On the Mourre estimates for three-body Floquet Hamiltonians

TL;DR

This paper establishes Mourre estimates for three-body Floquet Hamiltonians with time-periodic potentials, advancing spectral analysis techniques for such quantum systems.

Contribution

The paper introduces a conjugate operator within Mourre theory to prove spectral estimates for three-body Floquet Hamiltonians with time-periodic interactions.

Findings

Proved Mourre estimate for the Floquet Hamiltonian K.

Extended Mourre theory to three-body time-periodic systems.

Provided spectral analysis tools for Floquet Hamiltonians.

Abstract

In this paper, we consider the Floquet Hamiltonian $K$ associated with a three-body Schr\"odinger operator with time-periodic pair potentials $H(t)$. By introducing a conjugate operator $A$ for $K$ in the standard Mourre theory, we prove the Mourre estimate for $K$.

Equations

$$\tilde{H}(t)=\sum _{j=1}^N\left(-\frac{1}{2m_j}{\Delta }_j\right)+V(t),V(t)=\sum _{1\le j<k\le N}{V}_{jk}(t,r_j-r_k)$$

$${\Delta }_j=\sum _{l=1}^d{\partial }_{r_{j,l}}^2$$

$$V_{jk}(t+T,y)=V_{jk}(t,y),(t,y)\in \mathbf{R}\times {\mathbf{R}}^d.$$

$$r\cdot \tilde{r}=\sum _{j=1}^N{m}_j⟨r_j,{\tilde{r}}_j⟩,r=(r_1,\dots ,r_N),\tilde{r}=({\tilde{r}}_1,\dots ,{\tilde{r}}_N)\in {\mathbf{R}}^{d\times N},$$

$$X=\left\{r\in {\mathbf{R}}^{d\times N}|\sum _{j=1}^N{m}_j{r}_j=0\right\},$$

$$X_{\mathrm{cm}}=\left\{r\in {\mathbf{R}}^{d\times N}|r_1=\cdots =r_N=0\right\}.$$

$$H(t)=-\frac{1}{2}\Delta +V(t)$$

$$\tilde{H}(t)=H(t)\otimes \mathrm{Id}+\mathrm{Id}\otimes \left(-\frac{1}{2}{\Delta }_{\mathrm{cm}}\right)$$

$$-\Delta =p^2,-{\Delta }_{\mathrm{cm}}=(p_{\mathrm{cm}}{)}^2.$$

$$X^a=\left\{r\in X|\sum _{j\in C}{m}_j{r}_j=0\text{ for each cluster }C\text{ in }a\right\},$$

$$X_a=\left\{r\in X|r_j=r_k\text{ for each pair }(j,k)\subset a\right\}.$$

$$V_{(j,k)}(t,x^{(j,k)})=V_{jk}(t,r_j-r_k).$$

$$H_a(t)=-\frac{1}{2}\Delta +V^a(t),V^a(t)=\sum _{(j,k)\subset a}{V}_{(j,k)}(t,x^{(j,k)}),$$

$$H_a(t)=H^a(t)\otimes \mathrm{Id}+\mathrm{Id}\otimes \left(-\frac{1}{2}{\Delta }_a\right);H^a(t)=-\frac{1}{2}{\Delta }^a+V^a(t)$$

$$-{\Delta }^a=(p^a{)}^2,-{\Delta }_a=(p_a{)}^2.$$

$$I_a(t,x)=V(t,x)-V^a(t,x)=\sum _{(j,k)\not\subset a}{V}_{(j,k)}(t,x^{(j,k)}).$$

$$(\hat{U}(\sigma )\Phi )(t)=U(t,t-\sigma )\Phi (t-\sigma )$$

$$\hat{U}(\sigma )=e^{-i\sigma K}$$

$${\tilde{A}}_1=\frac{1}{2}\{x\cdot p(1+p^2/2{)}^{-1}+(1+p^2/2{)}^{-1}p\cdot x\}$$

$$\mathrm{Re}T=\frac{1}{2}(T+T^{\ast }).$$

$${\hat{A}}_0=\frac{1}{2}(x\cdot p+p\cdot x)=\mathrm{Re}(p\cdot x)$$

$$d_0(\lambda )=\mathrm{dist}(\lambda ,\omega \mathbf{Z}),d_1(\lambda )=\mathrm{dist}(\lambda ,\omega \mathbf{Z}\cap (-\infty ,\lambda ])$$

$$f_{\delta }(K-{\lambda }_0)i[K,{\tilde{A}}_1]f_{\delta }(K-{\lambda }_0)\ge \frac{2(d_1({\lambda }_0)-\delta )}{1+(d_1({\lambda }_0)-\delta )}{f}_{\delta }(K-{\lambda }_0{)}^2+C_{1,{\lambda }_0,f_{\delta }}$$

$$f_{\delta }(K-{\lambda }_0)i[K,{\tilde{A}}_1]f_{\delta }(K-{\lambda }_0)\ge \frac{2(d_0({\lambda }_0)-\delta )}{1+(d_0({\lambda }_0)-\delta )}{f}_{\delta }(K-{\lambda }_0{)}^2+C_{1,{\lambda }_0,f_{\delta }}^{\prime }$$

$$A_{{\lambda }_0,\delta }=({\lambda }_0-\delta -D_t{)}^{-1}\otimes {\hat{A}}_0$$

$$f_{\delta }(K-{\lambda }_0)i[K,A_{{\lambda }_0,\delta }]f_{\delta }(K-{\lambda }_0)\ge 2f_{\delta }(K-{\lambda }_0{)}^2+C_{{\lambda }_0,f_{\delta }}$$

$$({\hat{A}}_0{)}^a\otimes \mathrm{Id}+\mathrm{Id}\otimes ({\hat{A}}_0{)}_a$$

$$\begin{array}{rl}& ({\hat{A}}_0{)}^a=\frac{1}{2}(x^a\cdot p^a+p^a\cdot x^a)=\mathrm{Re}(p^a\cdot x^a),\\ & ({\hat{A}}_0{)}_a=\frac{1}{2}(x_a\cdot p_a+p_a\cdot x_a)=\mathrm{Re}(p_a\cdot x_a).\end{array}$$

$$({\lambda }_0-\delta -D_t{)}^{-1}\otimes \{({\hat{A}}_0{)}^a\otimes \mathrm{Id}+\mathrm{Id}\otimes ({\hat{A}}_0{)}_a\}$$

$${\tilde{A}}_{1,a}=\mathrm{Re}((1+(p_a{)}^2/2{)}^{-1}{p}_a\cdot x_a)$$

$$({\lambda }_0-\delta -D_t{)}^{-1}\otimes ({\hat{A}}_0{)}^a\otimes \mathrm{Id}+\mathrm{Id}\otimes \mathrm{Id}\otimes {\tilde{A}}_{1,a}$$

Full text

1 Introduction

In this paper, we consider a three-body quantum system with time-periodic pair interactions. Since we would like to introduce some notation in many body scattering theory, we denote the number of particles in the system by $N$ for a while. Of course, we mainly consider the case where $N=3$. The system under consideration is governed by the following Schrödinger operator with time-periodic potentials

[TABLE]

acting on $L^{2}(\boldsymbol{R}^{d\times N})$, where $m_{j}$ and $r_{j}\in\boldsymbol{R}^{d}$ are the mass and position vector of the $j$-th particle, respectively,

[TABLE]

is the Laplacian with respect to $r_{j}$, and $V_{jk}(t,r_{j}-r_{k})$’s are pair potentials. We suppose that $V_{jk}(t,y)$’s are real-valued functions on $\boldsymbol{R}\times\boldsymbol{R}^{d}$ which are periodic in $t$ with a period $T>0$:

[TABLE]

We would like to watch the motion of the system in the center-of-mass frame. To this end, we will introduce the following configuration spaces: We equip $\boldsymbol{R}^{d\times N}$ with the metric

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the standard inner product on $\boldsymbol{R}^{d}$. We usually write $r\cdot r$ as $r^{2}$. We put $|r|=\sqrt{r^{2}}$. We define two subspaces $X$ and $X_{\mathrm{cm}}$ of $\boldsymbol{R}^{d\times N}$ as

[TABLE]

Then $X$ and $X_{\mathrm{cm}}$ are perpendicular to each other, and satisfy $\boldsymbol{R}^{d\times N}=X\oplus X_{\mathrm{cm}}$. $\pi:\boldsymbol{R}^{d\times N}\to X$ and $\pi_{\mathrm{cm}}:\boldsymbol{R}^{d\times N}\to X_{\mathrm{cm}}$ denote the orthogonal projections onto $X$ and $X_{\mathrm{cm}}$, respectively. We put $x=\pi r$ and $x_{\mathrm{cm}}=\pi_{\mathrm{cm}}r$ for $r\in\boldsymbol{R}^{d\times N}$. Now we introduce the time-dependent Hamiltonian

[TABLE]

acting on $\mathscr{H}=L^{2}(X)$. Then $\tilde{H}(t)$ is represented as

[TABLE]

on $L^{2}(\boldsymbol{R}^{d\times N})=\mathscr{H}\otimes L^{2}(X_{\mathrm{cm}})$. Here $\Delta$ and $\Delta_{\mathrm{cm}}$ are the Laplace-Beltrami operators on $X$ and $X_{\mathrm{cm}}$, respectively. By introducing the velocity operators $p$ and $p_{\mathrm{cm}}$ on $X$ and $X_{\mathrm{cm}}$, respectively, $-\Delta$ and $-\Delta_{\mathrm{cm}}$ can be represented as

[TABLE]

We would like to study some scattering problems for this Hamiltonian $H(t)$ with $N=3$.

A non-empty subset of the set $\{1,\ldots,N\}$ is called a cluster. Let $C_{j}$, $1\leq j\leq m$, be clusters. If $\cup_{1\leq j\leq m}C_{j}=\{1,\ldots,N\}$ and $C_{j}\cap C_{k}=\emptyset$ for $1\leq j<k\leq m$, $a=\{C_{1},\ldots,C_{m}\}$ is called a cluster decomposition. $\#(a)$ denotes the number of clusters in $a$. Let $\mathscr{A}$ be the set of all cluster decompositions. Suppose $a$, $b\in\mathscr{A}$. If $b$ is obtained as a refinement of $a$, that is, if each cluster in $b$ is a subset of a cluster in $a$, we say $b\subset a$, and its negation is denoted by $b\not\subset a$. Any $a$ is regarded as a refinement of itself. The one and $N$-cluster decompositions are denoted by $a_{\mathrm{max}}$ and $a_{\mathrm{min}}$, respectively. The pair $(j,k)$ is identified with the $(N-1)$-cluster decomposition $\{(j,k),(1),\ldots,(\widehat{j}),\ldots,(\widehat{k}),\ldots,(N)\}$. If $N=3$, then $\{(1,2),\,(1,3),\,(2,3)\}$ is the set of all $2$-cluster decompositions.

Let $a\in\mathscr{A}$. We introduce two subspaces $X^{a}$ and $X_{a}$ of $X$:

[TABLE]

$\pi^{a}:X\to X^{a}$ and $\pi_{a}:X\to X_{a}$ denote the orthogonal projections onto $X^{a}$ and $X_{a}$, respectively. We put $x^{a}=\pi^{a}\,x$ and $x_{a}=\pi_{a}\,x$ for $x\in X$. Since $X^{(j,k)}$ is identified with the configuration space for the relative position of $j$-th and $k$-th particles, one can put

[TABLE]

We now define the cluster Hamiltonian

[TABLE]

which governs the motion of the system broken into non-interacting clusters of particles. Then $H_{a}(t)$ is represented as

[TABLE]

on $\mathscr{H}=\mathscr{H}^{a}\otimes\mathscr{H}_{a}=L^{2}(X^{a})\otimes L^{2}(X_{a})$, where $\Delta^{a}$ and $\Delta_{a}$ are the Laplace-Beltrami operators on $X^{a}$ and $X_{a}$, respectively. By introducing the velocity operators $p^{a}$ and $p_{a}$ on $X^{a}$ and $X_{a}$, respectively, $-\Delta^{a}$ and $-\Delta_{a}$ can be represented as

[TABLE]

The intercluster potential $I_{a}(t)$ is given by

[TABLE]

Under some suitable conditions on $V_{jk}(t)$, the existence and uniqueness of the unitary propagator $U(t,s)$ generated by $H(t)$ can be guaranteed, even if $N\geq 3$ (see e.g. Yajima [27, 28]). In the study of the asymptotic behavior of $U(t,s)\phi$, $\phi\in\mathscr{H}$, as $t\to\pm\infty$, we will frequently utilize the so-called Floquet Hamiltonian $K$ associated with $H(t)$: Let $\boldsymbol{T}=\boldsymbol{R}/(T\boldsymbol{Z})$ be the torus. Set $\mathscr{K}=L^{2}(\boldsymbol{T};\mathscr{H})\cong L^{2}(\boldsymbol{T})\otimes\mathscr{H}$, and introduce a strongly continuous one-parameter unitary group $\{\hat{U}(\sigma)\}_{{\sigma\in\boldsymbol{R}}}$ on $\mathscr{K}$ given by

[TABLE]

for $\Phi\in\mathscr{K}$. By virtue of Stone’s theorem, $\hat{U}(\sigma)$ is written as

[TABLE]

with a unique self-adjoint operator $K$ on $\mathscr{K}$. $K$ is called the Floquet Hamiltonian associated with $H(t)$, and is equal to the natural self-adjoint realization of $-i\partial_{t}+H(t)$. Here we denote by $D_{t}$ the operator $-i\partial_{t}$ with domain $AC(\boldsymbol{T})$, which is the space of absolutely continuous functions on $\boldsymbol{T}$ with their derivatives being square integrable (following the notation in Reed-Simon [21]). As is well-known, $D_{t}$ is self-adjoint on $L^{2}(\boldsymbol{T})$, and its spectrum $\sigma(D_{t})$ is equal to $\omega\boldsymbol{Z}$ with $\omega=2\pi/T$.

In this paper, we would like to propose the definition of a conjugate operator for $K$ with $N=3$. First we recall known results in the case where $N=2$ for reference. Yokoyama [29] introduced the self-adjoint operator

[TABLE]

on $\mathscr{K}$ as a conjugate operator for $K$. For the sake of brevity, we will use the notation $\mathrm{Re}\,T$ for an operator on $\mathscr{K}$ in this paper, which is defined by

[TABLE]

Then $\tilde{A}_{1}$ can be written as $\mathrm{Re}\,((1+p^{2}/2)^{-1}p\cdot x)$. Roughly speaking, $\tilde{A}_{1}$ is defined by multiplying the generator of dilations

[TABLE]

and the resolvent $(1+p^{2}/2)^{-1}$ of $p^{2}/2=-\Delta/2$. He established the following Mourre estimate under some suitable conditions on $V$: Put

[TABLE]

for $\lambda\in\boldsymbol{R}$. Suppose $\lambda_{0}\in\boldsymbol{R}\setminus\omega\boldsymbol{Z}$ and $0<\delta<\mathrm{dist}(\lambda_{0},\omega\boldsymbol{Z})=d_{0}(\lambda_{0})$. Then, for any $f_{\delta}\in C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ supported in $[-\delta,\delta]$, the Mourre estimate

[TABLE]

holds with some compact operator $C_{1,\lambda_{0},f_{\delta}}$ on $\mathscr{K}$. This estimate (1.8) is slightly better than the one obtained in [29]

[TABLE]

with some compact operator $C_{1,\lambda_{0},f_{\delta}}^{\prime}$ on $\mathscr{K}$, since $d_{0}(\lambda_{0})\leq d_{1}(\lambda_{0})$. Here we note that the positive constant of the Mourre estimate (1.8) depends on $\lambda_{0}$ strictly but the conjugate operator $\tilde{A}_{1}$ is independent of $\lambda_{0}$. However, its extension to the case where $N\geq 3$ has not been obtained yet, as far as we know (see also Møller-Skibsted [18]). Recently, Adachi-Kiyose [4] proposed an alternative conjugate operator for $K$ with $N=2$ at a non-threshold energy $\lambda_{0}$: Let $\lambda_{0}\in\boldsymbol{R}\setminus\omega\boldsymbol{Z}$. Then there exists a unique $n_{\lambda_{0}}\in\boldsymbol{Z}$ such that $\lambda_{0}\in I_{n_{\lambda_{0}}}$. Take $\delta$ as $0<\delta<\mathrm{dist}(\lambda_{0},\omega\boldsymbol{Z})$. Since $\lambda_{0}-\delta\in I_{n_{\lambda_{0}}}$, it is obvious that $\lambda_{0}-\delta\in\boldsymbol{R}\setminus\omega\boldsymbol{Z}\subset\rho(D_{t})$. Then we introduce the self-adjoint operator

[TABLE]

on $\mathscr{K}\cong L^{2}(\boldsymbol{T})\otimes\mathscr{H}$, by multiplying $\hat{A}_{0}$ and the resolvent $(\lambda_{0}-\delta-D_{t})^{-1}$ of $D_{t}$. Here we note that $(\lambda_{0}-\delta-D_{t})^{-1}$ is bounded and self-adjoint. Then the Mourre estimate

[TABLE]

holds with some compact operator $C_{\lambda_{0},f_{\delta}}$ on $\mathscr{K}$. Here we note that the positive constant of the Mourre estimate (1.9) is independent of $\lambda_{0}$ but the conjugate operator $A_{\lambda_{0},\delta}$ depends on $\lambda_{0}$ strictly. Its extension to the case where $N\geq 3$ has not been obtained generally yet, except in the case where all the pair potentials are independent of $t$.

The aim of this paper is that we will introduce a conjugate operator for $K$ with $N=3$ by utilizing the above conjugate operators for $K$ with $N=2$ due to both [29] and [4]. As is pointed out by Møller-Skibsted [18], it is important in obtaining the Mourre estimates for time-independent many body Schrödinger operators that the generator of dilations $\hat{A}_{0}$ in (1.7) can be decomposed into the sum

[TABLE]

acting on $\mathscr{H}\cong\mathscr{H}^{a}\otimes\mathscr{H}_{a}$, for $a\in\mathscr{A}$, where

[TABLE]

Unfortunately, the conjugate operator $\tilde{A}_{1}$ in (1.6) does not have such a property. This seems one of the reasons why its extension to the case where $N\geq 3$ has not been given yet. On the other hand, the conjugate operator $A_{\lambda_{0},\delta}$ in (1.10) can be decomposed into the sum

[TABLE]

acting on $\mathscr{K}\cong L^{2}(\boldsymbol{T})\otimes\mathscr{H}^{a}\otimes\mathscr{H}_{a}$, for $a\in\mathscr{A}$. If $N=3$ and $a\in\mathscr{A}$ is a pair, then one can recognize the operator $(\lambda_{0}-\delta-D_{t})^{-1}\otimes(\hat{A}_{0})^{a}$ as a conjugate operator for $K^{a}=D_{t}+H^{a}(t)$ acting on $\mathscr{K}^{a}=L^{2}(\boldsymbol{T};\mathscr{H}^{a})\cong L^{2}(\boldsymbol{T})\otimes\mathscr{H}^{a}$, by virtue of a result of Adachi-Kiyose [4]. $K^{a}$ is the Floquet Hamiltonian associated with the subsystem Hamiltonian $H^{a}(t)$. However, we cannot interpret the operator $(\lambda_{0}-\delta-D_{t})^{-1}\otimes(\hat{A}_{0})_{a}$ as a conjugate operator for the intercluster Hamiltonian $-\Delta_{a}/2$ acting on $\mathscr{H}_{a}$, unfortunately. We think that this is one of the reasons why any extension of $A_{\lambda_{0},\delta}$ to the case where $N\geq 3$ has not been given yet. In order to overcome the difficulty mentioned above, we will recognize the operator

[TABLE]

acting on $\mathscr{H}_{a}$ as a conjugate operator for $-\Delta_{a}/2$, and the sum

[TABLE]

as a conjugate operator $A_{a}$ for $K_{a}=D_{t}+H_{a}(t)$ acting on $\mathscr{K}$. $K_{a}$ is the Floquet Hamiltonian associated with the cluster Hamiltonian $H_{a}(t)$. We call $K_{a}$ a cluster Floquet Hamiltonian. After introducing $A_{a}$’s, we will glue these together by using a partition of unity of $X$. This is our strategy of introducing a conjugate operator $A$ for $K$ with $N=3$.

Now we will give the precise definition of $A$. We first note that without loss of generality, we may assume that a non-threshold energy $\lambda_{0}$ belongs to the interval $[0,\omega)$, because the spectrum $\sigma(K)$ of $K$ is $\omega$-periodic, as is well-known. Let $\delta\in(0,\omega/4)$ and $a\in\mathscr{A}$. We define a conjugate operator $A^{a}$ for $K^{a}=D_{t}+H^{a}(t)$ by

[TABLE]

acting on $L^{2}(\boldsymbol{T})\otimes\mathscr{H}^{a}$ (see (1.7) and (1.11) as for $\hat{A}_{0}$ and $(\hat{A}_{0})^{a}$). Here we note that $A^{a}$ is independent of $\lambda_{0}\in[0,\omega)$, unlike $A_{\lambda_{0},\delta}$ in (1.9), and that $K^{a_{\mathrm{min}}}=D_{t}$ and $A^{a_{\mathrm{min}}}=0$. We also define a conjugate operator $\tilde{A}_{\omega/4,a}$ for $-\Delta_{a}/2$ by

[TABLE]

acting on $\mathscr{H}_{a}$. Here we note that $\tilde{A}_{\omega/4,a_{\mathrm{max}}}=0$. Finally we put

[TABLE]

acting on $\mathscr{K}\cong\mathscr{K}^{a}\otimes\mathscr{H}_{a}$. $A_{a}$’s are self-adjoint. In order to glue $A_{a}$’s together, we will introduce a Graf partition of unity of $X$ (see e.g. Graf [10], Skibsted [24] and Dereziński-Gérard [7]): Given $\kappa_{0}>1$. Then there exist $r_{0},\,r_{1}>0$ and $\{j_{a}\}_{a\in\mathscr{A}}\subset C^{\infty}(X;\boldsymbol{R})$ such that the following is satisfied; $j_{a}$’s are all bounded smooth functions on $X$ with bounded derivatives satisfying $0\leq j_{a}(x)\leq 1$ and

[TABLE]

On $\mathrm{supp}\,j_{a}$, $|x^{a}|\leq r_{0}$ holds, and $|x^{(j,k)}|\geq r_{1}$ holds if $(j,k)\not\subset a$. If $\kappa\geq\kappa_{0}$, then $j_{a}(x)j_{b}(\kappa x)=0$ for $b\not\subset a$, and

[TABLE]

For the sake of brevity, we put $j_{a,R}(x)=j_{a}(x/R)$ for a parameter $R\geq 1$. By using $\{j_{a,R}\}_{a\in\mathscr{A}}$, we define

[TABLE]

with $j_{a,R}=j_{a,R}(x)$. The self-adjointness of $A(R)$ can be guaranteed by Nelson’s commutator theorem (see Theorem 2.1 in §2). We will see later that $A(R)$ with sufficiently large $R$ is a conjugate operator for $K$.

Now we impose the following condition $(V)_{3}$ on $V$ under consideration:

$(V)_{3}$ $V_{jk}(t,y)$, $(j,k)\in\mathscr{A}$, is a real-valued function on $\boldsymbol{R}\times\boldsymbol{R}^{d}$, is $T$-periodic in $t$, belongs to $C^{2}(\boldsymbol{R}\times\boldsymbol{R}^{d})$, and satisfies the decaying conditions

[TABLE]

with some $\rho>0$.

Here we give some remarks on the condition $(V)_{3}$. By a certain technical reason, which will be stated in §3, we do not allow $V_{jk}$’s to have any local singularity, although when $N=2$, some local singularity of $V_{12}$ can be allowed in [4] (see $(V)_{2}$ stated in §2). In keeping the application to some scattering problems under the AC Stark effect in mind, we mainly suppose that $V_{jk}$’s are given as

[TABLE]

where $\bar{V}_{jk}\in C^{2}(\boldsymbol{R}^{d};\boldsymbol{R})$ satisfying the decaying conditions

[TABLE]

and $c_{jk}\in C^{2}(\boldsymbol{T};\boldsymbol{R}^{d})$ (see §4 for details). By simple calculation, we have

[TABLE]

Hence it is obvious that $V_{jk}(t,y)$’s satisfy (1.16). In [4], the third condition in (1.16) with $(j,k)=(1,2)$ is replaced by

[TABLE]

This condition is stronger that the third one in (1.16). This causes that when we apply the results of [4] without modification to two-body scattering problems under the AC Stark effect, the short-range condition $\rho>1$ has to be assumed.

Now we state the main results of this paper.

Theorem 1.1**.**

Suppose $N=3$. Assume $V$ satisfies $(V)_{3}$. Put

[TABLE]

and

[TABLE]

for $\lambda\in\boldsymbol{R}$. Define $A(R)$ by (1.15). Then the following hold:

$(1)$* Let $\lambda_{0}\in[0,\omega)$, $\epsilon>0$ and $0<\delta_{0}<\omega/4$. Then there exists $R_{\epsilon}\geq 1$ and $0<\delta_{0,\epsilon}<\delta_{0}$ such that the following holds: Take $\delta$ such that $0<\delta<\delta_{0,\epsilon}$. If $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then for any $f_{\delta}\in C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ supported in $[-\delta,\delta]$,*

[TABLE]

holds with $A=A(R)$ for $R\geq R_{\epsilon}$ and some compact operator $C_{\lambda_{0},f_{\delta},\epsilon}$ on $\mathscr{K}$. On the other hand, if $\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then for any $f_{\delta}\in C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ supported in $[-\delta,\delta]$,

[TABLE]

holds. In particular, when $\lambda_{0}\not\in\varTheta$, by taking $\epsilon$ such that $\epsilon<d_{1}(\lambda_{0})$, and $\delta_{0}$ such that $2\delta_{0}\leq\epsilon$, the Mourre estimate

[TABLE]

can be obtained. Hence, for any $\hat{\delta}$ such that $0<\hat{\delta}<\delta$, $\sigma_{\mathrm{pp}}(K)\cap I_{\lambda_{0},\hat{\delta}}$ is finite, and the eigenvalues of $K$ in $I_{\lambda_{0},\hat{\delta}}$ are of finite multiplicity. Here we denote by $I_{\lambda,\delta^{\prime}}$ the open interval $(\lambda-\delta^{\prime},\lambda+\delta^{\prime})\subset\boldsymbol{R}$ centered around $\lambda\in\boldsymbol{R}$ with the radius $\delta^{\prime}>0$.

$(2)$* In addition, assume $\lambda_{0}\not\in\sigma_{\mathrm{pp}}(K)$. Take $\epsilon$ such that $2\epsilon<d_{1}(\lambda_{0})$, and $\delta_{0}$ such that $2\delta_{0}\leq\epsilon$ and $\delta_{0}\leq\widehat{d}_{0}(\lambda_{0})$, which implies $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$. Then there exists a small $\delta_{1,\epsilon}>0$ such that $\delta_{1,\epsilon}<\delta_{0,\epsilon}$ and*

[TABLE]

holds. Suppose $s>1/2$ and $0<\hat{\delta}<\delta_{1,\epsilon}$. Then

[TABLE]

holds, where $\overline{I_{\lambda,\delta^{\prime}}}=[\lambda-\delta^{\prime},\lambda+\delta^{\prime}]$. Moreover, $\langle A\rangle^{-s}(K-z)^{-1}\langle A\rangle^{-s}$ is a $\boldsymbol{B}(\mathscr{K})$-valued $\theta(s)$-Hölder continuous function on $z\in S_{\lambda_{0},\hat{\delta},\pm}$ with some $0<\theta(s)<1$, where

[TABLE]

And, there exist the norm limits

[TABLE]

in $\boldsymbol{B}(\mathscr{K})$ for any $\lambda\in\overline{I_{\lambda_{0},\hat{\delta}}}$. $\langle A\rangle^{-s}(K-(\lambda\pm i0))^{-1}\langle A\rangle^{-s}$ are also $\theta(s)$-Hölder continuous in $\lambda$.

Corollary 1.2**.**

Assume $V$ satisfies $(V)_{3}$. Then the following hold:

$(1)$* The eigenvalues of $K$ in $\boldsymbol{R}\setminus\varTheta$ can accumulate only at $\varTheta$. Moreover, $\widehat{\varTheta}$ is a countable closed set.*

$(2)$* Let $I$ be a compact interval in $\boldsymbol{R}\setminus\widehat{\varTheta}$. Suppose $1/2<s\leq 1$. Then*

[TABLE]

holds. Moreover, $\langle x\rangle^{-s}(K-z)^{-1}\langle x\rangle^{-s}$ is a $\boldsymbol{B}(\mathscr{K})$-valued $\theta(s)$-Hölder continuous function on $z\in S_{I,\pm}$, where

[TABLE]

And, there exist the norm limits

[TABLE]

in $\boldsymbol{B}(\mathscr{K})$ for $\lambda\in I$. $\langle x\rangle^{-s}(K-(\lambda\pm i0))^{-1}\langle x\rangle^{-s}$ are also $\theta(s)$-Hölder continuous in $\lambda$.

In order to obtain Corollary 1.2, we use the argument of Perry-Sigal-Simon [20], and the boundedness of

[TABLE]

which can be given by that $\langle D_{t}\rangle^{-1}(K-\lambda_{0}-i)^{-1}\langle p\rangle^{2}$ is bounded. By virtue of this, one can show that

[TABLE]

are also bounded. Then the limiting absorption principle

[TABLE]

may be expected as mentioned in [4], where $\mathcal{D}=\langle p\rangle+\langle D_{t}\rangle^{1/2}$ is equivalent to $\mathscr{D}^{1/2}=(\langle p\rangle^{4}+\langle D_{t}\rangle^{2})^{1/4}$ as weights, which was introduced in Kuwabara-Yajima [15] for the sake of obtaining a refined limiting absorption principle for $K$. But this has not been given by our analysis yet. It is caused by the unboundedness of

[TABLE]

Instead of the above limiting absorption principle, one can obtain

[TABLE]

from (1.22), as in [4]. As for general $N$-body Floquet Hamiltonians, a refined limiting absorption principle for $K$

[TABLE]

with $0\leq r<1/2<s\leq 1$ was obtained by Møller-Skibsted [18]. They used an extended Mourre theory due to Skibsted [24], and took a conjugate operator for $K$ in the extended Mourre theory as $\hat{A}_{0}$. However, we would like to stick to find a candidate of a conjugate operator for $K$ not in an extended but in the standard Mourre theory, because it seems much easier to obtain some useful propagation estimates for $K$ as will be seen in §4.

The plan of this paper is as follows: In §2, we will revisit the case where $N=2$. The construction of $A(R)$ in (1.15) is based on the arguments and results in §2. In §3, we will give the proof of Theorem 1.1, in particular, (1.18) and (1.19). In §4, we will make some remarks on our results.

Acknowledgement

The first author is partially supported by the Grant-in-Aid for Scientific Research (C) #17K05319 from JSPS.

2 The two-body case revisited

In this section, we revisit the proof of the Mourre estimate for $K$ with $N=2$. So we suppose $N=2$ throughout this section. We impose the following condition $(V)_{2}$ on $V$ under consideration:

$(V)_{2}$ $V_{12}(t,y)$ is a real-valued function on $\boldsymbol{R}\times\boldsymbol{R}^{d}$, is $T$-periodic in $t$, and is decomposed into the sum of $V_{12}^{\mathrm{sing}}(t,y)$ and $V_{12}^{\mathrm{reg}}(t,y)$, which are also $T$-periodic in $t$. If $d<3$, then $V_{12}^{\mathrm{sing}}=0$. If $d\geq 3$, then $V_{12}^{\mathrm{sing}}(t,\cdot)$ belongs to $C(\boldsymbol{R},L^{q_{0}}(\boldsymbol{R}^{d}))$ with some $q_{0}>d$, and $\mathrm{supp}\,V_{12}^{\mathrm{sing}}(t,\cdot)$’s are included in a common compact subset of $\boldsymbol{R}^{d}$. $(\partial_{t}V_{12}^{\mathrm{sing}})(t,\cdot)$ and $|(\nabla V_{12}^{\mathrm{sing}})(t,\cdot)|$ belong to $C(\boldsymbol{R},L^{q_{1}}(\boldsymbol{R}^{d}))$ with some $q_{1}>d/2$, where if $d=3$, then we define $q_{1}$ by $1/q_{1}=1/(2q_{0})+1/2$. On the other hand, $V_{12}^{\mathrm{reg}}(t,y)$ belongs to $C^{2}(\boldsymbol{R}\times\boldsymbol{R}^{d})$, and satisfies the decaying conditions

[TABLE]

with some $\rho>0$.

As for $V_{12}^{\mathrm{sing}}(t,y)$, we mainly suppose that it has a local singularity like $|y|^{-\gamma}$ with $\gamma>0$, as in [4] (see also e.g. Adachi-Kimura-Shimizu [3]). If $d\geq 3$, then the local singularity like $|y|^{-1+\epsilon}$ with $0<\epsilon<1$ can be permitted by $(V)_{2}$.

First we state some properties of $A^{a_{\mathrm{max}}}=(3\omega/2-D_{t})^{-1}\hat{A}_{0}$ for reference, although those of $A_{\lambda_{0},\delta}$ were given in [4]. One of the basic properties of $A^{a_{\mathrm{max}}}$ is that

[TABLE]

hold, where $K_{0}=K_{a_{\mathrm{min}}}=D_{t}+p^{2}/2$ is the free Floquet Hamiltonian. This yields the fact that

[TABLE]

are bounded. Under the condition $(V)_{2}$, $\langle K_{0}\rangle^{-1/2}i[V,A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ with $V=V_{(1,2)}$ is bounded. In fact, as for the regular part $V_{(1,2)}^{\mathrm{reg}}$ of $V_{(1,2)}$, it follows from

[TABLE]

that $i[V_{(1,2)}^{\mathrm{reg}},A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is bounded, where $\nabla=ip$. Here we used the fact that $\langle D_{t}\rangle^{-1/2}\langle p\rangle\langle K_{0}\rangle^{-1}$ is bounded, which can be shown in the same way as in the case of Stark Hamiltonians (see e.g. Simon [23]). Moreover, we see that $\langle K_{0}\rangle^{-1}i[V_{(1,2)}^{\mathrm{reg}},A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is compact, by virtue of the local compactness property of $K_{0}$. On the other hand, as for the singular part $V_{(1,2)}^{\mathrm{sing}}$ of $V_{(1,2)}$, by using the fact that for each $t\in\boldsymbol{R}$

[TABLE]

are bounded on $\mathscr{H}$, one can show firstly that $\langle K_{0}\rangle^{-1/2}i[V_{(1,2)}^{\mathrm{sing}},A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is bounded. Moreover, we see that $\langle K_{0}\rangle^{-1}i[V_{(1,2)}^{\mathrm{sing}},A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is compact. Next, by identifying $i[i[V_{(1,2)}^{\mathrm{sing}},A^{a_{\mathrm{max}}}],A^{a_{\mathrm{max}}}]$ with

[TABLE]

one can show that $\langle K_{0}\rangle^{-1}i[i[V_{(1,2)}^{\mathrm{sing}},A^{a_{\mathrm{max}}}],A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is also bounded. However, one cannot show generally that $i[i[V_{(1,2)}^{\mathrm{reg}},A^{a_{\mathrm{max}}}],A^{a_{\mathrm{max}}}]\langle K_{0}\rangle^{-1}$ is bounded, except in the case where $\rho\geq 1$. In fact, a simple calculation yields

[TABLE]

The first two terms of the right-hand side of this equality are $K_{0}$-bounded. But, in showing the $K_{0}$-boundedness of the third term of the right-hand side of this equality generally, the condition $\rho\geq 1$ is needed. In [4], a stronger condition (1.17) is assumed for the sake of avoiding this difficulty. In this paper, in order to avoid this difficulty, we will replace $A^{a_{\mathrm{max}}}$ by

[TABLE]

with $A_{a_{\mathrm{max}}}=A^{a_{\mathrm{max}}}+\tilde{A}_{\omega/4,a_{\mathrm{max}}}=A^{a_{\mathrm{max}}}$. Here we note that on $\mathrm{supp}\,j_{a_{\mathrm{max}},R}$, $|x|\leq r_{0}R$ holds. By virtue of this, one can show that

[TABLE]

are all bounded, and that $\langle K_{0}\rangle^{-1}i[V,\bar{A}_{a_{\mathrm{max}}}(R)]\langle K_{0}\rangle^{-1}$ is compact. Here we used $i[V,\bar{A}_{a_{\mathrm{max}}}(R)]=j_{a_{\mathrm{max}},R}i[V,A^{a_{\mathrm{max}}}]j_{a_{\mathrm{max}},R}$,

[TABLE]

and that $\langle D_{t}\rangle^{-1/2}\langle p\rangle\langle K_{0}\rangle^{-1}$ is bounded.

Next we state some properties of $\tilde{A}_{\omega/4,a_{\mathrm{min}}}=\mathrm{Re}\,\{(\omega/4+p^{2}/2)^{-1}p\cdot x\}$ for reference (see also [29]). One of the basic properties of $\tilde{A}_{\omega/4,a_{\mathrm{min}}}$ is that

[TABLE]

hold. Obviously these are bounded. Under the condition $(V)_{2}$, $i[V,\tilde{A}_{\omega/4,a_{\mathrm{min}}}]$ with $V=V_{(1,2)}$ is bounded. In fact, as for the regular part $V_{(1,2)}^{\mathrm{reg}}$ of $V_{(1,2)}$, it follows from

[TABLE]

with

[TABLE]

that $i[V_{(1,2)}^{\mathrm{reg}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}]$ is bounded. By virtue of the local compactness property of $K_{0}$, we also see that $\langle K_{0}\rangle^{-1}i[V_{(1,2)}^{\mathrm{reg}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}]\langle K_{0}\rangle^{-1}$ is compact. Similarly one can show that $i[i[V_{(1,2)}^{\mathrm{reg}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}],\tilde{A}_{\omega/4,a_{\mathrm{min}}}]$ is bounded. On the other hand, as for the singular part $V_{(1,2)}^{\mathrm{sing}}$ of $V_{(1,2)}$, by using the fact that

[TABLE]

are bounded on $\mathscr{H}$, one can show firstly that $i[V_{(1,2)}^{\mathrm{sing}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}]$ is bounded. We also see that $\langle K_{0}\rangle^{-1}i[V_{(1,2)}^{\mathrm{sing}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}]\langle K_{0}\rangle^{-1}$ is compact. One can also show that $i[i[V_{(1,2)}^{\mathrm{sing}},\tilde{A}_{\omega/4,a_{\mathrm{min}}}],\tilde{A}_{\omega/4,a_{\mathrm{min}}}]$ is also bounded, in the same way as in the case of $i[i[V_{(1,2)}^{\mathrm{sing}},A^{a_{\mathrm{max}}}],A^{a_{\mathrm{max}}}]$. However, in this paper, we will replace $\tilde{A}_{\omega/4,a_{\mathrm{min}}}$ by

[TABLE]

with $A_{a_{\mathrm{min}}}=A^{a_{\mathrm{min}}}+\tilde{A}_{\omega/4,a_{\mathrm{min}}}=\tilde{A}_{\omega/4,a_{\mathrm{min}}}$. Here we note that we do not have to deal with the local singularity of $V_{(1,2)}$ in the calculation of $i[K,\bar{A}_{a_{\mathrm{min}}}(R)]$ and $i[i[K,\bar{A}_{a_{\mathrm{min}}}(R)],\bar{A}_{a_{\mathrm{min}}}(R)]$ for large $R$, since on $\mathrm{supp}\,j_{a_{\mathrm{min}},R}$, $|x|\geq r_{1}R$ holds. By virtue of this, one can show that

[TABLE]

are all bounded, and

[TABLE]

as $R\to\infty$. Here we used $i[V,\bar{A}_{a_{\mathrm{min}}}(R)]=j_{a_{\mathrm{min}},R}i[V,\tilde{A}_{\omega/4,a_{\mathrm{max}}}]j_{a_{\mathrm{min}},R}$ and (2.3).

Now we will introduce

[TABLE]

as in (1.15). The following Nelson’s commutator theorem guarantees the self-adjointness of $A(R)$ (as for the proof, see e.g. Reed-Simon [21] and Gérard-Łaba [9]).

Theorem 2.1**.**

Let $\mathcal{K}$ be a Hilbert space. Suppose that $N_{0}\geq c>0$ is a self-adjoint operator on $\mathcal{K}$ and $A$ is a symmetric operator on $\mathcal{K}$ such that $D(N_{0})\subset D(A)$ and there exists a constant $C>0$ such that

[TABLE]

hold. Then $A$ is essentially self-adjoint on $D(N_{0})$. Denoting by $\bar{A}$ the unique self-adjoint extension of $A$, if $u\in D(\bar{A})$, then $(1+i\epsilon N_{0})^{-1}u$ converges to $u$ in the graph topology of $D(\bar{A})$ as $\epsilon\to 0$.

Applying Theorem 2.1 with $\mathcal{K}=\mathscr{K}$, $N_{0}=\langle D_{t}\rangle+p^{2}/2+x^{2}/2$ and $A=A(R)$, we see that $A(R)$ has its unique self-adjoint extension, which is also denoted by $A(R)$. Here we used (2.3) and

[TABLE]

with

[TABLE]

By virtue of the properties of $\{\bar{A}_{a}\}_{a\in\mathscr{A}}$, we see that

[TABLE]

are all bounded, and

[TABLE]

with some compact operator $C_{R}$ on $\mathscr{K}$.

In the usual proof of the Mourre estimate for $K$, one of the points to be checked is that the condition

[TABLE]

is satisfied by a conjugate operator $A(R)$ (see e.g. Mourre [16]). However, it seems not easy to verify directly that $A(R)$ defined by (2.5) satisfies (2.6). In order to overcome this difficulty, we need the following proposition (see e.g. Lemma 3.2.2 and Proposition 3.2.3 of [9]; see also Amrein-Boutet de Monvel-Georgescu [5]):

Proposition 2.2**.**

Let $\mathcal{K}$ be a Hilbert space. Suppose that $K$, $K_{0}$ and $N_{0}$ are self-adjoint operators on $\mathcal{K}$ such that $N_{0}\geq c>0$, $D(K)=D(K_{0})$ as Banach spaces, and for $z\in\boldsymbol{C}\setminus\sigma(K)$, $(K-z)^{-1}$ preserves $D(N_{0})$. Let $A$ be a symmetric operator on $\mathcal{K}$. Suppose that $K_{0}$ and $A$ satisfy $D(N_{0})\subset D(K_{0})$, $D(N_{0})\subset D(A)$,

[TABLE]

Denote the unique self-adjoint extension of $A$ also by $A$. Assume moreover that

[TABLE]

holds. Then the following hold:

$(1)$* $D(N_{0})$ is dense in $D(K)\cap D(A)$ with the norm $\|Ku\|+\|Au\|+\|u\|$.*

$(2)$* The commutator $i[K,A]$, defined as a quadratic form on $D(K)\cap D(A)$, is the unique extension of the quadratic form $i[K,A]$ on $D(N_{0})$.*

$(3)$* $K\in C^{1}(A)$, that is, for some $z\in\boldsymbol{C}\setminus\sigma(K)$, the map*

[TABLE]

is $C^{1}$ in the strong topology of $\mathscr{B}(\mathcal{K})$, which is the algebra of bounded linear operators in $\mathcal{K}$.

$(4)$* $D(K)\cap D(A)$ is a core for $K$, and the quadratic form $i[K,A]$ on $D(K)\cap D(A)$ extends uniquely to a bounded operator from $D(K)$ to its dual space $D(K)^{*}$, which is denoted also by $i[K,A]$.*

$(5)$* The virial relation holds:  For any $\lambda\in\boldsymbol{R}$,*

[TABLE]

holds. Here $E_{K}(S)$ stands for the spectral projection for $K$ onto $S\subset\boldsymbol{R}$.

$(6)$* For $z\in\boldsymbol{C}\setminus\sigma(K)$, $i[(K-z)^{-1},A]=-(K-z)^{-1}i[K,A](K-z)^{-1}$ holds.*

$(7)$* For $z\in\boldsymbol{C}\setminus\sigma(K)$, $(K-z)^{-1}$ preserves $D(A)$.*

By virtue of Proposition 2.2 with $\mathcal{K}=\mathscr{K}$, $N_{0}=\langle D_{t}\rangle+p^{2}/2+x^{2}/2$ and $A=A(R)$, one can show the following theorem and corollary without using (2.6):

Theorem 2.3**.**

Suppose $N=2$. Assume $V$ satisfies $(V)_{2}$. Put

[TABLE]

and

[TABLE]

for $\lambda\in\boldsymbol{R}$. Define $A(R)$ by (2.5). Then the following hold:

$(1)$* Let $\lambda_{0}\in[0,\omega)$, $\epsilon>0$ and $0<\delta_{0}<\omega/4$. Then there exists $R_{\epsilon}\geq 1$ such that the following holds: Take $\delta$ such that $0<\delta<\delta_{0}$. If $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then for any $f_{\delta}\in C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ supported in $[-\delta,\delta]$,*

[TABLE]

holds with $A=A(R)$ for $R\geq R_{\epsilon}$ and some compact operator $C_{\lambda_{0},f_{\delta},\epsilon}$ on $\mathscr{K}$. On the other hand, if $\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then for any $f_{\delta}\in C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ supported in $[-\delta,\delta]$,

[TABLE]

holds with $A=A(R)$ for $R\geq R_{\epsilon}$ and some compact operator $C_{\lambda_{0},f_{\delta},\epsilon}$ on $\mathscr{K}$. In particular, when $\lambda_{0}\not\in\varTheta$, by taking $\epsilon$ such that $\epsilon<d_{1}(\lambda_{0})$, and $\delta_{0}$ such that $2\delta_{0}\leq\epsilon$, the Mourre estimate

[TABLE]

can be obtained. Hence, for any $\hat{\delta}$ such that $0<\hat{\delta}<\delta_{0}$, $\sigma_{\mathrm{pp}}(K)\cap I_{\lambda_{0},\hat{\delta}}$ is finite, and that the eigenvalues of $K$ in $I_{\lambda_{0},\hat{\delta}}$ are of finite multiplicity.

$(2)$* In addition, assume $\lambda_{0}\not\in\sigma_{\mathrm{pp}}(K)$. Take $\epsilon$ such that $2\epsilon<d_{1}(\lambda_{0})$, and $\delta_{0}$ such that $2\delta_{0}\leq\epsilon$ and $\delta_{0}\leq\widehat{d}_{0}(\lambda_{0})$, which implies $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$. Then there exists a small $\delta_{1,\epsilon}>0$ such that $\delta_{1,\epsilon}<\delta_{0}$ and*

[TABLE]

holds. Suppose $s>1/2$ and $0<\hat{\delta}<\delta_{1,\epsilon}$. Then

[TABLE]

holds. Moreover, $\langle A\rangle^{-s}(K-z)^{-1}\langle A\rangle^{-s}$ is a $\boldsymbol{B}(\mathscr{K})$-valued $\theta(s)$-Hölder continuous function on $z\in S_{\lambda_{0},\hat{\delta},\pm}$ with some $0<\theta(s)<1$. And, there exist the norm limits

[TABLE]

in $\boldsymbol{B}(\mathscr{K})$ for any $\lambda\in\overline{I_{\lambda_{0},\hat{\delta}}}$. $\langle A\rangle^{-s}(K-(\lambda\pm i0))^{-1}\langle A\rangle^{-s}$ are also $\theta(s)$-Hölder continuous in $\lambda$.

Corollary 2.4**.**

Assume $V$ satisfies $(V)_{2}$. Then the following hold:

$(1)$* The eigenvalues of $K$ in $\boldsymbol{R}\setminus\varTheta$ can accumulate only at $\varTheta$. Moreover, $\widehat{\varTheta}$ is a countable closed set.*

$(2)$* Let $I$ be a compact interval in $\boldsymbol{R}\setminus\widehat{\varTheta}$. Suppose $1/2<s\leq 1$. Then*

[TABLE]

holds. Moreover, $\langle x\rangle^{-s}(K-z)^{-1}\langle x\rangle^{-s}$ is a $\boldsymbol{B}(\mathscr{K})$-valued $\theta(s)$-Hölder continuous function on $z\in S_{I,\pm}$. And, there exist the norm limits

[TABLE]

in $\boldsymbol{B}(\mathscr{K})$ for $\lambda\in I$. $\langle x\rangle^{-s}(K-(\lambda\pm i0))^{-1}\langle x\rangle^{-s}$ are also $\theta(s)$-Hölder continuous in $\lambda$.

We will sketch the proof of the estimates (2.7) and (2.8) only. (2.7) yields the Mourre estimate (2.9). Thus Theorem 2.3 and Corollary 2.4 can be shown by the standard argument in the Mourre theory. In particular, for the proof of Corollary 2.4, we use the argument due to Perry-Sigal-Simon [20], and the boundedness of

[TABLE]

which follows from that $\langle D_{t}\rangle^{-1}(K-\lambda_{0}-i)^{-1}\langle p\rangle^{2}$ is bounded.

Proof of (2.7) and (2.8).

Let $\lambda_{0}\in[0,\omega)$, $\epsilon>0$ and $0<\delta_{0}<\omega/4$. Denote by $f_{\delta_{0}}$ any function in $C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ such that $\mathrm{supp}\,f_{\delta_{0}}\subset[-\delta_{0},\delta_{0}]$. For the sake of simplicity, we write $f_{\delta_{0}}(K-\lambda_{0})$ as $f_{\delta_{0},\lambda_{0}}(K)$. By the assumption $(V)_{2}$, we see that

[TABLE]

holds with some compact operator $C_{R,1}$ on $\mathscr{K}$. By (2.2) and the IMS localization formula

[TABLE]

we have

[TABLE]

Here we note that $(3\omega/2-D_{t})^{-1}$ does commute with $j_{a,R}$’s. Since $f_{\delta_{0},\lambda_{0}}(K)-f_{\delta_{0},\lambda_{0}}(K_{0})$ is compact by the assumption $(V)_{2}$, we obtain

[TABLE]

with some compact operator $C_{R,2}$ on $\mathscr{K}$. Here we note that $\langle D_{t}\rangle^{-1/2}\langle p\rangle\langle K_{0}\rangle^{-1}$ and $\langle D_{t}\rangle^{-1}\langle p\rangle^{2}\langle K_{0}\rangle^{-1}$ are bounded as mentioned above, and

[TABLE]

with $H_{0}=p^{2}/2$. $f_{\delta_{0},\lambda_{0}}(K_{0})i[K_{0},A_{a_{\mathrm{max}}}]f_{\delta_{0},\lambda_{0}}(K_{0})$ can be decomposed into the direct integral

[TABLE]

with $f_{\delta_{0},\lambda_{0}-n\omega}(H_{0})=f_{\delta_{0}}(n\omega+H_{0}-\lambda_{0})$. Here we note that when $n\geq 2$, $f_{\delta_{0},\lambda_{0}-n\omega}(H_{0})=0$ holds since $\lambda_{0}-n\omega+\delta_{0}\leq\lambda_{0}-2\omega+\delta_{0}<0$. When $n\leq-1$,

[TABLE]

holds. We will consider the case where $n=0$. If $\lambda_{0}-\delta_{0}<0$, that is, $\lambda_{0}<\delta_{0}$, then

[TABLE]

while, if $\lambda_{0}\geq\delta_{0}$, then

[TABLE]

We will consider the case where $n=1$. If $\lambda_{0}-\omega+\delta_{0}>0$, that is, $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

On the other hand, if $\lambda_{0}-\omega+\delta_{0}\leq 0$, that is, $\lambda_{0}\leq\omega-\delta_{0}$, then $f_{\delta_{0},\lambda_{0}-\omega}(H_{0})=0$. By combining these, we see that if $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then

[TABLE]

while, if $0\leq\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

We next consider

[TABLE]

Noting $[f_{\delta_{0},\lambda_{0}}(K_{0}),j_{a_{\mathrm{min}},R}]\langle D_{t}\rangle^{-1/2}=O(R^{-1})$,

[TABLE]

can be shown easily by (2.15). Then we would like to use the estimate

[TABLE]

since $\lambda_{0}+\delta_{0}<5\omega/4$. This estimate yields

[TABLE]

It follows from these and (2.14) that if $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then

[TABLE]

with some compact operator $C_{R,3}$ on $\mathscr{K}$. Now we will take $\delta$ such that $0<\delta<\delta_{0}$. By sandwiching (2.16) in two $f_{\delta,\lambda_{0}}(K)$’s, one can obtain

[TABLE]

with some compact operator $C_{R,4}$ on $\mathscr{K}$, because of the arbitrariness of $f_{\delta_{0},\lambda_{0}}$. (2.17) yields (2.7), by taking $R\geq 1$ sufficiently large. Here we note $d_{1}(\lambda_{0})=\lambda_{0}$. Similarly, if $0\leq\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$,

[TABLE]

which yields (2.8) immediately. ∎

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1. As in §2, we will show the estimates (1.18) and (1.19) only. (1.18) yields the Mourre estimate (1.20). Thus Theorem 1.1 and Corollary 1.2 can be shown by the standard argument in the Mourre theory.

Proof of (1.18) and (1.19).

Let $\lambda_{0}\in[0,\omega)$, $\epsilon>0$ and $0<\delta_{0}<\omega/4$. Denote by $f_{\delta_{0}}$ any function in $C_{0}^{\infty}(\boldsymbol{R};\boldsymbol{R})$ such that $\mathrm{supp}\,f_{\delta_{0}}\subset[-\delta_{0},\delta_{0}]$. As in §2, we write $f_{\delta_{0}}(K-\lambda_{0})$ as $f_{\delta_{0},\lambda_{0}}(K)$. First of all, we note that the estimate

[TABLE]

holds with some compact operator $C_{R,1}$ on $\mathscr{K}$. Here we used the compactness of $f_{\delta_{0},\lambda_{0}}(K)j_{a_{\mathrm{max}},R}i[K,A_{a_{\mathrm{max}}}]j_{a_{\mathrm{max}},R}f_{\delta_{0},\lambda_{0}}(K)$. Put

[TABLE]

for $a\in\mathscr{A}\setminus\{a_{\mathrm{max}}\}$. We first estimate

[TABLE]

Here we note

[TABLE]

where $K_{0}=K_{a_{\mathrm{min}}}=D_{t}+p^{2}/2$ and $H_{0}=T_{a_{\mathrm{min}}}=p^{2}/2$. In fact, it is easy to see

[TABLE]

Now we will treat $i[K_{0},A_{a_{\mathrm{min}}}]$ with a partition of unity with respect to $H_{0}$. Let $\eta$ be a function in $C^{\infty}(\boldsymbol{R};\boldsymbol{R})$ such that $\mathrm{supp}\,\eta\subset[1,\infty)$, $0\leq\eta(\tau)\leq 1$, $\eta(\tau)=1$ on $[2,\infty)$, and

[TABLE]

where $\bar{\eta}(\tau)=1-\eta(\tau)$. For the sake of brevity, we put $\eta_{s}(\tau)=\eta(\tau/s)$ and $\bar{\eta}_{s}(\tau)=\bar{\eta}(\tau/s)$ for $s>0$. Thus $\eta_{s}(\tau)$ and $\bar{\eta}_{s}(\tau)$ satisfy $0\leq\eta_{s}(\tau)\leq 1$, $0\leq\bar{\eta}_{s}(\tau)\leq 1$,

[TABLE]

By the partition of unity $\{\eta_{\omega}(H_{0}),\,\bar{\eta}_{\omega}(H_{0})\}$, $(\omega/4+H_{0})^{-1}H_{0}$ can be decomposed into the sum

[TABLE]

Using the estimate

[TABLE]

we have

[TABLE]

Noting

[TABLE]

because of the boundedness of $\langle p\rangle\bar{\eta}_{\omega}(H_{0})$, we see that

[TABLE]

holds. We will use the direct integral

[TABLE]

as in §2. When $n\leq-1$,

[TABLE]

holds. Here we note that when $n\leq-3$, $\bar{\eta}_{\omega}(H_{0})^{2}f_{\delta_{0},\lambda_{0}-n\omega}(H_{0})^{2}=0$ holds. We will consider the case where $n=0$. In the same way as in §2, we see that if $\lambda_{0}<\delta_{0}$, then

[TABLE]

while, if $\lambda_{0}\geq\delta_{0}$, then

[TABLE]

We will also consider the case where $n=1$. If $\lambda_{0}\leq\omega-\delta_{0}$, then $f_{\delta_{0},\lambda_{0}-\omega}(H_{0})=0$. On the other hand, if $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

By combining these and using (3.3), we see that if $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then

[TABLE]

This and (3.2) yield

[TABLE]

Here we used $5\omega/4>\omega>\lambda_{0}-\delta_{0}$ and $\widehat{d}_{1,a_{\mathrm{min}}}(\lambda_{0})=\lambda_{0}$. Similarly, we see that if $0\leq\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

We next estimate $f_{\delta_{0},\lambda_{0}}(K)j_{a,R}i[K,A_{a}]j_{a,R}f_{\delta_{0},\lambda_{0}}(K)$ with $a\in\mathscr{A}\setminus\{a_{\mathrm{max}},a_{\mathrm{min}}\}$. We first note

[TABLE]

and $|x^{a}|\leq r_{0}R$ holds on $\mathrm{supp}\,j_{a,R}$. Here $T_{a}=(p_{a})^{2}/2$ and $K_{0}^{a}=D_{t}+(p^{a})^{2}/2$. In fact, it is easy to see

[TABLE]

Now we will treat $i[K_{a},A_{a}]$ with a partition of unity with respect to $-D_{t}$. For each $L\in\boldsymbol{N}$, we introduce a partition of unity $\{\eta_{L\omega}(-D_{t}),\,\bar{\eta}_{L\omega}(-D_{t})\}$. Then $i[K_{0},A_{a}]$ can be decomposed into the sum

[TABLE]

On the other hand, as for $i[V^{a},A_{a}]=i[V^{a},A^{a}]$, the relation

[TABLE]

as $L\to\infty$ can be obtained generally, in virtue of that $i[i[V^{a},A^{a}],D_{t}]$ is bounded by the assumption $(V)_{3}$. Except in the case where $V^{a}$ is time-independent, we have to deal with the above error term $O(L^{-1})$. This is one of the technical reasons why we also need some regularity of second derivatives of $V^{a}$, as mentioned in §1. Since

[TABLE]

and $(\omega/4+T_{a})^{-1}T_{a}\geq 0$, there exist a large $L_{0}\in\boldsymbol{N}$ such that if $L\geq L_{0}$, then

[TABLE]

holds. Here we used that $K_{0}^{a}$ and $(3\omega/2-D_{t})i[V^{a},A^{a}]j_{a,R}$ are $K$-bounded, and

[TABLE]

Since on $\mathrm{supp}\,j_{a,R}$, $|x^{a}|\leq r_{0}R$ holds, $f_{\delta_{0},\lambda_{0}}(K)j_{a,R}i[K^{a},A^{a}]j_{a,R}f_{\delta_{0},\lambda_{0}}(K)$ can be recognized as

[TABLE]

where $A_{R}^{a}$ is the conjugate operator for $K^{a}$ defined as in §2. Hence we have

[TABLE]

as $L\to\infty$, where

[TABLE]

Now we will treat

[TABLE]

Since $\langle p\rangle\bar{\eta}_{L\omega}(-D_{t})(K_{0}+i)^{-1}=O(L^{1/2})$, we see that

[TABLE]

holds by the assumption $(V)_{3}$. By using (3.8), we have

[TABLE]

Now, by following the argument of Froese-Herbst [8], we will show that there exists a small $\delta_{0,\epsilon}^{a}$ such that $0<\delta_{0,\epsilon}^{a}<\delta_{0}$, and

[TABLE]

for any $0<\delta\leq\delta_{1,\epsilon}^{a}$. The left-hand side of (3.10) can be decomposed into the direct integral

[TABLE]

By using the decomposition

[TABLE]

we have

[TABLE]

For a while we will treat

[TABLE]

for $\lambda_{a}\in J_{n}$ with some $n\in\boldsymbol{N}_{0}=\{0\}\cup\boldsymbol{N}$. It follows from the results in §2 that there exists a large $R_{\epsilon}^{a}\geq 1$ and a small $\delta_{1,\epsilon}^{a}$ such that for any $R\geq R_{\epsilon}^{a}$ and $0<\delta\leq\delta_{1,\epsilon}^{a}$, the following holds: When $n\geq 1$, $J_{n}$ is decomposed as $J_{n}=J_{n,1}\sqcup J_{n,2}$ with

[TABLE]

On the other hand, $J_{0}$ is decomposed as $J_{0}=J_{0,1}\sqcup J_{0,2}$ with

[TABLE]

Here we note that if $\lambda_{0}<\delta_{0}$, then $J_{0,1}=\emptyset$. If $\lambda_{a}\in J_{n,1}$, then

[TABLE]

holds because of $\lambda_{0}-\lambda_{a}\in[-n\omega,-(n-1)\omega-\delta_{0}]$; while, if $\lambda_{a}\in J_{n,2}$, then

[TABLE]

holds. Here we emphasize that $R_{\epsilon}^{a}$ and $\delta_{1,\epsilon}^{a}$ can be taken uniformly in $\lambda_{a}\in[0,\infty)$, by using the $\omega$-periodicity of $\sigma(K^{a})$ and following the argument of [8]. If $\lambda_{a}\in J_{n,1}$ with $n\geq 1$, then

[TABLE]

which yields

[TABLE]

because $\widehat{d}_{1,a}(\lambda_{0}-\lambda_{a})\geq 0$; while, if $\lambda_{0}\geq\delta_{0}$ and $\lambda_{a}\in J_{0,1}$, then

[TABLE]

because $\widehat{d}_{1,a}(\lambda_{0}-\lambda_{a})\geq\widehat{d}_{1,a}(\lambda_{0})-\lambda_{a}$ and $\omega/4+\lambda_{a}<3\omega/2$. On the other hand, if $\lambda_{a}\in J_{n,2}$ with $n\geq 1$, then

[TABLE]

which yields

[TABLE]

while, if $\lambda_{a}\in J_{0,2}$, then

[TABLE]

Finally we see that if $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$, then

[TABLE]

because of $\lambda_{0}\geq\widehat{d}_{1,a}(\lambda_{0})$; while, if $\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

Now we will consider the case where $\delta_{0}\leq\lambda_{0}\leq\omega-\delta_{0}$ for a while. By (3.7), (3.8), (3.9) and (3.13), the estimate

[TABLE]

can be obtained. Here we used

[TABLE]

By sandwiching (3.15) in two $f_{\delta,\lambda_{0}}(K)$ with $0<\delta<\delta_{1,\epsilon}^{a}$, one can obtain

[TABLE]

because of the arbitrariness of $f_{\delta_{1,\epsilon}^{a},\lambda_{0}}$. Then one can take $L_{\epsilon}\in\boldsymbol{N}$ so large that $L_{\epsilon}\geq L_{0}$ and

[TABLE]

By (3.1), (3.4), (3.17), we finally obtain the estimate

[TABLE]

with some compact operator $C_{R}$ on $\mathscr{K}$ for $0<\delta<\delta_{1,\epsilon}$, where

[TABLE]

Here we used

[TABLE]

(3.18) yields (1.18). Similarly one can show that if $\lambda_{0}<\delta_{0}$ or $\lambda_{0}>\omega-\delta_{0}$, then

[TABLE]

holds. (3.19) yields (1.19). ∎

Remark 3.1**.**

In the above proof, we have used (3.3) and (3.8). It has been well known since the work of Froese-Herbst [8] that the estimates like

[TABLE]

are very useful for the inductive argument in the proof of the Mourre estimates for Hamiltonians which govern many body quantum systems. However, in our case, we do not know whether (3.20) holds or not, as mentioned also in [18]. In our analysis, we need cut-offs like $\bar{\eta}_{\omega}(H_{0})$ or $\bar{\eta}_{L\omega}(-D_{t})$.

4 Concluding remarks

Let $N\geq 2$, and consider a system of $N$ particles moving in a given $T$-periodic electric field $\mathscr{E}(t)\in C^{0}(\boldsymbol{R};\boldsymbol{R}^{d})$. The total Hamiltonian $\hat{H}(t)$ in the center-of-mass frame is given as

[TABLE]

on $L^{2}(X)$, where

[TABLE]

is $T$-periodic, $q_{j}$ is the charge of the $j$-th particle, and $\bar{V}_{jk}$’s are time-independent pair potentials. $q_{j}/m_{j}$ is called the specific charge of the $j$-th particle. Suppose that there exists a pair $(j,k)$ such that $q_{j}/m_{j}\not=q_{k}/m_{k}$. Under this assumption, if $\mathscr{E}(t)\not=0$, then $E(t)\not=0$. Denote by $\hat{U}(t,s)$ the propagator generated by $\hat{H}(t)$, and put

[TABLE]

As in Møller [17] and Adachi [1], define $X$-valued $T$-periodic functions $b_{0}(t)$, $b(t)$ and $c(t)$ on $\boldsymbol{R}$ by

[TABLE]

and introduce the time-dependent Hamiltonian

[TABLE]

on $L^{2}(X)$. By introducing $\boldsymbol{R}^{d}$-valued $T$-periodic functions $\bar{\mathscr{E}}_{0}(t)$, $\bar{\mathscr{E}}(t)$ and $\bar{\bar{\mathscr{E}}}(t)$ as

[TABLE]

$b(t)$, $c(t)$ and $\bar{V}(x+c(t))$ can be represented as

[TABLE]

with $c_{jk}(t)=(q_{j}/m_{j}-q_{k}/m_{k})\bar{\bar{\mathscr{E}}}(t)$. Suppose $V_{jk}$’s belong to $C^{2}(\boldsymbol{R}^{d};\boldsymbol{R})$, and satisfy the decaying conditions

[TABLE]

with some $\rho>0$, and put $V_{jk}(t,y)=\bar{V}_{jk}(y+c_{jk}(t))$. Then $V_{jk}(t,y)$’s satisfy (1.16).

If $E_{\mathrm{m}}=0$, then $H_{0}$ is called the free $N$-body Schrödinger operator; while, if $E_{\mathrm{m}}\not=0$, then $H_{0}$ is called the free $N$-body Stark Hamiltonian. Denote by $U(t,s)$ the unitary propagator generated by $H(t)$. As is well-known, the following Avron-Herbst formula holds:

[TABLE]

with

[TABLE]

When $E_{\mathrm{m}}\not=0$, in [1] and [2], the author already obtained the result of the asymptotic completeness for the system under consideration, both in the short-range and the long-range cases, by introducing the Floquet Hamiltonian $K$ associated with $\hat{H}(t)$.

[TABLE]

can be taken as a conjugate operator for $K$ in the standard Mourre theory. Here we emphasize that in the case where $N=2$, in [17], Møller proposed this operator as a conjugate operator for $K$ before [1]. On the other hand, when $E_{\mathrm{m}}=0$, any candidates of a conjugate operator for $K$ in the standard Mourre theory have not been found up until now, except in the case where $N=2$. $\hat{H}(t)$ with $E_{\mathrm{m}}=0$ is called an $N$-body AC Stark Hamiltonian. As mentioned in §1, in the case where $N=2$, Yokoyama [29], and Adachi-Kiyose [4] proposed conjugate operators for $K$. Unfortunately, these operators seem not have any natural extension to $N$-body systems. Møller-Skibsted [18] used $\hat{A}_{0}$ as a conjugate operator for $K$ in an extended Mourre theory, in order to avoid this difficulty. Our construction of $A(R)$ in (1.15) seems the first attempt to give a conjugate operator for $K$ in the standard Mourre theory when $N\geq 3$.

As for the asymptotic completeness for $\hat{H}(t)$ with $N=2$, Yajima [26] proved it in the short-range case via the Howland-Yajima method, and Kitada-Yajima [13] proved it in the long-range case via the Enss method. On the other hand, for $\hat{H}(t)$ with $N=3$, Korotyaev [14] and Nakamura [19] gave some partial results on it in the short-range case via the Howland-Yajima and the Faddeev methods. As is well-known, the limiting absorption principle (1.23) yields the local $K$-smoothness of $\langle x\rangle^{-s}$ with $s>1/2$

[TABLE]

for $\lambda_{0}\not\in\widehat{\varTheta}$. (4.3) was already obtained by Møller-Skibsted even if $N\geq 3$. However, (4.3) is not enough for the proof of the asymptotic completeness in the case where $N\geq 3$, unlike in the case where $N=2$. We expect that the Mourre estimate (1.20) will be useful for the proof of the asymptotic completeness in the case where $N=3$. In fact, for $\lambda_{0}\not\in\widehat{\varTheta}$, the so-called minimal velocity estimate

[TABLE]

with some $c_{0}>0$ may be yielded by

[TABLE]

with some $0<c_{1}<c_{2}$. Here $F(x\in\Omega)$ denotes the characteristic function of the set of $\Omega$. The minimal velocity estimate is one of the most important propagation estimates for $N$-body Schrödinger operators, as is well-known (see e.g. Graf [10]). These propagation estimates can be proved in the same way as in Sigal-Soffer [22], by virtue of the Mourre estimate (1.20) or (1.21). The Mourre estimate for a general $N$-body Floquet Hamiltonian $K$ may be also obtained by our construction of a conjugate operator for $K$. We would like to study the problem of the asymptotic completeness for $\hat{H}(t)$ with $N\geq 2$ by using some useful propagation estimates like (4.4) in future research.