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Spectral theory, Scattering theory, Mourre theory
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\topmatter
\title{Two-Body short-range systems in a time-periodic electric field}\endtitle
\author{Jacob Schach M{\o}ller\footnote{Supported in parts by "Rejselegat for
matematikere" (Travelling grant for mathematicians)\newline
and TMR grant FMRX-960001.\newline}}\endauthor
\affil{Universit{\'e} Paris-Sud\\
D{\'e}partement de Math{\'e}matiques\\
Orsay, France.\\ email: Moeller.Jacob\@ lanors.math.u-psud.fr}\endaffil
\abstract{We apply a method developed by J.Howland and K.Yajima in
conjunction with ideas
from the analysis of Hamiltonians with constant electric fields to obtain
absence of bound states and asymptotic completeness for 2-body short-range
systems in an external time-periodic electric field.}\endabstract
\endtopmatter
\document
\subhead{Section 1: Introduction}\endsubhead
In the present paper we treat the scattering problem for two $\nu$-dimensional
particles interacting through a short-range potential and placed in an
external time-periodic electric field. The Hamiltonian for such a system is
$$
H(t)= \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} - q_1\Cal E(t)\cdot x_1
-q_2\Cal E(t)\cdot x_2 + v(x_2-x_1)\quad\text{on}\quad L^2(\Bbb
R^{2\nu}).\tag{1.1}
$$
Here $m_i$ and $q_i$, $i\in\{1,2\}$ are the masses and the charges of the two
particles and $x_1,x_2$ denotes their position. The electric field $\Cal E$
is periodic with some period $T>0$, that is $\Cal E(t+T) = \Cal E(t)$
almost everywhere. The short-range potential $v$ will be allowed to have an
explicit time-dependence as long as this dependence is periodic with the same
period as the field.
Recently asymptotic completeness for many-body systems in constant electric
fields has been proved for large classes of potentials, see [AT], [HMS1] and
[HMS2]. For a treatment of propagation estimates for such systems see [A].
All these results rely on well known techniques which uses local commutator
estimates to obtain spectral and scattering information.
By controlling the energy along the time evolution one can apply some of
these techniques to time-dependent problems. This has been done in [SS] and
applied in [Si1]. In [G1] and [Z] time-boundedness of the kinetic energy
plays an essential role and in [HL] the problem of bounding the kinetic
energy is treated for repulsive potentials using positive commutator
techniques.
In the present problem however one readily observes that the energy is
generally not bounded in time. In fact for very simple examples like $\Cal
E(t)=\frac12+\cos(t)$ ($\nu=1$ and $v=0$) one finds that the expectation
value of the energy oscillates with an amplitude that grows like $t^2$. On
the other hand the expectation of $x_2-x_1$ grows like
$(\frac{q_2}{m_2}-\frac{q_1}{m_1})t^2$ as one would expect from the
constant field problem. Consequently it is natural to suggest that
completeness and absence of bound states (the meaning of which will be
discussed later) hold here as well provided the particles have different
charge to mass ratio. In order to ensure the growth of $x_2-x_1$ we make
the crucial assumption that the field has non-zero mean.
In order to circumvent the problem of controlling the energy we adopt a
method due to Howland, [H1], which is motivated by a similar procedure in
Hamiltonian mechanics, where one also faces the problem of non-conservation
of energy in time-dependent problems.
The idea is to include time as a space variable and
introduce a new momentum variable $\tau$, conjugate to time. One defines a
new time-independent Hamiltonian $\hat{H} = \tau+H$ as a function on $\Bbb
R^\nu_q\times\Bbb R_t\times\Bbb R^\nu_p\times\Bbb R_\tau$, from the original
Hamiltonian, $H$, to obtain the following system
$$
\alignat2
\dot{q}(s) &= \nabla_p \hat{H}(t(s),\tau(s),q(s),p(s)),&\quad q(0)&=q_0\\
\dot{p}(s) &= -\nabla_q \hat{H}(t(s),\tau(s),q(s),p(s)),&\quad p(0)&=p_0\\
\dot{t}(s) &= 1,&\quad t(0)&=t_0\\
\dot{\tau}(s) &= -\partial_t \hat{H}(t(s),\tau(s),q(s),p(s)),&\quad\tau(0)&=\tau_0.
\endalignat
$$
Obviously the $(q,p)$ part of the solution to this extended system is nothing
more than time-translates of the solution to the physical system, with the
initial value $t_0$. In [H1] the scattering
problems of the quantized versions of $H$ and $\hat{H}$ are related and in
[Ya1] and [H2] the idea is applied to periodic systems.
It is shown that the ranges of wave operators are connected in the same
way the spectral subspaces of the monodromy operator $U(T,0)$ are
connected with those of $\hat{H}$. This observation makes it possible
to transfer asymptotic completeness statements between the two settings.
(The monodromy operator is the unitary operator that evolves the physical
system through one period.)
Assuming periodicity enables one to compactify the extra space variable.
The resulting Hamiltonian $\hat{H}$ is called the Floquet Hamiltonian.
Compactification makes it possible to show that the potential is relatively
compact with respect to the Floquet Hamiltonian, see Lemma 4.6.
This point was used first by Yajima in [Ya1] to treat two-body systems with
time-periodic potentials and pointed out to the author by his supervisor
E.Skibsted. It can be used to obtain positive commutators locally in energy.
This idea has recently been applied in [Yo] to obtain propagation estimates
for the Floquet Hamiltonian of time-periodic two-body Schr{\"o}dinger
operators.
The idea of this paper is thus to treat the spectral and scattering problems
for the time-independent Floquet Hamiltonian where one can apply local
commutator estimates to obtain results and then in the end make the
connection to the physical problem. More precisely we will prove that the
Floquet Hamiltonian has no bound states which due to an argument by
Yajima given in [Ya1] implies that the monodromy operator has empty
point spectrum. This is what we mean by absence of bound states.
We furthermore prove that the wave operators exist and are unitary.
The results hold under some regularity assumptions on the potential which
do not include singularities. We do however obtain obtain some partial results
for potentials with weak singularities.
For a treatment of the usual two-body problem with time-dependent potentials
that does not use local commutator methods see [KY] and for two-body systems
in a constant electric field see [G2] and [JY].
In the papers [Ya1] and [H2] the authors did not work explicitly with the
Floquet Hamiltonian but rather with its resolvent which can be
expressed in terms of the physical flow. In this paper however we wish to
utilize specific properties of the Stark Hamiltonian in our analysis of its
Floquet Hamiltonian.
In Section 2 we phrase the assumptions which we will impose on the potential
and the electric field and state the main results of this paper.
In Section 3 we elaborate on the work done by Howland
in [H2] and Yajima in [Ya1] and prove some abstract results on the structure
of Floquet Hamiltonians. Some
preliminary results are derived in Section 4 and in Section 5 we apply a
combination of ideas used in [Si2] and [HMS1] to obtain absence of bound
states for the Floquet Hamiltonian. In Section 6 we prove a Mourre estimate
and apply it to obtain absence of singular continuous spectrum for the
Floquet Hamiltonian as well as a pointwise propagation estimate for the
momentum operator which we use to get a minimal acceleration
estimate. Finally in Section 7 we prove existence and completeness of wave
operators for the Floquet Hamiltonian and argue as in [Ya1] and [H2] to
obtain the main result. Appendix A is devoted to a treatment of absolutely
continuous vector-valued functions and the derivative on an interval
and in Appendix B the time-dependent Sch{\"o}dinger equation for
Hamiltonians defined almost everywhere is dicussed.
The central ideas of this paper was developed during a stay at University
of Tokyo and the author wishes to thank his host Professor H.Kitada for
discussions on the absence of bound states.
The abstract theory and the removal of the time-dependency
from the electric field was done during a stay at University of Virginia.
A discussion the author had with his host, I.Herbst, inspired the
transformation which is used to move the time-dependency. The treatment of
singularities was added when the paper was included in the authors Ph.D.
thesis and comments from the committee led to the two appendices.
Finishing touches were made at Universit{\`e} Paris-Sud.
Finally the author would like to thank his supervisor E.Skibsted
and the committee, H.H.Andersen, J.Derezi{\`n}ski and A.Jensen, for
corrections and suggestions.
\vskip1cm
\subhead{Section 2: Assumptions, Notations and Results}\endsubhead
We will work within the following framework.
Let $X$ be a real, finite dimensional vector space equipped with an inner
product. We denote by $x$ the operator of multiplication with the identity
function on $X$ and $p$ is the momentum operator $\frac1{i}\nabla_x$.
We write $\weight$ for $(1+|x|^2)^\frac12$ and we will use the same
abbreviation
for parameters as well as for other self adjoint operators on $\Cal H=L^2(X)$.
Let $\nu = \dim X$. In this Section $T>0$ will denote a common period
for the field and the potential.
\proclaim{Assumption 2.1} The potentials $V_t\in C^2(X)$ form a family of
real-valued functions, periodic with period $T$.
The family and its distributional derivative with respect to $t$ satisfy
$$
\sup_{t\in\Bbb R} |V_t(x)|+
\sup_{t\in\Bbb R} |\nabla_x V_t(x)| = o(1)\text{ and }
\sup_{t\in\Bbb R} |\partial_t V_t(x)|+
\sup_{t\in\Bbb R} |\partial_x^\alpha V_t(x)| = O(1),
$$
for $|\alpha|=2$.
\endproclaim
We say that $V_t$ is short-range if it satisfies Assumption 2.1,
and the decay assumption
$$
\sup_{t\in\Bbb R}|V_t(x)| = O(\weight^{-\frac12-\epsilon}),
$$
for some $\epsilon>0$.
(The symbol $\epsilon$ will be used in this connection only.)
\proclaim{Assumption 2.2} The potentials $V_t=\vr^t+\vs^t$ are real valued,
$\vr^t$ satisfies Assumption 2.1 and $\vs^t=0$ if $\nu<3$.
The singular part is periodic with period $T$,
$\cup_{t\in\Bbb R}\supp(\vs^t)\subset X$ is
compact and there exist $p>\nu$ such that $\vs^t$ and its first order
distributional derivatives satisfy
$$
\sup_{t\in\Bbb R}\|\vs^t\|_{L^p(X)}
+\sup_{t\in\Bbb R}\|\partial_t\vs^t\|_{L^q(X)}
+\sup_{t\in\Bbb R}\|\nabla_x\vs^t\|_{L^q(X)}<\infty,
$$
where $q=\frac{2\nu p}{\nu+4p}$ if $\nu\geq 5$ and $q>\frac{2p}{p+1}$ if
$\nu\in\{3,4\}$.
\endproclaim
The specific form of this assumption is chosen such that the result
on existence of evolutions in [Ya2] applies and such that the class of
potentials satisfying Assumption 2.2 is invariant under the type of
transformations given in Lemma 4.4.
\proclaim{Assumption 2.3} The electric field $E\in\Lloc^1(\Bbb R;X)$,
$E(t+T) = E(t)$ a.e.\, and $\int_0^T E(t)\neq 0$.
\endproclaim
We consider the Hamiltonians
$$
H_0(t) = p^2-E(t)\cdot x\quad\text{and}\quad H(t) = H_0(t) + V_t.\tag{2.1}
$$
In Section 4 we prove that under Assumptions 2.2 and 2.3 the time-dependent
Schr{\"o}dinger equation corresponding to $H_0(t)$ and
$H(t)$ can be solved uniquely in the sense of Definition B.7
(with $\Cal H_1=\Cal D(p^2)\cap\Cal D(\weight)$).
We write $U_0(t,s)$ and $U(t,s)$ for the solutions.
The main results of this paper are
\proclaim{Theorem 2.4 (Absence of bound states)} Assume $V_t$ satisfies
Assumption 2.1 and $E$ satisfies Assumption 2.3. Then the monodromy operator
$U(T,0)$ has purely absolutely continuous spectrum.
\endproclaim
Under Assumption 2.2 we prove that the eigenfunctions of the Floquet Hamiltonian
vanishes in a half-space determined by the field.
\proclaim{Theorem 2.5 (Asymptotic Completeness)} Assume $V_t$ is short-range
and $E$ satisfies Assumption 2.3. Then the wave operators
$$
W_\pm(s) = \slim_{t\rightarrow\pm\infty} U^*(t,s)U_0(t,s)\tag{2.2}
$$
exist for all $s\in\Bbb R$ and are unitary. Furthermore
$$
W_\pm(s+T) = W_\pm(s)\mand U(s+T,s)W_\pm(s)=W_\pm(s)U_0(s+T,s)\tag{2.3}
$$
for all $s\in\Bbb R$.
\endproclaim
The Hamiltonian presented in (1.1) takes the form (2.1) in
the center of mass frame. The configuration space becomes $X=\{x\in\Bbb
R^{2\nu}:m_1x_1+m_2x_2=0\}$ with the inner product $x\cdot y =
2m_1x_1\cdot y_1+2m_2x_2\cdot y_2$. The orthogonal projection onto $X$ is
$$
\pi = \frac1{m_1+m_2}\pmatrix m_2I_\nu & -m_2I_\nu\\ -m_1I_\nu&m_1I_\nu\endpmatrix,
$$
where $I_\nu$ is the $\nu\times\nu$ identity matrix, and we find
$$
\align
E(t)&= \pi\pmatrix \frac{q_1}{2m_1}\Cal E(t)\\ \frac{q_2}{2m_2}\Cal
E(t)\endpmatrix\\
&=\frac1{2(m_1+m_2)}(\frac{q_1}{m_1}-\frac{q_2}{m_2})\pmatrix m_2\Cal
E(t)\\-m_1\Cal E(t)\endpmatrix.
\endalign
$$
We thus have the following corollary
\proclaim{Corollary 2.6} Assume $\frac{q_1}{m_1}\neq\frac{q_2}{m_2}$,
the potential $v$ is short-range (with $X=\Bbb R^\nu$) and
the electric field $\Cal E$ satisfies Assumption 2.3 (with $X=\Bbb R^\nu$).
Then the wave operators \rom{(2.2)} exist and they are unitary.
Furthermore the periodicity and intertwining relation
\rom{(2.3)} is satisfied.
\endproclaim
\vskip1cm
\define \hH{\hat{\Cal H}}
\define \Hh{\hat{\Cal H}}
\subhead{Section 3: The Floquet Hamiltonian}\endsubhead
Let $\{U(t,s)\}_{t,s\in\Bbb R}$ be a family of strongly continuous unitary
operators on a separable Hilbert space $\Cal H$. Assume the family satisfies
the Chapman-Kolmogorov equations
$$
U(t,r)U(r,s) = U(t,s),\mforall t,r,s,\in\Bbb R\tag{3.1}
$$
and is periodic, i.e. it satisfies the periodicity condition
$$
U(t+T,s+T) = U(t,s)\mforall t,s\in\Bbb R\tag{3.2}
$$
for some $T>0$ which we call the period. We note that $U(t,t)=I$ and
$U^*(t,s)=U(s,t)$ for all $t,s\in\Bbb R$.
For any unitary operator $V$ on $\Cal H$ we define the set
$$
\Cal D_V = \{\psi\in AC^2([0,T];\Cal H): \psi(0) = V\psi(T)\}
$$
and the selfadjoint operator $\tau_V$ as $\frac1{i}\frac{d}{dt}$ with domain
$\Cal D_V$. See Appendix A for a discussion of absolutely continuous functions
and the derivatives $\tau_V$.
We define a family of operators pointwisely on $C^0([0,T];\Cal H)$ by
$$
[\hat{U}(s)\psi](t)=U(t,t-s)\psi(t-s-[t-s]),\tag{3.3}
$$
where $[s]$ denotes the largest multiple of $T$ smaller than or equal to $s$.
By boundedness we extend to a family of unitary operators on
$$
\hH = L^2([0,T];\Cal H).
$$
It is easily verified that $\{\hat{U}(s)\}_{s\in\Bbb R}$
is a strongly continuous group with $\hat{U}(0)=I$. We have
\proclaim{Proposition 3.1 (The Floquet Hamiltonian)} The self adjoint
operator that generates $\hat{U}(s)$ is $\hat{H}=U\tau_{U(T,0)}U^*$ with
domain $U\Cal D_{U(T,0)}$, where $U$ is the unitary operator defined
pointwisely by $[U\psi](t) = U(t,0)\psi(t)$. For $\lambda\in\Bbb C$,
$\im\lambda\neq 0$, the resolvent of
$\hat{H}$ is given by
$$
\align
[(\hat{H}-\lambda)^{-1}\psi](t) =& iU(t,0)\{\int_0^t
e^{i(t-s)\lambda}U(0,s)\psi(s)ds\\
&+[e^{-i\lambda T}U(0,T)-I]^{-1}\int_0^T
e^{i(t-s)\lambda}U(0,s)\psi(s)ds\}
\endalign
$$
\endproclaim
\proof
Compute for $f\in C^0([0,T];\Cal H)$ using (3.1-3)
$$
\align
(\hat{U}(s)f)(t) &= U(t,0)U(0,t-s)f(t-s-[t-s])\\
&=U(t,0)U(-[t-s],0)(U^*f)(t-s-[t-s])\\
&=U(t,0) U(0,T)^{\frac{[t-s]}{T}}(U^*f)(t-s-[t-s]).
\endalign
$$
By Proposition A.8 we find that $\hat{U}(s)
=\exp(-is U\tau_{U(T,0)}U^*)$, which shows that the generator, $\hat{H}$,
of $\hat{U}(s)$ equals $U\tau_{U(T,0)}U^*$. The resolvent formula is now
a consequence of (A.2).
\endproof
Let $S(t)$ be a strongly continuous periodic family of unitary operators on
$\Cal H$. Then $S(t)U(t,s)S^*(s)$ satisfies (3.1-2) and has a Floquet
Hamiltonian. We say that $S(t)$ is a periodic change of coordinates.
\proclaim{Lemma 3.2} Let $U_1(t,s)$ and $U_2(t,s)$ be periodic families of
unitary operators.
Suppose there exists a periodic change of coordinates, $S(t)$, such that
$U_1(t,s)=S(t)U_2(t,s)S^*(s)$. Then the Floquet Hamiltonians
$\hat{H}_1$ and $\hat{H}_2$ satisfy $\hat{H}_1 = S\hat{H}_2S^*$, where
the unitary operator $S$ is given by $[Sf](t)=S(t)f(t)$.
\endproclaim
\proof We write $S_0$ for the unitary operator on $\hat{\Cal H}$ given by
$(S_0f)(t) = S(0)f(t)$ and
compute using the assumption, Proposition 3.1 and (A.3)
$$
\align
S\hat{H}_2S^* &= SU_2\tau_{U_2(T,0)}U_2 S^*\\
&=\{SU_2S_0^*\}\{S_0\tau_{U_2(T,0)}S_0^*\}\{S_0U_2S^*\}\\
&=U_1\tau_{S(0)U_2(T,0)S(0)^*}U_1\\
&=\hat{H}_1.
\endalign
$$
\endproof
We now present a result on the spectral structure of the Floquet
Hamiltonian, which was observed by Yajima in [Ya1]. It follows from
Propositions 3.1 and A.9.
\proclaim{Proposition 3.3} Let $U(t,s)$ be a periodic family of unitary
operators. The Floquet Hamiltonian satisfies
$$
\alignat2
\lambda\in\spp(\hat{H})&\iff e^{-i\lambda T}\in\spp(U(T,0)),&\quad
\Hpp(\hat{H}) & = UL^2([0,T];\Hpp(U(T,0)),\\
\lambda\in\sac(\hat{H})&\iff e^{-i\lambda T}\in\sac(U(T,0)),&\quad
\Hac(\hat{H}) & = UL^2([0,T];\Hac(U(T,0)),\\
\lambda\in\ssc(\hat{H})&\iff e^{-i\lambda T}\in\ssc(U(T,0)),&\quad
\Hsc(\hat{H}) &= UL^2([0,T];\Hsc(U(T,0)).
\endalignat
$$
\endproclaim
In the following $H(t)$ will denote a time-periodic family of Hamiltonians
which fits the
requirements given in Definition B.7 (for some $\Cal H_1\subset\Cal H$).
Time-periodicity means that the following identy holds a.e.
$$
H(t+T)\varphi = H(t)\varphi,\mforall \varphi\in\Cal H_1.\tag{3.4}
$$
Suppose we have a family of unitary operators $U(t,s)$, which
solves the time-dependent Schr{\"o}dinger equation in the sense of
Definition B.7.
For such a solution (3.1) is given by Corollary B.6 and (3.2) follows
from (3.4) and Theorem B.4. See Appendix B for details.
We introduce some assumptions which will enable us
to interpret $\hat{H}$ as the operator sum of $\tau_I$ and $H(t)$.
\proclaim{Assumption 3.4} Let $H(t)$ be a time-periodic family of
Hamiltonians.
Suppose there exists a self-adjoint operator $B$ on $\Cal H$ and a family
of unitary operators, $U(t,s)$, which solves the time-dependent
Schr{\"o}dinger equation whith $\Cal H_1=\Cal D(B)$ and suppose furthermore
that $H(t)(B-i)^{-1}\in L^2([0,T];\Cal B(\Cal H))$.
\endproclaim
\proclaim{Proposition 3.5} Let $H(t)$ be a time-periodic family of
Hamiltonians. Suppose Assumption 3.4 is satisfied for
some $B$. Then
$$
\Cal D_0 = \{\psi\in\Cal D_I:\psi([0,T])\subset\Cal D(B)
\text{ and }\sup_{t\in[0,T]}\|B\psi(t)\|<\infty\}
$$
is a core for $\hat{H}$. The operator $\tau_I+H$ defined on $\Cal D_0$ by
$[(\tau_I+H)\psi](t) = [\tau_I\psi](t) + H(t)\psi(t)$ is essentially
self-adjoint and its closure equals $\hat{H}$.
\endproclaim
\proof
Consider the set
$\Cal S=\spn{\{e^{im\frac{2\pi}{T}}\psi:m\in\Bbb Z, \psi\in\Cal D(B)\}}$ which is
dense in $\Hh$. Since $\Cal S\subset\Cal D_0$ we find that
$\Cal D_0$ is dense.
By (3.1), $U(t,t-s) = U(t,0)U(0,t-s)$ and this together with
Assumption 3.4 and (B.3) shows that the operator $(B+i)U(t,t-s)(B+i)^{-1}$
is bounded uniformly in $s,t\in [0,T]$. This implies by (3.3) that
$$
\sup_{s,t\in [0,T]} \|[B\hat{U}(s)\psi](t)\|<\infty,\tag{3.5}
$$
for $\psi\in\Cal D_0$. By (3.1) and Proposition B.3 ii) and iii) we find that
$$
t\rightarrow U(t,t-s)\psi(t-s-[t-s])\in AC([0,T];\Cal H)
$$
for $\psi\in\Cal D_0$ and furthermore we have
$$
\align
[U(t,t-s)-&U(s,0)]\psi(t-s-[t-s])\\
&=\int_s^t i[U(r,r-s)H(r-s)-H(r)U(r,r-s)]\psi(t-s-[t-s])\\
&\qquad\qquad+U(t,t-s)(\partial\psi)(t-s-[t-s])dr.
\endalign
$$
The last part of Assumption 3.4 and (3.5) now combines to prove that
the derivative of $\hat{U}(s)\psi$ is square-integrable and we thus get
$\hat{U}(s)\Cal D_0\subset\Cal D_0$. This inclusion shows that
$\hat{H}_{|\Cal D_0}$ has no other self-adjoint extension than $\hat{H}$
and is therefore essentially self-adjoint with closure equals $\hat{H}$.
We compute
$$
[\hat{H}\psi](t) = H(t)\psi(t)+[\tau_I\psi](t),
$$
for $\psi\in\Cal D_0$, which concludes the proof.
\endproof
{\bf Example 3.6} If $H(t)=H$ is independent of time one readily finds
Assumption 3.4 satisfied with $\Cal D=\Cal D(B)=\Cal D(H)$ and $B=H$.
In this case the conclusion of Proposition 3.5 is trivial since $\hat{H}$
equals the closure of $H\otimes I + I\otimes \tau_I$ on $\Cal D(H)\otimes
\{\psi\in AC([0,T];\Bbb C):\psi(0)=\psi(T)\}$.
{\bf Example 3.7} For $\alpha<1$ we consider $H(t) = (t-[t])^{-\alpha}$,
where $[t]$ is the integer part of $t$, as an
operator on $\Cal H=\Bbb C$ ($H(0)=0$). For $s,t\in[0,1]$ we find
$U(t,s)=e^{i\frac{s^{1-\alpha}-t^{1-\alpha}}{1-\alpha}}$ with $\Cal D=\Bbb C$.
For $\alpha<\frac12$ we find Assumption 3.4 satisfied with $B=1$. When
$\alpha\geq\frac12$ we find that any domain on which we should be able to
write $\hat{H}$ as a sum of $H(t)$ and $\tau_I$ has to have its functions
vanish at zero at some slow polynomial rate. By the periodicity requirements
for $\tau_I$ this will in turn fix functions in such a domain to be zero in
the endpoints. This is the classical example of an operator with deficiency
indices equal to $1$ and a whole range of self adjoint extensions indexed by
the unitary operators on $\Bbb C$. See also Proposition A.7.
\vskip1cm
\subhead{Section 4: Preliminary Results}\endsubhead
In this Section we solve the time-dependent Schr{\"o}dinger equation for the
Hamiltonians $H_0(t)$ and $H(t)$ and prove that multiplication by the
indicator function $F(|x|2R$ and $\chi(sR) = 1 - \chi(s0$ and $\lambda\in\Bbb C$ with $\im\lambda\neq 0$.
\endproclaim
\proof
The idea is to use the resolvent formula of Proposition 3.1, Avron-Herbst
formula and the known integral kernel for the free propagator to
approximate the operator
$$
\chi(|x|0$
(a.e with respect to $t$)
$$
\align
[\vs(\hat{H}_0-\lambda)^{-1}\vs \psi](t) &= i\int_0^\infty e^{is\lambda}[\vs\hat{U}_0(s)\vs \psi](t)ds\\
&=i\int_{-\infty}^te^{i(t-s)\lambda}\vs^tU_0(t,s)\vs^s\psi(s-[s])ds\\
&=I_1(t)+I_2(t),\tag{4.2}
\endalign
$$
where
$$
\align
I_1(t)&= i\sum_{n=-\infty}^{-2}\int_0^1 e^{i(t-s-n)\lambda}\vs^tU_0(t,s+n)\vs^s\psi(s)ds\\
I_2(t)&= i\int_{-1}^t e^{i(t-s)\lambda}\vs^tU_0(t,s)\vs^s\psi(s-[s])ds.
\endalign
$$
At this point we use Avron-Herbst formula to substitute
$U_0(t,s) = T(t)\exp(i(s-t)p^2)T^*(s)$ and rewrite
$$
\align
I_1(t)&= iT(t)\sum_{n=-\infty}^{-2}\int_0^1 e^{i(t-s-n)\lambda}\vs^{0,t}
\exp(i(s+n-t)p^2)\vs^{n,s}T^*(s)\psi(s)ds\\
I_2(t)&= iT(t)\int_{-1}^t e^{i(t-s)\lambda}\vs^{0,t}\exp(i(s-t)p^2)\vs^{0,s}T^*(s)\psi(s)ds,
\endalign
$$
where $\vs^{k,t}(x)=\vs^t(x-c(t+k))$ for $k\leq 0$.
Note that $\|\vs^{k,t}\|_{\tilde{p}} = \|\vs^t\|_{\tilde{p}}<\infty$
for all $2\leq \tilde{p}\leq p$ and $k\leq 0$. One can now proceed exactly as
in [Ya1] and use a result by Kato, namely
$$
\|f\exp(-itp^2)gu\|\leq (4\pi|t|)^{-\frac{\nu}{\tilde{p}}}\|f\|_{\tilde{p}}
\|g\|_{\tilde{p}}\|u\|,
$$
for $f,g\in L^{\tilde{p}}(X)$, $\tilde{p}\geq 2$, and $u\in\Cal D(g)$.
The unitary transformation $T(t)$ dissapears after an application of this
inequality to $I_1$ and $I_2$. The problem is now exactly the same as the one
considered in [Ya1] and we omit the remaining steps.
We thus obtain $I_1,I_2\in\Hper$ and
$$
\|I_1\|^2+\|I_2\|^2 \leq C(\im\lambda)\|f\|^2,
$$
where $C(r)\rightarrow 0$ as $|r|\rightarrow\infty$. The same procedure
applies to the case $\im\lambda<0$. The result now
follows from (4.2), Proposition 3.5 and the first resolvent formula.
\endproof
We are now in a position where we can begin the spectral analysis of
the Floquet Hamiltonian.
\vskip1cm
\subhead{Section 5: Absence of bound states}\endsubhead
\define \vptR{\varphi_{\theta,R}}
\define \ptR{\psi_{\theta,R}}
\define \eag{e^{\alpha G_\theta}}
In this Section we prove that the monodromy
operator $U(1,0)$ has no bound states (for non-singular potentials).
This is equivalent to proving
absence of bound states for $\hat{H}$ as noted by Yajima in [Ya1, Section 4],
see Proposition 3.3.
We restrict attention to constant electric fields, $E(t)=E_0$,
in view of Proposition 4.5 and use the notation $p_0 = \om\cdot p$,
where $\om = \frac{E_0}{|E_0|}$. We work under the assumption that
$V_t$ satisfies Assumption 2.2.
For $\theta>0$ we write $\chi_\theta(r)=\chi(\frac{r}{\theta}<1)$
and define
$$
G_\theta(r)=\int_0^r \chi_\theta(s) ds.
$$
This function will be used to reguralize an exponential weight.
The fact that $\weight^{-\frac12}p$ is relatively bounded with respect to the
Hamiltonian $H(t)$ is used in both [Si2] and [HMS1] to prove absence of bound
states in the case of constant fields. This is however not true for the
Floquet Hamiltonian. We introduce the following operator family in order to
circumvent the technical problems arising from this
$$
P_R = i(\frac{p_0}{R}+i)^{-1},\quad R>0,
$$
which satisfies
$$
\slim_{R\rightarrow\infty} P_R =
I\quad\text{and}\quad\slim_{R\rightarrow\infty}\frac{p_0}{R}P_R = 0.\tag{5.1}
$$
Pick $d>0$ such that
$$
\sup_{0\leq t\leq T}|\nabla\vr^t(x)|<\frac{|E_0|}2\mand x\nin
\bigcup_{0\leq t\leq T}\supp(\vs^t)\mfor \om\cdot x>d.
$$
We will in the following write, unless otherwise noted,
$\chi_\theta = \chi_\theta(\om\cdot x-d)$ and
$G_\theta = G_\theta(\om\cdot x-d)$ and introduce
$$
A_\theta = \frac12(\chi_\theta p_0+p_0\chi_\theta).
$$
We write $\hat{H}_1 = \hat{H}_0+\vr$ and compute the following commutators
(as forms on $\Cal D_0$)
$$
i[\hat{H}_1,P_R] = \frac{i}{R}P_R\om\cdot(E_0-\nabla\vr)P_R.\tag{5.2}
$$
and
$$
i[\hat{H}_1,\eag] = \{2\alpha A_\theta+i\alpha^2\chi_\theta^2\}\eag.\tag{5.3}
$$
\proclaim{Lemma 5.1} Suppose $V_t$ satisfies Assumption 2.2.
Let $\psi$ be an eigenfunction for $\hat{H}$. For all $\alpha>0$ we have
$e^{\alpha \om\cdot x}\psi\in \Hper$.
\endproclaim
\proof The proof is inspired by ideas of Sigal used in [Si2].
Assume there exists $\alpha>0$ such that
$e^{\alpha\om\cdot x}\psi\not\in\hat{\Cal H}$.
For functions $\varphi\in\Hper$ we abbreviate
$$
\varphi_{\theta,R} =
\frac{\eag P_R\varphi}{\|\eag P_R\varphi\|},\mand
\varphi_\theta = \frac{\eag\varphi}{\|\eag\varphi\|}.
$$
For expectation values we write
$$
\langle A\rangle_\varphi = \langle \varphi,A\varphi\rangle.
$$
Using the choice of $d$ we estimate for $\varphi\in\Cal D_0$
$$
\langle i[\hat{H}_1,p_0]\rangle_{\vptR}\geq \frac{|E_0|}2
-\|\nabla\vr\|_\infty\frac{\|\varphi\|^2}{\|\eag P_R\varphi\|^2}.\tag{5.4}
$$
On the other hand we can use (5.3) to compute
$$
\align
\langle i[\hat{H}_1,p_0]\rangle_{\varphi_{\theta,R}}
&=-4\alpha\re\{\langle A_\theta p_0\rangle_{\vptR}\}
+\alpha^2\langle i[\chi_\theta^2,p_0]\rangle_{\vptR}\\
&\quad-2\|\eag P_R\varphi\|^{-1}\re\{\langle i\eag
([\hat{H}_1,P_R]+P_R\{\hat{H}-\vs\})\varphi,
p_0\vptR\rangle\}.
\endalign
$$
To estimate the first term we notice that by construction
$$
A_\theta = p_0\chi_\theta +\frac{i}2\chi^\prime_\theta
$$
which in turn yields
$$
2\re\{A_\theta p_0\} = 2p_0\chi_\theta p_0 -
\frac12\chi_\theta^{\prime\prime}.
$$
This gives the identity
$$
\align
\langle &i[\hat{H}_1,p_0]\rangle_{\vptR}=
-4\alpha\langle p_0\chi_\theta p_0\rangle_{\vptR}+
\langle\alpha\chi_\theta^{\prime\prime}
-2\alpha^2\chi_\theta\chi_\theta^\prime\rangle_{\vptR}\\
&\quad - 2\|\eag P_R\varphi\|^{-1}\re\{\langle i\eag
([\hat{H}_1,P_R]+P_R\{\hat{H}-\vs\})\varphi,
p_0\vptR\rangle\}.\tag{5.5}
\endalign
$$
We rewrite the term containing $\vs$ using $\vs=\chi_\theta\vs$
$$
\langle \eag P_R\vs\varphi,p_0\vptR\rangle
=\langle \eag P_R\vs\varphi,\chi_\theta p_0\vptR\rangle
+\langle B_\theta(R)\vs\varphi,\frac1{R}p_0\vptR\rangle,
$$
where $B_\theta(R)$ denote bounded operators satisfying
$\sup_{R\geq 1}\|B_\theta(R)\|<\infty$. We estimate the first
term using Cauchy-Schwartz inequality and $\chi_\theta^2<\chi_\theta$
$$
\align
|\langle\eag P_R\vs\varphi,p_0\vptR\rangle| &\leq
\frac{\|\eag P_R\vs\varphi\|^2}{2\alpha\|\eag P_R\varphi\|}
+2\alpha\|\eag P_R\varphi\|\langle p_0 \chi_\theta p_0\rangle_{\vptR}\\
&\quad+\langle B_\theta(R)\vs\varphi,\frac1{R}p_0\vptR\rangle.
\endalign
$$
By this inequality, (5.2) and Lemma 4.7 we can estimate (5.5)
$$
\align
\langle i[\hat{H}_1,p_0]\rangle_{\vptR}\leq&
\langle\alpha\chi_\theta^{\prime\prime}
-2\alpha^2\chi_\theta\chi_\theta^\prime\rangle_{\vptR}\\
&-2\|\eag P_R\varphi\|^{-1}\re\{\langle i\eag P_R\hat{H}\varphi,p_0\vptR\rangle\}\\
&+\frac{\|\eag P_R\vs\varphi\|^2}{\alpha\|\eag P_R\varphi\|^2}
+\langle B_\theta(R)(\hat{H}_0+i)\varphi,\frac1{R}p_0\vptR\rangle.
\endalign
$$
First we combine this estimate with (5.4) and take the limit
$\varphi\rightarrow\psi$ in the graph norm of $\hat{H}$. This is possible
due to the fact that $\Cal D_0$ is core for $\hat{H}$
and by assumption the limit of $\re\{\langle i\eag P_R\hat{H}\varphi,
p_0\vptR\rangle\}$ is equal to zero.
Secondly we use both limits in (5.1) to take $R$ to infinity.
In this way we obtain the inequality
$$
\frac{|E_0|}2
\leq\langle\alpha\chi_\theta^{\prime\prime}
-2\alpha^2\chi_\theta\chi_\theta^\prime\rangle_{\psi_\theta}
+\frac{\|\nabla\vr\|_\infty\|\psi\|^2 + \frac1{\alpha}\|\vs\psi\|^2}{\|\eag\psi\|^2}.
$$
We have thus obtained a contradiction since $|\chi_\theta^{(k)}| =
O(\theta^{-k})$ and $\|\eag\psi\|\rightarrow\infty$ as $\theta\rightarrow\infty$
by assumption.
\endproof
\proclaim{Lemma 5.2} Let $\alpha>0$ and $R>2\alpha$. The following holds:
\roster
\item For any $\varphi\in\Hper$ we have $\|\eag P_R\varphi\|\leq
\frac1{1-\frac{\alpha}{R}}\|\eag\varphi\|$.
\item Let $\psi\in\Hper$ satisfy $e^{2\alpha\om\cdot x}\psi\in\Hper$. Then
$$
e^{\alpha\om\cdot x}P_R \psi\in \Hper\mand
\lim_{\theta\rightarrow\infty}\eag P_R \psi = e^{\alpha(\om\cdot x-d)}P_R\psi.
$$
\endroster
\endproclaim
\proof We first prove i). Compute the commutator
$$
i[\eag,P_R] = \frac{-i\alpha}{R}P_R\chi_\theta
\eag P_R.\tag{5.7}
$$
This gives the estimate
$$
\|\eag P_R\varphi\|\leq \|\eag\varphi\|+
\frac{\alpha}{R}\|\eag P_R\varphi\|,
$$
for any $\varphi\in\Hper$. Rearrangement gives i).
In order to prove ii) we first show that the family
$\{\psi_\theta\}_{\theta>0}=\{\eag P_R\psi\}_{\theta>0}$ is Cauchy.
Secondly we verify that its limit is as expected.
We use (5.7) again to estimate
$$
\|\psi_{\theta_1}-\psi_{\theta_2}\|\leq
\|(e^{\alpha G_{\theta_1}}-e^{\alpha G_{\theta_2}})\psi\|+
\frac{\alpha}{R}\{\|\psi_{\theta_1}-\psi_{\theta_2}\|
+\|(\chi_{\theta_1}-\chi_{\theta_2})\psi_{\theta_1}\|\}.
$$
We rewrite this to obtain
$$
\|\psi_{\theta_1}-\psi_{\theta_2}\|\leq
\frac1{1-\frac{\alpha}{R}}\{\|(e^{\alpha G_{\theta_1}}-e^{\alpha G_{\theta_2}})\psi\|+
\frac{\alpha}{R}\|(\chi_{\theta_1}-\chi_{\theta_2})\psi_{\theta_1}\|\}.
$$
We can estimate the second term using the Cauchy-Schwarz inequality and the
proof of i) with $G_\theta$ replaced by $2G_{\theta_1}$
$$
\|(\chi_{\theta_1}-\chi_{\theta_2})\psi_{\theta_1}\|^2\leq
\frac1{1-\frac{2\alpha}{R}}
\|(\chi_{\theta_1}-\chi_{\theta_2})^2 P_R\psi\|
\|e^{2\alpha G_{\theta_1}}\psi\|.
$$
The fact that the sets $\supp(\chi_{\theta_1}-\chi_{\theta_2})$ and
$\supp(e^{\alpha G_{\theta_1}}-e^{\alpha G_{\theta_2}})$ are contained in
$\{x:\om\cdot x\geq\min\{\theta_1,\theta_2\}+d\}$ together with the two bounds
$$
|\chi_{\theta_1}-\chi_{\theta_2}|\leq 2\mand
|(e^{\alpha G_{\theta_1}}-e^{\alpha G_{\theta_2}})e^{-\alpha\om\cdot x}|\leq 2e^{\alpha d}
$$
shows in conjunction with the choice of $\psi$ that $\{\psi_\theta\}_{\theta>0}$
is Cauchy.
The limit $\lim_{\theta\rightarrow\infty} \psi_\theta$ thus exists. The
Lebesgue Theorem on monotone convergence yields
$e^{\alpha\om\cdot x}P_R\psi\in \hat{\Cal H}$ and a simple argument
completes the proof of ii).
\endproof
We now apply Lemma 5.1 using an idea from [HMS1] to obtain absence of bound
states.
\proclaim{Proposition 5.3} Let $\psi$ be an eigenfunction for $\hat{H}$.
Then $\psi$ vanishes on the set $\{x\in X:\om\cdot x>d\}$.
If $\vs=0$ then $\spp(\hat{H}) = \emptyset$.
\endproclaim
\proof
Let $\lambda$ be an eigenvalue for $\hat{H}$ with corresponding
eigenfunction $\psi$. First we compute as in [HMS1] for
$\phi\in P_R\Cal D_0$ using (5.3)
$$
\align
\|\eag (\hat{H}_0-\lambda)\phi\|^2
&=\|2\alpha A_\theta\eag\phi\|^2 + \|[\hat{H}_0
-(\lambda+\alpha^2\chi_\theta^2)]\eag\phi\|^2\\
&\quad+2\alpha\langle i[p^2-E_0\cdot x
-\alpha^2\chi_\theta^2,A_\theta]\rangle_{\eag\phi}\\
&\geq 2\alpha |E_0|\langle
\chi_\theta\rangle_{\eag\phi}+
\langle 4\alpha^3\chi_\theta^2 \chi_\theta^\prime-\alpha\chi_\theta^{(3)}
\rangle_{\eag\phi}\\
&\quad + 2\alpha\langle p_0
\chi_\theta^\prime p_0\rangle_{\eag\phi}.\tag{5.8}
\endalign
$$
On the other hand substitute $\phi = P_R\varphi$, $\varphi\in\Cal D_0$
and compute
$$
\|\eag(\hat{H}_0-\lambda)P_R\varphi\|^2 =
\|\eag\{[\hat{H}_1,P_R] - \vr P_R + P_R(\hat{H}-\lambda-\vs)\}\varphi\|^2.
$$
We combine this with (5.8), take the limit $\varphi\rightarrow\psi$ in the
graph-norm of $\hat{H}$ and use (5.2), the choice of $d$ and
Lemma 5.2 i)
$$
\align
&2\left(\frac{\|E_0-\nabla\vr\|_\infty}{R-\alpha}+\|\vr\|_\infty\right)^2
\|\eag P_R\psi\|^2 + \frac2{(1-\frac{\alpha}{R})^2}\|\vs\psi\|^2\\
&\qquad\geq 2\alpha|E_0|\langle \chi_\theta\rangle_{\eag P_R\psi}
+ \langle 4\alpha^3\chi_\theta^2\chi_\theta^\prime
- \alpha\chi_\theta^{(3)}\rangle_{\eag P_R\psi}
+ 2\alpha\langle p_0 \chi_\theta^\prime p_0\rangle_{\eag P_R\psi}.
\endalign
$$
We apply Lemma 5.2 ii) and the fact that $|\chi_\theta^{(k)}|=O(\theta^{-k})$
and take the limit $\theta\rightarrow\infty$ on both sides to obtain
$$
\align
\left(\frac{\|E_0-\nabla\vr\|_\infty}{R-\alpha}+\|\vr\|_\infty\right)^2&
\|e^{\alpha\om\cdot x}P_R\psi\|^2+\frac{e^{\alpha d}}{(1-\frac{\alpha}{R})^2}\|\vs\psi\|^2\\
&\geq \alpha|E_0|\|e^{\alpha\om\cdot x}P_R\psi\|^2.\tag{4.8}
\endalign
$$
Here we used that $\|p_0e^{G_\theta}P_R\psi\|$ is bounded uniformly with
respect to $\theta$ as can be seen from (5.7), Lemma 5.1 and Lemma 5.2 i).
We estimate using Lemma 5.1 and Lemma 5.2 i) and ii)
$$
\|e^{\alpha\om\cdot x}(I-P_R)\psi\|\leq \|(I-P_R)e^{\alpha\om\cdot x}\psi\|
+\frac{\alpha}{R-\alpha}\|e^{\alpha\om\cdot x}\psi\|
$$
which by Lemma 5.1 and (5.1) implies
$$
e^{\alpha\om\cdot x}P_R\psi\rightarrow e^{\alpha\om\cdot
x}\psi\quad\text{as}\quad R\rightarrow\infty.
$$
Taking $R$ to infinity in (5.8) thus gives for all $\alpha>0$
$$
\|\vr\|_\infty^2\|e^{\alpha\om\cdot x}\psi\|^2
+e^{\alpha d}\|\vs\psi\|^2\geq\alpha|E_0|\|e^{\alpha\om\cdot x}\psi\|^2.
$$
From this inequality we conclude the statement of the Proposition.
\endproof
By Propositions 3.3 and 4.5 this proves that the monodromy operator
has no pure point spectrum.
If one could prove a unique continuation Theorem for the differential
operator $\tau+p^2$, see [ABG], one could conclude absence of bound states
for singular potentials as well.
\vskip1cm
\subhead{Section 6: Mourre Estimate}\endsubhead
We will write $\eta_\delta$ for any smooth function $\eta:\Bbb R\rightarrow
[0,1]$ satisfying that $\eta=0$ on the complement of $[-2\delta,2\delta]$ and
$\eta=1$ on $[-\delta,\delta]$.
We now combine the result on absence of bound states with Proposition 4.5
to obtain
the "squeezing rule" (cf [HMS1, Proposition 3.7] for a corresponding result
which played a crucial role in the treatment of the many-body constant field
problem). In this Section we work under Assumption 2.1.
\proclaim{Proposition 6.1 (Squeezing rule)} Suppose $E$ satisfies Assumption 2.3.
Let $\lambda\in\Bbb R$. Then we have for any $R>0$
$$
\lim_{\delta\rightarrow 0}\|F(|x|0$ such that
$$
\eta_\delta(\hat{H}-\lambda)i[\hat{H},p_0]\eta_\delta(\hat{H}-\lambda)\geq
e\eta_\delta^2(\hat{H}-\lambda).
$$
\endproclaim
From the abstract theory of Mourre [Mo] we get the limiting absorption
principle which implies the following corollary and
by Proposition 4.5 it holds for $E$ satisfying Assumption 2.3 (Note that
we have verified above the technical Assumptions used in [Mo]). In
conjunction with Proposition 3.3 this completes the proof of Theorem 2.4.
\proclaim{Corollary 6.3} The spectrum of $\hat{H}$ is purely absolutely
continuous.
\endproclaim
In the time-independent case one would now proceed in standard fashion
to obtain an
integral propagation estimate for $p_0$ from the limiting absorption principle
(local smoothness). Since $\langle x_0\rangle^{-\frac12}p_0$ is
bounded relative to $p^2-E_0\cdot x$ this implies an integral propagation
estimate for $x_0$ which in turn yields an easy proof of asymptotic
completeness. In the present case however $\langle x_0\rangle^{-\frac12}p_0$
is not $\hat{H}_0$-bounded and instead we choose to proceed via
pointwise propagation estimates.
\proclaim{Corollary 6.4} There exist $\kappa>0$ and $\rho>0$ such that
$$
F(\frac{p_0}{s}<\kappa)\exp(-is\hat{H})f(\hat{H})\langle p_0
\rangle^{-1} = O(\langle s\rangle^{-\rho})
\quad\text{as}\quad s\rightarrow\pm\infty,
$$
for any $f\in C_0^\infty(\Bbb R)$.
\endproclaim
\proof
Let $A(s) = p_0 - es$ for some $00$ such that
$$
F(\frac{|x_0|}{\weights^2}<\theta)\exp(-is\hat{H})
f(\hat{H})\langle p_0\rangle^{-1}\langle\tau_I\rangle^{-1} =
O(\weights^{-\rho})\quad\text{as}\quad s\rightarrow\pm\infty,
$$
for any $f\in C_0^\infty(\Bbb R)$.
\endproclaim
\proof
This proof is based on ideas used to solve an analogous problem in [HMS2,
Appendix A] combined with the regularization procedure of Section 5.
Let $\varphi\in\Hper$, $f\in C_0^\infty(\Bbb R)$, $\psi =
f(\hat{H})\langle p_0\rangle^{-1}\langle\tau_I\rangle^{-1}\varphi$
and $\psi_R = B_R\psi$, where
$$
B_R = i(\frac{p^2+\weight}{R}+i)^{-1},\mfor R>0.
$$
By Lemma 6.5 we find that
$$
\psi\in\Cal D_I\mand\psi_R\in\Cal D_0.
$$
We abbreviate
$\chi(s)=\chi(\frac{|x_0|}{\weights^2}<\theta)\exp(-is\hat{H})\psi\in
\Cal D(p_0)\cap\Cal D(\weight)$ and $\chi_R(s) =
\chi(\frac{|x_0|}{\weights^2}<\theta)\exp(-is\hat{H})\psi_R\in\Cal D_0$.
(Note that we verified in the proof of Proposition 3.5 that
$\exp(-is\hat{H})\Cal D_0\subset\Cal D_0$ and it is easy to check that
$\exp(-is\hat{H})\Cal D(p_0)\subset\Cal D(p_0)$.)
We can now compute (with the convention that all
constants $C>0$ are independent of $R>0$) writing $o(1)$ for errors
converging to $0$ as $R\rightarrow\infty$
$$
\align
\|p\chi_R(s)\|^2 & = \langle(\hat{H}+E_0\cdot
x-V-\tau_I)\chi_R(s),\chi_R(s)\rangle\\
&\leq \theta
C_1\weights^2\|\chi_R(s)\|^2+\|\tau_I\chi_R(s)\|\|\chi_R(s)\|+C_2\|\varphi\|^2
+o(1),\tag{6.1}
\endalign
$$
where there error term comes from the commutators $i[\hat{H},B_R]$
and $i[p_0,B_R]$ (see (5.1)).
Since $\|\tau_I\chi_R(s)\|\leq C\weights\|\varphi\|$ we get
$$
\|p\chi_R(s)\|^2
\leq \theta C_3\weights^2\|\chi_R(s)\|^2 + C_4\weights\|\varphi\|^2+o(1).
\tag{6.2}
$$
On the other hand we can estimate
$$
\align
\|p_0\chi_R(s)\|^2 &\geq
\delta^2\weights^2\|\chi(\frac{p_0^2}{\weights^2}>\delta^2)\chi_R(s)\|^2\\
&\geq\delta^2\weights^2\|\chi_R(s)\|^2-\delta^2\weights^2\|\chi(\frac{
p_0^2}{\weights^2}<\delta^2)\chi_R(s)\|\|\chi_R(s)\|\\
&\geq \frac12\delta^2\weights^2\|\chi_R(s)\|^2-C_5\weights^2\|\chi(\frac{
p_0^2}{\weights^2}<\delta^2)\chi_R(s)\|^2.
\endalign
$$
Combining this estimate with (6.2) and taking the limit $R\rightarrow\infty$
gives
$$
(\frac12\delta^2-\theta C_3)\weights^2\|\chi(s)\|^2\leq
C_5\weights^2\|\chi(\frac{
p_0^2}{\weights^2}<\delta^2)\chi(s)\|^2+C_4\weights\|\varphi\|^2.\tag{6.3}
$$
As in [HMS2, Appendix A] we have
$$
[\chi(\frac{p_0^2}{\weights^2}<\delta^2),\chi(\frac{|x|}{\weights^2}<\theta)]
= O(\weights^{-3}).
$$
By choosing $0<\delta<\frac{\kappa}4$ we can use Corollary 6.4 to estimate
(6.3) further
$$
(\frac12\delta^2-\theta C_3)\weights^2\|\chi(s)\|^2\leq
C_6\weights^{2-2\rho}\|\varphi\|^2,
$$
which yields the stated result by choosing $\theta<\frac{\delta^2}{2C_3}$.
\endproof
By the estimate (6.1) and Proposition 6.6 we get (since $p_0B_R\rightarrow p_0$
strongly on the domain of $p_0$ as $R\rightarrow\infty$)
\proclaim{Corollary 6.7} Let $f\in C_0^\infty(\Bbb R)$. We have the estimate
$$
p_0\chi(\frac{|x_0|}{\weights^2}<\theta)\exp(-is\hat{H})
f(\hat{H})\langle p_0\rangle^{-1}\langle\tau_I\rangle^{-1}=
O(\weights^{1-\rho}).
$$
\endproclaim
\vskip1cm
\subhead{Section 7: Asymptotic Completeness}\endsubhead
First we prove completeness for the Floquet Hamiltonians.
\proclaim{Proposition 7.1} Assume $V_t$ is short range and $E(t)$
satisfies Assumption 2.3. Then the wave operators
$$
\hat{W}^\pm = \slim_{s\rightarrow\pm\infty}\exp(is\hat{H})\exp(-is\hat{H}_0)
$$
exist and are unitary.
\endproclaim
\proof By Proposition 4.5 it is sufficient to prove the result for constant
non-zero fields. We will only prove existence of
$$
\slim_{s\rightarrow\pm\infty} \exp(is\hat{H}_0)\exp(-is\hat{H})
$$
since existence of wave operators follows in the same way. By a standard
argument this implies that the wave operators are unitary. Furthermore we
restrict attention to the limit at $+\infty$.
Let $\varphi\in \Hper$, $f\in C_0^\infty(\Bbb R)$ and $\psi =
f(\hat{H})\langle p_0\rangle^{-1}\langle\tau_I\rangle^{-1}\varphi$.
For $\delta>0$ we write
$\chi_\delta(s)$ for $\chi(\frac{|x_0|}{\weights^2}>\delta)$
and compute using Proposition 6.6
$$
\align
\exp(is&\hat{H}_0)\exp(-is\hat{H})\psi =
\exp(is\hat{H}_0)\chi_{\frac{\theta}2}(s)\exp(-is\hat{H})\psi+o(1)\\
&= \int_0^s \exp(ir\hat{H}_0)\{i[\hat{H}_0,\chi_{\frac{\theta}2}(r)] +
\chi_{\frac{\theta}2}(r)V
+\partial_r\chi_{\frac{\theta}2}(r)\}\exp(-ir\hat{H})\psi dr.
\endalign
$$
Here (as a form on $\Cal D_0$)
$$
i[\hat{H}_0,\chi_{\frac{\theta}2}(r)] = \weightr^{-2}b_1(r)
p_0\chi_\theta(r)+O(\weightr^{-4})\tag{7.1}
$$
and
$$
\partial_r \chi_{\frac{\theta}2}(r) = \weightr^{-1}b_2(r)\chi_\theta(r).
$$
where $b_1$ and $b_2$ denotes bounded functions from $\Bbb R_+$ into
$\Bbb R$. By [M{\o}, Lema 2.2] applied with $H=p_0$, $A=\hat{H}_0$,
$\tilde{H}=\chi_{\frac{\theta}2}(r)$ and $\Cal S=\Cal D_0$ we find that
(7.1) holds as a form on $\Cal D(\hat{H})\cap\Cal D(p_0)$ as well.
Proposition 6.6 and Corollary 6.7 now yields existence of the integral.
\endproof
\demo{Proof of Theorem 2.5}
We restrict attention to the case $s=0$
(since $W_\pm(s) = U(s,0)W_\pm(0)U_0(0,s)$).
We know from the Avron-Herbst formula and a standard stationary phase
argument that the physical wave operators
$$
W_\pm(0) = \slim_{t\rightarrow\pm\infty} U(0,t)U_0(t,0)
$$
exist. One can compute as in [Ya1]
$$
\hat{W}_\pm = U W_\pm(0) U_0^*,
$$
which imply
$$
\range(\hat{W}_\pm) = UL^2([0,1];\range(W_\pm(0)))
$$
and this shows by Theorem 7.1 that $W_\pm(0)$ are unitary.
(See also [H2, Corollary 4.1].)
The remaining statements follow from (3.1-2) and existence of wave operators.
\endproof
\vskip1cm
\subhead{Appendix A: Absolutely continuous vector-valued functions and the derivative}\endsubhead
In this Appendix we discuss absolutely continuous vector-valued functions
as well as different realizations of the derivative on an interval.
We restrict attention to separable Hilbert-spaces since that is all we need in
this paper and it makes the exposition simpler because the different
notions of measurability coincide. See [RSI] and [T] for some material
on measurability and integration. We just mention that integrals of
vector-valued and operator-valued functions are weak and strong respectively.
In the case where $\Cal H$ is finite-dimensional
being absolutely continuous is equivalent to being an indefinit integral,
see for example [R]. In general however this is not so (although being an
indefinite integral implies absolute continuouity in the usual sense).
In this section we work with indefinite integrals since these functions
form natural domains for the derivative.
\proclaim{Definition A.1} The space of absolutely continuous functions on the
real line is defined by
$$
AC(\Bbb R;\Cal H) = \{f:\Bbb R\rightarrow\Cal H:
\exists g\in\Lloc^1(\Bbb R;\Cal H)\text{ and }\psi\in\Cal H
\text{ s.\,t. } f(t) = \int_0^t g(s)ds + \psi\}
$$
\endproclaim
Note that $AC(\Bbb R;\Cal H)\subset C^0(\Bbb R;\Cal H)$ the space of continuous
$\Cal H$-valued functions. It is easy to check that the map
$$
(g,\psi)\rightarrow \int_0^tg(s)ds +\psi
$$
from $\Lloc^1(\Bbb R;\Cal H)\times\Cal H$ onto $AC(\Bbb R;\Cal H)$
is one to one and the following definition is therefore good
\proclaim{Definition A.2} The derivative $\partial:AC(\Bbb R;\Cal H)\rightarrow
\Lloc^1(\Bbb R,\Cal H)$ is given for $g\in\Lloc^1(\Bbb R,\Cal H)$
and $\psi\in\Cal H$ by
$$
\partial (\int_0^tg(s)ds+\psi)=g.
$$
\endproclaim
We have as in the case $\Cal H=\Bbb C$
\proclaim{Proposition A.3} Let $f\in AC(\Bbb R;\Cal H)$. Then the limit
$$
g(t) = \lim_{h\rightarrow 0}\frac1{h}(f(t+h)-f(t))
$$
exists almost everywhere and $g=\partial f$.
\endproclaim
\proof
Write $f(t)=\int_0^t \partial f (s)ds + f(0)$. Compute
$$
\frac1{h}(f(t+h)-f(t))-\partial f(t) =
\frac1{h}\int_t^{t+h}\partial f(s)-\partial f(t)ds
$$
and estimate
$$
\|\frac1{h}(f(t+h)-f(t))-\partial f(t)\|\leq \frac1{|h|}\int_t^{t+|h|}u_t(s)ds,
$$
where $u_t(s)=\|\partial f(s)-\partial f(t)\|$ is in $\Lloc^1(\Bbb R)$ for
almost every $t$. The limit on the right-hand side thus exists and equals
$u_t(t)=0$ almost everywhere. This concludes the proof.
\endproof
We will now consider abolutely continuous functions from $\Bbb R$
into the bounded operators between separable Hilbert-spaces $\Cal H_1$
and $\Cal H_2$. We write $\|\cdot\|_1$ and $\|\cdot\|_2$ for
their respective norms and $\|\cdot\|_{1,2}$ for the norm on $\Bet$.
\proclaim{Definition A.4} We say $B\in AC(\Bbb R;\Bet)$ if
$t\rightarrow (B\psi)(t):= B(t)\psi\in AC(\Bbb R;\Cal H_2)$ for all
$\psi\in\Cal H_1$
and $\sup_{\|\psi\|_1\leq 1}\|\partial(B\psi)(\cdot)\|_2 \in\Lloc^1(\Bbb R)$.
\endproclaim
If $B\in AC(\Bbb R;\Bet)$ there exists a family
$\partial B\in\Lloc^1(\Bbb R;\Bet)$ such that
$(\partial B\psi)(t)= (\partial B)(t)\psi$ almost everywhere.
Compute for $t>s$
$$
\|B(t)-B(s)\|_{1,2}\leq \int_s^t\sup_{\|\psi\|_1\leq 1}
\|(\partial B)(s)\psi\|_2 ds.
$$
By assumption the right-hand side converge to $0$ as $t\rightarrow s$
and the map $t\rightarrow B(t)$ is therefore continuous.
Again one can verify that the map
$$
(A,B_0)\rightarrow \int_0^t A(s)ds + B_0
$$
from $\Lloc^1(\Bbb R;\Bet)\times\Bet$ onto $AC(\Bbb R;\Bet)$ is one to one,
which justifies the following
\proclaim{Definition A.5}
The derivative $\partial:AC(\Bbb R;\Bet)\rightarrow
\Lloc^1(\Bbb R,\Bet)$ is given for $A\in\Lloc^1(\Bbb R,\Bet)$
and $B_0\in\Bet$ by
$$
\partial (\int_0^tA(s)ds+B_0)=A.
$$
\endproclaim
We similarly have, copying the proof of Proposition A.3,
\proclaim{Proposition A.6} Let $B\in AC(\Bbb R;\Bet)$. Then the limit
$$
A(t) = \lim_{h\rightarrow 0}\frac1{h}(B(t+h)-B(t))
$$
exists almost everywhere and $A=\partial B$.
\endproclaim
Let $f,g\in AC(\Bbb R;\Cal H)$. Since $f,g$ are indefinite integrals
one can verify that
$t\rightarrow \langle f(t),g(t)\rangle$ is an absolutely continuous functions
in the ordinary sense and hence itself an indefinite integral.
This argument shows
$$
f,g\in AC(\Bbb R;\Cal H)\Rightarrow
\langle f(\cdot),g(\cdot)\rangle\in AC(\Bbb R).\tag{A.1}
$$
We turn to the analysis of the Hilbert-space $L^2([0,T];\Cal H)$, $T>0$, and
consider the space of absolutely continuous functions with
square integrable derivative
$$
AC^2([0,T];\Cal H)=\{f\in AC([0,T];\Cal H): \partial f\in L^2([0,T];\Cal H)\}
$$
and the operators $\tau_0\subset\tau_V\subset\tau_*$, $V\in\Cal U(\Cal H)$
(the unitary operators on $\Cal H$), which are different realizations
of $-i\partial$ on the interval, namely with the respective domains
$$
\Cal D_* = AC^2([0,T];\Cal H),\quad\Cal D_0 =\{f\in\Cal D_*:f(0)=f(T)=0\}
$$
and
$$
\Cal D_V = \{f\in\Cal D_*:f(0)=Vf(T)\}.
$$
The following result can be verified as in [RSI]
(for the case $\Cal H=\Bbb C$).
\proclaim{Proposition A.7} We have
\roster
\item"i)" $\tau_*$ is closed.
\item"ii)" $\tau_0$ is closed and symmetric.
\item"iii)" $\tau_V$ is self-adjoint for any $V\in\Cal U(\Cal H)$.
\item"iv)" $\sigma(\tau_0)=\emptyset$ and $\spp(\tau_*)=\Bbb C$.
\item"v)" The adjoint of $\tau_0$ equals $\tau_*$.
\item"vi)" The spectrum of the $\tau_V$'s are periodic with period $\frac{2\pi}{T}$.\endroster
Furthermore, the resolvent of $\tau_V$ is given pointwisely by
$$
((\tau_V-\lambda)^{-1}f)(t) = i\int_0^te^{i\lambda(s-t)}f(s)ds
+ i(V^*-e^{i\lambda T})^{-1}\int_0^Te^{i\lambda (s-T)}f(s)ds\tag{A.4}
$$
for $\lambda\in\Bbb C$, $\im\lambda\neq 0$.
\endproclaim
Let $B\in\Cal B(\Cal H)$. We lift $B$ to a bounded operator on
$L^2([0,T];\Cal H)$
by $(Bf)(t)=Bf(t)$ for $f\in C^0([0,T];\Cal H)$ and extend it by continuity.
The following identity is easily verified.
$$
S^*\tau_VS =\tau_{S^*VS}\mfor S\in\Cal U(H).\tag{A.3}
$$
Here we have used the same symbol for the operator itself and its lifting.
If there is cause for confusion we will write $\oint B$ for the
lifted operator. We have for example
$$
\sigma(\oint B) = \sigma(B),\tag{A.4}
$$
which follows since resolvents of $\oint B$ maps the subspace
of constant functions into itself. In fact $(\oint B-z)^{-1}
=\oint(B-z)^{-1}$.
We now determine the flow generated by $\tau_V$. Let $f\in C^0([0,T];\Cal H)$
and define
$$
(U_V(s)f)(t) = V^{-\frac{[t-s]}{T}}f(t-s-[t-s]),
$$
where $[s]$ is the largest multiple of $T$ smaller than $s$.
This is clearly a one-parameter strongly continuous unitary group.
\proclaim{Proposition A.8} Let $V\in\Cal U(H)$ and $s\in\Bbb R$. Then
$\exp(-is\tau_V)=U_V(s)$.
\endproclaim
\proof
It is sufficient to prove that the generator of $U_V$ coincide with
$\tau_V$ on $f\in\Cal D_V\cap C^\infty([0,T];\Cal H)$ which by (A.2) is a core
for $\tau_V$. We compute
$$
\align
\|i\frac1{s}(U_V(s)f - f) - \tau_Vf\|^2
&=\int_0^s \|\frac1{s}(Vf(T+t-s)-f(t))+f^\prime(t)\|^2dt\\
&\quad +\int_s^T \|\frac1{s}(f(t-s)-f(t))+f^\prime(t)\|^2dt.
\endalign
$$
By the choice of $f$ and the Lebesgue Theorem on dominated convergence
we see that the right-hand side converge to zero as $s$ tends to zero,
which proves the result.
\endproof
We end the appendix with a structure result
\proclaim{Proposition A.9} Let $V\in\Cal U(\Cal H)$. Then
$$
\alignat2
\lambda\in\spp(\tau_V)&\iff e^{-i\lambda T}\in\spp(V),&\quad
\Hpp(\tau_V) &= L^2([0,T];\Hpp(V)),\\
\lambda\in\sac(\tau_V)&\iff e^{-i\lambda T}\in\sac(V),&\quad
\Hac(\tau_V) &= L^2([0,T];\Hac(V)),\\
\lambda\in\ssc(\tau_V)&\iff e^{-i\lambda T}\in\ssc(V),&\quad
\Hsc(\tau_V)&=L^2([0,T];\Hsc(V)).
\endalignat
$$
\endproclaim
\proof By Proposition A.8 we find that $\exp(-iT\tau_V) = \oint V$.
This shows the equivalence of pure point spectrums and it implies
the result for the pure point subspace. It also shows that
for a Borel set $\Omega\subset S^1$ (the unit circle)
and $f\in L^2([0,T];\Cal H)$
$$
\langle f,P_\Omega(\exp(-iT\tau_V))f\rangle=
\int_0^T\langle f(t),P_\Omega(V)f(t)\rangle,
$$
where $P_\Omega$ denotes the characteristic function for the set $\Omega$.
This identity shows the stated identity for the absolutely continuous
subspace and since the spectral subspaces decomposes the Hilbert space
we find the identity for the singular continuous subspaces as well.
The equivalence of the last two spectra now follows from this discussion
and (A.4).
\endproof
\vskip1cm
\subhead{Appendix B: The time-dependent Schr{\"o}dinger equation}\endsubhead
Let $\{H(t)\}_{t\in\Bbb R}$ be a family of self-adjoint operators on a
separable Hilbert-space $\Cal H$ which satisfies that there exists
a dense subspace $\Cal S$ with
$$
\Cal S\subset\cap_{t\in\Bbb R}\Cal D(H(t)).
$$
We wish to discuss the time-dependent Schr{\"o}dinger equation corresponding
to the family $H(t)$, in particular what is a solution supposed to
satisfy. The following suggestion is a natural one:
A (two-parameter) family of unitary
operators $\{U(t,s)\}_{t,s\in\Bbb R}$ is a solution to the time-dependent
Schr{\"o}dinger equation if
\roster
\item $U(t,s)\Cal S\subset\Cal S$ for all $t,s\in\Bbb R$.
\item For any $\varphi\in\Cal S$ the map $t\rightarrow U(t,s)\varphi$,
admits a pointwise derivative almost everywhere and its derivative satisfies
the vector-valued differential equation
$$
i\frac{d}{dt}U(t,s)\varphi = H(t)U(t,s)\varphi,\quad U(s) = I,\tag{B.1}
$$
almost everywhere.
\endroster
Solutions to this equation are not unique and we therefore need to discuss
which one we will consider (if any exist).
Let $\Cal S$ be as above. A natural class of unitary families would be those
for which $U\varphi\in AC(\Bbb R;\Cal H)$ for all $\varphi\in\Cal S$. It is
however not clear why this family should be stable under composition, which
makes it difficult to work with.
Instead we have a slightly weaker result which we choose to supply since it
covers what is needed in the present paper.
We consider the case where $\Cal H_1\subset\Cal H_2$
is a dense sub Hilbert-space equipped with a stronger norm.
$$
\|\psi\|_2\leq C\|\psi\|_1,\quad\psi\in\Cal H_1,
$$
for some $C>0$. (Compared to above $\Cal S=\Cal H_1$ and $\Cal H=\Cal H_2$.)
We write $\tilde{\Cal S}_{1,2}$ for the space of symmetric operators with
domain containing $\Cal H_1$ and we say $H_1\sim H_2$ if $H_{1|\Cal H_1}
=H_{2|\Cal H_1}$ and define $\Cal S_{1,2} = \tilde{\Cal S}_{1,2}/\sim$.
Since a symmetric operator $H\in\Cal S_{1,2}$ is closable we find that
$H$ as an operator from $\Cal H_1$ into $\Cal H_2$ is closed and hence
bounded by the Closed Graph Theorem.
There is thus a canonical inclusion of $\Cal S_{1,2}$ into $\Bet$.
In the norm $\sup_{\|\psi\|_1\leq 1} \|H\psi\|_2$
the space $\Cal S_{1,2}$ is complete and the inclusion into
$\Bet$ is an isometry.
We will consider operator families $H$ from the space
$\Lloc^1(\Bbb R;\Cal S_{1,2})$ which we identify
with a subspace of $\Lloc^1(\Bbb R;\Bet)$. We are interested
in solutions $U(\cdot,s)$ to the evolution equation
$$
i\partial U = H U,\quad U(s)=I.\tag{B.2}
$$
Solutions will be sought in the set $\Cal U_{1,2}$.
\proclaim{Definition B.1} The set $\Cal U_{1,2}$
consists of $U:\Bbb R\rightarrow\Cal U(\Cal H_2)$ which are measurable
and satisfy that $U_{|\Cal H_1}\in AC(\Bbb R;\Bet)$ and
$U\psi,U^*\psi\in \Lloc^\infty(\Bbb R;\Cal H)$
for all $\psi\in\Cal H_1$.
\endproclaim
Note that being a solution to (B.2) is consistent with being an element
of $\Cal U_{1,2}$.
All operators are identified with operators on the
large Hilbert-space $\Cal H_2$ and adjoints are taken with respect
to the inner product $\langle\cdot,\cdot\rangle_2$.
Note that families in $\Cal U_{1,2}$ are strongly continuous and
by the Uniform Boundedness Principle we have
$$
\sup_{|t|\leq T} (\|U(t)\|_{1,1}+\|U^*(t)\|_{1,1})<\infty,\tag{B.3}
$$
for any $T>0$.
Instead of (B.2) one could seek solutions $U(s,\cdot)$ from $\Cal U_{1,2}$
to the equation
$$
i\partial U = -UH,\quad U(s,s)=I\tag{B.4}
$$
and in fact we have the following result
\proclaim{Lemma B.2} We have
\roster
\item"i)" Suppose $U(\cdot,s)\in\Cal U_{1,2}$ solves \rom{(B.2)}.
Then $U^*(\cdot,s)\in\Cal U_{1,2}$ and it solves \rom{(B.4)}.
\item"ii)" Suppose $U(s,\cdot)\in\Cal U_{1,2}$ solves \rom{(B.4)}.
Then $U^*(s,\cdot)\in\Cal U_{1,2}$ and it solves \rom{(B.2)}.
\endroster
\endproclaim
\proof
We only verify i) since ii) can be proved in similar fashion.
Since $U(\cdot,s)$ is in $AC(\Bbb R;\Bet)$ and solves (B.2) we find
$$
iU(t,s) = I + \int_s^t H(r)U(r,s)dr.
$$
Due to the construction of the integral (see Appendix A) we thus find,
for $\psi,\varphi\in\Cal H_1$,
$$
\langle \psi,iU(t,s)\varphi\rangle = \langle \psi,\varphi\rangle
+\int_s^t\langle U^*(r,s)H(r)\psi,\varphi\rangle dr,
$$
which implies that
$$
-iU^*(t,s) = I +\int_s^t U^*(r,s)H(r)dr.
$$
Hence $U^*(\cdot,s)\in\Cal U_{1,2}$ and it solves (B.4).
\endproof
The solution set is a group, more precisely
\proclaim{Proposition B.3} Let $U_1,U_2\in\Cal U_{1,2}$ and
$\psi\in AC(\Bbb R;\Cal H_2)\cap\Lloc^\infty(\Bbb R;\Cal H_1)$.
Then we have
\roster
\item"i)" $U_1^*\in\Cal U_{1,2}$, $\Cal H_1\subset \Cal D((\partial U)^*)$ and
$\partial U^* = (\partial U)^*_{|\Cal H_1}$.
\item"ii)" $U_1\psi\in AC(\Bbb R;\Cal H_2)\cap\Lloc^\infty(\Bbb R;\Cal H_1)$ and
$\partial U_1\psi = (\partial U_1)\psi + U\partial\psi$.
\item"iii)" $U_2U_1\in\Cal U_{1,2}$ and $\partial(U_2U_1)=(\partial U_2)U_1
+U_2\partial U_1$.
\endroster
\endproclaim
\proof
Compute for $\psi,\varphi\in\Cal H_1$ using (A.1) and Proposition A.6
$$
0 = \partial\langle \psi,\varphi\rangle_2 =
\partial\langle U_1(\cdot)\psi,U_1(\cdot)\varphi\rangle_2 =
\langle(\partial U_1)(\cdot)\psi,U_1(\cdot)\varphi\rangle_2
+\langle U_1(\cdot)\psi,(\partial U_1)(\cdot)\varphi\rangle_2,
$$
which shows that $U_1^*(i\partial U_1)\in\Lloc^1(\Bbb R;\Cal S_{1,2})$.
For $\psi,\varphi\in\Cal H_1$ we thus have
$$
\langle\psi,(\partial U_1)(\cdot)\varphi\rangle_2 =
\langle U_1^*(\cdot)\psi,U_1^*(\cdot)(\partial U_1)(\cdot)\varphi\rangle_2 =
-\langle U_1^*(\cdot)(\partial U_1)(\cdot) U_1^*(\cdot)\psi,\varphi\rangle_2.
$$
This implies that $\Cal H_1\subset\Cal D((\partial U_1)^*(t))$ for almost
all $t$ and
$$
(\partial U_1)^*_{|\Cal H_1} = -U_1^*(\partial U_1)U^*_{1|\Cal H_1}.\tag{B.5}
$$
As in the proof of Lemma B.2, this shows that $U_1^*\in\Cal U_{1,2}$.
Let $U_1$ and $\psi$ be as in the statement of the Proposition.
Compute for $\varphi\in\Cal H_1$ using (A.1) and the above
$$
\langle\varphi,U_1(t)\psi(t)\rangle=
\int_0^t h(s) ds + \langle\varphi,U_1(0)\psi(0)\rangle,
$$
where by Proposition A.6
$$
\align
h&=\partial\langle U_1^*(\cdot)\varphi,\psi(\cdot)\rangle\\
&= \langle (\partial U_1)^*(\cdot)\varphi,\psi(\cdot)\rangle
+\langle U_1^*(\cdot)\varphi,(\partial\psi)(\cdot)\rangle\\
&=\langle \varphi,(\partial U_1)(\cdot)\psi(\cdot)
+U_1(\cdot)(\partial\psi)(\cdot)\rangle.
\endalign
$$
By construction of the integral and (B.3) this concludes the proof of ii) and iii) follows
directly from ii).
\endproof
Notice that by (B.5) the maps
$$
\Phi_L(U)=(i\partial U) U^*\mand \Phi_R(U) = U^*(i\partial U)
$$
both map $\Cal U_{1,2}$ into $\Lloc^1(\Bbb R;S_{1,2})$ and $U$ is hence
a solution to (B.2) with $H=\Phi_L(U)$ and to (B.4) with $H=\Phi_R(U)$.
\proclaim{Theorem B.4 (Uniqueness)} Let $s\in\Bbb R$ and
$H\in \Lloc^1(\Bbb R;\Cal S_{1,2})$.
\roster
\item"i)" There can be at most one solution
$U(\cdot,s)\in\Cal U_{1,2}$ to \rom{(B.2)}.
\item"ii)" There can be at most one solution
$U(s,\cdot)\in\Cal U_{1,2}$ to \rom{(B.4)}.
\endroster
\endproclaim
\proof
We restrict attention to i).
Assume there exist $U_1(\cdot,s),U_2(\cdot,s)\in\Cal U_{1,2}$ such that
$\Phi_L(U_1(\cdot,s))=\Phi_L(U_2(\cdot,s))$.
Compute using Proposition B.3 i), iii) and (B.5)
$$
\align
\partial(U_2^*U_1)&=(\partial U_2^*)U_1 + U_2^*(\partial U_1)\\
&= - U_2^*(\partial U_2) U_2^*U_1+U_2^*(\partial U_1)U_1^* U_1\\
&= -i U_2^*\Phi_L(U_2)U_1 + i U_2^*\Phi_L(U_1)U_1\\
&= 0.
\endalign
$$
This shows that $U_2(\cdot,s)^*U_1(\cdot,s) = U_0(s)$,
a unitary operator, and completes the proof since the left hand
side equals the identity at $t=s$.
\endproof
We have the following easy consequences of Lemma B.2, Proposition B.3
and Theorem B.4.
\proclaim{Corollary B.5} Let $U(t,s)$ be a family of unitary operators.
The following three statements are equivalent
\roster
\item"i)" $U(\cdot,s),U(s,\cdot)\in\Cal U_{1,2}$ and solves both \rom{(B.2)}
and \rom{(B.4)}.
\item"ii)" $U(\cdot,s)\in\Cal U_{1,2}$ solves \rom{(B.2)} and $U^*(t,s)=U(s,t)$.
\item"iii)" $U(s,\cdot)\in\Cal U_{1,2}$ solves \rom{(B.4)} and $U^*(t,s)=U(s,t)$.
\endroster
\endproclaim
\proclaim{Corollary B.6 (Chapman-Kolmogorov)} Suppose the family $U(t,s)$
satisfies either one of the three conditions in Corollary B.5. Then
$$
U(t,s)U(s,r) = U(t,r)\mforall t,s,r\in\Bbb R.
$$
\endproclaim
In the light of Theorem B.4 and Corollary B.5 we will in the present paper
employ the following definition of what a solution to the time-dependent
Schr{\"o}dinger equation is
\proclaim{Definition B.7} Let $H(t)$ be a family of Hamiltonians for which
there exists a Hilbert-space $\Cal H_1\subset\Cal H_2=\Cal H$ such that
$H\in \Lloc^1(\Bbb R;\Cal S_{1,2})$.
A family of unitary operators $U(t,s)$ is said to solve the
time-dependent Schr{\"o}dinger equation if
\roster
\item"i)" $U(\cdot,s),U(s,\cdot)\in\Cal U_{1,2}$.
\item"ii)" $U(\cdot,s)$ solves \rom{(B.2)} and $U(s,\cdot)$ solves \rom{(B.4)}.
\endroster
\endproclaim
For other purposes one might want to assume the Hamiltonians $H(t)$ to be
essentially self-adjoint on $\Cal H_1$ for almost all $t$ instead of
just symmetric. This is however not nescessary in our situation.
In construction procedures one often considers a corresponding
integral equation and invariance of some domain is a typical
byproduct, see [Ta] and [Ya1]. In [DG, Appendix B.3] a situation similar
to ours is considered and it is shown how one can use control of certain
commutators to obtain invariance of a domain under the flow.
Their approach is independent of a construction procedure.
These two observations indicate that Definition B.7 is not to restrictive.
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\enddocument
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