Vilasi G. Hamiltonian Dynamics
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424
The Ge~f~n~~~~5~~a~henk~ Equation
and appear in the asymptotic behavior
of
the
5’s.
Thus, we have shown that
~(ixj)
=
~c~e-X3~#(%,
ixjf.
Therefore,
by
using
for
#
the
expression given
by
Eq.
(G.5),
we finally
have
By setting
and
we can write Eq.
(G.11);
i.e.
B
=
D,
in the following
form:
The above equation, which
is
an integro-differential equation of Volterra type,
is
known
as
the
~arc~en~u
eq~at~on
for
x
E
R
and
as
the
G~~~a~~-Le~~tan
equation
for
3:
E
R+
u
(0).
The
Gel’fand-Levitan-Marchenlco
equation allows
us
to recover the inter-
action potential
U
once the scattering data
S
E
{xJ,
cj,R(k)}
are given, since
The
Gelfand-Levitan-Marchenko
Equation
425
the
following
relation
can
be provent:
d
v(z)
=
-2-A(x, dx
z)
,
In order to prove that the above relation holds, let
us
compu~e, from
Eq.
(G.5),
the derivative
of
$(x,
k)
with respect to
x.
We
have
so
that
-
[Az(x,
y)Jy=s
+
A,(%,
gflv=xfeik"
-
ikA(z,
~)h=~e
ikx
.
On
the other hand, performing the integral in
Eq,
(G.5)
by
part twice, with
the assumption
lim
A(z,
y)
=
0,
5-00
we obtain
tIt
is
worth recalling that
A(z,y)
is
defined only for
y
>
z.
Thus,
Of
course,
A(z,y)
can be defined by continuity at
zf
=
x,
and
if
it
is
differentiable at the
point, the potential will be given by
426
By
replac~ng
the
previous expressions
of
$
and
4”
on
the
eft-hand
side
of
Schrodinger’s equation,
we
obtain
-2A,(z,
z)eikx
i.
[Azx(z,
y)
-
A,,(x,
y>]eik”dy
=
U+(z,
k)
,
s,”
and,
by
using once again Eq.
(G.5)
on the right-hand side, also
which
finally gives
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