JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 2
FEBRUARY 2002
Inverse scattering theory of the heat equation
for a perturbed one-soliton potential
M. Boiti and F. Pempinelli
Dipartimento di Fisica dell’Università and Sezione INFN, 73100 Lecce, Italy
A. K. Pogrebkov
Steklov Mathematical Institute Moscow, 117966, GSP-1, Russia
B. Prinari
Dipartimento di Fisica dell’Università and Sezione INFN, 73100 Lecce, Italy
共Received 20 June 2001; accepted for publication 24 October 2001兲
The inverse scattering theory of the heat equation is developed for a special subclass of potentials nondecaying at space infinity—perturbations of the one-soliton
potential by means of decaying two-dimensional functions. Extended resolvent,
Green’s functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the
inverse problem is formulated in an exact distributional sense. © 2002 American
Institute of Physics. 关DOI: 10.1063/1.1427410兴
I. INTRODUCTION
The equation of the heat conduction, or heat equation for short,
L⌽ 共 x 兲 ⫽0,
共1.1兲
where the operator
L共 x, ⳵ x 兲 ⫽⫺ ⳵ x 2 ⫹ ⳵ x2 ⫺u 共 x 兲 ,
1
x⫽ 共 x 1 ,x 2 兲 ,
共1.2兲
for more than 25 years has been known1,2 to be associated to the Kadomtsev–Petviashvili 共more
precisely, KPII兲 equation
共 u t ⫺6uu x 1 ⫹u x 1 x 1 x 1 兲 x 1 ⫽⫺3u x 2 x 2 .
共1.3兲
The scattering theory for the equation of heat conduction with a real potential u(x) was developed
in Refs. 3–5, but only the case of potentials rapidly decaying at large distances on the x-plane was
considered. On the other side, it is well known that 共1.3兲 is a (2⫹1)-dimensional generalization of
the famous KdV equation: if the function u 1 (t,x 1 ) obeys KdV, then
u 共 t,x 1 ,x 2 兲 ⫽u 1 共 t,x 1 ⫹ ␮ x 2 ⫺3 ␮ 2 t 兲
共1.4兲
solves 共1.3兲 for an arbitrary constant ␮ 苸R. Thus it is natural to consider solutions of 共1.3兲 that are
not decaying in all directions at space infinity but have one-dimensional rays with behavior of the
type 共1.4兲. The scattering theory for the operator 共1.2兲 with such potentials is absent in the
literature. Moreover, it is easy to observe that, like in the KPI case 共see Ref. 6兲, the standard
integral equation for the Jost solution3 is meaningless for this situation and does not determine the
solution itself. In trying to solve this problem for the nonstationary Schrödinger operator, associated to the KPI equation, a new general approach to the inverse scattering theory was introduced,
which was called resolvent approach 共see Refs. 6 –10 and references therein兲. In Ref. 11 we
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© 2002 American Institute of Physics
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
1045
developed the scattering theory for the N-soliton solutions given in terms of Bäcklund transformations of the decaying background potential. These results for the simplest case N⫽1 are
essentially used below.
Here we apply the resolvent approach to the heat equation 共1.2兲 with a potential u(x) that is
a perturbation of a one-dimensional potential u 1 (x) of the kind 共1.4兲 by means of a potential u 2 (x)
rapidly decaying in all directions:
u 共 x 兲 ⫽u 1 共 x 兲 ⫹u 2 共 x 兲 .
共1.5兲
We introduce and study properties of the resolvent, dressing operators, Jost solutions and scattering data, and formulate the inverse problem relevant to this case. In fact, we consider here the
simplified version of 共1.4兲 in which ␮ ⫽0. The generic case is reconstructed by means of the
Galilean invariance of 共1.3兲. Thus in what follows u 1 (x)⬅u 1 (x 1 ) and, moreover, we consider for
simplicity the case where u 1 is the one-dimensional soliton potential 关see 共3.1兲兴.
Thus here we apply the inverse scattering theory to a nonscattering situation since the ‘‘obstacle’’ is infinite. Such extension of the inverse scattering theory results in the new and unexpected properties of familiar objects, like the Jost solutions and the spectral data. We show that
they get specific singularities in the complex domain of the spectral parameter. Derivation and
description of these singularities are our main results here. The article is organized as follows. In
Sec. II we sketch some general aspects of the resolvent approach that are necessary for our
construction. In Sec. III we present results of embedding the theory of the one-dimensional
one-soliton potential in two dimensions. Presentation here is based on work Ref. 11. We describe
in detail properties of the extended resolvent and Green’s functions of operator 共1.2兲 with u(x)
⫽u 1 (x 1 ). On this basis in Sec. IV the resolvent of the operator 共1.2兲 now with the generic
potential u(x) given in 共1.5兲 is introduced and its properties are described. The departure from
analyticity of the resolvent leads us to definitions of the Jost solutions and spectral data and
description of their properties 共Sec. V兲. In this way we supply all terms of the inverse problem
with proper meaning in terms of distributions. In the Conclusion some generalizations and future
developments of these results are discussed. The main results of this article were announced in our
earlier work, Ref. 12.
II. EXTENSION OF DIFFERENTIAL OPERATORS AND RESOLVENT
In the framework of the resolvent approach we work in the space S ⬘ of tempered distributions
A(x,x ⬘ ;q) of the six real variables x⫽(x 1 ,x 2 ),x ⬘ ,q苸R2 . It is convenient to consider q as the
imaginary part of a two-dimensional complex variable q⫽qR⫹iqI⫽(q1 ,q2 )苸C2 and to introduce
the ‘‘shifted’’ Fourier transform
A 共 p;q兲 ⫽
1
共 2␲ 兲2
冕 冕
dx
dx ⬘ e i(p⫹qR)x⫺iqRx ⬘ A 共 x,x ⬘ ;qI兲 ,
共2.1兲
where p苸R2 , px⫽p 1 x 1 ⫹p 2 x 2 and qRx⫽q1Rx 1 ⫹q2Rx 2 . We consider the distributions
A(x,x ⬘ ;q) and A(p;q) as kernels in two different representations, the x- and p-representation,
respectively, of the operator A(q) 共A for short兲. The composition law in the x-representation is
defined in the standard way, that is,
共 AB 兲共 x,x ⬘ ;q 兲 ⫽
冕
dx ⬙ A 共 x,x ⬙ ;q 兲 B 共 x ⬙ ,x ⬘ ;q 兲 .
共2.2兲
Since the kernels are distributions, this composition is neither necessarily defined for all pairs of
operators nor associative. In terms of the p-representation 共2.1兲, this composition law is given by
a sort of a ‘‘shifted’’ convolution, (AB)(p;q)⫽ 兰 d p ⬘ A( p⫺ p ⬘ ;q⫹ p ⬘ )B(p ⬘ ;q). On the space of
these operators we define the conjugation A * , which in the x-representations reads as
A * 共 x,x ⬘ ;q 兲 ⫽A 共 x,x ⬘ ;q 兲 ,
共2.3兲
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1046
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
where bar denotes complex conjugation, or as A * (p;q)⫽A(⫺ p;⫺q̄) in the p-representation.
Below we say that the operator A(q) is real if A * (q)⫽A(q), which in terms of p-representation
means that A(p;q)⫽A(⫺p;⫺q̄). The set of differential operators D(x, ⳵ x )⫽ 兺 d n (x) ⳵ nx is embedded in the introduced space of operators by considering the operators D with kernel D(x,x ⬘ )
⫽D(x, ⳵ x ) ␦ (x⫺x ⬘ ), where ␦ (x)⫽ ␦ (x 1 ) ␦ (x 2 ) is the two-dimensional ␦-function and, then, by
mapping them in the operators D(q) with kernel
D 共 x,x ⬘ ;q 兲 ⬅e ⫺q(x⫺x ⬘ ) D 共 x,x ⬘ 兲 ⫽D共 x, ⳵ x ⫹q 兲 ␦ 共 x⫺x ⬘ 兲 ,
共2.4兲
to which we refer as the extended version of the differential operator D. The notion of reality for
a differential operator D is exactly the condition that its coefficients d n (x) are real.
For the operator 共1.2兲 the extension L(q) is given by
L⫽L 0 ⫺U,
共2.5兲
where L 0 is the extension of L(x, ⳵ x ) in the case of zero potential, i.e., it has kernels
L 0 共 x,x ⬘ ;q 兲 ⫽ 关 ⫺ 共 ⳵ x 2 ⫹q 2 兲 ⫹ 共 ⳵ x 1 ⫹q 1 兲 2 兴 ␦ 共 x⫺x ⬘ 兲 ,
L 0 共 p;q兲 ⫽ 共 iq2 ⫺q21 兲 ␦ 共 p 兲 ,
共2.6兲
and the multiplication operator U can be called the potential since it has kernel
U 共 x,x ⬘ ;q 兲 ⫽u 共 x 兲 ␦ 共 x⫺x ⬘ 兲 .
共2.7兲
Below we always suppose that u(x) is real, which by 共2.3兲 means that the operator 共1.2兲 is real
also: L * ⫽L.
The main object of our approach is the extended resolvent M (q) of the operator L(q), which
is defined as the inverse of the operator L, that is,
LM ⫽M L⫽I,
共2.8兲
in the space of operators. Here I is the unity operator, I(x,x ⬘ ;q)⫽ ␦ (x⫺x ⬘ ), I(p;q)⫽ ␦ (p). In
order to make this inversion uniquely defined we impose the condition that the product
冕
dp ⬘ M 共 p⫺ p ⬘ ;q⫹s⫹ p ⬘ 兲 M 共 p ⬘ ;q兲
共2.9兲
exists as distribution in p and q and that it is a continuous function of s in a neighborhood of s
⫽0 when s⫽0.
Thanks to definitions 共2.5兲, 共2.6兲, and 共2.8兲, M is real and, in particular, the resolvent M 0 of
the bare operator L 0 has in the p-representation kernel
M 0 共 p;q兲 ⫽ ␦ 共 p 兲
1
iq2 ⫺q21
共2.10兲
.
As function of q it is singular when q⫽ l (q1 ), where the special two-component vector
l 共 k 兲 ⫽ 共 k,⫺ik 2 兲
共2.11兲
was introduced. The kernel of M 0 in the x-representation
M 0 共 x,x ⬘ ;q 兲 ⫽
1
2␲
冕
d ␣ 关 ␪ 共 q 21 ⫺ ␣ 2 ⫺q 2 兲 ⫺ ␪ 共 x 2 ⫺x ⬘2 兲兴 e ⫺(i l
( ␣ ⫹iq 1 )⫹q)(x⫺x ⬘ )
共2.12兲
is obtained from 共2.10兲 by using 共2.1兲.
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
1047
For a generic operator A with kernel A(x,x ⬘ ;q) the operation inverse to the extension procedure, defined in 共2.4兲 for a differential operator, is given by
 共 x,x ⬘ ;q 兲 ⫽e q(x⫺x ⬘ ) A 共 x,x ⬘ ;q 兲 .
共2.13兲
In contrast with the case of the extended differential operators for which D̂(x,x ⬘ ;q)⫽D(x,x ⬘ )
⬅D(x, ⳵ x ) ␦ (x⫺x ⬘ ), in general Â(x,x ⬘ ;q) does depend on q and, moreover, can have an exponential growth at space infinity. Therefore Â(x,x ⬘ ;q) does not necessarily belong to the space S ⬘
of tempered distributions. The fact that Â(x,x ⬘ ;q) can depend on q will play a crucial role in the
following. For instance, also in the case of the simplest resolvent 共2.12兲, we have that the function
M̂ 0 (x,x ⬘ ;q) depends effectively on the variable q and is exponentially growing at space infinity.
More generally from 共2.8兲 we have
L共 x, ⳵ x 兲 M̂ 共 x,x ⬘ ;q 兲 ⫽L d共 x ⬘ , ⳵ x ⬘ 兲 M̂ 共 x,x ⬘ ;q 兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ,
共2.14兲
where L d is the operator dual to L. The function M̂ (x,x ⬘ ;q) can be considered a parametric (q
苸R2 ) family of Green’s functions of the operator L. In what follows we use special notations for
the equalities of the type 共2.14兲, writing them as
ជ M̂ 共 q 兲 ⫽M̂ 共 q 兲 L
ឈ ⫽I,
L
共2.15兲
ជ denotes the operator L applied to the x-variable of the function M̂ (x,x ⬘ ;q) and L
ឈ denotes
where L
the operator dual to L applied to the x ⬘ -variable of the same function. Operation 共2.13兲 has no
analog in terms of the p-representation. Nevertheless, local properties of the kernels in the
x-representation are preserved, and we use the kernels with the hat in what follows intensively.
Thanks to our definitions 共2.1兲 and 共2.4兲 it is easy to see that in terms of the p-representation
the dependence on the q-variables of the kernels of the extension of a differential operator is
polynomial 关like in the example 共2.6兲兴. Correspondingly, the essential role in the study of the
properties of the resolvent is played by the investigation of its departure from analyticity, in
particular, by its d-bar derivatives with respect to the q-variables. Thus to a generic operator A
with kernel A(p;q) in the p-representation we associate two operators ¯⳵ j A with kernels
共¯⳵ j A 兲共 p;q兲 ⫽
⳵ A 共 p;q兲
,
⳵ q̄ j
j⫽1,2,
共2.16兲
where the derivatives are considered in the sense of distributions. In terms of the objects introduced in 共2.13兲 we get by inversion of 共2.1兲 that
⳵ j A 兲共 q 兲 ⫽
共¯d
i ⳵ Â 共 q 兲
.
2 ⳵q j
共2.17兲
Multiplying equalities in 共2.8兲 from the left and right, correspondingly, by M 0 we get, thanks
to 共2.5兲,
M ⫽M 0 ⫹M 0 UM ,
M ⫽M 0 ⫹M UM 0 .
共2.18兲
Since the resolvent M 0 is explicitly given, these are integral equations determining the solution M
of 共2.8兲. In the literature 共see, say, Refs. 4 and 5兲 on the Jost solutions of the heat equation some
small norm conditions on the potential u are known to guarantee the existence of the Jost solutions. So it is natural to assume that under such conditions the solution M of the above integral
equations exists and is unique 共the same for both integral equations兲. In this case the resolvent M
can be considered as a small perturbation of the resolvent M 0 and this bare resolvent determines
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1048
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
the properties of M by means of 共2.18兲. The main problem of construction of the inverse scattering
transform for the operator 共1.2兲 is that the potential u(x) in 共1.5兲 does not obey any small norm
condition.
In order to overcome this difficulty we use a so-called inverse scattering transform on a
nontrivial background.10 Let us consider a kind of Hilbert identity, known in the standard spectral
theory of operators. Precisely, if M (q) is the extended resolvent of the operator L(q) with potential u and M ⬘ (q) the extended resolvent of the operator L ⬘ (q) with a different potential u ⬘ , then,
by 共2.8兲 we have
M ⬘ ⫺M ⫽⫺M ⬘ 共 L ⬘ ⫺L 兲 M .
共2.19兲
Strictly speaking, this follows under the assumption that the product on the rhs is associative. This
is a natural assumption since L ⬘ (x,x ⬘ ;q)⫺L(x,x ⬘ ;q)⫽(u(x)⫺u(x ⬘ )) ␦ (x⫺x ⬘ ) and M satisfies
condition 共2.9兲. Let now L1 denote the operator 共1.2兲 in the special case where the potential u(x)
in 共1.5兲 is purely one-dimensional, i.e., u 2 (x)⬅0. Let L 1 denote its extension and M 1 its resolvent,
that is, let 关cf. 共2.5兲兴
L 1 ⫽L 0 ⫺U 1 ,
L1 共 x, ⳵ x 兲 ⫽⫺ ⳵ x 2 ⫹ ⳵ x2 ⫺u 1 共 x 兲 ,
1
L⫽L 1 ⫺U 2 ,
L 1 M 1 ⫽M 1 L 1 ⫽I,
共2.20兲
共2.21兲
where as in 共2.7兲 U j (x,x ⬘ ;q)⫽u j (x) ␦ (x⫺x ⬘ ). Choosing now in 共2.19兲 L ⬘ ⫽L 1 we get
M ⫽M 1 ⫹M 1 U 2 M ,
M ⫽M 1 ⫹M U 2 M 1 ,
共2.22兲
where the second equality is derived in analogy. These equations generalize 共2.18兲 for the case
where M 1 is nontrivial and, if the resolvent M 1 is known, they also can be considered as defining
the resolvent M . If we choose U 2 obeying the small norm condition mentioned above, we can
assume that the solution M of both Eqs. 共2.22兲 exists and is unique. Then, thanks to 共2.21兲, M
obeys 共2.8兲. Contrary to 共2.18兲, now M can be considered a perturbation of the resolvent M 1 . So,
in the next section, we study the properties of the resolvent M 1 in detail.
III. ONE-DIMENSIONAL POTENTIAL
We already mentioned in the Introduction that in this article we deal with the case where u 1
in 共1.5兲 is the one-dimensional soliton potential
u 1共 x 兲 ⫽
⫺2a 2
,
cosh2 关 a 共 x 1 ⫺x 0 兲兴
共3.1兲
with a⬎0 and x 0 real constants. In this section we consider the case where the perturbation is
absent, u 2 ⬅0. We reformulate in the two-dimensional space the well known facts about Jost
solutions of this one-soliton potential and introduce and study the properties of the resolvent and
Green’s functions in this case.
The differential equations L1 (x, ⳵ x )⌽ 1 (x,k)⫽0, L d1 (x ⬘ , ⳵ x ⬘ )⌿ 1 (x ⬘ ,k)⫽0 for the Jost solution
⌽ 1 (x,k) and its dual ⌿ 1 (x ⬘ ,k), by using the notation introduced in 共2.15兲 and by considering
⌽ 1 (x,k) and ⌿ 1 (x ⬘ ,k) as kernel operators independent, respectively, of x ⬘ and x, can be rewritten
in the operatorial form
ជ 1 ⌽ 1 共 k 兲 ⫽0,
L
ឈ 1 ⫽0,
⌿ 1共 k 兲 L
共3.2兲
which we will use frequently in the following. These solutions are given explicitly by
⌽ 1 共 x,k 兲 ⫽
k⫺ia tanh关 a 共 x 1 ⫺x 0 兲兴 ⫺i l
e
k⫺ia
(k)x
,
共3.3兲
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J. Math. Phys., Vol. 43, No. 2, February 2002
⌿ 1 共 x,k 兲 ⫽
Inverse scattering theory
k⫹ia tanh关 a 共 x 1 ⫺x 0 兲兴 i l
e
k⫹ia
(k)x
1049
共3.4兲
,
where k苸C and the two-component vector l (k) is defined in 共2.11兲. They obey the conjugation
properties
⌽ 1 共 x,k 兲 ⫽⌽ 1 共 x,⫺k̄ 兲 ,
⌿ 1 共 x,k 兲 ⫽⌿ 1 共 x,⫺k̄ 兲
共3.5兲
that are equivalent to the reality condition for the potential u 1 , and are normalized at k-infinity as
follows:
lim e i l
(k)x
lim e ⫺i l
⌽ 1 共 x,k 兲 ⫽1,
k→⬁
(k)x
⌿ 1 共 x,k 兲 ⫽1.
共3.6兲
k→⬁
The functions ⌽ 1 (x,k) and ⌿ 1 (x,k) are meromorphic in the complex domain of the spectral
parameter k with poles at k⫽ia and k⫽⫺ia, correspondingly. Thus, these functions obey the
d-bar equations
⳵ ⌽ 1 共 x,k 兲
⳵ k̄
⳵ ⌿ 1 共 x,k 兲
⫽i ␲ ⌽ 1,a 共 x 兲 ␦ 共 k⫺ia 兲 ,
⳵ k̄
⫽i ␲ ⌿ 1,⫺a 共 x 兲 ␦ 共 k⫹ia 兲 ,
共3.7兲
where we introduced the notations
⌽ 1,a 共 x 兲 ⫽⫺i res ⌽ 1 共 x,k 兲 ,
⌿ 1,⫺a 共 x 兲 ⫽⫺i res ⌿ 1 共 x,k 兲 .
k⫽ia
共3.8兲
k⫽⫺ia
Explicitly we have
2
ae ax 0 ⫹a x 2
,
⌽ 1,a 共 x 兲 ⫽
cosh关 a 共 x 1 ⫺x 0 兲兴
2
ae ax 0 ⫺a x 2
⌿ 1,⫺a 共 x 兲 ⫽⫺
.
cosh关 a 共 x 1 ⫺x 0 兲兴
共3.9兲
Let
共3.10兲
c⫽2ae 2ax 0
and ⌽ 1,⫺a (x) and ⌿ 1,a (x) be the values of the Jost solutions in the conjugated points,
⌽ 1,⫺a 共 x 兲 ⫽⌽ 1 共 x,⫺ia 兲 ,
⌿ 1,a 共 x 兲 ⫽⌿ 1 共 x,ia 兲 .
共3.11兲
Then, thanks to 共3.3兲, 共3.4兲, and 共3.9兲, the Jost solutions obey in the complex domain of the
spectral parameter the following scalar products:
冕
dx 1 ⌿ 1 共 x,k⫹p 兲 ⌽ 1 共 x,k 兲 ⫽2 ␲ ␦ 共 p 兲 ,
c
冕
冕
冕
p苸R,
dx 1 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x 兲 ⫽1,
dx 1 ⌿ 1,a 共 x 兲 ⌽ 1 共 x,k 兲 ⫽0,
dx 1 ⌽ 1,⫺a 共 x 兲 ⌿ 1 共 x,k 兲 ⫽0,
k I2 ⬍a 2 ,
k I2 ⬍a 2 ,
共3.12兲
共3.13兲
共3.14兲
共3.15兲
and the completeness relation
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1050
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
1
2␲
冕
x 2 ⫽x ⬘2
dk R ⌽ 1 共 x,k 兲 ⌿ 1 共 x ⬘ ,k 兲 ⫹c ␪ 共 a 2 ⫺k I2 兲 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x ⬘ 兲
冏
⫽ ␦ 共 x 1 ⫺x ⬘1 兲 .
x 2 ⫽x ⬘2
共3.16兲
Equations 共3.7兲 can be considered as two inverse problems defining the Jost solution and the
dual Jost solution. The formulation of these problems is closed by giving the normalization
conditions 共3.6兲 and the following relations:
⌽ 1,a 共 x 兲 ⫽c⌽ 1,⫺a 共 x 兲 ,
⌿ 1,⫺a 共 x 兲 ⫽⫺c⌿ 1,a 共 x 兲 ,
共3.17兲
where ⌽ 1,⫺a (x) and ⌿ 1,a (x) are defined in 共3.11兲.
These formulas show that the embedding in two dimensions of the Jost solutions of the
one-soliton potential is trivial and just mimics the one-dimensional construction up to the warning
that, due to their exponential dependence on x 2 , the functions ⌽ 1,a (x) and ⌿ 1,⫺a (x) are not
square integrable with respect to the x-variables and, therefore, are not eigenfunctions of the
operator L1 .
On the contrary, the two-dimensional resolvent M 1 (q) as well as the Green’s function G 1 of
the two-dimensional operator L1 with the one-dimensional potential u 1 are not trivial extensions
of the corresponding one-dimensional objects associated to the operator (k 2 ⫹ ⳵ x2 ⫺u 1 (x 1 )). In
1
terms of the Jost solutions introduced above, we can write the kernel of this resolvent obtained in
Ref. 11 as
M̂ 1 共 x,x ⬘ ;q 兲 ⫽
1
2␲
冕
k I⫽q 1
2
dk R 关 ␪ 共 q 21 ⫺q 2 ⫺k R
兲 ⫺ ␪ 共 x 2 ⫺x 2⬘ 兲兴 ⌽ 1 共 x,k 兲 ⌿ 1 共 x ⬘ ,k 兲
⫹c ␪ 共 a 2 ⫺q 21 兲关 ␪ 共 a 2 ⫺q 2 兲 ⫺ ␪ 共 x 2 ⫺x 2⬘ 兲兴 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x ⬘ 兲 ,
共3.18兲
where the hat over the kernel is used in the sense of notation 共2.13兲.
Thanks to the equalities 共3.2兲 and 共3.16兲 it is easy to check directly that M̂ 1 (q) obeys the
equations
ជ 1 M̂ 1 共 q 兲 ⫽M̂ 1 共 q 兲 L
ឈ 1 ⫽I,
L
共3.19兲
which means 关cf. 共2.15兲兴 that M 1 (q) obeys 共2.21兲 and is indeed the inverse of the operator L 1 (q).
Moreover, using the explicit formulas 共3.3兲, 共3.4兲 and 共3.9兲 we get that M 1 (x,x ⬘ ;q)苸S ⬘ (R 6 ), i.e.,
it belongs to the space of operators under consideration. It can also be proved directly that M 1
obeys condition 共2.9兲, so it is the extended resolvent according to our definition. By means of 共3.5兲
we get also that M 1 is real, M 1* ⫽M 1 according to definition 共2.3兲.
We emphasize that in order to prove these results it is not necessary to use the explicit
formulas for ⌽ 1 and ⌿ 1 but only their general properties. In fact, if one considers a onedimensional potential u 1 which, in addition, has a nontrivial continuous spectrum, one gets the
same formula for the resolvent M 1 . If the discrete 共one-dimensional兲 spectrum of u 1 contains N
solitons with parameters a j and c j ( j⫽1,2, . . . ,N), then the last term in 共3.18兲 must be substituted
by the sum of similar terms each corresponding to a value of j.
Now we describe in detail the properties of M 1 (x,x ⬘ ;q). The first term on the rhs of 共3.18兲 is
a continuous function of q⫽(q 1 ,q 2 ) on the q-plane with discontinuities on the lines q 1 ⫽⫾a due
to the pole singularities of ⌽ 1 (k) and ⌿ 1 (k). The second term, thanks to the ␪ functions, has
discontinuities on the lines q 1 ⫽⫾a and on the cut q 2 ⫽a 2 , 兩 q 1 兩 ⬍a. The singularities on the lines
q 1 ⫽⫾a are exactly compensated among the two terms. Thus the kernel M 1 (x,x ⬘ ;q) is a continuous function of q with a discontinuity on the cut q 2 ⫽a 2 , 兩 q 1 兩 ⬍a. This discontinuity is
specific of the potential u 1 (x), or more generally of a potential with discrete spectrum and it gives
the essential difference of M 1 with respect to the bare resolvent M 0 共2.12兲. Let us underline that,
in spite of the fact that L1 applied to the term with ␪ (a 2 ⫺q 2 ) that causes this discontinuity gives
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
1051
zero, this term cannot be omitted in 共3.18兲, since, only thanks to the fact that ␪ (x 2 ⫺x ⬘2 ) and
␪ (a 2 ⫺q 2 ) have opposite signs, the kernel M 1 (x,x ⬘ ;q)⬅e ⫺q(x⫺x ⬘ ) M̂ 1 (x,x ⬘ ;q) is a tempered distribution with respect to the x-variables.
The kernel M 1 (p;q) in the p-representation is not an analytic function of q. By 共2.17兲 the
d-bar derivatives of M 1 with respect to q j are proportional to ⳵ M̂ 1 / ⳵ q j and for the latter we get
from 共3.18兲 equalities
⳵ M̂ 1 共 q 兲 i
⫽
⳵q1
␲
冕
k I⫽q 1
⳵ M̂ 1 共 q 兲 ⫺1
⫽
⳵q2
2␲
冕
dk R¯k ␦ 共 l
2I共 k 兲 ⫺q 2 兲
⌽ 1共 k 兲 丢 ⌿ 1共 k 兲 ,
共3.20兲
dk R ␦ 共 l
2I共 k 兲 ⫺q 2 兲
⌽ 1共 k 兲 丢 ⌿ 1共 k 兲 ,
共3.21兲
k I⫽q 1
where (⌽ 1 (k) 丢 ⌿ 1 (k))(x,x ⬘ )⬅⌽ 1 (x,k)⌿ 1 (x ⬘ ,k) is the standard direct product, l 2I(k) is the
imaginary part of the second component of the vector l (k) defined in 共2.11兲 and where by the
above discussion we consider q 2 ⫽a 2 . For the discontinuity along this line we get
M̂ 1 共 q 兲 兩 q 2 ⫽a 2 ⫹0 ⫺M̂ 1 共 q 兲 兩 q 2 ⫽a 2 ⫺0 ⫽⫺c ␪ 共 a 2 ⫺q 21 兲 ⌽ 1,⫺a 丢 ⌿ 1,a .
共3.22兲
We have to study now the behavior of M 1 (q) at the end points of the cut, i.e., when q
⬃(⫾a,a 2 ). First, it is convenient to subtract from M 1 (q) its value, say, on the upper or lower
edges of the cut:
g⫾
1 ⫽
共3.23兲
lim M̂ 1 共 q 兲 兩 兩 q 1 兩 ⬍a .
q 2 ⫽a 2 ⫾0
Since ⌽ 1 (k) and ⌿ 1 (k) are analytic for 兩 k I兩 ⬍a, we deduce from 共3.18兲 that g ⫾
1 are independent
also of q 1 and their kernels equal
g⫾
1 共 x,x ⬘ 兲 ⫽⫺
␪ 共 x 2 ⫺x 2⬘ 兲
2␲
冕
d ␣ ⌽ 1 共 x, ␣ 兲 ⌿ 1 共 x ⬘ , ␣ 兲 ⫿c ␪ 共 ⫾ 共 x 2 ⫺x ⬘2 兲兲 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x ⬘ 兲 ,
共3.24兲
where 兰 d ␣ denotes integration along the whole real axis. Now extracting explicitly from the first
term on the rhs of 共3.18兲 the contribution coming from the poles of ⌽ 1 (k) and ⌿ 1 (k) we get that,
2
say, difference M̂ 1 (q)⫺g ⫺
1 behaves at points q⫽(⫾a,a ) as
M̂ 1 共 q 兲 ⫺g ⫺
1 ⫽⫺c
冉
冊
␪ 共 q 21 ⫺q 2 兲
a⫺ 兩 q 1 兩
arccot 2
⫹ ␪ 共 q 2 ⫺q 21 兲 ␪ 共 q 2 ⫺a 2 兲 ⌽ 1,⫺a 丢 ⌿ 1,a ⫹o 共 1 兲 ,
␲
冑q 1 ⫺q 2
q⬃ 共 ⫾a,a 2 兲 .
共3.25兲
Thus M̂ 1 (q) is bounded but discontinuous at q⫽(⫾a,a 2 ), while its regular part, g ⫺
1 , is the same
for both these points.
Now it is easy to see that the discontinuity of the resolvent along the cut q 2 ⫽a 2 , 兩 q 1 兩 ⬍a and
the ill definiteness at the points q⫽(⫾a,a 2 ) are the result of embedding the one-dimensional
potential in the two-dimensional space. Indeed, the resolvent of the Sturm–Liouville operator
⳵ x2 ⫺u 1 (x 1 )⫺q 2
is
obtained
from
M 1 (q)
by
means
of
the
operation
1
兰 dx 2 e ⫺q 2 (x 2 ⫺x 2⬘ ) M̂ 1 (x,x ⬘ ;q). By 共3.18兲 and 共3.3兲, 共3.4兲, and 共3.9兲 we get the standard expression
for the one-dimensional Green’s function with a pole at q 2 ⫽a 2 .
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1052
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
We already noted that M̂ 1 (x,x ⬘ ;q) defines a family of Green’s functions. Among them that we
expect should play a special role are those obtained considering the values of q belonging to the
support of the defects of analyticity given in 共3.20兲, 共3.21兲 and 共3.22兲. We consider, therefore, the
Green’s functions
G 1 共 x,x ⬘ ,k 兲 ⫽M̂ 1 共 x,x ⬘ ;q 兲 兩 q⫽ l
I(k)
共3.26兲
,
G⫾
1 共 x,x ⬘ ;k 兲 ⫽M̂ 1 共 x,x ⬘ ;q 兲 兩 q 1 ⫽k I , q 2 ⫽a 2 ⫾0 ,
共3.27兲
where k苸C is the spectral parameter and we denote q 1 ⫽k I 关see 共2.11兲兴 in order to meet the
standard notation. From these definitions it follows directly that
ជ 1 G 1 共 k 兲 ⫽G 1 共 k 兲 L
ឈ 1 ⫽I,
L
⫾
ជ 1G ⫾
ឈ
L
1 共 k 兲 ⫽G 1 共 k 兲 L1 ⫽I,
G 1 共 k 兲 ⬅G 1 共 ⫺k̄ 兲 ⬅G 1 共 k 兲 ,
G 1 共 k 兲 兩 k R⫽0 ⫽G 1 共 0 兲 ,
共3.28兲
共3.29兲
⫾
G⫾
1 共 k 兲 ⫽G 1 共 ik I 兲 ,
共3.30兲
⫾
i.e., G ⫾
1 (k) are independent on k R and then inside the strip they coincide with g 1 introduced in
共3.23兲,
⫾
G⫾
1 共 x,x ⬘ ;k 兲 兩 兩 k I兩 ⬍a ⫽g 1 共 x,x ⬘ 兲 .
共3.31兲
As well from 共3.18兲 we get the representations
G 1 共 x,x ⬘ ,k 兲 ⫽
1
2␲
冕
k I⬘ ⫽k I
⬘ 兩 兲 ⫺ ␪ 共 x 2 ⫺x 2⬘ 兲兴 ⌽ 1 共 x,k ⬘ 兲 ⌿ 1 共 x ⬘ ,k ⬘ 兲
dk ⬘ 关 ␪ 共 兩 k R兩 ⫺ 兩 k R
⫹c ␪ 共 a⫺ 兩 k I兩 兲 ␪ 共 x ⬘2 ⫺x 2 兲 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x ⬘ 兲 ,
G⫾
1 共 x,x ⬘ ,k 兲 ⫽
1
2␲
冕
k I⬘ ⫽k I
共3.32兲
⬘ 关 ␪ 共共 k I兲 2 ⫺a 2 ⫺ 共 k R
⬘ 兲 2 兲 ⫺ ␪ 共 x 2 ⫺x ⬘2 兲兴 ⌽ 1 共 x,k ⬘ 兲 ⌿ 1 共 x ⬘ ,k ⬘ 兲
dk R
⫿c ␪ 共 a 2 ⫺k I2 兲 ␪ 共 ⫾ 共 x 2 ⫺x ⬘2 兲兲 ⌽ 1,⫺a 共 x 兲 ⌿ 1,a 共 x ⬘ 兲 .
共3.33兲
The first of these equalities shows that the cut of the resolvent at q 2 ⫽a 2 , 兩 q 1 兩 ⬍a is not inherited
by G 1 (k) 共in contrast to the case of the nonstationary Schrödinger equation, as mentioned in the
Introduction兲 and that G 1 (k) is discontinuous only at the points k⫽⫾ia. Its behavior in the
neighborhoods of these points follows from 共3.25兲 and reads as
G 1 共 k 兲 ⫽g ⫺
1 ⫺
再
冎
c
a⫺ 兩 k I兩
arccot
⌽ 1,⫺a 丢 ⌿ 1,a ⫹o 共 1 兲 ,
␲
兩 k R兩
k⬃⫾ia.
共3.34兲
Also the Green’s functions G ⫾
1 (k) are discontinuous only at k I⫽⫾a and one gets, thanks to
共3.25兲, that for k⬃ia or k⬃⫺ia
⫺
G⫾
1 共 k 兲 ⫽g 1 ⫺c
1⫾ ␪ 共 a⫺ 兩 k I兩 兲
⌽ 1,⫺a 丢 ⌿ 1,a ⫹o 共 1 兲 .
2
共3.35兲
Notice that these functions G ⫾
1 (k) coincide when 兩 k I兩 ⬎a and are independent of k I 关and then of
k by 共3.30兲兴 when 兩 k I兩 ⬍a. On the borders of these strips they have the discontinuity
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
c
⫾
G⫾
1 共 k 兲 兩 outside⫺G 1 共 k 兲 兩 inside⫽⫾ ⌽ 1,⫺a 丢 ⌿ 1,a .
2
1053
共3.36兲
Taking into account the discontinuous behavior of the Green’s functions we see that equalities of
⫾
the type G ⫾
1 (ia)⫽G 1 (ia) and G 1 (⫺ia)⫽G 1 (⫺ia) have no meaning in our case. Thanks to
共3.32兲 and 共3.35兲 we have only that
lim G 1 共 k 兲 ⫽g ⫺
1 ,
lim
兩 k I兩 →a⫺0 k R→0
共3.37兲
where the limiting procedure must be performed in such a way that 兩 k R兩 /(a⫺ 兩 k I兩 )→⫹0.
In order to complete the study of the Green’s functions we mention that G 1 (k) obeys the
standard equalities
lim 共 ⫺2ik 兲
k→⬁
⳵ il
e
⳵x1
⳵ G 1 共 x,x ⬘ ,k 兲
⳵ k̄
⫽
(k)(x⫺x ⬘ )
sgn k R
2␲
G 1 共 x,x ⬘ ,k 兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ,
⌽ 1 共 x,⫺k̄ 兲 ⌿ 1 共 x ⬘ ,⫺k̄ 兲 .
共3.38兲
共3.39兲
The first of them follows either from the differential equations 共3.28兲, or from the integral representation 共3.32兲 and properties 共3.6兲. The second one also follows from 共3.28兲, or it can be derived
from 共3.26兲 by means of 共3.20兲 and 共3.21兲. This equality must be understood in the sense of
distributions and we see that the discontinuity of G 1 (k) at points k⫽⫾ia leads 关by 共3.3兲 and
共3.4兲兴 to the pole singularities of the rhs at these points. In view of 共3.39兲 in what follows we refer
to G 1 (k) as the Green’s function of the Jost solutions.
IV. RESOLVENT OF THE PERTURBED L -OPERATOR
Now we consider the general case of the operator 共1.2兲 with potential given in 共1.5兲, where
u 2 (x) is a real function of two space variables, smooth and rapidly decaying at space infinity. The
extended resolvent M (q) is determined by 共one of兲 Eqs. 共2.22兲 and we need to study its analyticity
properties first. The increment M (p;q⫹s)⫺M (p;q) of M can be obtained from the Hilbert
identity 共2.19兲 where prime means the increment s of q. We have M ⬘ ⫺M ⫽⫺M ⬘ (L 1⬘ ⫺L 1 )M and,
then, using 共2.21兲,
M ⬘ ⫺M ⫽M ⬘ L 1⬘ 共 M 1⬘ ⫺M 1 兲 L 1 M .
共4.1兲
Thus for the d-bar derivatives with respect to q j we get
¯⳵ j M ⫽ 共 M L 1 兲共¯⳵ j M 1 兲共 L 1 M 兲 ,
j⫽1,2,
共4.2兲
in the region where M 1 is continuous, i.e., for q 2 ⫽a 2 . In terms of the objects introduced in 共2.13兲,
we obtain
¯d
ឈ 1 共¯d
ជ 1 M̂ ,
⳵ j M ⫽M̂ L
⳵ jM 1兲L
j⫽1,2,
共4.3兲
where we used that L̂ 1 (x,x ⬘ ;q)⫽L1 ␦ (x⫺x ⬘ ) and took into account that when kernels with hats
are considered, the multiplication by L1 is no longer associative and it is necessary to use the
arrows to indicate the correct order of operations 关cf. 共2.15兲兴. Now, thanks to 共2.17兲 and using
共3.20兲 and 共3.21兲 we get for q 2 ⫽a 2
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1054
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
⳵ M̂ 共 q 兲 i
⫽
⳵q1
␲
冕
k I⫽q 1
⳵ M̂ 共 q 兲 ⫺1
⫽
⳵q2
2␲
冕
dk R¯k ␦ 共 l
2I共 k 兲 ⫺q 2 兲
⌽共 k 兲 丢 ⌿共 k 兲,
共4.4兲
dk R ␦ 共 l
2I共 k 兲 ⫺q 2 兲
⌽共 k 兲 丢 ⌿共 k 兲,
共4.5兲
k I⫽q 1
where ⌽(k) and ⌿(k) are defined by
ឈ 1⌽ 1共 k 兲 ,
⌽ 共 k 兲 ⫽G 共 k 兲 L
ជ 1G共 k 兲,
⌿ 共 k 兲 ⫽⌿ 1 共 k 兲 L
共4.6兲
with
G 共 x,x ⬘ ,k 兲 ⫽M̂ 共 x,x ⬘ ;q 兲 兩 q⫽ l
I(k)
.
共4.7兲
More explicitly, say, the first of Eqs. 共4.6兲 stands for ⌽(x,k)⫽ 兰 dx ⬘ (L d1 (x ⬘ , ⳵ x ⬘ )
⫻G 1 (x,x ⬘ ,k))⌽ 1 (x ⬘ ,k). The function G(k) with kernel G(x,x ⬘ ,k) defined in 共4.6兲 satisfies the
differential equations
ជ G 共 k 兲 ⫽G 共 k 兲 L
ឈ ⫽I,
L
共4.8兲
which can be obtained as a direct reduction of 共2.15兲. Therefore, G(k) is a Green’s function. Since
the reduction is the same used in 共3.26兲 for getting G 1 (k) from M̂ 1 we derive from 共2.22兲 that this
Green’s function obeys the integral equations
G 共 k 兲 ⫽G 1 共 k 兲 ⫹ G 1 共 k 兲 U 2 G 共 k 兲 ,
G 共 k 兲 ⫽G 1 共 k 兲 ⫹ G 共 k 兲 U 2 G 1 共 k 兲 .
共4.9兲
Again, as in Sec. III, thanks to 共4.7兲 and 共4.4兲 and 共4.5兲 we get the d-bar derivative of the Green’s
function in the form
⳵G共 k 兲
⳵ k̄
⫽
sgn k R
2␲
⌽ 共 ⫺k̄ 兲 丢 ⌿ 共 ⫺k̄ 兲 ,
共4.10兲
where ⌽(k) and ⌿(k) are defined in 共4.6兲. These objects, due to their definition and 共4.9兲, obey
the integral equations
⌽ 共 k 兲 ⫽⌽ 1 共 k 兲 ⫹ G 1 共 k 兲 U 2 ⌽ 共 k 兲 ,
⌿ 共 k 兲 ⫽⌿ 1 共 k 兲 ⫹⌿ 共 k 兲 U 2 G 1 共 k 兲 ,
共4.11兲
where again the first equation more explicitly reads as ⌽(x,k)⫽⌽ 1 (x,k)⫹ 兰 dx ⬘ G 1 (x,x ⬘ ,k)
⫻u 2 (x ⬘ )⌽(x ⬘ ,k). It is clear that the differential equations
ជ ⌽ 共 k 兲 ⫽0,
L
ឈ ⫽0
⌿共 k 兲L
共4.12兲
hold and, therefore, we can consider ⌽(x,k) and ⌿(x,k) as the generalization of the Jost solutions
to the case where the perturbation u 2 (x) is different from zero. Let us mention that thanks to these
definitions we succeeded in avoiding the indeterminacy in the definition of the Jost solutions
discussed in the Introduction. Later we study the properties of the Green’s function and the Jost
solutions in more detail and discuss the singular structure of the terms involved in 共4.10兲. Now let
us mention the following standard properties
⫺2i lim k ⳵ x 1 共 e i l
(k)(x⫺x ⬘ )
G 共 x,x ⬘ ,k 兲兲 ⫽ ␦ 共 x⫺x ⬘ 兲 ,
共4.13兲
k→⬁
G 共 k 兲 ⫽G 共 ⫺k̄ 兲 ⫽G 共 k 兲 ,
共4.14兲
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
⌽ 共 x,k 兲 ⫽⌽ 共 x,⫺k̄ 兲 ,
⌿ 共 x,k 兲 ⫽⌿ 共 x,⫺k̄ 兲 ,
1055
共4.15兲
which can be obtained by means of the integral equations 共4.9兲 and properties 共3.5兲, 共3.29兲, and
共3.38兲 for the Green’s function G 1 (k).
Till now we studied the departure from analyticity of the resolvent in the case q 2 ⫽a 2 . Since
the resolvent M 1 (q) is discontinuous along the line q 2 ⫽a 2 关see 共3.22兲兴, the integral equations
共2.22兲 suggest that also M (q) has a discontinuity. Let us denote the limiting values on the two
edges of the line by
M ⫾ 共 q 兲 ⫽M 共 q 兲 兩 q 2 ⫽a 2 ⫾0 .
共4.16兲
Then from the Hilbert identity 共4.1兲 we derive that
⫺
⫿
M ⫹ 共 q 兲 ⫺M ⫺ 共 q 兲 ⫽M ⫾ 共 q 兲 L 1 共 q 兲共 M ⫹
1 共 q 兲 ⫺M 1 共 q 兲兲 L 1 共 q 兲 M 共 q 兲 ,
q 2 ⫽a 2 ,
共4.17兲
where the lhs is independent of the choice of the sign on the rhs. In analogy with 共3.27兲 we
introduce the two Green’s functions
G ⫾ 共 x,x ⬘ ;k 兲 ⫽M̂ 共 x,x ⬘ ;q 兲 兩 q 1 ⫽k I, q 2 ⫽a 2 ⫾0
共4.18兲
⫺
ឈ 1 )(G ⫹
and rewrite 共4.17兲 in these terms as G ⫹ (k)⫺G ⫺ (k)⫽(G ⫾ (k)L
1 (k)⫺G 1 (k))
ជ 1 G ⫿ (k)). Then by 共3.22兲 and 共3.27兲 we get
⫻(L
G ⫹ 共 k 兲 ⫺G ⫺ 共 k 兲 ⫽⫺c ␪ 共 a 2 ⫺k I2 兲 ⌽ ⫾ 共 k 兲 丢 ⌿ ⫿ 共 k 兲 ,
共4.19兲
where the new solutions 关cf. 共4.6兲兴 were introduced:
ឈ 1 ⌽ 1,⫺a ,
⌽ ⫾ 共 k 兲 ⫽G ⫾ 共 k 兲 L
ជ 1G ⫾共 k 兲 .
⌿ ⫾ 共 k 兲 ⫽⌿ 1,a L
共4.20兲
Following properties of G ⫾
1 (k) it is easy to show that these Green’s functions obey the following
differential and integral equations and reality condition:
ជ G ⫾ 共 k 兲 ⫽G ⫾ 共 k 兲 L
ឈ ⫽I,
L
⫾
⫾
G ⫾ 共 k 兲 ⫽G ⫾
1 共 k 兲⫹ G 1 共 k 兲U 2G 共 k 兲,
⫾
⫾
G ⫾ 共 k 兲 ⫽G ⫾
1 共 k 兲⫹ G 共 k 兲U 2G 1 共 k 兲,
G ⫾ 共 k 兲 ⫽G ⫾ 共 k 兲 .
共4.21兲
共4.22兲
共4.23兲
By definition they are independent of k R and by the corresponding properties of G ⫾
1 (k) we have
that G ⫹ (k)⫽G ⫺ (k) when 兩 k I兩 ⬎a and they are independent of k I when 兩 k I兩 ⬍a. By 共4.20兲 and
共4.21兲 we get that ⌽ ⫾ (k) and ⌿ ⫾ (k) are solutions of the heat equation with potential 共1.5兲,
ជ ⌽ ⫾ 共 k 兲 ⫽0,
L
ឈ ⫽0.
⌿ ⫾共 k 兲 L
共4.24兲
Integral equations for these solutions follow by applying operations 共4.20兲 to the equations 共4.22兲:
⫾
⌽ ⫾ ⫽⌽ 1,⫺a ⫹ G ⫾
1 U 2⌽ ,
⌿ ⫾ ⫽⌿ 1,a ⫹⌿ ⫾ U 2 G ⫾
1 .
共4.25兲
Let us also mention that, thanks to 共4.23兲, these solutions are real and are independent of k inside
the strip 兩 k I兩 ⬍a, due to the corresponding property of G ⫾ (k) and 共4.20兲. Since in the following
we use intensively the Green’s functions and these solutions inside the strip, it is convenient to
introduce the following specific notations:
g ⫾ 共 x,x ⬘ 兲 ⫽G ⫾ 共 x,x ⬘ ,k 兲 兩 兩 k I兩 ⬍a ,
共4.26兲
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1056
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
and also
␾ ⫾ 共 x 兲 ⫽⌽ ⫾ 共 x,k 兲 兩 兩 k I兩 ⬍a ,
␺ ⫾ 共 x 兲 ⫽⌿ ⫾ 共 x,k 兲 兩 兩 k I兩 ⬍a .
共4.27兲
Equality 共4.19兲 enables us to find relations between solutions 共4.20兲. Let 兩 k I兩 ⬍a. Then apឈ 1 ⌽ 1,⫺a to this equality from the right and using 共4.20兲 we derive that
plying, say, L
共 1⫹␭ 兲 ␾ ⫹ ⫽ ␾ ⫺ ,
共4.28兲
ជ 1g ⫺L
ឈ 1 ⌽ 1,⫺a 兲 .
␭⫽c 共 ⌿ 1,a L
共4.29兲
where
Explicitly ␭⫽c 兰 dx 兰 dx ⬘ ⌿ 1,a (x)(L1 (x, ⳵ x )L d1 (x ⬘ , ⳵ x ⬘ )g ⫺ (x,x ⬘ )⌽ 1,⫺a (x ⬘ )). By 共4.23兲 this conជ 1 ␾ ⫺ )⫽c( ␺ ⫺ L
ឈ 1 ⌽ 1,⫺a ). Inserting
stant is real and, thanks to 共4.20兲, it is also equal to ␭⫽c(⌿ 1,a L
here L1 ⫽L⫹U 2 we get by 共4.21兲 and 共4.24兲 that
␭⫽c 兵 共 ⌿ 1,a U 2 ⌽ 1,⫺a 兲 ⫹ 共 ⌿ 1,a U 2 g ⫺ U 2 ⌽ 1,⫺a 兲 其 ,
共4.30兲
ជ 1 ⌽ 1,⫺a ⫽0 and ⌿ 1,a L
ឈ 1 ⫽0, which
or ␭⫽c(⌿ 1,a U 2 ␾ ⫺ )⫽c( ␺ ⫺ U 2 ⌽ 1,⫺a ), where we also used L
ជ
follow from 共3.2兲 and 共3.8兲. Next, applying to 共4.28兲 ⌿ 1,a L1 from the left and again by 共4.20兲 we
get
ជ 1g ⫹L
ឈ 1 ⌽ 1,⫺a 兲兴 ⫽1,
共 1⫹␭ 兲关 1⫺c 共 ⌿ 1,a L
共4.31兲
ជ 1g ⫹L
ឈ 1 ⌽ 1,⫺a )⫽(⌿ 1,a U 2 ⌽ 1,⫺a )⫹(⌿ 1,a U 2 g ⫹ U 2 ⌽ 1,⫺a ) 关cf. 共4.30兲兴
where a new constant (⌿ 1,a L
appeared. Since we chose u 2 to be rapidly decaying at infinity, all terms must be finite. Then
1⫹␭⫽0 and, more precisely, taking into account that for u 2 →0 also ␭→0, we have that
共4.32兲
1⫹␭⬎0.
Summarizing, we get the following relations:
ជ 1g ⫹L
ឈ 1 ⌽ 1,⫺a 兲 ⫽
c 共 ⌿ 1,a L
␾ ⫹⫽
␾⫺
,
1⫹␭
G ⫹ 共 k 兲 ⫽G ⫺ 共 k 兲 ⫺
␺ ⫹⫽
␭
,
1⫹␭
␺⫺
,
1⫹␭
c ␪ 共 a⫺ 兩 k I兩 兲 ⫺
␾ 丢 ␺ ⫺.
1⫹␭
共4.33兲
共4.34兲
共4.35兲
Here 共4.33兲 is just 共4.31兲, the first equality in 共4.34兲 is 共4.29兲 and the second equality is derived by
analogy, and 共4.35兲 follows from 共4.19兲 thanks to 共4.34兲. In their turn 共4.33兲 and 共4.34兲 follow
from 共4.35兲 thanks to 共4.20兲 and 共4.28兲.
We have shown in 共3.36兲 that the Green’s functions G ⫾
1 (k) are discontinuous at k I⫽a and
k I⫽⫺a. By 共4.22兲 we deduce that G ⫾ (k) have the same behavior. In order to study this discontinuity we use, as above, the Hilbert identity 共4.1兲 where M ⫽M (q), M ⬘ ⫽M (q ⬘ ), etc. We choose
q 2 ⫽q ⬘2 ⫽a 2 ⫾0, q 1 ⫽a⫺␧, q 1⬘ ⫽a⫹␧, and in the limit ␧→⫹0 we use the hat notation 共2.13兲 and
definitions 共3.27兲 and 共4.18兲 of the Green’s functions. Then we get G ⫾ (i(a⫹0))⫺g ⫾ ⫽G ⫾ (i(a
⫾ ជ
⫾
ឈ 1 (G ⫾
⫹0))L
1 (i(a⫹0))⫺g 1 )L1 g , where again 共3.31兲 and 共4.26兲 were used. Now by 共3.36兲 for
the discontinuity of the unperturbed Green’s functions we obtain
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
c
G ⫾ 共 i 共 a⫹0 兲兲 ⫺g ⫾ ⫽⫾ ⌽ ⫾ 共 i 共 a⫹0 兲兲 丢 ␺ ⫾ ,
2
1057
共4.36兲
ឈ 1 ⌽ 1,⫺a from the right and ⌿ 1,a L
ជ1
where notations 共4.20兲, 共4.26兲 and 共4.27兲 were used. Applying L
⫾
ជ
from the left in analogy with the derivation of 共4.35兲 we get by 共4.28兲 that c(⌿ 1,a L1 G (i(a
ឈ 1 ⌽ 1,⫺a )⫽2␭(2⫹␭) ⫺1 , which is finite due to 共4.32兲. Then omitting details we derive the
⫹0))L
equalities G ⫾ (i(a⫹0))⫽G ⫾ (⫺i(a⫹0))⫽g ⫺ ⫺c(2⫹␭) ⫺1 ␾ ⫺ 丢 ␺ ⫺ , that, say, for the bottom
sign can also be rewritten in the form
G ⫺ 共 k 兲 ⫽g ⫺ ⫺
c ␪ 共 兩 k I兩 ⫺a 兲 ⫺
␾ 丢 ␺ ⫺ ⫹o 共 1 兲 ,
2⫹␭
共4.37兲
k⬃⫾ia,
where we took 共4.26兲 into account.
V. PROPERTIES OF THE JOST SOLUTIONS AND INVERSE PROBLEM
In this section we complete the investigation of the properties of the Jost solutions by describing their behavior at the points k⫽⫾ia. Formulas 共4.6兲 suggest to study first the behavior of the
Green’s function G(k). We expect that it is ill defined at these points, so in order to describe this
behavior we compare G(k) with some well defined Green’s function, say, g ⫺ . For this aim, as we
have already shown, relations of the type 共4.19兲 can be very useful. In order to derive them we
start again from the Hilbert identity 共4.1兲 where M ⬘ ⫽M (q ⬘ ) and M ⫽M (q) and we choose q ⬘
⫽ l I(k), q 1 ⫽k I , q 2 ⫽a 2 ⫺0 关see 共2.11兲, 共4.7兲 and 共4.16兲兴. Then, passing to the objects with hats
by 共2.13兲, recalling definitions 共4.7兲 and 共4.18兲 and keeping only the leading term in the neighborhood of k⬃⫾ia, we get
ឈ 1 共 G 1 共 k 兲 ⫺G ⫺
ជ ⫺
G 共 k 兲 ⫺G ⫺ 共 k 兲 ⫽G 共 k 兲 L
1 共 k 兲兲 L1 G 共 k 兲 ⫹o 共 1 兲 ,
k⬃⫾ia.
⫺
Inserting the explicit singular behaviors of G 1 (k), G ⫺
1 (k) and G (k) at k⫽⫾ia given in 共3.34兲,
共3.35兲 and 共4.37兲, we have
G 共 k 兲 ⫺g ⫺ ⫽⫺
冉
冊
c ␪ 共 兩 k I兩 ⫺a 兲 ⫺
1
a⫺ 兩 k I兩 ␪ 共 兩 k I兩 ⫺a 兲
ឈ 1 ⌽ 1,⫺a
␾ 丢 ␺ ⫺ ⫹c ⫺ arccot
⫹
G共 k 兲L
2⫹␭
␲
兩 k R兩
2
丢 ⌿ ⫺ 共 k 兲 ⫹o 共 1 兲 ,
where in the last multiplier the definition of ⌿ ⫺ (k) in 共4.20兲 was used. Again by 共4.20兲 and 共4.37兲
⌿ ⫺ (k)⫽ 兵 关 2⫹␭ ␪ (a⫺ 兩 k I兩 ) 兴 /(2⫹␭) 其 ␺ ⫺ , where as always ␺ ⫺ denotes ⌿ ⫺ (k) for 兩 k I兩 ⬍a by
共4.27兲. Then
再
G 共 k 兲 ⫺g ⫺ ⫽ ⫺
⫻
冉
c ␪ 共 兩 k I兩 ⫺a 兲 ⫺
1
a⫺ 兩 k I兩 ␪ 共 兩 k I兩 ⫺a 兲
␾ ⫹c ⫺ arccot
⫹
2⫹␭
␲
兩 k R兩
2
2⫹␭ ␪ 共 a⫺ 兩 k I兩 兲
ឈ 1 ⌽ 1,⫺a
G共 k 兲L
2⫹␭
冎
丢 ␺ ⫺ ⫹o 共 1 兲 .
冊
共5.1兲
ឈ 1 ⌽ 1,⫺a , which follows by
Thus in order to get the behavior of G(k) we need to find that of G(k)L
ឈ
applying to 共5.1兲 operation L1 ⌽ 1,⫺a from the right and using again 共4.20兲, 共4.27兲 and 共4.29兲. Then
ឈ 1 ⌽ 1,⫺a ⫽
G共 k 兲L
␲␾⫺
⫹o 共 1 兲 ,
A共 k 兲
k⬃⫾ia,
共5.2兲
where we denoted for brevity
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1058
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J. Math. Phys., Vol. 43, No. 2, February 2002
A 共 k 兲 ⫽ ␲ ⫹␭ arccot
a⫺ 兩 k I兩
.
兩 k R兩
共5.3兲
This function is real positive thanks to 共4.32兲 and discontinuous at k⫽⫾ia. Now inserting 共5.2兲
in 共5.1兲 we derive finally that
G 共 k 兲 ⫽g ⫺ ⫺
冉
冊
c
a⫺ 兩 k I兩
arccot
␾ ⫺ 丢 ␺ ⫺ ⫹o 共 1 兲 ,
A共 k 兲
兩 k R兩
k⬃⫾ia.
共5.4兲
ជ 1 and recalling the definitions 共4.20兲 and 共4.27兲
Applying to 共5.4兲 from the left the operation ⌿ 1,a L
we derive
ជ 1G共 k 兲⫽
⌿ 1,a L
␲␺⫺
⫹o 共 1 兲 ,
A共 k 兲
k⬃⫾ia,
共5.5兲
and by 共4.29兲 also
ជ 1G共 k 兲L
ឈ 1 ⌽ 1,⫺a ⫽
c⌿ 1,a L
␲␭
⫹o 共 1 兲 ,
A共 k 兲
k⬃⫾ia.
共5.6兲
Correspondingly, we get for the behavior of the Jost solutions in the neighborhood of k⫽⫾ia,
thanks to 共3.3兲 and 共5.2兲,
⌽共 k 兲⫽
i␲c␾⫺
⫹O 共 1 兲 ,
A 共 k 兲共 k⫺ia 兲
⌽共 k 兲⫽
␲ ␾⫺
⫹o 共 1 兲 ,
A共 k 兲
k⬃ia,
k⬃⫺ia
共5.7兲
共5.8兲
and analogous relations for ⌿(k).
Now we are ready to consider the d-bar derivative in the sense of distributions of the Jost
solution, say, ⌽(k). Let first k⫽⫾ia. Then we use 共3.3兲, 共4.6兲, and 共4.10兲 to derive
⳵⌽共 k 兲
⳵ k̄
⫽⌽ 共 ⫺k̄ 兲 r 共 k 兲 ,
k⫽⫾ia,
共5.9兲
where the spectral data are defined as follows:
r共 k 兲⫽
sgn k R
ជ 1G共 k 兲L
ឈ 1 ⌽ 1 共 k 兲兲 .
共 ⌿ 1 共 ⫺k̄ 兲 L
2␲
共5.10兲
Thanks to 共3.3兲, 共3.4兲 and 共5.6兲 we get the singular behavior of these spectral data in the form
r共 k 兲⫽
i␭ sgn k R
⫹o 共 1 兲 ,
2 共 k R⫹i 兩 k I兩 ⫺ia 兲 A 共 k 兲
k⬃⫾ia,
共5.11兲
i.e., in both points it has a pole singularity multiplied by the discontinuous function A(k). Taking
into account that the singular behavior of ⌽(⫺k̄) is given by the denominator (k⫺ia)A(k) at
point k⫽ia and by A(k) at point k⫽⫺ia we see that the rhs in 共5.9兲 is integrable at the latter
point but it has a singularity sgn kR兩 k⫺ia 兩 ⫺2 A(k) ⫺2 at point k⫽ia, which is not integrable. On
the other hand, ⌽(k) is locally integrable for any k, so ⌽(x,k)e i l (k)x is a Schwartz distribution
with respect to k. Thus its d-bar derivative in the sense of distributions exists and can be defined
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
1059
in the standard way. Let f (k) be a test function that properly decays at infinity 共we are now not
interested in the exponential growth due to the multiplier e ⫺i l (k)x ). Then the d-bar derivative of
⌽(k) is defined as
冕
d 2k
⳵⌽共 k 兲
⳵ k̄
f 共 k 兲 ⫽⫺
冕
d 2k ⌽共 k 兲
⳵ f 共k兲
⳵ k̄
⫽⫺ lim
␧→0
冕
兩 k⫾ia 兩 ⬎␧
d 2k ⌽共 k 兲
⳵ f 共k兲
⳵ k̄
,
where in the last equality we again used the property of local integrability of ⌽(k). Integrating by
parts for ␧⬎0 we can use 共5.9兲 and we have
⫺ lim
␧→0
冕
兩 k⫾ia 兩 ⬎␧
d 2k ⌽共 k 兲
⳵ f 共k兲
⳵ k̄
⫽
f 共 ia 兲
2i
冖
lim
兩 k⫺ia 兩 ⫽␧
␧→0
dk ⌽ 共 k 兲 ⫹ lim
␧→0
冕
兩 k⫺ia 兩 ⬎␧
d 2k
⫻⌽ 共 ⫺k̄ 兲 r 共 k 兲 f 共 k 兲 ,
where we omitted the term 养 兩 k⫹ia 兩 ⬎␧ since thanks to 共5.8兲 it gives zero in the limit ␧→0. Thanks
to 共5.7兲 and 共5.8兲 both limits on the rhs exist. To be more precise, let us introduce the distribution
p.v.
冕
d 2k
sgn k R f 共 k 兲
⫽ lim
兩 k⫺ia 兩 2 A 共 k 兲 2 ␧→0
冕
兩 k⫺ia 兩 ⬎␧
d 2k
sgn k R f 共 k 兲
.
兩 k⫺ia 兩 2 A 共 k 兲 2
共5.12兲
Notice the presence in the numerator of sgn kR that guaranties existence of the limit. We used the
principal value 共p.v.兲 notation in analogy with the one-dimensional case. It can be checked directly
that
p.v.
冕
d 2k
sgn k R f 共 k 兲
⫽
兩 k⫺ia 兩 2 A 共 k 兲 2
冕
d 2k
⫽
1
2
冕
sgn k R
关 f 共 k 兲 ⫺ ␪ 共 ␦ ⫺ 兩 k⫺ia 兩 兲 f 共 ia 兲兴
兩 k⫺ia 兩 2 A 共 k 兲 2
f 共 k 兲 ⫺ f 共 ⫺k̄ 兲
d 2 k sgn k R
,
兩 k⫺ia 兩 2 A 共 k 兲 2
共5.13兲
where ␦ is some real positive parameter and the second term in 共5.13兲 is independent of the choice
of ␦. In the case where a distribution has singularities of this form at some finite number of points
a 1 , a 2 , etc., we use the same notation for the integral assuming that either the cutoff procedure in
共5.12兲 or the subtraction procedure in 共5.13兲 is performed at each point. Of course, the parameters
␧ j and ␦ j must be chosen in such a way that corresponding discs do not overlap.
Let us denote
⌽ a ⫽⫺
1
lim
2 ␲ ␧→0
冖
兩 k⫺ia 兩 ⫽␧
dk ⌽ 共 k 兲 ,
共5.14兲
so that i⌽ a can be considered as an extension of the definition of residuum to the case in which
the pole singularity is multiplied by a function discontinuous at the same point. Thanks to 共5.7兲 we
get that this limit also exists and equals
⌽ a ⫽c
log共 1⫹␭ 兲 ⫺
␾ .
␭
共5.15兲
Thus, summarizing all above definitions we get that
⳵⌽共 k 兲
⳵ k̄
⫽⌽ 共 ⫺k̄ 兲 r 共 k 兲 ⫹i ␲ ⌽ a ␦ 共 k⫺ia 兲 ,
共5.16兲
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1060
Boiti et al.
J. Math. Phys., Vol. 43, No. 2, February 2002
where ⌽(x,⫺k̄)r(k) is now a distribution in k defined by the p.v. prescription given above. By
共5.7兲, 共5.8兲 and 共5.11兲 it is integrable at k⫽⫺ia, but it behaves as 兩 k⫺ia 兩 ⫺2 A ⫺2 (k) in the
neighborhood of the point k⫽ia.
Equation 共5.16兲 supplies us with the first equation of the inverse problem. In order to close it
we need the analog of the first relation in 共3.17兲, where it is stated that the residuum of the
function is proportional to its value in the conjugated point. But in our case ⌽(k) is discontinuous
at point k⫽⫺ia, so again some modification of the notion of ‘‘value’’ at this point must be given.
Following the procedure used in 共5.14兲 we can define it as
⌽ ⫺a ⫽
1
lim
2 ␲ i ␧→0
冖
dk
⌽共 k 兲.
兩 k⫹ia 兩 ⫽␧ k⫹ia
共5.17兲
Thanks to 共5.8兲 this limit also exists and equals
⌽ ⫺a ⫽
log共 1⫹␭ 兲 ⫺
␾ ,
␭
共5.18兲
so that by 共5.15兲 we have
⌽ a ⫽c⌽ ⫺a ,
共5.19兲
which shows that the parameter c is not modified by the perturbation. This equality closes the
formulation of the inverse problem 共5.16兲. Finally, taking into account the asymptotic behavior of
⌽(x,k) and 共5.17兲 and 共5.19兲, we can formulate the inverse problem as the following system of
integral equations:
⌽ 共 x,k 兲 ⫽e ⫺i l
1
⌽ 共 x 兲 ⫽e ⫺i l
c a
(k)x
⫹
(⫺ia)x
1
p.v.
␲
⫺
冕
1
p.v.
␲
d 2 k ⬘ i( l (k )⫺ l
⬘
e
k⫺k ⬘
冕
d 2k
(k))x
⌽ 共 x,⫺k ⬘ 兲 r 共 k ⬘ 兲 ⫹i
⌽ 共 x,⫺k̄ 兲 r 共 k 兲 i( l
e
k⫹ia
(k)⫺ l (⫺ia))x
⫹
e i( l
(ia)⫺ l (k))x
k⫺ia
e i( l
⌽ a共 x 兲 ,
共5.20兲
(ia)⫺ l (⫺ia))x
2a
⌽ a共 x 兲 .
共5.21兲
The integrands on the rhs of these two equations are not locally integrable, respectively, the first
at k⫽ia and the second at k⫽⫾ia. Correspondingly, their integrals are regularized by means of
the principal value prescription, as in 共5.12兲 or 共5.13兲, at k⫽ia and at k⫽⫾ia.
The potential is reconstructed by means of
u 共 x 兲 ⫽⫺
2i
p.v.
␲
冕
d 2k
⳵
共 e il
⳵x1
(k)x
⌽ 共 x,⫺k̄ 兲 r 共 k 兲兲 ⫹2
⳵
共 e il
⳵x1
(ia)x
⌽ a 共 x 兲兲 .
共5.22兲
VI. CONCLUSION
In this article on the basis of the resolvent approach we gave a detailed presentation of an
extension of the inverse scattering theory for the heat operator to the case where the potential 共1.5兲
is a perturbation of the one-dimensional one-soliton potential u 1 (x 1 ) 共3.1兲 by means of a smooth,
decaying at infinity function u 2 (x) of two space variables. To our knowledge this is the first time
that inverse scattering theory is applied to a nonscattering situation, i.e., a situation with an infinite
obstacle. As a result of our investigation we proved that under such a perturbation the Jost
solutions get specific singularities 共5.7兲 and 共5.8兲 on the complex plane of the spectral parameter
k. We demonstrated that the d-bar problem 共5.16兲 and 共5.19兲, while looking familiar for a potential
whose spectrum has a discrete and continuous part, needs a substantially modified approach due to
the singularity structure of the spectral data given in 共5.11兲. It was necessary to establish the
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J. Math. Phys., Vol. 43, No. 2, February 2002
Inverse scattering theory
1061
meaning in the sense of distributions of all terms involved in this problem, in order to be able to
formulate the inverse problem as the system of integral equations 共5.20兲 and 共5.21兲. It is easy to
check that the singular behavior of the spectral data and Jost solution as given in 共5.11兲 and 共5.7兲
and 共5.8兲 is compatible with this inverse problem. On the other side, it is necessary to prove that
the potential u(x) reconstructed by means of 共5.22兲 is of the type 共1.5兲. We plan to address this
problem in a forthcoming work.
Another open problem is the application of these results to the KPII equation 共1.3兲 itself. In
particular, investigation of the time asymptotics of solutions with initial data of the type 共1.5兲 must
be performed. Let us mention only that the singular behavior 共5.11兲 of the spectral data is preserved under evolution 共1.3兲. Indeed,3 the time dependence of the spectral data is given as
r 共 k,t 兲 ⫽e 4i(k
3 ⫹k̄ 3 )
.
共6.1兲
Thus we get that
a⫽const,
␭⫽const,
共6.2兲
also with respect to time.
In Sec. III we mentioned that the above construction can be easily generalized to the case
where the potential u 1 (x 1 ) is a pure N-soliton one-dimensional potential. At the same time our
approach also admits straightforward generalization to the case where u 1 (x) is not a function of
one space variable but the result of application of the Bäcklund transformation to a generic
background two-dimensional potential u 0 (x) decaying on the x-plane. Then the inverse problem is
again given by Eqs. 共5.20兲 and 共5.21兲, where the spectral data r(k) are replaced with
r共 k 兲⫹
共 k⫹ia 兲共 k̄⫹ia 兲
共 k̄⫺ia 兲共 k⫺ia 兲
r 0共 k 兲 ,
共6.3兲
where r(k) is of the type 共5.11兲 and r 0 (k) are the spectral data of the potential u 0 (x) 共see Ref. 11兲.
The theory of the heat equation with respect to the nonstationary Schrödinger equation is in
some respects simpler and in some other respects unexpectedly more difficult. As we have shown,
under perturbation the Jost solution get singularities more complicated than poles, but this solution
has no additional cut in the complex domain, in contrast with the nonstationary Schrödinger case
as discovered in Ref. 13. On the other hand, the generalization of this scheme to the case of
multi-ray structure of the potential u(x) meets with essential problems, first of all due to the fact
that the resolvent 共or Green’s function兲 of the heat equation even of a two-soliton 共generic兲
potential is unknown in the literature. This problem also needs future development.
ACKNOWLEDGMENTS
A.K.P. thanks his colleagues at the Department of Physics of the University of Lecce for kind
hospitality and S. V. Manakov and V. E. Zakharov for fruitful discussions. This work was supported in part by INTAS 99-1782, by Russian Foundation for Basic Research 99-01-00151 and
00-15-96046 and by COFIN 2000 ‘‘Sintesi.’’
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