Content-Type: multipart/mixed; boundary="-------------9907301139129" This is a multi-part message in MIME format. ---------------9907301139129 Content-Type: text/plain; name="99-289.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-289.keywords" Spectral theory, Scattering theory, Mourre theory ---------------9907301139129 Content-Type: application/x-tex; name="mparcper.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mparcper.tex" \input amstex \documentstyle{amsppt} \magnification=\magstep1 \baselineskip=18truept \voffset=0.3truein % adjustment for 1i margin \hsize=6.0truein % printed page horizontal size \vsize=8.5truein % printed page vertical size \parskip=5truept % increase distance between paragraphs \overfullrule=0pt % suppress black bars generated by overfull hboxes \NoRunningHeads % pagenumbers bottom centered %\NoPageNumbers % definerer nu ‘›†, ’¯¸ \catcode`‘=\active \def‘{{\ae}} \catcode`›=\active \def›{{\o}} \catcode`†=\active \def†{{\aa}} \catcode`’=\active \def’{{\AE}} \catcode`¯=\active \def¯{{\O}} \catcode`¸=\active \def¸{{\AA}} \TagsOnRight \define \Lloc{L_{\text{\rom{loc}}}} \define \weight{\langle x\rangle} \define \weights{\langle s\rangle} \define \weightt{\langle t\rangle} \define \weightA{\langle A\rangle} \define \weightz{\langle z\rangle} \define \weighty{\langle y\rangle} \define \weightr{\langle r\rangle} \define \mand{\quad\text{and}\quad} \define \mfor{\quad\text{for}\quad} \define \mforall{\quad\text{for all}\quad} \define \mwhere{\quad\text{where}\quad} \define \im{operatorname{Im}} \define \re{\operatorname{Re}} \define \imply{\Rightarrow} %\define \equiv{\Leftrightarrow} \define \veci{{\vec{i}}} \define \vecj{{\vec{j}}} \define \pa{\partial^\alpha_x} \define \pb{\partial^\beta_\xi} \define \pd#1,#2{\partial^{#1}_{#2}} \define \ad{\operatorname{ad}} \define \mdiv{\operatorname{div}} \define \tr{\operatorname{tr}} \define \spn{\operatorname{span}} \define \range{\operatorname{Range}} \define \supp{\operatorname{supp}} \define \nsubset{\not\subset} \define \nin{\not\in} \define \dprime{{\prime\prime}} \define \endproof{$\hfill\square$\enddemo} \define \proof{\demo{Proof}} \define \slim{\text{\rom{s}}-\lim} \define \wlim{\text{\rom{w}}-\lim} \define \spp{\sigma_{\text{\rom{pp}}}} \define \ssc{\sigma_{\text{\rom{sc}}}} \define \sac{\sigma_{\text{\rom{ac}}}} \define \ses{\sigma_{\text{\rom{ess}}}} \define \Hpp{\Cal H_{\text{\rom{pp}}}} \define \Hac{\Cal H_{\text{\rom{ac}}}} \define \Hsc{\Cal H_{\text{\rom{sc}}}} \font\myfont=cmbx10 scaled 1300 \widestnumber\key{HMS2} \define \Hper{\hat{\Cal H}} \define \weightp{\langle p\rangle} \define \om{\omega_0} \define \vr{V_{\text{\rom{reg}}}} \define \vs{V_{\text{\rom{sing}}}} \define \hc{H^{\text{\rom{const}}}} \define \hch{\hat{H}^{\text{\rom{const}}}} \define \Bet{\Cal B(\Cal H_1;\Cal H_2)} \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. 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