-&

1-

(119)

Let us assume at first that C(z) has only finitely many simple zeros for 1 E B C(A f i00)= lim C(A * ic) 0 exists where ~10 means that 6 > 0 and tends to zero monotonically. the set B and the set of all zeros of C(z). We make the ansatz k P R(Z)= I&+ I

i=

,

B$dd

z,,, . . , zk outside

B and that ( 120)

The singularities of R(z) are

Wl)

s

The PTiare the residues of R(z) in ziy i.e., 1 zi - D

(122)

with “tzi>

=

(v

I czi_ oj* I u>

(123)

In order to calculate the pi observe 1

x&i0

P = - T i&(X). x

where P denotes the principal value. Hence we have for 1 E B R[A - io) - R(1+

io) = 2 p i=l

The f&t term cancels because the z, 4 B. We obtain

Using for A E R

(124)

71

Radiative transfer equation for a slab of finite optical depth

we have C(A f io) = c,(n) * i76;,uu,

(128)

and after some easy calculations 1 C&)l4,.> + 4j55 p; = c,(n)* + n2zS2u; [

In>

I[

G(~)(4,l+

P

%.GI -1-D

1

(129)

where 4;,(~) = cS(A -k). The results can be written in the following way: we 1h;ive a discrete spectrum formed out of the zeros z, , . . . , z, of C(z) with the eigenvectors (130)

u> and the dual eigenvectors

(131) with

Nf=(v( The continuous

(zilD)*lu”

(132)

spectrum consists of B. The eigenvectors are

iy;.> =$y G(A)l4;~>+ 6:&u,] I.[

(133)

@;.I =&

(134)

and the dual eigenvectors are

*[

GQK4,l + k.
N; = C,(A)’ + n’B:u; We have to check the orthogonality

(135)

and the completeness of the eigenvectors. We want to have

(@‘,ly/c> = Bik5

(136)


(137)

@;.lYJ

(138)

(!F;.IY,)

= 0,

= s(n -1’)

(139)

and

$, IyU,>
(140)

These relations can either be checked by hand using the identity P P ---=A x-2.x-l’

+n*d(x

-1’

or obtained by using general properties so-called resolvent equation 1 1 ---_-=2, -Mz,-A4

- n)s(x - A’)

(141)

of the resolvent. Inserting the ansatz above into the 1 z2 -

ZI

>

(142)

12

we find the orthogonality

G. V. Efimov et al

relations p:, p:k =

and hence the orthogonality that

Bi.k

pz,

(143)

3

pip;: = 6 (A - 1 ‘IPi, >

(149

Pj,pi = O,

(145)

pnPj.= O

(146)

conditions for the eigenvectors. Assuming C(z)-+ 1 for z + cc we see

R(z) dz = 1

(147)

where K, is a circle of radius p. Insertion yields CPZ, +

dApj,= 1. sB

(148)

This is the completeness. For the degenerate eigenvalues we want to consider only the case specific to this paper. We assume 0 $ B and C(0) and C’(0) = C”‘(O)= 0 and C”(0) # 0. We have C(O)=l+
(149)

1; 124)= 0,

(150)

ic”(o)=(oI~/u)=cozo,

(151)

C’(0) = (u

1; 1u)

f C’(O) = (v

= 0.

(152)

The function R(z) has the Laurent expansion in z = 0 R(z)=$+;+o(l)

(153)

with

lu)(ol;,

A =g 0

(154)

PO=& $Iu><4~+~l,>(ul; 0 (

.

(155)

>

Call IY,)=--

I’u,>=

l ’ lu>,

(156)

l

(157)

Jc,D zs

l lu>,

(!&I= -w-$9 l JG

(!&I=

l -(ol$ Jc,

(158)

(159)

73

Radiative transfer equation for a slab of finite optical depth

Then /Y, ) and (p,I are eigenvectors and /Y, ) and (9, ( are principal vectors for the eigenvalue 0. The spectrum of M consists of the zeros of C(z) and B. The eigenvectors of the eigenvalues # 0 are defined as above. Completeness and orthogonality can be proven in a similar way. If M is finite dimensional with simple eigenvalues we have

and more general for any complex function

f(M) = Cf(& lPk . Similarly, if for R(z) the ansatz Eq. (121) holds +

s

4fU Ip;..

(162)

E

If R(z) has one double eigenvalue at 0

+ C f(Zi)Pz, +

f(M) = Pof(O)+ Aof’

Z< #O

APPENDIX

1dilf(~lP;

(163)

B

2

Estimate for tanh M

We want to give an estimate how well zj tanh MA/2 is approximated M in a more symmetrical way by introducing u,,. = Ik )-“%(k -k’)

by tanh KA/2. We write (164)

Then d=UMU-‘=D-pIq)(xI

(165)

with Y/k= 6,lk ]-“2,

(166)

Xk=jk]m’i*

(167)

so 73l?>

=

Ix>

(168)

We have [Eq. (73)] U(z, tanh Ms)U-’

= z) tanh As = tanh KS + W

(169)

with (170)

and 712m+l yl?I=Tj

s

(171)

W=(l+tanhKs)(l+W,)

(172)

Then l+r,tanhMs=l+tanhKs+ with

w, =

1

1 + tanh KS

(173)

74

G. V. Efimov et al

We show that

28

11w, 11 <

f

w

+uiI> Yi

~(1 + tanhs),=,

1 - 2/3

arctan Ym -I Ym

(1741

The approximation gets better for large s. We understand W and W, as operators in L’(B) where B = (x czR : Ix[ >, If and 11W, [I is the operator norm. We have

II WII G

l

1 + tanh(Ks)

II WII -

(175)

At first

Then (177) with (178) and

(179) As

s

1

1

sdkrj;7y:,+k’=2

Eqs (173, (176) and (179) show Eq. (174).

s

“dk 1 --=3og(l , k y:+k’

y;

+yi)

(180)