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Inverse scattering. Time-dependent method
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\begin{document}
\baselineskip=23.6pt
\title{The Time-Dependent Approach to Inverse Scattering\thanks{{\sc ams}
classification 35P25, 35Q40, 35R30 and 81U40. Research partially supported by
Proyecto PAPIIT IN 105799. DGAPA-UNAM}}
\author{Ricardo Weder\thanks{Fellow Sistema Nacional de Investigadores}\\
Instituto de Investigaciones en Matem\'aticas Aplicadas y en Sistemas,\\
Universidad Nacional Aut\'onoma de M\'exico, \\Apartado Postal 20-726, M\'exico D.F.
01000\\E-Mail: weder@servidor.unam.mx\\}
\date{}
\maketitle
\begin{center}
\vspace{2cm}
Notes of the lectures given at the Pan-American Advanced Studies Institute (PASI)
at the Mathematical Sciences Research Institute, Berkeley. October 29-November 2,
2001
\vspace{2cm}
\begin{minipage}{5.75in}
\centerline{{\bf Abstract}}\bigskip
In these lectures I give an introduction to the time-dependent approach to inverse scattering, that has been developed recently.
The aim of this approach is to solve various inverse scattering problems with time-dependent methods that closely follow the physical (and geometrical)
intuition of the scattering phenomena. This method has been applied to many linear
and nonlinear scattering problems.
We first discuss the case of quantum mechanical potential scattering.
We give explicit limits for the high-energy behaviour of the scattering
operator that offer us formulae for the unique reconstruction of the potential.
Then, we consider the case of the Aharonov-Bohm effect ( Schr\"{o}dinger operators with
singular magnetic potentials and exterior domains). This is a particularly interesting inverse
scattering problem that shows that in quantum mechanics a magnetic field acts on a
charged particle -by means of the magnetic potential- even in regions where it is
identically zero.
The key issue for these two problems is that at high-energies translation of
the wave packet dominates over spreading during the interaction time.
\end{minipage}
\newpage
\begin{minipage}{5.75in}
In fact, in this limit it is sufficient for the calculation of the scattering
operator to consider translation of wave packets rather than
their correct free evolution.
Finally, we study the nonlinear Schr\"{o}dinger equation with a potential. In
this case, from the scattering operator we uniquely reconstruct the potential and
the nonlinearity. For this purpose, we observe that in the small amplitude limit
the nonlinear effects become negligible and scattering is dominated by the linear
term. Using this idea we prove that the derivative at zero of the nonlinear
scattering operator is the linear one. With the aid of this fact we first uniquely
reconstruct the potential from the associated linear inverse scattering problem and
in a second step we uniquely reconstruct the nonlinearity.
\end{minipage}
\end{center}
%\newpage
\section{Potential Scattering}\sss
First we briefly discuss time-dependent direct scattering theory in the
particular case of potential scattering for the Schr\"{o}dinger equation.
It is important to keep in mind that physical scattering is a time-dependent
phenomenon
that studies the interaction of
a finite-energy wave packet with a target. Initially, for large negative times,
the wave packet is far from the target and since the interaction is very small
its evolution is well approximated by an {\it incoming asymptotic state},
$\phi_-$, that propagates according to the free dynamics, with the interaction
set to zero. During the
interaction time the wave packet is close to the target and, as the interaction is
strong, the evolution of the wave packet is given by an
{\it interacting state} that evolves according to the interacting dynamics.
But eventually, the wave packet flies away from the target and its evolution
is again well approximated by an {\it outgoing asymptotic state}, $\phi_+$, that
evolves according to the free dynamics. The scattering operator,\, $S$,\, is the
operator that
sends $\phi_-$ to $\phi_+$. The aim of scattering experiments is to measure
the transition probabilities, $(S \phi, \psi)$.
One objective of the time-dependent
approach to inverse scattering theory is to use in an essential way the physical
propagation aspects to solve the inverse scattering problem and to obtain
mathematical proofs that closely follow physical intuition. It is hoped that a good
physical understanding of the inversion mechanisms will be reflected in more
transparent mathematical methods. In the stationary (frequency domain) method the
physical solution is
idealized as a time-periodic solution with infinite energy. By doing so, the
propagation aspects of physical scattering are lost. This loss of physics
is then reflected in mathematical methods that do not give much
information about the physics of the inversion. Moreover, in the stationary method
the wave packets (finite-energy solutions) are obtained using a generalized
Fourier transfom that integrates over the infinite-energy, time-periodic, solutions.
Here the linearity of the direct scattering problem plays an essential role,
because it is only
in this case that linear combinations of solutions are solutions. On the contrary,
the time-dependent approach does not use the linearity of the
direct scattering problem in an essential way and, as we will see below,
it has a natural extension to the case where the direct scattering problem is
nonlinear.
But, let us be more specific.
We consider a quantum-mechanical particle in $\ER^n$ whose dynamics is
described by the Schr\"{o}dinger equation,
\beq
i \frac{\partial}{\partial t} \Phi (t,x) =\frac{\p^2}{2m} \Phi (t,x)+ V(x)
\Phi (t,x), \,\,\Phi (0,x)= \Phi_0(x)\in \l,
\label{1.1}
\ene
where $ t \in \ER, \,\,x \in \ER^n, n=2,\cdots$ , and $\Phi$ is complex valued,
$\p := -i \nabla$ is the momentum operator, and $ m >0$ is the mass of the particle.
We take
Planck's constant equal to one. The target is given by the potential, $V$, that is a
real-valued function. For simplicity
of the presentation, we consider the following class of bounded continuous
short-range
potentials. For the general case including singular and long-range potentials
see \cite{ew2} and \cite{ew3}.
\beq
{\cal V}_{SR}:= \left\{ V \in C\left( \ER^n\right):
\hbox{sup}_{|x| \geq R}\, \left| V(x)\right| \in L^1\left( [0, \infty )\right)
\right\}.
\label{1.2}
\ene
Both the free Hamiltonian, $H_0:= \frac{\p^2}{2m}$ and the interacting Hamiltonian,
$H:= H_0 +V$ are self-adjoint in $\l$ with domain the Sobolev space,
$W_{2,2}$ \cite{8}. In what follows we denote by $\left\| \cdot \right\|$ the norm in $\l$.
As is well known (see \cite{rs} for a general reference in scattering theory) if
$V \in {\cal V}_{SR}$ -and also under much more general conditions- the wave operators,
\beq
\w := \hbox{s -} \lim_{t \rightarrow \pm \infty} e^{i t H}\, e^{-it H_0},
\label{1.3}
\ene
exist and are complete. That is to say, the strong limits in (\ref{1.3}) exist and
their range
is equal to the subspace of absolute continuity of
$H$, ${\cal H}_{ac}$. Moreover, $\w$ are unitary from $L^2\left(\ER^n \right)$ onto
$ {\cal H}_{ac}$. These facts give a mathematical basis to the physical
description of scattering given above, as we explain now.
The solutions to the free Schr\"{o}dinger equation
(\ref{1.1}) with $V=0$ are given by $ e^{-itH_0}\Phi_-, \,\,\Phi_- \in \l$.
They correspond
to free particles that travel in space without being perturbed by a potential.
Since the potential is localized near zero an incoming
particle that is localized at spatial infinity for very large negative times
will
be -to a good approximation- a solution to the free Schr\"{o}dinger
equation, $e^{-itH_0}\Phi_- $, with
{ \it incoming asymptotic configuration } $\Phi_-$.
As time goes by, and the incoming solution is near the scattering center (zero)
it will
feel the
influence of the potential and we expect that it will be close to a solution
of the interacting Schr\"{o}dinger equation (\ref{1.1}) with the potential,
$e^{-itH} \Psi$, with {\it interacting state}, $\Psi$ . But (\ref{1.3})
with $t \rightarrow -\infty$ tells us
that there is a unique
$\Psi:= W_- \Phi_- \in {\cal H}_{ac}$ such that this is true,
\beq
\lim_{t \rightarrow - \infty} \left\| e^{-itH} \Psi -e^{-itH_0} \Phi_-
\right\|=0.
\label{1.4}
\ene
Furthermore, we expect that at later times the particle will escape the influence
of the potential and that as $ t \rightarrow \infty$ it will travel to
spatial infinity where again it will be close to an outgoing solution of the
free Schr\"{o}dinger equation. In fact
(\ref{1.3}) with $t \rightarrow \infty$ tells us that this is true
and that there is a unique {\it outgoing asymptotic configuration}
$\Phi_+:= W_+^{\ast}\Psi$
such that,
\beq
\lim_{t \rightarrow \infty}\left\| e^{-itH} \Psi - e^{-itH_0 }\Phi_+
\right\|=0.
\label{1.5}
\ene
In a scattering experiment, given the {\it incoming free solution} one
seeks to obtain information about the {\it outgoing free solution}. This is actually parametrized by the
corresponding Cauchy data at $t=0$, $\Phi_{\mp}$. The scattering operator is
then, the operator that assigns $\Phi_+$ to $\Phi_-$:
\beq
S:= W_+^{\ast}\, W_-.
\label{1.6}
\ene
We prove below that the
high-energy limit of the scattering operator gives the Radon (or X-ray) transform
of the potential. Inverting this transform we uniquely reconstruct the potential.
The mathematical proof closely follows physical intuition. The key issue is that at
high energies {\it translation of the wave packets} dominates over {\it spreading}
during the interaction time. In fact, in the high-energy limit it is sufficient
for the calculation of the scattering operator to consider {\it translation of wave
packets} rather than their correct free evolution. Since on this limit
{\it spreading} occurs only when and where the interaction is negligible, i.e. when the
free and the interacting time evolutions are almost the same, the effect of
{\it spreading} does not appear on the scattering operator. For this reason
scattering
simplifies on the high-energy limit and we can uniquely reconstruct the potential.
We also obtain error bounds.
Let us consider states, $\Phi_0$, with compact
momentum support on the open ball, $B_{m \eta}$ of radius $ m \eta$ and center zero,
\beq
\hat{\Phi_0} \in C^{\infty}_0 \left( B_{m \eta}(0)\right),
\label{1.7}
\ene
where $\hat{\Phi_0}$ denotes the Fourier transform of $\Phi_0$. The boosted state,
\beq
\Phi_{\bf v}:= e^{im {\bf v}\cdot x} \Phi_0 \,\leftrightarrow \,
\hat{\Phi}_{\bf v}= \hat{\Phi_0}(p-m -\v)\in C^{\infty}_0\left( B_{m \eta}(m {\bf v})\right),
\label{1.8}
\ene
has velocity support of radius $\eta$ around ${\bf v}$. Above we denote by
$B_{m \eta}(m \v)$ the open ball of center $m \v$ and radius $m \eta$. In the theorem below we
use the
high-velocity limit in an arbitrary fixed direction $\hat{\bf v}:=
\frac{\bf v}{|{\bf v}|},\,\,\, v:= |{\bf v}| \rightarrow \infty$.
\begin{theorem}
Suppose that $V \in {\cal V}_{S R}$ and that for some $ 0 \leq \rho \leq 1$,
\beq
(1+R)^{\rho} \hbox{\rm sup}_{|x| \geq R}\, \, \left| V(x)\right| \in
L^1\left( [0, \infty )\right).
\label{1.9}
\ene
Then, for all $\Phi_{\v}, \Psi_{\v}$ as defined in (\ref{1.8})
$$
iv \left( \left(S-I\right)\Phi_{\v}, \Psi_{\v} \right)
\equiv iv \left( e^{-im\v \cdot x}\,\left(S-I\right)\, e^{im\v \cdot x}\Phi_{0},
\Psi_{0} \right)
$$
\beq
= \left( \int_{-\infty}^{\infty} \, d \tau V(x+ \hat{\v} \tau ) \Phi_0,
\Psi_0 \right)+\cases{ o\left( v^{-\rho} \right),& $ 0 \leq \rho < 1$,\cr\cr
O\left( v^{-1 } \right),& $\rho =1$.}
\label{1.10}
\ene
Moreover, the scattering operator, $S$, determines uniquely the potential
$V \in {\cal V}_{S R}$.
\end{theorem}
\bull
Theorem 1.1 is proven in \cite{ew1} and \cite{ew3}. We give an idea of the proof.
By Duhamel's formula,
\beq
\w = I + i\int_0^{\pm \infty} dt \,e^{itH} V e^{-itH_0},
\label{1.11}
\ene
Then,
\beq
W_+ - W_- =i \int_{-\infty}^{\infty} dt\, e^{iHt}\, V\, e^{-itH_0}.
\label{1.12}
\ene
Since the wave operators $\w$ are unitary, $\a \, \w =I$, and moreover, as they satisfy
the intertwinig relations,
$ \w H_0 = H \w$, it follows from (\ref{1.6}) and (\ref{1.12}) that
$$
i (S-I)= i \left( W_+^{\ast} W_- - W_-^{\ast} W_-\right) = i
\left( W_+ -W_- \right)^{\ast}
W_-
= \int dt\, e^{iH_0t}\, V\, e^{-iHt}\, W_-
$$
\beq
= \int dt\, e^{iH_0t}\, V\, W_-\,
e^{-iH_0t}.
\label{1.13}
\ene
Taking a boost with velocity $\v$ and making the substitution $\tau =v t$, we
obtain that,
\beq
e^{im \v \cdot x}iv (S-I) e^{-im\v\cdot x}= L_{\v}+R_{\v},
\label{1.14}
\ene
where,
\beq
L_{\v}:=\int \left[e^{-im\v\cdot x}\, e^{iH_0 \tau / v} e^{im\v\cdot x}\right]\,V(x)\,
\left[ e^{-im\v\cdot x}\,
e^{-i H_0 \tau /v} e^{im\v\cdot x}\right] \, d \tau,
\label{1.15}
\ene
and
\beq
R_{\v}:= \int d \tau\, e^{-im\v\cdot x}\, e^{iH_0 \tau / v}
\, V\, (W_- -I)
\, e^{-iH_0 \tau / v}
e^{im\v\cdot x}.
\label{1.16}
\ene
$L_{\v}$ - that is the first Born approximation-
is the leading term which tends to a finite limit if the velocity $v$ goes to
infinity. We will use it to reconstruct the potential. The remainder, $R_{\v}$,
represents multiple scattering. It is intuitively clear that each instance
of scattering by a short-range potential yields a factor $v^{-1}$: the strength
of the interaction as measured by $(S-I)$ or $(\w -I)$ is proportional to the time
which the particle spends in the interacting region where the potential is strong.
This time is inversely proportional to the speed $v$ and since we have rescaled
multiplying by $v$, we expect $R_{\v}$ to decay as $1/v$, and that we can
neglect it as $v \rightarrow \infty$.
For any Borel function $f$
let us define the operator $f(\p)$ by functional calculus, or equivalently, as
$f(\p):= {\cal F}^{-1} f(\cdot ) {\cal F}$, where ${\cal F}$
denotes the Fourier transform.
Under translation in momentum or configuration space, generated by $x$ or
$\p:= -i \nabla $, respectively, we obtain
\beq
e^{-im \v \cdot x} f(\p) e^{im\v \cdot x} = f(\p +m \v),
\label{1.17}
\ene
in particular,
\beq
e^{-im \v \cdot x} e^{-iH_0 \tau /v} e^{im\v\cdot x}= e^{-i \p \cdot \uv \tau}\,
e^{-i H_0 \tau /v} e^{-im v \tau/2}.
\label{1.18}
\ene
Moreover,
\beq
e^{i\p \cdot \uv \tau } f(x) e^{-i\p \cdot \uv \tau} = f(x + \uv \tau).
\label{1.19}
\ene
By (\ref{1.18}) the boosted free evolution consists of the classical translation in
configuration space, $e^{-i\p \cdot \uv \tau}$ that is independent of $v$,
the term, $ e^{-i H_0 \tau /v}$ -that is responsible for the spreading of the
wave packet- and an unimportant phase. In the limit when $v \rightarrow \infty$ with
$\tau$ fixed, $ e^{-i H_0 \tau /v}$ dissapears and it follows that we can replace the boosted
free evolution by the classical translation $ e^{-i\p \cdot \uv \tau} \Phi=
\Phi (x- \uv \tau)$. The point is that the classical translation is independent of
$v$ and the term responsible for the spreading goes to zero like $1/v$ and then it
is irrelevant in the high velocity limit.
Then, by (\ref{1.19}) the pointwise limit under the integral in (\ref{1.15}) gives us,
\beq
\lim_{v \rightarrow \infty} L_{\v}= \int e^{i\p \cdot \uv \tau} V(x)
e^{-i\p \cdot \uv \tau} \, d \tau = \int V(x + \uv \tau) \, d \tau.
\label{1.20}
\ene
This is the the desired limit in Theorem 1.1. To make this argument rigorous
we have to prove that the integrals above exist when applied to states as in
Theorem 1.1 and we need an integrable uniform bound for all large enough $v$ that allows us to
use the dominated convergence theorem to exchange limit and integration.
At the same time, we will also prove that the
remainder, $R_{\v}$, goes to zero as $1/v$.
We first prove a propagation estimate
that expresses in a convenient way the fact that the solutions to the free
Schr\"{o}dinger equation have rapid decay away from the classically allowed region,
i.e. away from the region in space where a classical particle that travels in
straight lines with constant velocity would be. In fact the result follows easily
from the standard stationary phase estimate (see the Corollary to Theorem XI.14 in
\cite{rs}). Let us denote by $ F( x \in {\cal M})$\ the operator of
multiplication by the characteristic function of ${\cal M}$.
We designate by $\hat{f}$ the Fourier transform of $f$,
$\hat{f}(p):= 1 /(2 \pi)^{n/2} \int e^{-ip\cdot x}\, f(x)\, dx$.
\begin{lemma}
For any $f \in C^{\infty}_0( B_{m \eta})$, for some $ \eta > 0$, and any $ l=1,2, \cdots $, there is a constant
$C_l$ such that the following estimate holds:
\beq
\left\| F( x \in \tilde{{\cal M}})\, e^{-it H_0}\, f\left( \frac{{\p} - m {\v}}{v^{\rho}} \right)
F( x \in {\cal M}) \right\| \leq C_l (1+r v^{\rho}+ \eta \,v^{2 \rho}|t|)^{-l},
\label{1.21}
\ene
for every $ {\bf v} \in \ER^n, \, t \in \ER , \,\,v > 0, \,\,\rho \in \ER$, and any measurable sets
${\cal M}$ and ${\tilde{\cal M}}$ in $\ER^n$
such that
$r:={\rm dist}\left( \tilde{{\cal M}}, {\cal M}+ {\bf v}t\right) -\eta v^{\rho } |t| \geq 0$.
\end{lemma}
\noindent {\it Proof:}\,\,by (\ref{1.17})-(\ref{1.19}) it is enough to prove the
estimate,
\beq
\left\| F( x \in \tilde{{\cal M}})\, e^{-it H_0}\, f\left( \frac{\p}{v^{\rho}} \right)
F( x \in {\cal M}) \right\| \leq C_l \, (1+r v^{\rho}+ \eta \,v^{2 \rho}|t|)^{-l} ,
\label{1.22}
\ene
provided that, $ r:={\rm dist} \left( \tilde{{\cal M}}, {\cal M}\right)-\eta v^{\rho } |t| \geq 0$.
We have that,
\beq
\left\| F( x \in \tilde{{\cal M}})\, e^{-itH_0}\, f\left(\frac{\p}{v^{\rho}}\right) \, F(x\in {{\cal M}}) \phi \right\|^2
= \frac{1}{(2 \pi)^{n} } \int_{\tilde{{\cal M}}} dx \int_{{\cal M}} dy \int_{{\cal M}}
dz \,\tilde{f}_t (x-y)\,\phi (y)\, \overline{\tilde{f}_t (x-z) \,\phi(z)},
\label{1.23}
\ene
where,
\beq
\tilde{f}_t (x):= \frac{1}{(2 \pi)^{n/2} } \int e^{ip\cdot x} e^{-i p^2 t/2m } \, f(p/ v^{\rho})
\, dp.
\label{1.24}
\ene
But, as $ 2 |\phi (y)|\,|\phi (z)|\leq |\phi (y)|^2 +|\phi (z)|^2 $,
\beq
\left\| F( x \in \tilde{{\cal M}})\, e^{-itH_0}\, f\left(\frac{\p}{v^{\rho}}\right) \, F(x\in {\cal M}) \phi \right\|^2
\leq \frac{1}{(2 \pi)^{n}} \left[ \int_{|x|\geq r + \eta v^{\rho} |t|} |\tilde{f}_t (x)|\, dx \right]^2
\, \|\phi \|^2.
\label{1.25}
\ene
Finally, by the Corollary to Theorem XI.14 in \cite{rs}, for any $N=1,2, \cdots ,$ there is a
constant $C_N$ such that,
\beq
|\tilde{f}_t (x)| \leq C_N \, v^{n \rho} \, (1+|x| v^{\rho}+ v^{2 \rho}|t|)^{-N},
\label{1.26}
\ene
for $|x| \geq \eta v^{\rho} |t|$. The lemma follows inserting (\ref{1.26}) in
(\ref{1.25}).
\begin{corollary}
For any $ f \in C^{\infty}_0 \left( B_{m \eta}\right)$, any $v > 0$ and for any
$l=1,2, \cdots $, there is a
constant $C_l$ such that,
\beq
\left\| F(| x|\geq |\tau|/4 + \eta |\tau |/v )\, e^{-iH_0 \tau /v} \,
f(p)
F(| x |\leq |\tau |/ 8) \right\| \leq C_l (1+ |\tau |)^{-l}.
\label{1.27}
\ene
\end{corollary}
\noindent {\it Proof :}\,\, the Corollary follows from Lemma 1.2 with $ \rho =0$,
$\v =0$ and
$\tilde{{\cal M}}=\{ |x| > |\tau|/4 + \eta |\tau |/ v \}, \,
{\cal M}= \{|x| \leq |\tau |/8 \}$. Observe that
$r:={\rm dist}\left( \tilde{{\cal M}}, {\cal M} \right) - \eta | \tau |/ v \geq
|\tau|/8 \geq 0$ .
\bull
By (\ref{1.15}) and(\ref{1.18}) for any $\Phi_0$, as in (\ref{1.7}),
\beq
L_{\v} \Phi_0= \int e^{iH_0 \tau / v} \,V(x+ \uv \tau )\,
e^{-i H_0 \tau /v} \, \Phi_0\, d \tau.
\label{1.28}
\ene
We prove below the bound,
\beq
\left\| e^{iH_0 \tau /v} V(x +\hat{\v} \tau ) e^{-iH_0 \tau /v}
\Phi_0 \right\| \leq h(|\tau |),
\label{1.29}
\ene
where $h$ is integrable. The left-hand side of (\ref{1.29}) side is dominated by
\beq
C \left\| V(x + \hat{\v} \tau ) F\left( |x| \leq \frac{|\tau |}{2}\right)
\right\| + C \left\| F\left(|x| \geq \frac{|\tau|}{2}\right) e^{-iH_0
\tau /v} \Phi_0 \right\|.
\label{1.30}
\ene
The first summand is bounded by
$h_1(|\tau |):= \sup_{| y| \geq |\tau |/2} |V ( y)|$ which is independent
of $v$ and integrable by (\ref{1.2}). If $v \geq 4 \eta $ ( with $\eta $
the radius of the velocity support of $\Phi_0$), the second term describes
free propagation into the {\it classically forbidden region } and it is
rapidly decaying. Let $g \in C^{\infty}_0 \left(B_{m \eta}\right)$
be such that
$g(\p )\Phi_0= \Phi_0$. Then, for $v \geq 4 \eta$,
$$
\left\| F\left(|x | \geq \frac{|\tau |}{2}\right) e^{-iH_0 \tau
/v} \Phi_0 \right\| \leq \left\| F\left( |x | \geq \frac{|\tau |}{8}\right) \Phi_0
\right\|
$$
\beq
+ C \left\| F\left(|x |\geq \frac{|\tau |}{4}+ \eta \frac{|\tau |}{v}
\right) e^{-iH_0 \tau /v} g(\p ) F\left( |x |
\leq \frac{|\tau |}{8}\right) \right\|:= h_2 (\tau ) +h_3(\tau ).
\label{1.31}
\ene
The two terms on the right-hand side of (\ref{1.31}) have rapid decay as
$|\tau| \rightarrow \infty$ uniformly on $v$, for $v \geq 4 \eta $ . In the case of the first
term this is obvious because $\Phi_0$ belongs to Schwartz space. For the
second term it follows from Corollary 1.3. Defining, $h:= h_1+h_2+h_3$ we obtain
the integrable bound (\ref{1.29}) for all $ v \geq 4 \eta$. By the dominated convergence
theorem we can take pointwise limit under the integral in (\ref{1.28}) and we obtain,
\beq
\hbox{s -}\lim_{v \rightarrow \infty} L_{\v} \Phi_0 =
\int_{-\infty}^{\infty} \, d \tau V(x+ \uv \tau ) \Phi_0.
\label{1.32}
\ene
We now prove that the remainder $R_{\v}$ ({\it multiple scattering})
is one order smaller, i.e., that it goes to zero as $1/v, \,\,v \rightarrow \infty$.
By (\ref{1.11}), (\ref{1.16}) and (\ref{1.18}) have that,
$$
\left|\left( R_{\v} \Phi_0, \Psi_0 \right)\right|
\leq \left|\int_{-\infty}^{\infty} d \tau \left(\int_{-\infty}^0 \, dt \, e^{it H}\, V e^{-i(t+ \tau /v)H_0} e^{im \v \cdot x}
\Phi_0, V e^{-iH_0 \tau /v} e^{im \v \cdot x} \Psi_0 \right) \right|
$$
\beq \leq \frac{1}{v}
\left(\int_{-\infty}^{\infty} d\tau \left\| V e^{-iH_0 \tau/v} e^{im \v \cdot x} \Phi_0 \right\|
\right)\,
\left( \int_{-\infty}^{\infty}d\tau \left\| V e^{-iH_0 \tau /v}
e^{im \v \cdot x}\Psi_0\right\|\right).
\label{1.33}
\ene
Using (\ref{1.17})-(\ref{1.19}) and (\ref{1.29}) we prove that,
\beq
\int_{-\infty}^{\infty}d\tau \left\| V(x )e^{-iH_0 \tau/v} e^{im \v \cdot x}
\Psi_0\right\|=
\int_{-\infty}^{\infty}d\tau \left\| V(x +\uv \tau) e^{-iH_0 \tau /v} \Psi_{0}
\right\| \leq C
\label{1.34}
\ene
uniformly in $v$. By (\ref{1.33}) and (\ref{1.34})
\beq
\left|\left( R_{\v} \Phi_0, \Psi_0 \right)\right| \leq C / v, v \geq 4 \eta.
\label{1.35}
\ene
By (\ref{1.14}), (\ref{1.32}) and (\ref{1.35})
$$
\lim_{v \rightarrow \infty} iv \left( \left(S-I\right)\Phi_{\v}, \Psi_{\v} \right)
= \lim_{v \rightarrow \infty} iv \left( e^{-im\v \cdot x}\,\left(S-I\right)\,
e^{im\v \cdot x}\Phi_{0},
\Psi_{0} \right)
$$
\beq
= \left( \int_{-\infty}^{\infty} \, d \tau V(x+ \hat{\v} \tau ) \Phi_0,
\Psi_0 \right).
\label{1.36}
\ene
This is the correct limit as required in Theorem 1.1. We estimate the error term in a
similar way. See \cite{ew3} for details.
Let us denote: $ x^{\perp}:= x -(x \cdot \hat{\v}) \hat{\v}\equiv
x -x^{\parallel} \hat{\v}$. Then, the integral
\beq
W(x^{\perp}; \hat{\v}):= \int d \tau V(x +\hat{\v} \tau),
\label{1.37}
\ene
exists and is continuous by (\ref{1.1}). The set of all $\Phi_0,
\Psi_0$ is rich enough to determine for any $\hat{\v}$ the continuous function
$ W(\cdot , \hat{\v})$ from the r.h.s. of (\ref{1.10}). For $n=2$,
$W(x^{\perp}; \hat{\v})$ is the Radon transform of the square integrable potential $V(x )$.
It is well known that the Radon transform determines $V$ uniquely (\cite{he}, p. 115). If $n \geq 3$
it is the $X$-ray transform. However, one can fix arbitrarily
$(x_3,x_4, \cdots x_n )$ and apply the same to the resulting two-dimensional
function. Then, varying $\hat{\v}$ in a plane is actually sufficient to reconstruct
$V(x)$ from $W(x^{\perp}; \hat{\v})$. This completes the proof of Theorem 1.1.
The time-dependent approach is quite flexible. It has
been applied to many inverse scattering problems. In \cite{ew2}, \cite{ew3}
to N-body systems with singular and long-range potentials, in \cite{we2} to
the N-body Stark effect, and in \cite{ew4} to two-cluster scattering.
In \cite{we3} the case of N-body systems with time-dependent potentials was treated.
The case of regular magnetic fields on $\ER^n$
was considered in \cite{a1}, \cite{a2}, and \cite{a3}. The relativistic
Schr\"{o}dinger operator, and the Dirac and Klein-Gordon equations were studied
in \cite{j1} and \cite{j2}. In \cite{it} the Dirac equation with time-dependent electromagnetic
potentials was considered. For the case of the Aharonov-Bohm effect, see Section 2.
For references on the stationary theory see
\cite{we1} and \cite{ew3}. In all of these papers the direct scattering problem is
linear. For the case when the direct scattering problem is nonlinear see Section 3.
\section{The Aharonov-Bohm Effect}\sss
We now discuss the Aharonov-Bohm effect \cite{ab}. Aharonov and Bohm
considered the scattering of an electron off the magnetic
field of a tiny solenoid, idealized as having infinite length and zero radius
(scattering off a thread of magnetic flux). We assume that the solenoid is located on the
vertical axis of the coordinate system, and in consequence it is enough to consider scattering in the plane orthogonal to
the solenoid, as the problem is invariant under translation along the vertical direction.
The magnetic field of an unshielded solenoid at zero is given by, $ B_s \,\delta (x)
\linebreak $ with $ x=(x_1,x_2) \in \ER^2$ (this
actually corresponds to a magnetic field in $\ER^3$ with components $(0,0, B_s \,\delta (x))$.
We also assume that there is a regular magnetic field $B_{R} \in C^1_0 (\ER^2)$ that is
continuously differentiable and has compact support. The total magnetic
field is written as,
\beq
B:= B_s \,\delta (x) + B_R,
\label{2.1}
\ene
where, of course, $B_s$ is a real constant and $B_R$ is a real-valued function. In order to define the
Schr\"{o}dinger operator for an electron in the presence of $B$ we have to introduce a magnetic
potential. The magnetic potential for $B_s \,\delta (x)$ in the Coulomb
gauge is given by,
\beq
A_0:= \frac{\alpha_{\0} }{ |x|^2} \left[\begin{array}{c} -x_2 \\ x_1 \end{array}
\right],
\label{2.2}
\ene
where we denote $\alpha_{\0}:=\frac{B_s }{2 \pi}$. Observe that $\alpha_{\0}$ is the flux ( across zero) of the singular magnetic field normalized by $2\pi$. It is easily checked that
$\nabla \times A_s:= \frac{\partial}{\partial x_1} A_{s,2}
-\frac{\partial}{\partial x_2} A_{s,1}= B_s \,\delta (x)$ and that $ \nabla \cdot A_s=0$,
with the derivatives taken in distribution sense in ${\cal D}'$.
The magnetic potential in the Coulomb gauge for $B_R$ is given by,
\beq
A_R= \frac{1}{2 \pi} \int B_R(x-y) \left[\begin{array}{c} -\hat{y}_2 \\ \hat{y}_1
\end{array}\right]\, \frac{dy}{|y|},
\label{2.3}
\ene
where, $ \hat{y}:= \frac{y}{|y|}$. Clearly, $A_R\in C^1 \left(\ER^2, \ER^2\right)$.
The magnetic potential for $B$ in the Coulomb gauge is given by,
\beq
A_c:= A_s+ A_R.
\label{2.5}
\ene
As is well known, the magnetic potential is not uniquely defined by the magnetic field; there is
always the possibility of a gauge transformation. We introduce
below a general class of magnetic potentials that is convenient for our purposes.
\begin{definition}
We denote by ${\cal A}_{\0}(\alpha_{\0},B_R)$ the set of all real-valued
$A \in C^1\left( \ER^2 \setminus 0,
\ER^2 \right) \cap L^1_{{\rm loc}} \left( \ER^2 , \ER^2 \right)$, with $ \nabla \times A =
2 \pi \alpha_{\0} \,\delta (x) + B_R$, in ${\cal D}'$. Moreover, we assume that
$A(x)=O\left(|x|^{-1}\right), |x| \rightarrow \infty$ and that,
\beq
a(r):= \sup_{|x| \geq r} |A(x)\cdot \hat{x}| \in L^1 ([0, \infty )).
\label{2.6}
\ene
\end{definition}
Let us now study the case where a general singular magnetic field
is contained inside an infinite cylinder,
with axis along the vertical direction, and transversal section $K$, where $K$ is a compact
subset of $ \ER^2 $. The purpose of the cylinder is to shield the singular magnetic field from
the incoming electrons. As we will see below, we cannot hope that the scattering operator
determines uniquely the magnetic field inside $K$. In fact, we can only determine the
(normalized ) flux of the magnetic field across $K$ modulo 2. This suggests that instead of
specifying the magnetic field inside $K$ we only fix the magnetic flux across $K$,
normalized by $2 \pi$, that by Stokes' theorem is given by the circulation of the magnetic potential, $A$, along $ \partial K$,
\beq
\alpha_K := \frac{1}{2 \pi} \int_{\partial K} A(x)\cdot dx,
\label{2.7}
\ene
where we integrate in counter-clockwise sense.
Of course, we also specify a regular magnetic field
$B_R \in C^1_0 \left( \ER^2 \setminus K \right)$ outside of $K$.
The magnetic flux $\alpha_K$ could be produced, for example, by a finite number of
solenoids inside $K$, and also by a regular magnetic field contained inside
$K$, or by a combination of both. The considerations above suggest the definition of the following
class of magnetic potentials. In what follows
we denote, $\Omega:= \ER^2 \setminus K$, where for the unshielded solenoid, $K=\{0\}$.
\begin{definition}
Let $K$ be a compact set such that $0 \in K$ and that its boundary,
$\partial K$, is a simple, closed, $C^1$ curve. Then, for any $\alpha_K \in \ER$ and any real-valued
$B_R \in C^1_0 \left( \overline{\Omega}\right)$ we denote by
${\cal A}_K \left( \alpha_K, B_R\right)$ the set of all real-valued
$A \in C^1\left( \overline{\Omega},
\ER^2 \right)$ with $ \nabla \times A = B_R$ and such that,
\beq
\alpha_K = \frac{1}{2 \pi} \int_{\partial K} A(x)\cdot dx,
\label{2.8}
\ene
where we integrate in counter-clockwise sense.
Moreover, we assume that
$A(x)=O\left(|x|^{-1}\right), |x| \rightarrow \infty$ and that,
\beq
a(r):= \sup_{x \in \Omega ,|x| \geq r} |A(x)\cdot \hat{x}| \in L^1 ([0, \infty )).
\label{2.9}
\ene
\end{definition}
The formal Hamiltonian is the operator,
\beq
h_A:= \frac{\left( \p -A \right)^2}{2 m},
\label{2.10}
\ene
with domain $C^2_0 \left( \Omega \right)$, $ \p := -i \nabla$ is the momentum
operator, and $ m> 0$ is the mass of the electron. We take Planck's constant and the speed of light all equal to one and the charge of the electron
equal to minus one.
The quadratic form associated to $h_A$ is given by,
\beq
q_A( \phi ,\psi ):= \left( (\p -A) \phi , (\p -A) \psi \right),
\label{2.11}
\ene
with domain $C^1_0 \left( \Omega \right)$. The form $q_A$ is non-negative and
closable.
The Hamiltonian, $H_A$, is the self-adjoint operator in $L^2 ( \Omega)$ associated to the closure
of $q_A$ (see \cite{ka}). $H_A$ is the extension of $h_A$ with Dirichlet boundary condition on
$\partial \Omega$.
Let $J$ be the identification operator from $L^2 \left( \ER^2 \right)$ onto
$L^2 \left( \Omega \right)$ given by multiplication by the characteristic function of $\Omega$. In the case $K= \0$ we take $J=I$.
The unperturbed Hamiltonian is given by $H_0:= \frac{\p^2}{2m}$, with domain the Sobolev space
$W_{2,2}\left( \ER^2 \right)$. The wave operators are defined as,
\beq
\w (A):= \hbox{s -}\lim_{t \rightarrow \pm \infty} e^{it H_A}\, J\, e^{-it H_0}.
\label{2.12}
\ene
We prove in \cite{we11} that if $A \in {\cal A}_K (\alpha_K,B_R)$,
the strong limits exist and are isometric. The scattering operator is given by,
\beq
S(A):= W_+^{\ast}\left( A \right)\,\,W_- \left( A \right).
\label{2.13}
\ene
Note that to define $H_A$ we only use the values of $A$ in $\Omega$.
This means that as long as $A$ is fixed in $\Omega$, we can change the magnetic
potential in the interior of $K$ without changing $H_A$. Note however that as
$ A\in C^1 \left( \overline{\Omega}\right)$ the flux
$\alpha_K$ is uniquely defined by the values of $A$ in $\Omega$.
This explains why we cannot hope to uniquely reconstruct the magnetic field
inside $K$ from the scattering operator and makes it plausible that we can
reconstruct $\alpha_K$.
As we said above, the only purpose of the obstacle, $K$, is to shield the
incoming electron from the magnetic
field, and in order to separate the scattering effect of the magnetic potential from that
of the obstacle, we consider asymptotic configurations that have negligible interaction
with $K$ for all times in the high-velocity limit. For this purpose, given $\hat{\v}\in S^1$,
let us denote,
\beq
\Omega_{\uv}:= \left\{ x \in \Omega : x+ \hat{\v} \tau \in \Omega,\, \hbox{for all}\, \tau \in
\ER \right\}.
\label{2.14}
\ene
Given $\v \in \ER^2$ we take asymptotic configurations $\Phi \in C^{\infty}_0 ( \Omega_{\hat{\v}})$,
where $ \hat{\v}:= \v / v$, with $ v:= |\v|$. The free evolution boosted by $\v$ is given by
$ e^{-im \v \cdot x} \, e^{-itH_0} \, e^{im \v \cdot x}= e^{-im v^2 t/2}
e^{-i \p \cdot \v t} e^{-itH_0},$ and -to a good approximation- in the limit when $ v \rightarrow
\infty $ with $\hat{\v}$ fixed this can be replaced (modulo an
unimportant phase) by the classical translation $e^{-it\p \cdot \v}$. Then, in the
high-velocity limit it is a good approximation to assume that the free evolution of our
asymptotic configuration is given by
$e^{-it\p \cdot \v} \Phi_0 = \Phi_0 (x- \v t) $, and as
$\Phi_0 \in C^{\infty}_0 ( \Omega_{\hat{\v}})$, it has negligible interaction with $K$
for all times. In the following theorem we evaluate the high-velocity asymptotics of the
scattering operator.
\begin{theorem}
Suppose that $A \in {\cal A}_K(\alpha_K, B_R)$ and that $\Phi_0, \Psi_0 \in C^{\infty}_0\left(\Omega_{\uv}
\right)$. Let
$ \Phi_{\v}, \Psi_{\v}$ be the boosted asymptotic configurations,
\beq
\Phi_{\v}:= e^{im \v \cdot x} \Phi_0, \,\, \Psi_{\v}:= e^{im \v \cdot x} \Psi_0.
\label{2.15}
\ene
Then,
\beq
\left( S(A) \Phi_{\v}, \Psi_{\v}\right)= \left( e^{i \int_{-\infty}^{\infty} \uv\cdot A(x+
\uv \tau )
\, d\tau} \Phi_0, \Psi_0 \right)+ O\left(\frac{1}{|v|}\right),\,\,\, |v| \rightarrow \infty.
\label{2.16}
\ene
\end{theorem}
We prove in \cite{we11} that (\ref{2.16}) determines uniquely the
Radon transforms,
\beq
\int_{-\infty}^{\infty}B_R (x + \uv \tau) \, d \tau, \,\,\, x \in \Omega_{\hat{v}}.
\label{2.17}
\ene
By the support theorem for the Radon transform \cite{na}, $B_R$ is uniquely
determined in $\Omega$, provided that $K$ is convex. The fact that
the magnetic flux, $\alpha_K$, is determined modulo 2 follows from an explicit
calculation. This gives us
the following theorem.
\begin{theorem}
Suppose that $A^{(j)}\in {\cal A}_K\left(\alpha^{(j)}_K, B^{(j)}_R\right),
j=1,2$ and that $K$ is convex. Then, if $S\left(A^{(1)}\right)= S\left(A^{(2)}\right)$, we have that,
$\alpha^{(1)}_K= \alpha^{(2)}_K$ modulo $2$ and that $B^{(1)}_R(x)=
B^{(2)}_R(x), x \in \Omega$.
\end{theorem}
For a complete study of this problem including proofs and an analysis of gauge transformations see
\cite{we11}.
Observe that since (\ref{2.17}) is obtained from (\ref{2.16}) we actually only
need to know the
high-velocity limit of the scattering operator. Moreover, our proof gives a method
for the
reconstruction of $B_R$ in $\Omega$. Note that in spite of the fact that the
scattering of the
electron takes place outside of $K$, we determine the magnetic flux in
$K$- modulo 2- from the scattering
operator. This is the Aharonov-Bohm effect \cite{ab} that shows that in quantum
mechanics the magnetic field acts on a charged particle -by means of the magnetic
potential-
even in regions where it is identically zero.
Nicoleau has proven in \cite{ni} the following result using stationary methods.
Suppose that the magnetic field, $B$, is infinitely differentiable in $\ER^2$
and has
compact support, and that $K$ is compact, convex, $ 0 \in K$ and
$\partial K$ is smooth. Then, if $S\left(A^{(1)}\right)= S\left(A^{(2)}\right)$,
with $A^{(j)},\,\, j=1,2$, the Coulomb gauge potentials,
then the two magnetic fluxes across $\ER^2$ are equal modulo $2$ and
$B^{(1)}_R(x)=
B^{(2)}_R(x), x \in \Omega$. The case of an unshielded solenoid with $K=\{0\}$ is
not covered by the result of \cite{ni}. Note that this is actually the problem
considered by Aharonov and Bohm \cite{ab}.
In the case where the interior of $K$ is non-empty, \cite{ni} considers the
situation where the magnetic flux across $K$ is produced by a magnetic field in
$C^{\infty}_0 \left( \ER^2 \right)$. Our result is considerably more general
in the sense that we study the case where the magnetic flux across $K$ is produced
by any magnetic field inside $K$. The only restriction is that the magnetic
flux across $K$ has to be finite.
We could as well consider the case where there is also a scalar potential -as
is done in \cite{ni}- however, we do not pursue that direction here.
\section{The Non-Linear Schr\"{o}dinger Equation} \sss
We discuss now the non-linear Schr\"{o}dinger equation with a potential on
the line. The goal is to give a method to uniquely reconstruct the potential
and the nonlinearity. This problem was solved in \cite{we4}, \cite{we5},
\cite{we6}, \cite{we7} and
\cite{we8}. In these papers the multidimensional case was also considered.
For the nonlinear Klein-Gordon equation see \cite{we12}, \cite{we9} and \cite{we10}.
Let us consider the following non-linear Schr\"{o}dinger equation with a potential
\beq
i \frac{\partial }{\partial t}u(t,x)= -\frac{\partial^2}{\partial x^2} u(t,x)+V_0(x)u(t,x) +
F(x,u),\,\,u(0,x)= \phi (x),
\label{3.1}
\ene
where $t,x \in \ER$, the potential, $V_0$, is a real-valued function, and $F(x,u)$ is a
complex-valued function.
Let $H_0$ denote the
unique self-adjoint realization of $-\frac{d^2}{d x^2}$ with domain the Sobolev
space $W_{2,2}$ \cite{8}.
For any $\gamma \in \ER$, $L^1_{\gamma}$ denotes the Banach space of all
complex-valued measurable functions, $\phi $, defined on $\ER$ and such that
\beq
\left\|\phi \right\|_{L^1_{\gamma}}:= \int |\phi (x)| \, (1+|x|)^{\gamma} \, dx
< \infty .
\label{3.4}
\ene
If $V_0 \in L^1_1$ the differential expression $\tau :=-\frac{d^2}{d x^2}+V_0(x)$ is
essentially self-adjoint.
We denote by $H$ the unique self-adjoint realization of $\tau$. The $H$ has a finite
number of negative eigenvalues, it
has no positive or zero eigenvalues, it has no singular-continuous spectrum and
the absolutely-continuous spectrum is $[0, \infty )$ (for these results see \cite{wei}).
The wave operators are defined as in the multidimensional case,
\beq
W_{\pm}:= \hbox{s}-\lim_{t \rightarrow \pm \infty} e^{itH} e^{-itH_0}.
\label{3.4b}
\ene
The limits in (\ref{3.4b}) are taken in the strong topology in $L^2$.
As is well known (see \cite{s}), the $W_{\pm}$ exist and are complete.
The scattering operator for the linear Schr\"{o}dinger equation ((\ref{3.1}) with
$F=0$) is defined as follows:
\beq
S_L:=W_+^{\ast} \, W_-.
\label{3.7b}
\ene
The key issue for the scattering theory for equation (\ref{3.1}) is the
following time-dependent $L^p-L^{\acute{p}}$ estimate that we proved in \cite{we5},
\begin{equation}
\left\|e^{-itH} P_c \right\|_{{\cal B}\left( L^p,L^{\acute{p}}\right)} \leq
\frac{C}{t^{(\frac{1}{p}-\frac{1}{2})}},\, t > 0,
\label{p}
\end{equation}
for some constant $C,\,\, 1 \leq p \leq 2$, and $\frac{1}{p}+\frac{1}{\acute{p}}=1$,
and where $P_c$ denotes the projector onto the space of continuity of $H$.
The $L^p-L^{\acute{p}}$ \,estimate \,(\ref{p}) expresses the dispersive nature of
the solutions to the linear
Schr\"{o}dinger equation with initial data on the continuous subspace of
$H$. It gives a quantitative meaning to the
{\it spreading of the wave packets}. In typical applications the nonlinearity, F, is
proportional to a high-enough power of $u$. This
type of nonlinearity makes the solutions to (\ref{3.1}) even larger where they
are already large. On the other hand, the {\it spreading } of the associated
linear equation prevents the solution from becoming
to large, provided that the initial data was small enough. It is the
balance from these two
phenomena that is at the heart of small-amplitude scattering theory. Eventually, the
{\it spreading} prevents the solution from blowing up in a finite time, and for
large times the evolution is dominated by the linear part in the sense that the
solution is asymptotic to a solution of the linearized equation. This is the
physical content of Theorem 3.1 below. By the same argument, on the small amplitude
limit
the nonlinear effects become negligible and scattering is dominated by the linear
term. This fact is expressed in a quantitave way by Theorem 3.2 that allows us to
reconstruct the linear scattering operator from the derivative at zero of the
nonlinear scattering operator. It is interesting to remark that the {\it spreading
of the wave packets} -that is irrelevant on the high-energy limit in the linear
case-
is actually essential on the low-energy (small amplitude) limit
in the non-linear case.
Before we state our results we introduce some standard
definitions and notations.
We say that $F(x,u)$ is a $C^k$ function of $u$ in the real sense if
for each fixed $x \in \ER$,
$\Re F$ and $\Im F$ are $C^k$ functions of the real and
imaginary parts of $u$. We will assume that $F$ is $C^2$ in the real sense and
that
$\left(\frac{\partial}{\partial x} F\right)(x,u)$ is $C^1$ in the real sense.
If $F=F_1+iF_2$ with $F_1, F_2$
real-valued, and $u=r+is,\,\, r,s \in \ER$ we denote,
\beq
F^{(2)}(x,u):=\sum_{j=1}^2\left[\, \left| \frac{\partial^2}{\partial r^2} F_j (x,u)
\right|+ \left| \frac{\partial^2}{\partial r \partial s} F_j (x,u)\right|
+\left| \frac{\partial^2}{\partial s^2} F_j (x,u)\right|\,\right],
\label{3.2}
\ene
\beq
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,u):=\sum_{j=1}^2\left[\, \left|
\frac{\partial}{\partial r} \left(\frac{\partial}{\partial x}F_j\right)(x,u)\right|
+\left| \frac{\partial}{\partial s} \left(\frac{\partial}{\partial x}F_j\right)
(x,u)\right|\,\right].
\label{3.3}
\ene
For any pair $u,v$
of solutions to the stationary Schr\"{o}dinger equation:
\beq
-\frac{d^2}{d x^2} u+V_0u = k^2 u,\, k \in {\bf C}^+,
\label{3.8}
\ene
let $[u,v]$ denote the Wronskian of $u,v$:
\beq
[u,v]:= \left(\frac{d}{d x}u\right) v- u \frac{d}{d x}v.
\label{3.9}
\ene
Let $f_1(x,k)\approx e^{ikx}, x \rightarrow \infty, f_2(x,k)\approx e^{-ikx}, x \rightarrow -
\infty$, be the Jost solutions to (\ref{3.8}) (see
for example \cite{DT}). A potential $V_0$ is said
to be {\it generic} if $[f_1(x,0),f_2(x,0)]\neq 0$ and $V_0$ is said to be
{\it exceptional} if $[f_1(x,0),f_2(x,0)]=0$. If $V_0$ is {\it exceptional} there is
a bounded solution to (\ref{3.8}) with $k^2=0$
(a half-bound state or a zero-energy resonance). The trivial
potential, $V_0=0$, is {\it exceptional}.
Let us denote,
$$
M:= \left\{ u \in C(\ER, W_{1,p+1}): \sup_{t \in \ER} (1+|t|)^d \|u\|_{W_{1,p+1}}
< \infty \right\},
$$
\beq
\hbox{with norm} \|u\|_M:= \sup_{t \in \ER}
(1+|t|)^d \|u\|_{W_{1,p+1}},
\label{3.10}
\ene
where $ p \geq 1$,\, and \, $d:= \frac{1}{2} \frac{p-1}{p+1}$. For functions
$u(t,x)$ defined in $\ER^2$
we simply write $u(t)$, instead $u(t,\cdot)$. The $W_{k,p}$ are the Sobolev spaces
\cite{8}.
The small-amplitude scattering operator is given in
the following theorem.
\begin{theorem}
Suppose that $V_0 \in L^1_{\gamma}$, where in the {\rm generic case} $ \gamma > 3/2$
and in the {\rm exceptional case} $ \gamma > 5/2$, that $H$ has no negative
eigenvalues, and that
\beq
N(V_0):= \sup_{x \in \ER} \int_x^{x+1} |V_0(y)|^2\, dy < \infty.
\label{3.10b}
\ene
Furthermore, assume that $F$ is $C^2$ in the real sense, that
$F(x,0)=0$, and that for each fixed $x \in \ER$ all the first order derivatives,
in the real sense, of $F$ vanish at zero. Moreover, suppose that
$ \frac{\partial}{\partial x}F$ is $C^{1}$
in the real sense. We assume that the following estimates hold:
\beq
F^{(2)}(x,u)= O\left(|u|^{p-2}\right),\,\,\,
\left(\frac{\partial}{\partial x}F\right)^{(1)}(x,u)= O\left(|u|^{p-1}\right),\,
\,u \rightarrow 0,
\label{3.11}
\ene
uniformly for $x \in \ER$, for some $ \rho < p < \infty$, and where $\rho$ is the positive root of
$\frac{1}{2}\frac{\rho -1}{\rho +1}=\frac{1}{\rho}$.
Then, there is a $\delta > 0$ such that for all
$\phi_- \in W_{2,2} \cap W_{1,1+\frac{1}{p}} $ with
$\|\phi_-\|_{W_{2,2}}+ \|\phi_-\|_{W_{1,1+\frac{1}{p}} } \leq \delta$ there is a
unique solution, $u$, to (\ref{3.1})
such that $u \in C(\ER, W_{1,2}) \cap M$ and,
\beq
\lim_{t \rightarrow -\infty}\|u(t)- e^{-itH} \phi_- \|_{W_{1,2}}=0.
\label{3.12}
\ene
Moreover, there is a unique $\phi_+ \in W_{1,2} $ such that
\beq
\lim_{t \rightarrow \infty}\|u(t)- e^{-itH} \phi_+ \|_{W_{1,2}}=0.
\label{3.13}
\ene
Furthermore, $e^{-itH} \phi_{\pm} \in M$ and
\beq
\left\| u- e^{-itH} \phi_{\pm} \right\|_M \leq C \left\|e^{-itH} \phi_{\pm}
\right\|_M^p,
\label{3.14}
\ene
\beq
\left\|\phi_+ - \phi_- \right\|_{W_{1,2}} \leq C \left[\left\|\phi_-
\right\|_{W_{2,2}} +\left\|\phi_- \right\|_{W_{1, 1+\frac{1}{p}}}\right]^p.
\label{3.15}
\ene
The scattering operator, $S_{V_0} : \phi_- \hookrightarrow \phi_+$ is injective on
$ W_{1,1+\frac{1}{p}} \cap W_{2,2}$.
\end{theorem}
Observe that, $ \rho \approx 3.56$. Remark that we do not to restrict $F$ in such a way that
energy is conserved.
To reconstruct the potential, $V_0$, we introduce below the scattering operator associated with
asymptotic states that are solutions to the linear Schr\"{o}dinger equation
with potential zero:
\beq
S:= W_+^{\ast}\, S_{V_0}\, W_-.
\label{3.16}
\ene
\begin{theorem}
Suppose that the assumptions of Theorem 3.1 are satisfied. Then for every
$\phi \in W_{2,2} \cap W_{1,1+\frac{1}{p}}$
\beq
\left.\frac{d}{d \epsilon} \,S(\epsilon \phi)\right|_{\epsilon=0}= S_L \phi,
\label{3.17}
\ene
where the derivative on the left-hand side of (\ref{3.17}) exists in the sense of
strong convergence in $W_{1,2} \cap W_{1,p+1}$.
\end{theorem}
\begin{corollary}
Under the conditions of Theorem 3.1 the scattering operator, $S$,
determines uniquely
the potential $V_0$.
\end{corollary}
In the case where $F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)}u$, with fixed integer
$j_0$, we can also
reconstruct the $V_j,\,\,\, j=1,2, \cdots$.
\begin{lemma}
Suppose that the conditions of Theorem 3.1 are satisfied, and moreover, that
$F(x,u)= \sum_{j=1}^{\infty} V_j(x) |u|^{2(j_0+j)} u,$\, where $j_0$ is an
integer such that, $j_0 \geq (p-3)/2$,
\, for $|u| \leq \eta$, for
some $\eta > 0$, and where $V_j \in W_{1, \infty}$ with $\|V_j\|_{W_{1, \infty}} \leq
M^j, j=1,2, \cdots$, for some positive constant $M$. Then, for
any $\phi \in W_{2,2} \cap W_{1,1+\frac{1}{p}} $ there is an
$\epsilon_0 > 0$ such that for all $ 0 < \epsilon < \epsilon_0$:
\beq
i \left( (S_{V_0}-I )(\epsilon \phi),\, \phi \right)_{L^2}=
\sum_{j=1}^{\infty}\epsilon^{2(j_0+j)+1}\left[ \int \int \, dt\, dx \, V_j(x)
\left| e^{-itH}\phi \right|^{2(j_0+j+1)} +Q_j\right],
\label{3.18}
\ene
where $Q_1=0$ and $Q_j, j > 1$, depends only on $\phi$ and on $V_k$ with $k< j$.
Moreover, for any $\acute{x} \in \ER$, and any
$ \lambda > 0$, we denote, $\phi_{\lambda}(x):=
\phi ( \lambda (x- \acute{x}))$.
Then, if $ \phi \neq 0$:
\beq
V_j(\acute{x})=\frac{\lim_{\lambda \rightarrow \infty}\lambda^3 \int \int\, dt \,
dx \, V_j(x) \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)}}{\int \int\, dt \,
dx \, \left|e^{-itH_0} \phi \right|^{2(j_0+j+1)}}.
\label{3.19}
\ene
\end{lemma}
\begin{corollary}
Under the conditions of Lemma 3.4 the scattering operator, S, determines uniquely the
potentials $V_j, j=0,1,\cdots$.
\end{corollary}
The method to reconstruct the potentials $V_j, j=0,1,\cdots$, is as follows. First
we obtain $S_L$ from $S$ using (\ref{3.17}). By any standard method for inverse
scattering for the linear Schr\"{o}dinger equation on the line we reconstruct $V_0$
(recall that $H$ has no eigenvalues).
We then reconstruct $S_{V_0}$ from $S$ using (\ref{3.16}). Finally (\ref{3.18}) and
(\ref{3.19}) give us, recursively, $V_j, j=1,2, \cdots$.
Theorems 3.1, 3.2, Lemma 3.4 and Corollaries 3.3, 3.5 are proven in \cite{we7} where
also a discussion of the literature is given. We give below an idea of the proofs.
In Theorem 1.1 of \cite{40} we proved that $\w$ and $\a$ are bounded
operators on $ W_{k,p},k=0,1,\, 1 < p < \infty$. By Theorem 3 in page 135 of
\cite{9},
\beq
\|{\cal F}^{-1} (1+q^2)^{k/2}({\cal F} f)(q)\|_{L^p},
\label{4.2}
\ene
is a norm that is equivalent to the norm of $W_{k,p}, 1 < p < \infty$.
In (\ref{4.2}) ${\cal F}$ denotes the
Fourier transform. Then, by the continuity of the $\w$ and $\a$ on $W_{k,p}$
(see Corollary 1.2 of \cite{40})
\beq
\| (I+H)^{k/2}\,f \|_{L^p},
\label{4.3}
\ene
defines a norm that is equivalent to the norm of $W_{k,p}, k=0,1, 1 < p < \infty$.
Condition (\ref{3.10b}) and Theorem 2.7.1 in page 35 of \cite{s} imply that,
$D(H)=D(H_0)= W_{2,2}$, and that the following norm is equivalent to the norm of
$W_{2,2}$:
\beq
\left\| (H+I) \phi \right\|_{L^2}.
\label{4.9b}
\ene
The weight $(I+H)^{k/2}, k=1,2$ has the advantage that it commutes with
$e^{-it H}$ and moreover, in the case $p=2$ the equivalent norm is invariant under
the time evolution given by $e^{-it H}$. We will use these equivalences without further comments.
In particular, it
follows from (\ref{4.3}) that estimate (\ref{p}) holds in the norm on
${\cal B}\left( W_{1,p} ,W_{1,\acute{p}}\right),\,\, 1 < p \leq 2$.
By Sobolev's imbedding theorem \cite{8}, $W_{1,2}$ is continuously imbedded
in $L^{\infty}$. It follows that $F$ is locally Lipschitz continuous on $W_{1,2}$.
Then, by
standard arguments, $u \in C(\ER ,W_{1,2}) \cap M $ is a solution
to (\ref{3.1}) with $\lim_{t \rightarrow - \infty} \|u(t) -
e^{-itH} \phi \|_{\h}=0$, for some $ \phi \in W_{1, 2}$, if and only if
$u$ is a solution to the following integral equation:
\beq
u= e^{-itH} \phi + \frac{1}{i} \int_{-\infty}^t e^{-i(t -\tau )H} F(x,u(\tau ))\, d \tau.
\label{4.4}
\ene
As we prove below the integral in the right-hand side of (\ref{4.4})
converges absolutely in ${\h}$ and in $M$. For $u \in M$ we denote
\beq
{\cal Q}u(t):=\frac{1}{i} \int_{-\infty}^t e^{-i(t-\tau )H} F(x,u(\tau ))\, d \tau .
\label{4.5}
\ene
It follows from (\ref{p}),
and since$W_{1, p+1}$ is continuously imbedded in $L^{\infty}$, that
\beq
\left\| {\cal Q}u(t) - {\cal Q}v(t) \right\|_{\X} \leq C \,
(1+|t|)^{-d}( \|u\|_M + \|v \|_M )^{p-1}
\|u-v\|_M,
\label{4.6}
\ene
where we used that $p\, d > 1$. The constants $C$ in (\ref{4.6})
can be taken uniform in closed balls in $M$.
By (\ref{4.6}) with $v(t)=0$ :
$$
\left\|{\cal Q}u(t) \right\|_{\h}^2\leq C \Re \int_{-\infty}^t \,d \tau \,
\left(\sqrt{I+H} F(x, u(\tau)), \sqrt{I+H}{\cal Q}u(\tau) \right)_{L^2} \leq \,C
\int_{-\infty}^t d \tau \,
\|F(x, u)(\tau )\|_{W_{1,{1+1/p}}}\times
$$
$$
(1+ |\tau |)^{-d}\, \|u\|_M^p
\leq C
\int_{-\infty}^t\, d \tau \, \|u\|_{W_{1,p+1}}^p\, \, (1+|\tau |)^{-d}
\|u\|^p_M \leq
C \int_{-\infty}^t \,d \tau \, (1+|\tau |)^{-d(p+1)}\, \|u\|^{2p}_M
$$
\beq
\leq C
(1+\max[0,-t ])^{-(d+dp-1)}\, \|u\|^{2p}_M .
\label{4.7}
\ene
By (\ref{4.6}) with $v(t)=0$, the integral in the right-hand side of (\ref{4.4})
converges in $M$
and by (\ref{4.7}) the converge holds also in $\h$.
By (\ref{4.9b}) and Sobolev's imbedding theorem,
\beq
\|e^{-itH} \phi_- \|_{W_{1,p+1}} \leq C \| e^{-itH}\phi_- \|_{W_{2,2}}\leq C
\left\| (H+I) e^{-itH} \phi_- \right\|_{L^2}= C
\left\| (H+I) \phi_- \right\|_{L^2}\leq C \left\| \phi_- \right\|_{W_{2,2}}.
\label{4.10}
\ene
Then, (\ref{p}) and (\ref{4.10}) imply that,
\beq
\left\|e^{-itH} \phi_- \right\|_M \leq C \left[ \left\| \phi_-\right\|_{W_{2,2}}+
\| \phi_- \|_{W_{1,1+\frac{1}{p}}}\right].
\label{4.11}
\ene
For $R > 0$ let us denote: $ M_{R}:=\{ u \in M: \|u\|_M \leq R\}$. Let us take $R$ so
small that $C (2 R)^{p-1} \leq 1/2$, with $C$ as in (\ref{4.6}), and $ \delta > 0$ so
small that $C \delta \leq R/4$, with $C$ as in (\ref{4.11}).
It follows from (\ref{4.6})
and (\ref{4.11}) that the map $ u \hookrightarrow
e^{-itH} \phi_- + {\cal Q} \, u$ is a contraction from $M_{R}$ into $M_{R}$
for all $\phi_- \in W_{2,2} \cap W_{1,1+\frac{1}{p}}$ with
$\| \phi_- \|_{W_{2,2}}+\|\phi_-\|_{W_{1,1+\frac{1}{p}}} \leq \delta$. The
contraction mapping theorem implies that there is an unique solution to
(\ref{4.4}) in $M_{R}$. This is the solution $u(t)$ of Theorem 3.1. Moreover,
\beq
\left\|u\right\|_M \leq \left\|e^{-itH} \phi_- \right\|_M +\frac{1}{2}
\left\|u\right\|_M.
\label{4.13}
\ene
Then,
\beq
\left\|u\right\|_M \leq C \left\|e^{-itH} \phi_- \right\|_M.
\label{4.14}
\ene
We define:
\beq
\phi_+ =\phi_- + \frac{1}{i} \int_{-\infty}^{\infty} e^{i\tau H} F(x, u(\tau ))\, d \tau.
\label{4.15}
\ene
For further
details on the proof of Theorem 3.1 see \cite{we7}.
\noindent {\it Proof of Theorem 3.2 :} Since, $S(0)=0$,
and $W_{\pm}$ are bounded on $W_{2,2} \cap W_{1,1+\frac{1}{p}}$ \cite{40},
it is enough to prove that
\beq
s-\lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\,
(S_{V_0}(\epsilon \phi)-\epsilon \phi )
=0.
\label{4.22}
\ene
By (\ref{4.11}) and (\ref{4.14}) with $\phi_-$ replaced by $\epsilon \phi$:
\beq
\|u\|_M \leq C |\epsilon| \left[ \left\| \phi \right\|_{W_{2,2}}+
\| \phi \|_{W_{1,1+\frac{1}{p}}}\right].
\label{4.23}
\ene
Using and (\ref{p}) and (\ref{4.15}) we obtain that,
$$
\left\|S_{V_0}(\epsilon \phi)-\epsilon \phi \right\|^2_{W_{1,2}}
\leq C \int_{-\infty}^{\infty} \,d \tau \,
\left(\sqrt{I+H} F(x, u(\tau)), \sqrt{I+H} \int_{-\infty}^{\infty} d \rho \, e^{-i(\tau - \rho )H}
F(x, u(\rho )) \right)_{L^2}
$$
$$
\leq \,C
\int_{-\infty}^{\infty} d \tau \,
\|F(x, u)(\tau )\|_{W_{1,{1+1/p}}}\times
$$
$$
(1+ |\tau |)^{-d}\, \|u\|_M^p
\leq C
\int_{-\infty}^{\infty}\, d \tau \, \|u\|_{W_{1,p+1}}^p\, \, (1+|\tau |)^{-d}
\|u\|^p_M \leq
C \int_{-\infty}^{\infty} \,d \tau \, (1+|\tau |)^{-d(p+1)}\, \|u\|^{2p}_M
$$
\beq
\leq C \|u\|^{2p}_M .
\label{4.23b}
\ene
Equation (\ref{4.22}) follows from (\ref{4.23}) and (\ref{4.23b}).
\noindent {\it Proof of Corollary 3.3:}\,By Theorem 3.2 $S$ determines uniquely
$S_L$. From $S_L$ we
get the
reflection coefficients for linear Schr\"odinger scattering on the line. As $H$ has
no bound states we
uniquely reconstruct $V_0$ from one of the reflection coefficients by using any
method for inverse scattering on the line.
\noindent {\it Proof of Lemma 3.4 :}\,\, By the contraction mapping theorem,
\beq
u(t)= e^{-itH}\epsilon \phi+ \sum_{n=1}^{\infty} {\cal Q}^n e^{-itH}\epsilon
\phi.
\label{4.24}
\ene
Equation (\ref{3.18}) follows from (\ref{4.15}) and (\ref{4.24}).
By Sobolev's imbedding theorem \cite{8}, $W_{2,2} \subset L^{q},\, 2 \leq q \leq
\infty$. Then, estimating as in (\ref{4.10}) we prove that,
$\left\|e^{-itH} \phi \right\|_{L^q} \leq C_q \left\|
e^{-itH} \phi \right\|_{W_{2,2}} \leq C_q\left\|
\phi \right\|_{W_{2,2}}, 2 \leq q \leq \infty $ , and as $ 2(j_0 +j +1) \geq p+1$ it follows from (\ref{p})
that:
\beq
\int \int\, dt \,
dx \, \left|e^{-itH} \phi \right|^{2(j_0+j+1)}\leq \left\|e^{-itH}\phi
\right\|_{L^{\infty}}^{2(j_0 +j +1)- p-1} \int\, dt \,
dx \, \left|e^{-itH} \phi \right|^{p+1} < \infty,\,\,\, j=1,2, \cdots.
\label{4.25}
\ene
For
$ \lambda > 0$ and $ \acute{x} \in \ER$ we denote
by $H_{\lambda}$ the following self-adjoint operator in $L^2$:
\beq
H_{\lambda}:= H_0 + V_{\lambda}(x),\, \hbox{where}\, V_{\lambda}(x)=
\frac{1}{\lambda^2}V_0\left(\frac{x}{\lambda}+ \acute{x}\right).
\label{3.17b}
\ene
Since $H$ has no eigenvalues, we have that $H_{\lambda}$ has no eigenvalues, i.e.,
$ H_{\lambda} > 0$.
It follows from (\ref{3.10b}) and from Theorem 2.7.1 on page 35 of \cite{s}
that
\beq
C_1 \left\| \phi \right\|_{W_{2,2}} \leq \left\| (H_{\lambda}+I) \phi \right\|_{L^2}
\leq C_2 \left\| \phi \right\|_{W_{2,2}},
\label{4.26}
\ene
for some constants $C_1, C_2$. Moreover, since $N(V_{\lambda}) \leq
\frac{1}{\lambda^3}\,N (V_0),\,\,\, \lambda \geq 1 $, the proof of
Theorem 2.7.1 on page 35 of \cite{s} implies that
we can take fixed $C_1$ and $C_2$ for all $ \lambda \geq 1$.
To prove (\ref{3.19}) we denote: $\tilde{t}:= \lambda^2 t$ and
$\tilde{x}:= \lambda (x-\acute{x})$. Then, we observe that,
\beq
\left( e^{-i\tilde{t} H_{\lambda}} \phi\right)(\tilde{x})=
\left( e^{-itH} \phi_{\lambda}\right)(x).
\label{4.27}
\ene
This can be seen as follows,
\beq
i \frac{\partial}{\partial t} \left( e^{-i\tilde{t} H_{\lambda}} \phi\right)=
H \left( e^{-i\tilde{t} H_{\lambda}} \phi\right),\,\,\, \hbox{and}\,\,\,
\left( e^{-i\tilde{t} H_{\lambda}} \phi\right)\left.\right|_{t=0}= \phi_{\lambda}.
\label{4.27b}
\ene
Since the solution to the linear Schr\"{o}dinger equation is unique, (\ref{4.27}) is proved.
It follows from (\ref{4.27}) that,
\beq
I_j:= \lambda^3 \int \int\, dt \,
dx \, V_j(x) \left|e^{-itH} \phi_{\lambda} \right|^{2(j_0+j+1)}=
\int \int\, d\tilde{t} \,
d\tilde{x} \, V_j(\frac{\tilde{x}}{\lambda}+\acute{x})
\left|e^{-i\tilde{t}H_{\lambda}} \phi \right|^{2(j_0+j+1)}
(\tilde{x}).
\label{4.28}
\ene
By (\ref{4.26})
\beq
s- \lim_{\lambda \rightarrow \infty}
e^{-i\tilde{t}H_{\lambda}} \phi =
e^{-i\tilde{t}H_0} \phi,
\label{4.29}
\ene
where the limit exists in the strong topology on $W_{2,2}$. By Sobolev's imbedding
theorem, the limit in (\ref{4.29}) also exists in the strong topology on $L^q, \,
2 \leq q\leq \infty$, and moreover,
\beq
\left\|e^{-i\tilde{t}H_{\lambda}} \phi\right\|_{L^q} \leq
C_q \left\| \phi \right\|_{W_{2,2}}, \,\,\,2 \leq q \leq \infty,\,\,\, \lambda \geq 1.
\label{4.30}
\ene
Furthermore, by (\ref{p}) and(\ref{4.27})
\beq
\left\|e^{-i\tilde{t} H_{\lambda}}\phi \right\| _{L^{p+1}}^{p+1}= \lambda \left\|e^{-i t H}\phi_{\lambda}
\right\| _{L^{p+1}}^{p+1}
\leq C \frac{1}{ t^{d(p+1)}} \lambda \left\| \phi_{\lambda} \right\|_{L^{1+1/p}}^{p+1} = C
\frac{1}{ \tilde{t}^{d(p+1)}} \left\|\phi \right\|_{L^{1+1/p}}^{p+1},
\label{4.31}
\ene
with $d:= \frac{1}{2}\frac{p-1}{p+1}$. Equation (\ref{3.19}) follows from
(\ref{4.28}), (\ref{4.29}), (\ref{4.30}),
(\ref{4.31}) and the dominated convergence theorem, observing that $2(j_0+j+1) \geq
p+1$, that $d (p+1) > 1$ and that $V_j$ is continuous.
\noindent {\it Proof of Corollary 3.5:}\,\, By Corollary 3.3, $S$ determines
uniquely $V_0$. Then the
wave operators, $W_{\pm}$, are uniquely determined, and by (\ref{3.16}), $S$
determines uniquely $S_{V_0}$. Finally by (\ref{3.18}) and (\ref{3.19}) $S_{V_0}$
determines
uniquely $V_j, j=1,2, \cdots$.
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