- Email: [email protected]
Inverse problem and singularity of the integration kernel
s -.g Kl
11 September 199.5 PHYSICS
ELSEVIER
Physics Letters
A 205 (1995)
LETTERS
A
130-136
Inverse problem and singularity of the integration kernel Hu Gang a-b,C.Z. Ning arc,*, H. Haken a a Institut fiir Theoretische
Physik und Synergetik b Department of Physics, ’ Arizona Center for Mathematical
Uniuersitiit Stuttgart, Pfaffenwaldring 57/J, D-70550 Stuttgart, Beijing Normal University, Beijing 100875, China 1 Sciences, Uniuersig ofArizona, Tucson, AZ 85721, USA ’
Received 31 March 1995; accepted for publication Communicated
Germany
6 July 1995
by A.R. Bishop
Abstract
Many important problems in physics and other sciences can be formulated in terms of the inverse problem of type n(y) = /K( y 1x)g( x> dx, where g(x) is unknown. We show that this problem can be completely solved for a quite general class of kernel K(y / x) by analytically dilating n(y) and K(y 1x) to the complex z plane, and by the analysis of the singularity of the dilated kernel K(z 1x). The formalism is also extended to multi-dimensional cases.
The inverse problem is an old problem which, however, attracts continued interest due to its theoretical importance and wide variety of applications [l-6]. Recently, Chen and his coworkers have made significant progressin this field [6-81. In particular, they relate the Fermi distribution to the Dirac 6 function and solved the inverse problem of Fermi systemsin a closed form by an elegant computation involving the 6 function, In Ref. [9], Zhao and Liu have improved the approach by introducing a limiting procedure and have put the formalism on a solid mathematical ground. In Ref. [8] Chen et al. showed that their approach can be applied to solve a number of inverse problems of practical importance of Fermi or Fermi-like systems by using a surprisingly simple calculation. In this Letter, we will develop a systematicalformalism, basedon the analytical continuation of n(y) and K(y 1x) to the complex z plane and the analysis of the singularities of the kernel K(z 1x), to completely solve the following general inverse problem,
* Corresponding author. ’ Present address. 2 Present address. 0375.9601/95/$09.50
0 1995 Ekevier
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H. Gang et al./Physics
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130-136
131
where the kernel K and the function n are known while the function g is to be determined. The problem ‘defined by (1) is called an integral equation of the first kind [lo]. First we consider the one-dimensionalcase 4~)
=
IbK(y a
I4dx)
(2)
dx
In detail, and then extend the formalism to the multi-dimensional problem of Eq. (1). We first specify some conditions required for our approach. As in most practical problems, g(x) and K( y 1x) are real and analytic functions of real variables x and y in their supportsdefined by the corresponding physical problems. Thus, n(y) must be a real and analytic function, too, for real y. We assumethat K(y / x) and n(y) can be analytically dilated to complex z and denote the dilated right-hand side of Eq. (2) as S(z). Since K( y / x> is a known analytical function, its analytical continuation can be directly done by replacing real ,V with the complex variable z. We further assumethat the analytical continuation of the known n(y) to a complex z (denoted as Z(z)> in various Riemann sheetsis entirely known, too. Hence, the first crucial point in our approach is that Eq. (2) (and also Eq. (1)) is valid not only for real y but also for the complex z variable. This point can be easily verified by the above assumptionsof analytic continuation. K( y 1x) is assumedto be analytical for the variables x and y on its physically meaningful supports. .4ccording to the Liouville theorem [ll], there exist, however, singularities outside the physical region, if K(z / x> is bounded. We make the physically reasonable assumption that K(z I x) + 0 as I z I -j m. For simplicity, we assumethat K(z I x) has a unique singularity denoted by 2=/l(f).
(3)
The functions h(i) may be multi-valued. We consider only the case of single-valuedness.The singularity is assumed,in this Letter, to be simple in the sensethat the function 4(x,
z) = [x-+)]K(z/x)
(4)
is analytic and nonzero at (i, 2). The following approach can be extended to the more general caseswhere I x) has more than one singularities and Eq. (3) has multiple solutions. More general casesof the higher arder singularities could be similarly treated. We will present these detailed results elsewhere.It is well-known that
K(z
1 lim ~ 8’0 x-yii.5
1 =P----X-Y
fi7r6(x-y),
(5)
where P denotesthe Cauchy principal value. It is interesting to note that this formula can be extended to the following rather general and practical form, 1 K[‘l(ilx)
= iI”,
x&
&(x,
2) =4(x>
u(2) =qb(Z;, 2).
;)P
x-h(i)
+i7r~(i)8(~-h(i)), -
(6)
(7)
The extension from Eq. (5) to Eq. (6) is the most important step in our approach from which our main result follows. Analytically dilating Eq. (2) to complex z and applying Eqs. (6) and (7) we have n’(2)
=/bKr+1(2/~)g(~)
a
dx
1 =p / b$(x> Z>g(x> x-h(i) a
(8) dx&iru(i)g(P),
(9)
132
H. Gang et al. /Physics
Letters A 205 (1995)
130-136
leading to ;2+(2)
-A-(f)
=2inu(i)g(x),
(10) where f should be replaced by the x through the inverse function of (3), i.e., i = h-r(x). The result (10) is surprisingly simple. The inverse problem is now reduced to a simple algebraic computation. The main work is then to seek the singular point of K( z 1x), (2, J?), and to specify the function U( 2). It is to be emphasized that both A+ and ii- satisfy Eq. (2). Specifically, when K(z 1x) has a singular cut given by (3), Z(z) should have a multi-sheet structure. In the following calculation we will directly use ?I’(,?> - Z-(Z) in place of the left-hand side of (10). In certain cases expression (10) can be further simplified. (A) 4(x, 2) is real. Then Eq. (10) can be reduced to
K[+‘(il x) =I+‘(21 x>*,
(11)
g(x)
(12)
= [7rzQ(x))]-’
(B) 4(x,
Im[fi+(Z(x))].
2) is purely imaginary. Then we have
K[f’(z^l
x) = -I&‘(21
g(x)
X)‘)
= [rri+(i(x))]-1
Re[fi+(i(x))].
Chen et al. [7,8] have dealt with the inverse problem of the Fermi distribution which belongs to the case (A) with i=i,, + ri, 2 =iRe, and u(Z) = - 1. Generally, 4(x, 2) is complex with both nonzero real and imaginary parts, and one thus has to use formula (10). Now we analyze some examples to verify our results. The first two and the fourth examples are related to well-known practical physical problems. The third example is constructed to examine certain aspects of the general result Eq. 410). Example 1.
II(?.)=j;l
mdx) dx l+yx’ ’
n(y)=aex~[b(~/w,+w~/~)],
y>O,
(13)
which is a typical example of the problem of finding the relaxation-time distribution g(x) by measuring the permittivity of a material n(y). We have, from the dilated kernel l/(I + zx’), i=
--s
(s>O>,
i=h(i)
= T-l/&.
In this case we have u(i) = - \il z^1/2i = 2/2. This example belongs to case(A). Based on (12) and (13), we have g(x)
= gsin[
b(l/o,x
- wax)],
which is given in Refs. [I] and [8]. Our approach is, however, much simpler than Ref. [l] and more systematic than Ref. [8]. The dilated form of (13), n(z), has an obvious two-sheet structure. Taking, e.g., zr = y exp(4nrri) and z2 = y exp[(2n + 1) X 25-i)] in the integral of (13) we obtain the sameresult. Inserting zi and z2 into the known function ?I(z) leads to different values, namely ?I+ and %- respectively. Example 2. n( y) = Lm[ 1 + ~-‘a( T) exp( -X/H)] ?L(y) = [l +y-la(T)]
-=,
4.q
-I g( x) dx, > 0,
O
(15)
H. Gang et al. /Physics
Letters A 205 (1995)
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133
This model is often used for describing the adsorption on heterogeneous substrates. The task is to find the distribution function g(x) from the measured total isotherm and the theoretically known local isotherm K(y 1x) (the pressure P in the problem is denoted by y in our expression). Now we have h n z = ZRe < 0, & = 0, 2 = h( 2) = RT ln[ -i,,/a( T)] , (16) leading to u(f) = RT. This model belongs to case (A) again. A direct investigation may immediately clarify the multi-sheet structure of the known function in this case. Here we will not go into details. The simplified formula (12) provides us with the solution Im[l
s(x)=&
1 sin( rc) = TRT [exp( x/RT) - l] ’ ’
+ a(T)/?]
(17)
which is again identical to the results obtained in Refs. [2], [3] and [s]. We have tested a number of practical inverse problems that have been investigated be others, and got correct results by using our much more direct and simple approach. Example 3.
= n(y)=L
yg(x> dx n(y)
l+exp(y-x)’
from which we have A ,. z=zRe + 7ri,
P=i,,,
u(2)
exp(-Y> ldl + exp(y)l)
=Y
=i=i,,+ri.
(19) Now u(T) has both nonzero real and imaginary parts, and we have to use the general formula (10). The difference of n(z) in various Riemann sheets is due to the function ln[l + exp(i)]. In particular, the difference in two adjacent sheets at f is given by E’(2)
-F(2)
=2rrii
exp(-i).
(20) Inserting Eqs. (19) and (20) into (lo), we obtain g(x) = exp(-x). The correctness of the result can be easily verified. Up to now, we have considered certain particular examples. To more convincingly confirm formula (lo), let us examine the following rather general and practical Fermi-like problem. Example 4. n(Y)
= lb
g(x) dx
a 1 +exp(x-y)
’
m>b>a>O,
(21)
which can be cast, by a transformation, X=exp(x),
Y=exp(-y)
(22)
into a simpler form, N(Y)
q”i:);,
(23)
with A=exp(a),
B=exp(b),
G(X)
=exp( -x)g(x).
(24)
Considering Eq. (31, (4), and (7) and denoting the singularity by 2 and f, we have --oo< -l/A
-l/B
R = l/f,
U(f)
= l/Y?
(25)
H. Gang ef al. /Physics
134
Letters
A 205 (1995)
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Now let us directly evaluate the integral (23). Setting V= 1 +XY, we have G(X) = G(XY/Y) V/Y). According to the analyticity of G(X), we can make a standard expansion to get G(X)
= G( -l/Y)
with T(V,
= G(-
+ V?P(V, Y),
l/Y + (26)
Y) being analytical at V= 0. Inserting Eq. (26) into (23), we obtain
l+BY In EY (27) +@ty), l i where O(Y) is analytical at Y since the prefactor V in front of the function 9 in Eq. (26) rules out the singularity of the denominator in Eq. (23). The multi-sheet structure of function N(Y) in the complex Y plane is entirely induced by ln[(l + BY)/(l +AY)]. Therefore, among N(Y) the first term in (27) contributes to g(x) in Eq. (lo), which produces N(Y)
= GG(-l/Y)
N+(P)
25-i = ~ r; G(l/I;).
-Ii-
Considering
(28)
Eqs. (25) and (28), we find
N+(F) -N-(F) 2riU(
P)
(29)
= G(X),
which is nothing but Eq. (10). By transforming n’(9)
-n-(9> 2rriu( 9)
(X, Y> back to (x, y), it is an easy matter to verify
(30)
=&T(x).
Actually the general multi-sheet structure of Z(z) and the relation between the jump of the integral at the singular point of the kernal can be given in a more direct manner. Assuming 4(x, y) and g(x) analytical with respect to both x and y, we can make the expansion
(31) where T’“‘( 2) = d”( +g)/dx”
I ~=h(i),
(32)
V=x-h(2).
(33)
Inserting this expansion into Eq. (2), we get H(Y) = &(Y)T
YldqY)l
i
b-h(Y) Ln a -h(y)
with o=
c n=l
Lb-NYP
I
+ @(Y),
[4YrT’“‘(y) nn!
where 0 is analytic on the whole complex y plane. The Riemann-sheet term in (341,
(35) structure of n(y) comes from the first
(36)
H. Gang et al. / Physics Letters A 205 (1995)
130-136
135
where cr takes the principal value with - rr < (Y < rr. Obviously CY= 0 corresponds to the case without analytical continuation. ii+ and iz-, satisfying Eq. (2) on both sides of the singular cut of the kernel, correspond to m = 0 and m = 1, respectively. From (34) one can see clearly that the residue of Eq. (2) at the singular point of the kernel is related to .+-nby Eq. (10). Eqs. (31)-(36) represent the general result that includes examples 3 and 4 as special cases. However, the forms of the multi-sheet structure of n(z) for the first two examples are different from that given by (31). The reason is that in these examples, g(x) itself is singular at some points. Nevertheless the following three points hold for all examples: (1) The dilated form of n(z) has a multi-sheet structure as K(y 1x) has a singular cut in the complex plane. (2) The integral equation holds on both sides of the singular cut. (3) The difference of n(z) between adjacent sheets at the singular cut is identical to the residue of the integral at this point. In this way, the solution of the inverse problem can be given in terms of Eq. (10). All the above analysis can be extended to multi-dimensional problems. The extension is, however, nontrivial. In the usual treatment the multi-dimensional inverse problem is much more complicated than the one-dimensional one. It would therefore be extremely useful to systematically overcome the difficulty caused by increasing the dimension. Let us first study the two-dimensional problem. We consider a two-dimensional version of (l), 4Yl>
Y2) =ldlbqYli
Y21 Xl,
x2)&7
4
dx1
dx2
(37)
and suppose n(y,, y2) and K(y,, y, / x1, x2) are known for real x1, x2 and for complex ~1, ~2. For simplicity, we will use y,, y, to denote the complex variables in the following. The function K has its singularity at (i-,, Pa; $r, F2), j1 =91Re+ii)llm,
92=i)2Re+ii)21my
where jjr and F2 are constrained by fi(jjl, 4 = hl( 91, 92)> The singularity
&=h,(j,,
(38)
E2) =f2(i),,
it,) = 0, and
P,).
(39)
is simple in the sense that the function
x2 - WqYl, Yz I x13 x2) 44x1, 4=(x1-4H (40) is analytical and nonzero at (.?r, Rz; Qr, jX2). There is a single complex variable y in the one-dimensional case, now two complex variables y, and y, are involved in our problem. Consequently, instead of K[+] and Krpl in Eq. (7) we should define K[++l, K[+-I, K[-+], and Kcppl for Eq. (31) as
K[+*](f,,
i)2 1 x1,
x2)
=
lim
1
lim
1
(41) Correspondingly, .[**I
we have
($,, &) =f~bK[+,(xl.
Applying the sameprocedures as those from Eqs. (7)-(lo), n1++1+
.[--I
- &+I
(42)
x2) dx, dx,.
- nc++1=
-4AL(9,,
j2)g(
we have x1, x2),
(43)
where we have ~(y,, jj2) = +(a,, .+Z2;jl, it,)- Eq. (37) yields n[++l dx17
x2)
+ .[--I
- n[+-l
-
&+I
= -
47r2z@,,
92)
’
(44)
136
H. Gang et al. /Physics
Letters A 205 (1995)
130-136
where the left-hand side becomes a function of x1, and x2 through the identities x1 2 = h, 2(i)l, i)2). All .[* * I are defined in adjacent Riemann hypersurfaces for both complex y, and y2 at the singular point of the kernel, (91, ii,>. The computations for the two-dimensional problem can be extended to multi-dimensional problems. Here we give the final solution of Eq. (1) only,
‘S(X1,.“’ xq)=
bl
(43
(25ri)“u(j,,...,jj,)’
where the notation [n] indicates an algebraic summation of 2” terms of all n[ * ... * I. Each term takes a coefficient (- 1)” with p being the number of minus signs in the superscript, namely [n]
=n [+
+] _ n[-+
+J _ n[+-+
+n[-+-+...+]+n[+--+...+]-n[---+.
+] + n[--+
+] _ n[++-+
+]
.+]+...(--1)4n[~...-].
(46)
In conclusion we would like to emphasize the following. The essential new points in this Letter are: We successfully reduce the inverse problem to the study of a singularity of the kernel and the discontinuity of the given function n(y) after analytical continuation; and get a unified and strikingly simple result which is valid for rather general inverse problems, Eqs. (1) and (2). This solution is expected to have very wide applications in the problems of inverse scattering, signal treatments, time evolutions of Hamiltonian and dissipative systems, and so on. The authors thank Dr. Chen for kindly informing them of his recent work and for directing the authors’ attention to the inverse problem, and thank Dr. Liu for showing them his recent work. G.H. was supported in part by the University of Stuttgart, and in part by the National Natural Foundation of China and the Nonlinear Science Project of China. C.Z.N. was supported by the DFG through SFB 329.
References [ll [2] 131 [4] 151 [6] [7] [8] [9] 101 111
V.A. Ligachev and V.A. Filikov, Fiz. Tverd. Tela (Leningrad) 33 (1991) 3292 [Sov. Phys. Solid State 33 (1991) U. Landman and E.W. Montroll, J. Chem. Phys. 64 (1976) 1762. R. Sips, J. Chem. Phys. 15 (1948) 490; 18 (1950) 1024. R.L. W&field, J. Appl. Phys. 54 (1983) 6401. J. Sworakowski and S. Nespurek, J. Appl. Phys. 65 (1989) 1559. N.X. Chen and G.B. Ren, Phys. Lett. A 160 (1991) 319. N.X. Chen, Phys. Rev. A 46 (1992) 3538. N.X. Chen, C.F. Zhang, M. Zhou, G.B. Ren, and W.B. Zhao, Phys. Rev. E 48 (1993) 3567. X.G. Zhao and J.X. Liu, Phys. Rev. E 48, October (1993). S. FenyG and H.W. Stolle, Theorie und Praxis der linearen Integralgleichungen 3 (Birkhauser, Basel, 1983). P.M. Morse and H. Feshbach, Methods of theoretical physics, Vol. 1 (McGraw-Hill, New York, 1953) p. 381.
18571
11 September 199.5 PHYSICS
ELSEVIER
Physics Letters
A 205 (1995)
LETTERS
A
130-136
Inverse problem and singularity of the integration kernel Hu Gang a-b,C.Z. Ning arc,*, H. Haken a a Institut fiir Theoretische
Physik und Synergetik b Department of Physics, ’ Arizona Center for Mathematical
Uniuersitiit Stuttgart, Pfaffenwaldring 57/J, D-70550 Stuttgart, Beijing Normal University, Beijing 100875, China 1 Sciences, Uniuersig ofArizona, Tucson, AZ 85721, USA ’
Received 31 March 1995; accepted for publication Communicated
Germany
6 July 1995
by A.R. Bishop
Abstract
Many important problems in physics and other sciences can be formulated in terms of the inverse problem of type n(y) = /K( y 1x)g( x> dx, where g(x) is unknown. We show that this problem can be completely solved for a quite general class of kernel K(y / x) by analytically dilating n(y) and K(y 1x) to the complex z plane, and by the analysis of the singularity of the dilated kernel K(z 1x). The formalism is also extended to multi-dimensional cases.
The inverse problem is an old problem which, however, attracts continued interest due to its theoretical importance and wide variety of applications [l-6]. Recently, Chen and his coworkers have made significant progressin this field [6-81. In particular, they relate the Fermi distribution to the Dirac 6 function and solved the inverse problem of Fermi systemsin a closed form by an elegant computation involving the 6 function, In Ref. [9], Zhao and Liu have improved the approach by introducing a limiting procedure and have put the formalism on a solid mathematical ground. In Ref. [8] Chen et al. showed that their approach can be applied to solve a number of inverse problems of practical importance of Fermi or Fermi-like systems by using a surprisingly simple calculation. In this Letter, we will develop a systematicalformalism, basedon the analytical continuation of n(y) and K(y 1x) to the complex z plane and the analysis of the singularities of the kernel K(z 1x), to completely solve the following general inverse problem,
* Corresponding author. ’ Present address. 2 Present address. 0375.9601/95/$09.50
0 1995 Ekevier
SSLM0375-9601(95)00544-7
Science
B.V. AI1 rights reserved
H. Gang et al./Physics
Letters
A 205 (1995)
130-136
131
where the kernel K and the function n are known while the function g is to be determined. The problem ‘defined by (1) is called an integral equation of the first kind [lo]. First we consider the one-dimensionalcase 4~)
=
IbK(y a
I4dx)
(2)
dx
In detail, and then extend the formalism to the multi-dimensional problem of Eq. (1). We first specify some conditions required for our approach. As in most practical problems, g(x) and K( y 1x) are real and analytic functions of real variables x and y in their supportsdefined by the corresponding physical problems. Thus, n(y) must be a real and analytic function, too, for real y. We assumethat K(y / x) and n(y) can be analytically dilated to complex z and denote the dilated right-hand side of Eq. (2) as S(z). Since K( y / x> is a known analytical function, its analytical continuation can be directly done by replacing real ,V with the complex variable z. We further assumethat the analytical continuation of the known n(y) to a complex z (denoted as Z(z)> in various Riemann sheetsis entirely known, too. Hence, the first crucial point in our approach is that Eq. (2) (and also Eq. (1)) is valid not only for real y but also for the complex z variable. This point can be easily verified by the above assumptionsof analytic continuation. K( y 1x) is assumedto be analytical for the variables x and y on its physically meaningful supports. .4ccording to the Liouville theorem [ll], there exist, however, singularities outside the physical region, if K(z / x> is bounded. We make the physically reasonable assumption that K(z I x) + 0 as I z I -j m. For simplicity, we assumethat K(z I x) has a unique singularity denoted by 2=/l(f).
(3)
The functions h(i) may be multi-valued. We consider only the case of single-valuedness.The singularity is assumed,in this Letter, to be simple in the sensethat the function 4(x,
z) = [x-+)]K(z/x)
(4)
is analytic and nonzero at (i, 2). The following approach can be extended to the more general caseswhere I x) has more than one singularities and Eq. (3) has multiple solutions. More general casesof the higher arder singularities could be similarly treated. We will present these detailed results elsewhere.It is well-known that
K(z
1 lim ~ 8’0 x-yii.5
1 =P----X-Y
fi7r6(x-y),
(5)
where P denotesthe Cauchy principal value. It is interesting to note that this formula can be extended to the following rather general and practical form, 1 K[‘l(ilx)
= iI”,
x&
&(x,
2) =4(x>
u(2) =qb(Z;, 2).
;)P
x-h(i)
+i7r~(i)8(~-h(i)), -
(6)
(7)
The extension from Eq. (5) to Eq. (6) is the most important step in our approach from which our main result follows. Analytically dilating Eq. (2) to complex z and applying Eqs. (6) and (7) we have n’(2)
=/bKr+1(2/~)g(~)
a
dx
1 =p / b$(x> Z>g(x> x-h(i) a
(8) dx&iru(i)g(P),
(9)
132
H. Gang et al. /Physics
Letters A 205 (1995)
130-136
leading to ;2+(2)
-A-(f)
=2inu(i)g(x),
(10) where f should be replaced by the x through the inverse function of (3), i.e., i = h-r(x). The result (10) is surprisingly simple. The inverse problem is now reduced to a simple algebraic computation. The main work is then to seek the singular point of K( z 1x), (2, J?), and to specify the function U( 2). It is to be emphasized that both A+ and ii- satisfy Eq. (2). Specifically, when K(z 1x) has a singular cut given by (3), Z(z) should have a multi-sheet structure. In the following calculation we will directly use ?I’(,?> - Z-(Z) in place of the left-hand side of (10). In certain cases expression (10) can be further simplified. (A) 4(x, 2) is real. Then Eq. (10) can be reduced to
K[+‘(il x) =I+‘(21 x>*,
(11)
g(x)
(12)
= [7rzQ(x))]-’
(B) 4(x,
Im[fi+(Z(x))].
2) is purely imaginary. Then we have
K[f’(z^l
x) = -I&‘(21
g(x)
X)‘)
= [rri+(i(x))]-1
Re[fi+(i(x))].
Chen et al. [7,8] have dealt with the inverse problem of the Fermi distribution which belongs to the case (A) with i=i,, + ri, 2 =iRe, and u(Z) = - 1. Generally, 4(x, 2) is complex with both nonzero real and imaginary parts, and one thus has to use formula (10). Now we analyze some examples to verify our results. The first two and the fourth examples are related to well-known practical physical problems. The third example is constructed to examine certain aspects of the general result Eq. 410). Example 1.
II(?.)=j;l
mdx) dx l+yx’ ’
n(y)=aex~[b(~/w,+w~/~)],
y>O,
(13)
which is a typical example of the problem of finding the relaxation-time distribution g(x) by measuring the permittivity of a material n(y). We have, from the dilated kernel l/(I + zx’), i=
--s
(s>O>,
i=h(i)
= T-l/&.
In this case we have u(i) = - \il z^1/2i = 2/2. This example belongs to case(A). Based on (12) and (13), we have g(x)
= gsin[
b(l/o,x
- wax)],
which is given in Refs. [I] and [8]. Our approach is, however, much simpler than Ref. [l] and more systematic than Ref. [8]. The dilated form of (13), n(z), has an obvious two-sheet structure. Taking, e.g., zr = y exp(4nrri) and z2 = y exp[(2n + 1) X 25-i)] in the integral of (13) we obtain the sameresult. Inserting zi and z2 into the known function ?I(z) leads to different values, namely ?I+ and %- respectively. Example 2. n( y) = Lm[ 1 + ~-‘a( T) exp( -X/H)] ?L(y) = [l +y-la(T)]
-=,
4.q
-I g( x) dx, > 0,
O
(15)
H. Gang et al. /Physics
Letters A 205 (1995)
130-136
133
This model is often used for describing the adsorption on heterogeneous substrates. The task is to find the distribution function g(x) from the measured total isotherm and the theoretically known local isotherm K(y 1x) (the pressure P in the problem is denoted by y in our expression). Now we have h n z = ZRe < 0, & = 0, 2 = h( 2) = RT ln[ -i,,/a( T)] , (16) leading to u(f) = RT. This model belongs to case (A) again. A direct investigation may immediately clarify the multi-sheet structure of the known function in this case. Here we will not go into details. The simplified formula (12) provides us with the solution Im[l
s(x)=&
1 sin( rc) = TRT [exp( x/RT) - l] ’ ’
+ a(T)/?]
(17)
which is again identical to the results obtained in Refs. [2], [3] and [s]. We have tested a number of practical inverse problems that have been investigated be others, and got correct results by using our much more direct and simple approach. Example 3.
= n(y)=L
yg(x> dx n(y)
l+exp(y-x)’
from which we have A ,. z=zRe + 7ri,
P=i,,,
u(2)
exp(-Y> ldl + exp(y)l)
=Y
=i=i,,+ri.
(19) Now u(T) has both nonzero real and imaginary parts, and we have to use the general formula (10). The difference of n(z) in various Riemann sheets is due to the function ln[l + exp(i)]. In particular, the difference in two adjacent sheets at f is given by E’(2)
-F(2)
=2rrii
exp(-i).
(20) Inserting Eqs. (19) and (20) into (lo), we obtain g(x) = exp(-x). The correctness of the result can be easily verified. Up to now, we have considered certain particular examples. To more convincingly confirm formula (lo), let us examine the following rather general and practical Fermi-like problem. Example 4. n(Y)
= lb
g(x) dx
a 1 +exp(x-y)
’
m>b>a>O,
(21)
which can be cast, by a transformation, X=exp(x),
Y=exp(-y)
(22)
into a simpler form, N(Y)
q”i:);,
(23)
with A=exp(a),
B=exp(b),
G(X)
=exp( -x)g(x).
(24)
Considering Eq. (31, (4), and (7) and denoting the singularity by 2 and f, we have --oo< -l/A
-l/B
R = l/f,
U(f)
= l/Y?
(25)
H. Gang ef al. /Physics
134
Letters
A 205 (1995)
130-136
Now let us directly evaluate the integral (23). Setting V= 1 +XY, we have G(X) = G(XY/Y) V/Y). According to the analyticity of G(X), we can make a standard expansion to get G(X)
= G( -l/Y)
with T(V,
= G(-
+ V?P(V, Y),
l/Y + (26)
Y) being analytical at V= 0. Inserting Eq. (26) into (23), we obtain
l+BY In EY (27) +@ty), l i where O(Y) is analytical at Y since the prefactor V in front of the function 9 in Eq. (26) rules out the singularity of the denominator in Eq. (23). The multi-sheet structure of function N(Y) in the complex Y plane is entirely induced by ln[(l + BY)/(l +AY)]. Therefore, among N(Y) the first term in (27) contributes to g(x) in Eq. (lo), which produces N(Y)
= GG(-l/Y)
N+(P)
25-i = ~ r; G(l/I;).
-Ii-
Considering
(28)
Eqs. (25) and (28), we find
N+(F) -N-(F) 2riU(
P)
(29)
= G(X),
which is nothing but Eq. (10). By transforming n’(9)
-n-(9> 2rriu( 9)
(X, Y> back to (x, y), it is an easy matter to verify
(30)
=&T(x).
Actually the general multi-sheet structure of Z(z) and the relation between the jump of the integral at the singular point of the kernal can be given in a more direct manner. Assuming 4(x, y) and g(x) analytical with respect to both x and y, we can make the expansion
(31) where T’“‘( 2) = d”( +g)/dx”
I ~=h(i),
(32)
V=x-h(2).
(33)
Inserting this expansion into Eq. (2), we get H(Y) = &(Y)T
YldqY)l
i
b-h(Y) Ln a -h(y)
with o=
c n=l
Lb-NYP
I
+ @(Y),
[4YrT’“‘(y) nn!
where 0 is analytic on the whole complex y plane. The Riemann-sheet term in (341,
(35) structure of n(y) comes from the first
(36)
H. Gang et al. / Physics Letters A 205 (1995)
130-136
135
where cr takes the principal value with - rr < (Y < rr. Obviously CY= 0 corresponds to the case without analytical continuation. ii+ and iz-, satisfying Eq. (2) on both sides of the singular cut of the kernel, correspond to m = 0 and m = 1, respectively. From (34) one can see clearly that the residue of Eq. (2) at the singular point of the kernel is related to .+-nby Eq. (10). Eqs. (31)-(36) represent the general result that includes examples 3 and 4 as special cases. However, the forms of the multi-sheet structure of n(z) for the first two examples are different from that given by (31). The reason is that in these examples, g(x) itself is singular at some points. Nevertheless the following three points hold for all examples: (1) The dilated form of n(z) has a multi-sheet structure as K(y 1x) has a singular cut in the complex plane. (2) The integral equation holds on both sides of the singular cut. (3) The difference of n(z) between adjacent sheets at the singular cut is identical to the residue of the integral at this point. In this way, the solution of the inverse problem can be given in terms of Eq. (10). All the above analysis can be extended to multi-dimensional problems. The extension is, however, nontrivial. In the usual treatment the multi-dimensional inverse problem is much more complicated than the one-dimensional one. It would therefore be extremely useful to systematically overcome the difficulty caused by increasing the dimension. Let us first study the two-dimensional problem. We consider a two-dimensional version of (l), 4Yl>
Y2) =ldlbqYli
Y21 Xl,
x2)&7
4
dx1
dx2
(37)
and suppose n(y,, y2) and K(y,, y, / x1, x2) are known for real x1, x2 and for complex ~1, ~2. For simplicity, we will use y,, y, to denote the complex variables in the following. The function K has its singularity at (i-,, Pa; $r, F2), j1 =91Re+ii)llm,
92=i)2Re+ii)21my
where jjr and F2 are constrained by fi(jjl, 4 = hl( 91, 92)> The singularity
&=h,(j,,
(38)
E2) =f2(i),,
it,) = 0, and
P,).
(39)
is simple in the sense that the function
x2 - WqYl, Yz I x13 x2) 44x1, 4=(x1-4H (40) is analytical and nonzero at (.?r, Rz; Qr, jX2). There is a single complex variable y in the one-dimensional case, now two complex variables y, and y, are involved in our problem. Consequently, instead of K[+] and Krpl in Eq. (7) we should define K[++l, K[+-I, K[-+], and Kcppl for Eq. (31) as
K[+*](f,,
i)2 1 x1,
x2)
=
lim
1
lim
1
(41) Correspondingly, .[**I
we have
($,, &) =f~bK[+,(xl.
Applying the sameprocedures as those from Eqs. (7)-(lo), n1++1+
.[--I
- &+I
(42)
x2) dx, dx,.
- nc++1=
-4AL(9,,
j2)g(
we have x1, x2),
(43)
where we have ~(y,, jj2) = +(a,, .+Z2;jl, it,)- Eq. (37) yields n[++l dx17
x2)
+ .[--I
- n[+-l
-
&+I
= -
47r2z@,,
92)
’
(44)
136
H. Gang et al. /Physics
Letters A 205 (1995)
130-136
where the left-hand side becomes a function of x1, and x2 through the identities x1 2 = h, 2(i)l, i)2). All .[* * I are defined in adjacent Riemann hypersurfaces for both complex y, and y2 at the singular point of the kernel, (91, ii,>. The computations for the two-dimensional problem can be extended to multi-dimensional problems. Here we give the final solution of Eq. (1) only,
‘S(X1,.“’ xq)=
bl
(43
(25ri)“u(j,,...,jj,)’
where the notation [n] indicates an algebraic summation of 2” terms of all n[ * ... * I. Each term takes a coefficient (- 1)” with p being the number of minus signs in the superscript, namely [n]
=n [+
+] _ n[-+
+J _ n[+-+
+n[-+-+...+]+n[+--+...+]-n[---+.
+] + n[--+
+] _ n[++-+
+]
.+]+...(--1)4n[~...-].
(46)
In conclusion we would like to emphasize the following. The essential new points in this Letter are: We successfully reduce the inverse problem to the study of a singularity of the kernel and the discontinuity of the given function n(y) after analytical continuation; and get a unified and strikingly simple result which is valid for rather general inverse problems, Eqs. (1) and (2). This solution is expected to have very wide applications in the problems of inverse scattering, signal treatments, time evolutions of Hamiltonian and dissipative systems, and so on. The authors thank Dr. Chen for kindly informing them of his recent work and for directing the authors’ attention to the inverse problem, and thank Dr. Liu for showing them his recent work. G.H. was supported in part by the University of Stuttgart, and in part by the National Natural Foundation of China and the Nonlinear Science Project of China. C.Z.N. was supported by the DFG through SFB 329.
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