# Comment on “Traversable wormhole dynamics on a quantum processor”

Bryce Kobrin  Department of Physics, University of California, Berkeley, CA
94720, USA  Thomas Schuster  Department of Physics, University of California,
Berkeley, CA 94720, USA  Norman Y. Yao  Department of Physics, University of
California, Berkeley, CA 94720, USA  Department of Physics, Harvard
University, MA 02138, USA

###### Abstract

A recent article [Nature 612, 51–55 (2022)] [1] claims to observe traversable
wormhole dynamics in an experiment. This claim is based upon performing a
teleportation protocol using a Hamiltonian that consists of seven Majorana
fermions with five fully-commuting terms. The Hamiltonian is generated via a
machine-learning procedure designed to replicate the teleportation behavior of
the Sachdev-Ye-Kitaev (SYK) model. The authors claim that the learned
Hamiltonian reproduces gravitational dynamics of the SYK model and
demonstrates gravitational teleportation through an emergent wormhole. We
find: (i) in contrast to these claims, the learned Hamiltonian does not
exhibit thermalization; (ii) the teleportation signal only resembles the SYK
model for operators that were used in the machine-learning training; (iii) the
observed perfect size winding is in fact a generic feature of small-size,
fully-commuting models, and does not appear to persist in larger-size fully-
commuting models or in non-commuting models at equivalent system sizes.

The holographic principle posits that certain quantum mechanical Hamiltonians
are dual to quantum theories of gravity [2, 3, 4]. Recently, there has been
tremendous interest towards _experimentally_ realizing such quantum mechanical
models on a quantum processor. To this end, an important development was the
discovery of a quantum teleportation protocol that is related to traversable
wormholes (Fig. 2a) [5, 6, 8, 7, 10, 9]. Specifically, when the teleportation
protocol is implemented using a Hamiltonian that is holographically dual to
gravity, successful teleportation is described from the dual perspective as a
particle traveling through a traversable wormhole.

A recent article [1] claims to observe traversable wormhole dynamics in an
experimental setting. The most direct way to observe traversable wormhole
dynamics would be to experimentally implement the SYK model, which is dual to
gravity [11, 12]. However, it is extremely challenging to experimentally
implement even a small-size version of the SYK model. To this end, [1] uses a
machine-learning procedure to construct a sparse Hamiltonian that aims to
preserve gravitational physics. More specifically, the machine-learning
procedure is based upon reproducing the teleportation behavior of the SYK
model (at system size $N=10$) with only a small number of Hamiltonian terms.
The result is the following Hamiltonian, henceforth “Model 1”,

$\displaystyle H=$
$\displaystyle-0.36\psi^{1}\psi^{2}\psi^{4}\psi^{5}+0.19\psi^{1}\psi^{3}\psi^{4}\psi^{7}-0.71\psi^{1}\psi^{3}\psi^{5}\psi^{6}$
$\displaystyle+0.22\psi^{2}\psi^{3}\psi^{4}\psi^{6}+0.49\psi^{2}\psi^{3}\psi^{5}\psi^{7}.$
(1)

Here, $\psi^{i}$ are Majorana fermions satisfying
$\\{\psi^{i},\psi^{j}\\}=\delta_{ij}$.

The authors claim that Model 1 “is consistent with gravitational dynamics of
the dense SYK Hamiltonian” and demonstrates “gravitational teleportation…by
means of an emergent wormhole”. They analyze five key properties of
traversable wormhole physics:

  1. (i)

scrambling and thermalization dynamics

  2. (ii)

a teleportation signal that is consistent with a negative energy shockwave

  3. (iii)

perfect size winding

  4. (iv)

causal time-ordering of teleported signals

  5. (v)

a Shapiro time delay

These claims are surprising given that:

  * Model 1 is fully-commuting. Each of the five Hamiltonian terms commutes with every other term.

which is alluded to in the Supplemental Material of [1]. This property is
distinct from the SYK model, and fully-commuting models are known to exhibit
markedly different dynamics from non-commuting models.

Given the interest in realizing non-trivial models of quantum gravity in
experiment, it seems worthwhile to investigate the extent to which the model
and strategy in [1] indeed capture the stated features of gravitational
physics. Our central findings are:

  * •

In contrast to the claims of [1], Model 1 does not thermalize. It exhibits
strong oscillations in the correlation functions that characterize scrambling
and thermalization (Fig. 1). The observation of thermalization in [1] is an
artifact of averaging over these oscillations.

  * •

In order to generate Model 1, the machine-learning procedure in [1] trains on
teleportation involving two specific operators, $\psi^{1}$ and $\psi^{2}$. The
authors characterize the teleportation signal [properties (ii, iv, v)] and
size winding [property (iii)] only for those operators that were trained on.
We find that the teleportation signal only resembles that of the SYK model for
the specific operators that were trained on, and not for general operators
that were not involved in the training (Fig. 2).

  * •

The observed perfect size winding in Model 1, is in fact, a widespread
property of fully-commuting Hamiltonians at small system sizes (Fig. 3).
Putting random numerical coefficients in front of the terms in Eq. (1) or
taking random commuting terms, also produces perfect size winding. In the
cases that we have examined, perfect size winding does not persist to larger
system sizes or to non-commuting models at equivalent system sizes.

Figure 1: Lack of thermalization in Model 1. (a) Two-point correlation
functions averaged over Majorana operators, $G_{\textrm{avg}}(t)$, for Model 1
(green; replicating Fig. 3b of [1]) and several disorder realizations of the
$N=10$ SYK model (orange). As observed in [1], the average correlation
function displays similar behavior between the two models. For both models,
$\beta=4$, and, for the SYK model, the couplings are drawn from a normal
distribution with mean zero and variance $6J^{2}/N^{3}$, where $J=1.125$. (b)
In Model 1, the _individual_ two-point correlation functions, $G_{i}(t)$,
display large oscillations. (c) In the SYK model (taking a single disorder
instance), the individual correlation functions all exhibit decay. This
behavior is independent of the disorder realization. (d-f) Analogous results
for the average and individual four-point correlation functions,
$F_{\textrm{avg}}(t)$ and $F_{i}(t)$, in Model 1 and the $N=10$ SYK model.
Again, the agreement between the two models holds only for the averaged
correlation functions (d), and not for the individual correlation functions
(e-f).

We emphasize an inherent tension between the first and third observations:
Small-size, fully-commuting Hamiltonians do not thermalize but generally
exhibit perfect size winding, while the opposite is true for larger or non-
commuting systems. None of the systems considered in [1] satisfy both
properties simultaneously. Nevertheless, both thermalization [13, 14] and size
winding [7, 10] are central to the holographic correspondence and are known to
occur in the SYK model at large system sizes [10].

Before proceeding, we note that two additional Hamiltonians are numerically
studied in the Supplemental Material of [1]. We address these in Appendix A.
The Hamiltonian Eq. (S16) in [1], henceforth “Model 2”, is produced by the
same machine-learning procedure as Model 1. We find that Model 2 is nearly
fully-commuting, and that the above observations similarly hold. In contrast,
the Hamiltonian Eq. (S17) in [1], henceforth “Model 3”, is produced by an
alternate machine-learning procedure designed to maximize the difference in
the teleportation signal between negative and positive couplings. Model 3 is
not fully-commuting and exhibits clearer signatures of thermalization.
However, as noted in [1], it does not exhibit perfect size winding.

Figure 2:  Teleportation signal of Model 1. (a) Teleportation circuit from
[1]. The qubit to be teleported is swapped with a pair of Majorana operators
in the left system L. The success of teleportation from L to R is measured by
the mutual information between a reference qubit P and a readout qubit T. (b)
Mutual information, $I_{\textrm{PT}}$, of the symmetric teleportation protocol
with $\mu=-12$ for Model 1 (orange; replicating Fig. 2a of [1]) and several
realizations of the $N=10$ SYK model with $J=1.25$ (grey). The machine-
learning procedure in [1] trains Model 1 to reproduce the mutual information
(as a function of time) of the SYK model for a specific pair of input
operators, $\psi^{1}$ and $\psi^{2}$. For this pair of operators, the mutual
information indeed shows good agreement between the two models. (c) In Model
1, when the teleportation protocol is performed with input operators that were
not involved in the training procedure, i.e. $\psi^{i}$ and $\psi^{j}$ where
$i<j\in[3,7]$, the mutual information as a function of time exhibits
significant variations. (d) For comparison, in the $N=10$ SYK model, the
mutual information for all pairs of input operators is consistent. The dashed
line indicates the mutual information at $t=0$ for reference.

Thermalization and scrambling—Ref. [1] claims that Model 1 “scrambles and
thermalizes similarly to the original SYK model as characterized by the four-
and two-point correlators.” To support this, they plot the _average_ of each
correlator over local Majorana operators. For example, they plot (Fig. 3b of
[1]) the two-point correlator
$G_{\textrm{avg}}(t)=\frac{1}{8}\sum_{i=1}^{8}G_{i}(t)$, where
$G_{i}(t)=\textrm{Re}\left[\left<\psi^{i}(t)\psi^{i}(0)\right>_{\beta}\right]$
and the sum is over the seven operators in Eq. (1) and an additional operator
$\psi^{8}$ that does not enter the Hamiltonian (reproduced in Fig. 1a). Here,
$\left<\cdot\right>_{\beta}=\textrm{Tr}[(\cdot)\rho_{\beta}]$ with
$\rho_{\beta}=e^{-\beta H}/\textrm{Tr}[e^{-\beta H}]$.

The decay of two-point correlation functions is indicative of thermalization
[15]. We note that it is not typical to average over a system’s two-point
correlation functions when exploring thermalization, since this averaging can
lead to a decay that is not representative of the individual correlation
functions. As shown in Fig. 3b of [1] and we reproduce in Fig. 1a above, for
both the SYK model and Model 1, the averaged correlation function,
$G_{\textrm{avg}}(t)$, indeed exhibits decay. For the SYK model, this decay is
consistent with the behavior of individual two-point correlation functions,
and thus, thermalization (Fig. 1c). However, for Model 1, the individual two-
point correlators, $G_{i}(t)$, exhibit strong revivals as a function of time
(Fig. 1b). This indicates that the agreement in the thermalization behavior
between the SYK model and Model 1 observed in Fig. 3b of [1] is an artifact of
averaging over the two-point correlation functions, and that in fact, Model 1
does not thermalize.

In Fig. 1d-f, we turn to the behavior of four-point correlation functions,
$F_{i}(t)=-\textrm{Re}\left[\big{\langle}\left[\psi^{i}(t),\psi^{i}(0)\right]^{2}\big{\rangle}_{\beta}\right]$,
as well as their average, $F_{\textrm{avg}}(t)=\sum_{i=1}^{8}F_{i}(t)$. Much
like the two-point correlators, the agreement between the four-point
correlation functions of Model 1 and the SYK model (Fig. 3b of [1]), is an
artifact of averaging.

In the holographic correspondence, the persistent decay of the two- and four-
point correlators corresponds to a perturbation falling toward a black hole
[14, 16]. The strong revivals in the correlators of Model 1 contrast with this
physics.

Teleportation signal—We now explore the claim that Model 1 “is consistent with
gravitational dynamics of the dense SYK Hamiltonian beyond its training data”.
Specifically, the authors claim that the teleportation signal of Model 1
demonstrates behavior compatible with a qubit emerging from a traversable
wormhole. These claims are based upon analyzing the teleportation signal for
the pair of operators, $\psi^{1}$ and $\psi^{2}$, that were involved in the
machine-learning training (Fig. 2 of [1]). To further test whether Model 1 is
consistent with the gravitational dynamics of the dense SYK model, we examine
the teleportation signal for operators that were not involved in the training
procedure.

Fig. 2 of [1] presents the mutual information of the teleportation protocol as
a function of two times: the injection time, $t_{0}$, and the readout time,
$t_{1}$ (see e.g. the circuit in Fig. 2a). For the symmetric teleportation
protocol, with $t=t_{0}=t_{1}$, traversable wormhole dynamics lead to the
presence of a single peak in the mutual information as a function of time [1].
We note that Model 1 is trained to reproduce the symmetric teleportation
signal of the SYK model for a specific pair of input Majorana operators,
$\psi^{1}$ and $\psi^{2}$. As depicted in Fig. 2a in [1] and in Fig. 2b above,
the mutual information indeed exhibits a single peak for the trained operators
in Model 1 and for various instances of the SYK model. However, for a generic
pair of untrained operators in Model 1, the mutual information does not
exhibit single-peak behavior, but rather, displays large oscillations as a
function of time that strongly vary for different input operators (Fig. 2c).
This sharply contrasts with teleportation in the SYK model, where the mutual
information exhibits a single consistent peak in time for any pair of input
operators (Fig. 2d).

We note that Ref. [1] also examines the teleportation protocol at a fixed
injection time, while varying the readout time (Fig. 2b in [1]); our analysis
of this protocol is shown in Appendix C. At short times, for Model 1, the
mutual information exhibits a single peak for all pairs of operators, albeit
with large variations in peak height. At longer times, generic pairs of
operators exhibit multiple peaks in the mutual information, which are not
observed in the SYK model.

Figure 3: Comparison of size winding behavior in Model 1 and other random
small-size fully-commuting Hamiltonians. (a) Scatter plots depicting the eight
non-zero coefficients, $c_{P}^{2}$, for $\psi^{1}$ of Model 1 at $t=2.8$. The
x-values are the coefficient magnitudes, $|c_{P}|^{2}$, and the y-values are
the coefficient phases, $\phi_{P}\equiv\textrm{arg }c_{P}^{2}-\textrm{arg
}q(1)$. Perfect phase alignment occurs when the phase of all coefficients at a
given size $|P|$ matches the phase of their sum (dashed line). This occurs
trivially for the single coefficient with $|P|=1$, and via the alignment of
$\sim$2-3 coefficients for $|P|=3,5$. (b) A comparison of the phase alignment,
$\bar{r}$, for each operator in: Model 1, Model 1 with random coefficients,
Model 1 with random terms and coefficients, a random all-to-all Ising model,
the $N=10$ SYK model, Model 2, and Model 3. For Models 1,2,3, the phase
alignment for the trained operators, $\psi^{1}$ and $\psi^{2}$, is indicated
with a star. As in [1], we take $\beta=4$, and time $t=2.8$ for Models 1,2 and
$t=2$ for Model 3. For the random models, three different disorder
realizations are shown, with small horizontal offsets for clarity. (c) An
analogous comparison for the linear slope metric, $\chi$. (d) The size winding
phase, $\textrm{arg }q(\left|P\right|)-\textrm{arg }q(1)$, as a function of
the operator size, $\left|P\right|$, for Model 1 (top) and Model 1 with random
coefficients (bottom). The size of each marker is scaled proportional to
$\left|q(\left|P\right|)\right|$. The stars in Model 1 correspond to operators
$\psi^{1}$ (replicating Fig. 3d in [1]) and $\psi^{2}$.

Size winding—Beyond a comparison to the $N=10$ SYK model, the authors claim
that Model 1 satisfies general behavior predicted by gravity. They focus on
the property of size winding (Fig. 3d and S14 in [1]), which is defined by
decomposing a time-evolved operator as follows:

$\rho_{\beta}^{1/2}\psi^{i}(t)=\sum_{P}c_{P}\psi^{P}$ (2)

where $c_{P}$ is a complex coefficient, and $\psi^{P}$ is a Majorana string
with support $P\in\\{0,1\\}^{N}$. Here, $P_{i}=1$ indicates that $\psi^{P}$
has support on site $i$. Size winding is the condition that the phases of the
squared coefficients, $c_{P}^{2}$, depend linearly on the size $|P|$ of the
Majorana string, i.e. $c_{P}=e^{i(\alpha|P|/N+\phi)}r_{P}$ for some real
values, $\alpha,\phi,r_{P}$. As in [1], it is convenient to separate size
winding into two distinct properties:

  1. 1.

_Phase alignment_ —The phase of $c_{P}^{2}$ is equal for all $P$ of the same
size.

  2. 2.

_Linear slope_ —The phase of the sum of coefficients of size $l$,
$q(l)=\sum_{|P|=l}c_{P}^{2}$, follows a linear slope with respect to $l$.

Analytic calculations show that the SYK model exhibits both properties in the
limit of large system sizes [10]. The authors emphasize that perfect size
winding is a necessary criteria of general holographic systems [17].

We begin by noting a difference in the structure of time-evolved operators
between fully-commuting models such as Model 1, and non-commuting models, such
as the SYK model. In fully-commuting Majorana Hamiltonians, a time-evolved
operator has non-zero coefficients for up to $2^{\lfloor N/2\rfloor}$
different strings. In a non-commuting Majorana Hamiltonian, a time-evolved
operator has non-zero coefficients for up to $2^{N-1}$ strings. This
difference is particularly pronounced at small system sizes: In Model 1 there
are $8$ non-zero coefficients (Fig. 3a), while in the $N=7$ SYK model there
are $64$ non-zero coefficients and in the $N=10$ SYK model studied in [1]
there are $512$ non-zero coefficients.

_Phase alignment—_ Ref. [1] quantifies the degree of phase alignment by
considering the ratio:

$r_{l}=\frac{\left|\sum_{|P|=l}c_{P}^{2}\right|}{\sum_{|P|=l}|c_{P}|^{2}},$
(3)

which is unity when phase alignment is perfect. Using $\psi^{1}$ (which is one
of the operators that was trained upon), the authors show that Model 1
exhibits $r_{l}\gtrsim 0.95$ for all $l$ (Fig. S14 in [1]). The authors refer
to this as perfect size winding. In comparison, Ref. [1] finds that the $N=10$
SYK model exhibits $r_{l}\sim 0.75$ (Fig. S19 in [1]), which the authors refer
to as “damped” size winding.

To avoid characterizing the phase alignment ratio for each $l$, we consider
the average ratio, $r=\sum_{l}\left|q(l)\right|$. Since $r$ is lower bounded
by a two-point correlation function,
$W=\textrm{tr}(\psi^{i}\rho_{\beta}^{1/2}\psi^{i}\rho_{\beta}^{1/2})=\sum_{P}c_{P}^{2}$,
it is natural to define a rescaled phase alignment ratio as
$\bar{r}=\frac{r-W}{1-W}$ (see Appendix D for details).

In Fig. 3b, $\bar{r}$ is plotted for Model 1 and the SYK model, for all
operators. Consistent with the authors’ observations (Figs. S14, S19 in [1]),
Model 1 exhibits a significantly higher $\bar{r}$ than the SYK model.

We begin by examining to what extent the large phase alignment, $\bar{r}$, is
a result of the authors’ machine learning procedure. In particular, we
consider Model 1 with _random_ numerical coefficients in front of the five
Hamiltonian terms. In all cases, we find that the phase alignment, $\bar{r}$,
is similar to Model 1 and significantly higher than the SYK model. Three
specific instances are shown in Fig. 3b. Next, we further randomize Model 1,
by considering Hamiltonians with five random, commuting, four-body Majorana
terms (in addition to random coefficients). Again, we find that the phase
alignment, $\bar{r}$, is similar to Model 1 and significantly higher than the
SYK model (Fig. 3b).

The above observations suggest that a large phase alignment, $\bar{r}$, is a
generic feature of many fully-commuting models at small system sizes. As a
further example, we plot the phase alignment, $\bar{r}$, for a random all-to-
all Ising model with four spins, and find similar behavior to Model 1; as
shown in Appendix E, we find that the phase alignment, $\bar{r}$, decreases
with increasing system size.

_Linear slope—_ The authors claim that Model 1 exhibits a linear size-winding
slope. To demonstrate this, in Fig. 3d of [1], the authors plot the phase of
$q(l)$ as a function of $l$ for the Majorana operator $\psi^{1}$ (which was
trained upon). The phase indeed exhibits a linear slope with respect to $l$.

We examine how generic this behavior is, and the extent to which it results
from the authors’ machine learning procedure. In order to better compare
across models, we introduce a metric, $\chi$, for quantifying the linear slope
property (see Appendix D for details). As depicted in Fig. 3c, $\psi^{1}$ is
indeed characterized by a large value of $\chi$. For the other operator that
was trained upon, $\psi^{2}$, we also find that $\chi$ is large. However, for
the untrained operators, some Majoranas exhibit large values of $\chi$, while
others exhibit small values of $\chi$ implying that they do not satisfy the
linear slope property. This behavior is illustrated in Fig. 3d and contrasts
with the SYK model. For the SYK model, all operators exhibit the linear slope
property and $\chi$ is clustered near unity, as shown in Fig. 3c.

As in the preceding discussion about phase alignment, the fact that some
operators in Model 1 exhibit a linear slope with large $\chi$, is a generic
feature of small-size, fully-commuting Hamiltonians. The distribution of
$\chi$ across all operators is illustrated in Fig. 3c, for the three types of
models considered in the preceding subsection: (i) Model 1 with random
coefficients, (ii) Model 1 with random terms and coefficients, and (iii)
fully-commuting random Ising models. In each case, the behavior is comparable
to Model 1—certain operators exhibit the linear slope property with large
$\chi$, while others do not.

We note that for all three models studied by the authors (Model 1 in the main
text and Models 2,3 in the Supplemental Material), the trained operators
exhibit a relatively high degree of linearity compared to other operators.
This suggests that the authors’ machine-learning procedure may have introduced
a bias among the trained operators. Nevertheless, the distribution of $\chi$
over all operators resembles that of generic random small-size fully-commuting
models.

Figure 4: (a) Two-point correlation functions, $G_{i}(t)$, for all Majorana
operators in Model 2. (b) (left) Mutual information for the symmetric
teleportation protocol with the trained operators, $\psi^{1}$ and $\psi^{2}$,
and $\mu=-12$ for: Model 2 (blue), and multiple instances of $N=10$ SYK model
(grey). (right) Mutual information for the symmetric teleportation protocol
with all pairs of untrained operators, and $\mu=-12$. (c) Size-winding phase
for each of the untrained operators at $t=2.8$. (d-f) Depicts the analogous
results for Model 3. As in [1], the teleportation protocol is performed with
$\mu=-17$ for Model 3 (replicating Fig. S25 of [1]), and the size-winding
phase is evaluated at $t=2$.

Size winding has recently emerged as a prominent feature of the holographic
correspondence for systems with a nearly AdS2 bulk [7], leading to speculation
that the presence of perfect size winding could be a strong signature of
gravity [10, 9]. The fact that the perfect size winding observed in [1] seems
reliant on small-size, fully-commuting models—which defy other features of
holography such as thermalization [13, 14], complexity [18], and chaos
[19]—raises the question of whether the observed perfect size winding is
indeed connected to gravitational physics in a substantive manner.

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##  Appendix A Other learned models

In the Supplemental Material of [1], two additional learned Hamiltonians are
studied numerically.

_Model 2_ —The first of these, which we refer to as Model 2, is given in Eq.
(S16) of [1]:

$\begin{split}H=&-0.35\psi^{1}\psi^{2}\psi^{3}\psi^{6}+0.11\psi^{1}\psi^{2}\psi^{3}\psi^{8}-0.17\psi^{1}\psi^{2}\psi^{4}\psi^{7}\\\
&-0.67\psi^{1}\psi^{3}\psi^{5}\psi^{7}+0.38\psi^{2}\psi^{3}\psi^{6}\psi^{7}-0.05\psi^{2}\psi^{5}\psi^{6}\psi^{7}.\end{split}$
(4)

Model 2 is produced from the same machine-learning procedure as Model 1, i.e.
designed to match the teleportation signal of the $N=10$ SYK model. The
authors claim that Model 2 demonstrates perfect size winding and “is
consistent with other gravitational signatures”.

As noted in [1], Model 2 is not fully commuting. Nevertheless, we observe that
Model 2 becomes fully-commuting if: (i) the two _smallest_ terms in Eq. (4)
are removed, and (ii) one performs a basis rotation:

$\begin{split}\psi^{1}&\rightarrow\cos(\theta)\psi^{1}+\sin(\theta)\psi^{7},\\\
\psi^{7}&\rightarrow\cos(\theta)\psi^{7}-\sin(\theta)\psi^{1},\end{split}$ (5)

with $\theta=\tan^{-1}(-0.35/0.38)$. At the timescale of teleportation
($t=2.8$), the two smallest terms provide relatively small corrections to
physical observables. Thus, Model 2 can be considered weakly perturbed from a
fully-commuting limit.

Consistent with this observation, we find that our main observations regarding
Model 1 also apply to Model 2 (Fig. 4a-d). In particular, the individual two-
point correlation functions exhibit strong revivals, the teleportation signal
does not resemble the SYK model for untrained operators, and the size winding
behavior resembles that of a random fully-commuting Hamiltonian (Fig. 3c). In
addition, we note that the teleportation signal for the _trained_ operators in
Model 2 displays a significant revival within the timescale on which it was
trained (Fig. 4b). This contrasts with the $N=10$ SYK model and indicates that
the training procedure was not fully successful; such disagreement is not
shown or commented on in [1].

_Model 3_ —The second additional model, which we refer to as Model 3, is given
in Eq. (S17) of [1]:

$\begin{split}H&=0.60\psi^{1}\psi^{3}\psi^{4}\psi^{5}+0.72\psi^{1}\psi^{3}\psi^{5}\psi^{6}+0.49\psi^{1}\psi^{5}\psi^{6}\psi^{9}\\\
&+0.49\psi^{1}\psi^{5}\psi^{7}\psi^{8}+0.64\psi^{2}\psi^{4}\psi^{8}\psi^{10}-0.75\psi^{2}\psi^{5}\psi^{7}\psi^{8}\\\
&+0.58\psi^{2}\psi^{5}\psi^{7}\psi^{10}-0.53\psi^{2}\psi^{7}\psi^{8}\psi^{10}.\end{split}$
(6)

Model 3 is produced via a different machine-learning procedure, which is
designed to optimize the asymmetry in the teleportation signal between
positive and negative couplings. Unlike Models 1 and 2, Model 3 is not fully-
commuting or near fully-commuting.

Referring to the average two-point correlator, the authors demonstrate that
“no periodicities are present despite the small number of terms in the
Hamiltonian” (Fig. S26 of [1]). In Fig. 4d, we observe that the individual
two-point correlators also exhibit thermalizing behavior at long time scales
($t\sim 30$). This is consistent with Model 3 being non-commuting. At earlier
times, the correlators exhibit oscillations that are smaller than those of
Model 1 and 2, but larger than fluctuations in the $N=10$ SYK model.

The teleportation signal for Model 3 exhibits a single-peak structure for
nearly all operators, albeit with large variations in peak height (Fig. 4e).

The authors note that Model 3 does not exhibit perfect size winding, but
rather features a “consistently large ratio [of phase alignment], suggesting
slightly damped size winding”. Indeed, we find that the phase alignment,
$\bar{r}$, for Model 3 is comparable to that of the $N=10$ SYK model and lower
than that of small-size fully-commuting models (Fig. 3b). This is consistent
with our observation that perfect phase alignment at small system sizes is a
generic feature of fully-commuting Hamiltonians and not of non-commuting
Hamiltonians. We note that only some operators in Model 3 (including the
trained operators) exhibit a high degree of linearity $\chi$ (Fig. 3c). In
this respect, Model 3 resembles the behavior of fully-commuting or nearly
fully-commuting models (including Model 1 and 2) and not the SYK model.

Figure 5: (a) Four-point correlation functions, $F_{ij}(t)$, for Model 1,
shown for all pairs of Majorana operators, $i<j\in[1,7]$. (b) The same
correlation functions for a specific instance of the $N=10$ SYK model with
$J=1.125$ and $i<j\in[1,10]$.

##  Appendix B Four-point correlators with $i\neq j$

Scrambling is quantified in [1] via the behavior of the four-point correlation
functions, $F_{\textrm{avg}}(t)=\sum_{i=1}^{8}F_{i}(t)$, with
$F_{i}(t)=-\textrm{Re}\left[\big{\langle}\left[\psi^{i}(t),\psi^{i}(0)\right]^{2}\big{\rangle}_{\beta}\right]$.
We note that such correlation functions, consisting of the same Majorana
$\psi^{i}$ for the time-evolved and static operators, are not the most direct
probe of scrambling dynamics, since their initial growth occurs on the same
timescale as the decay of two-point correlation functions (i.e. the
thermalization time). In the SYK model at large system sizes, the initial
growth reaches a value of unity and is followed by a slower decay to value
$1/2$ on the timescale of the scrambling time; such non-monotonic behavior is
evident in the time traces of the $N=10$ SYK model shown in Fig. 1d. A more
typical probe of scrambling is the four-point correlator,
$F_{ij}(t)=-\textrm{Re}\big{[}\big{\langle}\left[\psi^{i}(t),\psi^{j}(0)\right]^{2}\big{\rangle}_{\beta}\big{]}$,
for different operators, $i\neq j$. In the SYK model at large system sizes,
this correlator decays monotonically from unity to value $1/2$ on the
timescale of the scrambling time.

Figure 6: (a) Mutual information of the teleportation protocol with fixed
injection time as a function of the readout time (i.e. $t_{0}=2.8$ and
$t=t_{1}$). The mutual information for Model 1 and the trained operators,
$\psi^{1}$ and $\psi^{2}$, is in reasonable agreement with that of multiple
instances of the $N=10$ SYK model (grey). (b) The mutual information for Model
1 and all pairs of untrained operators, $\psi^{i}$ and $\psi^{j}$ with
$i<j\in[2,7]$, exhibits variations and revivals as a function of time. (c) The
mutual information for all pairs of operators in the $N=10$ SYK model exhibits
a single consistent peak.

In Fig. 5, we plot the four-point correlation functions, $F_{ij}(t)$ with
$i\neq j$, for both Model 1 and the $N=10$ SYK model. Much like the four-point
correlation functions with $i=j$ (i.e. $F_{i}(t)$, see Fig. 1), we find that
the four-point correlation functions in Model 1 exhibit strong oscillations in
time for all $i\neq j$. In fact, the oscillations for many pairs of operators
have unit amplitude. In contrast, in the $N=10$ SYK model, all correlation
functions exhibit a smooth decay to value $1/2$.

##  Appendix C Teleportation at fixed injection time

As previously discussed, two versions of the teleportation protocol are
analyzed in [1]: using symmetric injection / readout times and fixed injection
time. In Fig. 6, we present results for latter protocol for Model 1 and the
$N=10$ SYK model. For Model 1, when the protocol is performed with the pair of
operators that were trained on, the mutual information displays a single peak
as a function of time. For other pairs of operators, the mutual information
displays an initial peak, whose height varies significantly for different
pairs of operators, followed by revivals at later times. This contrasts with
the SYK model, in which the mutual information displays a single consistent
peak for all pairs of operators, with small and infrequent fluctuations at
late times.

##  Appendix D Size-winding metrics

Here, we elaborate on the phase alignment, $\bar{r}$, and the linear slope
metric, $\chi$, which are plotted in Fig. 3b and Fig. 3c.

_Phase alignment_ —We recall that in [1], the phase alignment is quantified by
plotting the ratio,
$r_{l}=\left|\sum_{|P|=l}c_{P}^{2}\right|/\sum_{|P|=l}|c_{P}|^{2}$, for
different sizes $l$ (Figs. S14 and S19 of [1]). The denominator of this
quantity is the operator size distribution, $p(l)=\sum_{|P|=l}|c_{P}|^{2}$,
which is normalized to one, $\sum_{l}p(l)=1$. To facilitate comparison between
different operators and models, we consider the weighted average of $r_{l}$,
$r=\sum_{l}p(l)\,r_{l}=\sum_{l}\left|\sum_{|P|=l}c_{P}^{2}\right|=\sum_{l}\left|q(l)\right|$.
For a given Hamiltonian, $r$ is lower bounded by the two-point function,
$W=\tr(\psi^{i}(t)\rho^{1/2}\psi^{i}(t)\rho^{1/2})=\sum_{P}c_{P}^{2}$. We note
that this two-point function is constant in time, and therefore the sum of the
squared coefficients is also constant in time. Taking into account this lower
bound motivates us to rescale $r$ as $\bar{r}=\frac{r-W}{1-W}$, which ranges
from zero to one.

_Linear slope_ —We seek to quantify the degree to which the phases of $q(l)$
follow a linear slope with respect to the size $l$. The fit of a line of slope
$\mu$ can be quantified via $C(\mu)=\left|\sum_{l}q(l)e^{-i\mu l}\right|$.
When deviations from a linear slope are small, this reduces to unity minus a
weighted sum of squared errors; when deviations are large, it takes into
account the periodicity of the phases. The best fit, $C^{*}$, is found by
maximizing over $\mu$, $C^{*}=\max_{\mu}C(\mu)$.

We define the metric, $\chi$, to interpolate between zero and one as $C^{*}$
interpolates between its minimum and maximum values. The maximum value of
$C(\mu)$ is given by the weighted average, $r$, of the phase alignment ratio.
The minimum value is lower bounded by the two-point function, $W=C(\mu=0)$. In
addition, at small system sizes it is relevant to consider a second lower
bound, corresponding to fitting a line between the two coefficients,
$q(l_{1})$ and $q(l_{2})$, with the largest magnitude. This consideration is
necessary to avoid concluding that functions $q(l)$ with support on only two
values of $l$ have non-trivial size winding. This fit produces a $C$ of value
at least
$M=|q(l_{1})|+|q(l_{2})|-(r-|q(l_{1})|+|q(l_{2})|)=2|q(l_{1})|+2|q(l_{2})|-r$.
We thus define the metric,

$\chi=\frac{C^{*}-L}{r-L},$ (7)

where $L$ is the larger of the two lower bounds, $L=\max(W,M)$.

##  Appendix E Other fully-commuting models

Here we include details on the random fully-commuting models presented in Fig.
3. For all random models in Fig. 3, we take $\beta=4$ and $t=2.8$, identical
to Model 1.

_Majorana models_ —In Model 1 with randomized coefficients, we draw each
coefficient from a normal distribution with mean zero and standard deviation
equal to the root-mean-square of the coefficients of Model 1. In Model 1 with
randomized terms and coefficients, we generate five random fully-commuting
terms by successively drawing random four-Majorana terms (from $N=7$ total
Majorana operators) and keeping each term only if it commutes with all terms
already kept.

_Ising models_ —We consider random all-to-all Ising models with Hamiltonian,
$H=\frac{1}{\sqrt{N}}\sum_{i<j}J_{ij}Z^{i}Z^{j}$. The coefficients $J_{ij}$
are drawn from a normal distribution with mean zero and standard deviation
$J=0.17$.

_Finite-size scaling_ —To explore whether the size winding behavior of fully-
commuting models persists at larger system sizes, in Fig. 7 we plot the phase
alignment, $\bar{r}$, for random all-to-all Ising models as a function of the
system size $N$. We focus on Ising models to avoid subtleties with scaling
random fully-commuting Majorana models to larger system sizes (namely, there
is no canonical choice of which fully-commuting terms to include). We scale
the evolution time $t$ with the square root of the system size,
$t=2.8\sqrt{N/4}$, to ensure that operators grow to the same fraction of the
system size for each $N$. We find that the phase alignment, $\bar{r}$,
exhibits a decreasing trend with the system size.

Figure 7: Phase alignment, $\bar{r}$, of the random all-to-all Ising model as
a function of the system size $N\in[4,8]$, with $J=0.17$ and $\beta=4$. Three
disorder realizations are shown at each system size, with small horizontal
offsets for clarity.

##  Appendix F Missing parameters

Several parameters are omitted from [1] which are necessary for reproducing
the numerical results shown. For the ease of future studies, we list these
parameters below:

  * •

In plots showing mutual information (e.g. Figs. 1a, 3a, 3e of [1]), the mutual
information is divided by $\log(2)$.

  * •

For the teleportation plots (e.g. Figs. 1a, 3a, 3e of [1]), the two-sided
coupling is $V=\frac{1}{qN}\sum_{i}\psi_{L}^{i}\psi_{R}^{i}$ with $N=10$ and
$q=4$, for both Model 1 and the SYK models. The teleportation protocol is
performed using $\psi^{1}$ and $\psi^{2}$.

  * •

In Fig. 3f of [1], the instantaneous coupling plot uses $\mu=-12$, however
with $V$ normalized using $N=8$. This is distinct from the other plots in
Figs. 1-3, which normalize $V$ with $N=10$. The Trotterized coupling plot uses
$\mu=-18$ and $N=8$, and is Trotterized into three time steps, $t=-1.6,0,1.6$.

  * •

As previously emphasized, the correlation functions in Fig. 3b of [1] are
averaged over all local Majorana operators (for Model 1, this consists of $8$
operators: the 7 operators in Eq. (1) and an additional uncoupled operator).
Also, the authors plot the real part of two- and four-point correlation
functions. To summarize, the plots correspond to
$G_{\textrm{avg}}(t)=\frac{1}{8}\sum_{i=1}^{8}G_{i}(t)$ with
$G_{i}(t)=\textrm{Re}[\left<\psi^{i}(t)\psi^{i}(0)\right>_{\beta}]$ and
$F_{\textrm{avg}}(t)=\sum_{i=1}^{8}F_{i}(t)$ with
$F_{i}(t)=-\textrm{Re}[\big{\langle}\left[\psi^{i}(t),\psi^{i}(0)\right]^{2}\big{\rangle}_{\beta}]$.

  * •

We are not able to exactly replicate the SYK dynamics shown in Figs. 1 and 3b
in [1]. We find qualitatively good agreement using an ensemble of $N=10$ SYK
models with $J=1.25$ for Fig. 1 in [1] and $J=1.125$ for Fig. 3b in [1].


111Supported by JSPS grant in aid for scientific research No. 19K03589

.

# Boundedness of energy for N-body Schrödinger equations with time dependent
small potentials

Kenji Yajima  Department of Mathematics
Gakushuin University
1-5-1 Mejiro
Toshima-ku
Tokyo 171-8588 (Japan<EMAIL_ADDRESS>
###### Abstract.

We prove that Sobolev norms of solutions to time dependent Schrödinger
equations for $d$-dimensional $N$-partcles interacting via time dependent two
body potentials are bounded in time if certain Lebesgue norms of the
potentials are small uniformly in time. The proof uses the scattering theory
in the extended phase space which proves that all particles scatter freely in
the remote past and far future.

##  1\. Introduction, Results

We consider Schrödinger equations for $N$ particles in ${\mathbb{R}}^{d}$,
$d\geq 3$,

$i{\partial}_{t}u(t,x)=\Big{(}-\sum_{j=1}^{N}\frac{1}{2m_{j}}\Delta_{j}+\kappa
V(t,x)\Big{)}u(t,x)=\colon H_{\kappa}(t)u(t,x),$ (1.1)

interacting via (complex) time dependent short range two-body potentials:

$V(t,x)=\sum_{1\leq j<k\leq N}V_{jk}(t,x_{j}-x_{k})$

with small coupling constant $\kappa$, where $x_{j}\in{\mathbb{R}}^{d}$ and
$m_{j}$, $1\leq j\leq N$, are the position and the mass of $j$-th particle
respectively, $x=(x_{1},\dots,x_{N})\in{\mathbb{R}}^{Nd}$, $\Delta_{j}$ is the
$d$-dimensional Laplacian with respect to $x_{j}$ and we have set $\hslash=1$.
The purpose of the present paper is to show that the Sobolev norm
$\|u(t,x)\|_{H^{m}({\mathbb{R}}^{Nd})}$, $m=0,1,\dots$ of solutions (1.1),
hence the energy $(H_{\kappa}(t)u(t),u(t))_{L^{2}({\mathbb{R}}^{Nd})}$ of the
system, is bounded in time $t\in{\mathbb{R}}$ if

$\sum_{|\alpha|\leq
m}\sup_{t\in{\mathbb{R}}}\|{\partial}_{x}^{\alpha}V_{jk}(t,\cdot)\|_{(L^{p}\cap
L^{q})({\mathbb{R}}^{d})}\leq C\quad 1\leq j<k\leq N\,,$ (1.2)

for some $p$ and $q$ such that $1\leq p<d/2<q\leq\infty$.

If $V_{jk}(t,y)=V_{jk}(x)$, $1\leq j<k\leq N$ are real and independent of
$t\in{\mathbb{R}}$, then $H_{\kappa}$ on the right of (1.1) with
$\kappa\in{\mathbb{R}}$ is selfadjoint in
${\mathscr{H}}=L^{2}({\mathbb{R}}^{Nd})$ with domain
$D(H_{\kappa})=H^{2}({\mathbb{R}}^{Nd})$ and the energy $\langle
u(t),H_{\kappa}u(t)\rangle$ is conserved, hence, a fortiori bounded as
$t\to\pm\infty$. However, if $V(t,x)$ is genuinly $t$-dependent, it is a suble
question whether or not the energy of the system remains bounded in time. When
$N=2$ and $V(t,x)$ is real, smooth and rapidly decreasing as $|x|\to\infty$
uniformly with respect to $t\in{\mathbb{R}}$, Bourgain ([4]) has shown that
Sobolev norms $\|u(t)\|_{H^{m}({\mathbb{R}}^{d})}$, $m\geq 0$, of solutions of
(1.1) remain bounded as $t\to\pm\infty$ if $\kappa$ is sufficiently small and
that, without the smallness condition, they satisfy
$\|u(t)\|_{H^{m}({\mathbb{R}}^{d})}\leq C_{\varepsilon}\langle
t\rangle^{\varepsilon}\|u(0)\|_{H^{m}({\mathbb{R}}^{d})}$ for any
$\varepsilon>0$, however, the factor $\langle t\rangle^{\varepsilon}$ cannot
be removed when $m\geq 1$ in general. Thus, we extend in this paper the first
part of [4] to $N$-body Hamiltonians with time dependent complex singular
potentials.

There are many works on the large $t$ behavior of Sobolev norms of the
solutions of Schrödinger equations. However, except [4] mentioned above, they
deal with the case that the operator $H_{\kappa}(t)$ on the right of (1.1) has
for each $t$ discrete spectrum or $V(t,x)$ is periodic in $x$ and prove that
Sobolev norms can increase only as slowly as $C|t|^{\varepsilon}$ for arbitray
small $\varepsilon>0$ or $C\log|t|$ as $t\to\infty$ (see [3], [5], [6], [7],
[8], [9], [15], [18] and references therein) and, as far as the author is
aware of, there are no results so far for $N$-body system with genuinely time
dependent potentials.

For the reason to be explained below Theorem 1.1, we consider (1.1) for vector
valued functions $u(t,x)\in{\mathbb{C}}^{n}$, $n=1,2,\dots$ with matrix
potentials $V_{jk}(t,y)\in M(n)$:

$i{\partial}_{t}u=-\Delta_{X}u+{\kappa}V(t,x)u,\quad V(t,x)=\sum_{1\leq
j<k\leq N}V_{jk}(t,x_{j}-x_{k})$ (1.3)

where $-\Delta_{X}=\sum_{j=1}^{N}(2m_{j})^{-1}\Delta_{j}$ (see §2) and
$M(n)=M(n,n)$, $M(m,n)$ being the space of $m\times n$-matrices. For
$M(m,n)$-valued function $A=(a_{jk})$ on a measure space $\Omega$, $A\in
L^{p}(\Omega)$ means that $a_{jk}\in L^{p}(\Omega)$ for all $j,k$,
$\|A\|_{L^{p}}=\left(\sum\|a_{jk}\|_{L^{p}}^{p}\right)^{\frac{1}{p}}$ and
$\|A\|_{p}=\|A\|_{L^{p}}$. For Banach spaces ${\mathscr{X}}$ and
${\mathscr{Y}}$ which are subspaces of a linear topological space
${\mathscr{T}}$, ${\mathscr{X}}\cap{\mathscr{Y}}$ and
${\mathscr{X}}+{\mathscr{Y}}$ are Banach spaces with respective norms

$\|u\|_{{\mathscr{X}}\cap{\mathscr{Y}}}=\|u\|_{{\mathscr{X}}}+\|u\|_{{\mathscr{Y}}},\quad\|u\|_{{\mathscr{X}}+{\mathscr{Y}}}=\inf\\{\|a\|_{{\mathscr{X}}}+\|b\|_{{\mathscr{Y}}}\colon
u=a+b\\};$

${\mathscr{L}}^{p}=L^{p}({\mathbb{R}}^{d}:{\mathbb{C}}^{n})$ and
${\mathscr{L}}_{p,q}=(L^{p}\cap L^{q})({\mathbb{R}}^{d}\colon M(n))$;
$\dot{A}_{jk}(t,y)={\partial}_{t}A_{jk}(t,y)$, etc. are $t$-derivatives.

We remark before stating the theorem that, since ${\mathscr{L}}_{p,q}\subset
L^{d}({\mathbb{R}}^{d}\colon M(n))$ if $p<d<q$, $V_{jk}(t,y)$ of the following
Theorem 1.1 satisfy the conditions of Theorems 2.2, 2.3 and 2.4 below with
$p=d/2$ and (1.3) generates a unique propagator $U(t,s,\kappa)$ on
${\mathscr{H}}\colon=L^{2}({\mathbb{R}}^{Nd}\colon{\mathbb{C}}^{n})$ (see
Definition 2.1). Theorem 1.1 holds for more general multi-particle
interactions, however, we restrict ourselves to two-body interactions for
notational simplicity.

######  Theorem 1.1.

Let $n=1,2,\dots$ and $m\in{\mathbb{N}}\cup\\{0\\}$. Suppose that
$V_{jk}(t,y)\in M(n)$, ${1\leq j<k\leq N}$, are factorized by $A_{jk}(t,y)\in
M(n)$ and $B_{jk}(t,y)\in M(n)$,

$V_{jk}(t,y)=B_{jk}(t,y)^{\ast}A_{jk}(t,y),$

which satisfy the following conditions for any $|\alpha|\leq m$:

  1. (1)

There exist $p$ and $q$ such that $2\leq p<d<q\leq\infty$ and $q\geq 4$ if
$d=3$ and such that ${\partial}^{\alpha}_{y}A_{jk}(t,\cdot)$ and
${\partial}^{\alpha}_{y}B_{jk}(t,\cdot)$ are ${\mathscr{L}}_{p,q}$-valued
functions of $t\in{\mathbb{R}}$ and

$\sup_{t\in{\mathbb{R}}}\left(\|{\partial}^{\alpha}_{y}A_{jk}(t,\cdot)\|_{{\mathscr{L}}_{p,q}}+\|{\partial}^{\alpha}_{y}B_{jk}(t,\cdot)\|_{{\mathscr{L}}_{p,q}}\right)<\infty\,.$
(1.4)

  2. (2)

For a.e. $y\in{\mathbb{R}}^{d}$ they are locally absolutely continuous with
respect to $t$ and

${\partial}^{\alpha}_{y}\dot{A}_{jk}(t,y),{\partial}^{\alpha}_{y}\dot{B}_{jk}(t,y)\in
L^{2}_{\textrm{loc}}({\mathbb{R}},(L^{r}+L^{\infty})({\mathbb{R}}^{d}\colon
M(n)))$

for $r=2$ if $d=3$ and $r=d/2$ if $d\geq 4$.

Then, for $|\kappa|<\kappa_{m}$, $\kappa_{m}>0$ being a small constant, there
exits a $C_{\kappa,m}<\infty$ such that

$\sup_{t,s\in{\mathbb{R}}}\|U(t,s,\kappa){\varphi}\|_{H^{m}({\mathbb{R}}^{Nd}\colon{\mathbb{C}}^{n})}\leq
C_{\kappa,m}\|{\varphi}\|_{H^{m}({\mathbb{R}}^{Nd}\colon{\mathbb{C}}^{n})}.$
(1.5)

For the proof of Theorem 1.1 we use the scattering theory for (1.1) and the
induction argument of Bourgain([4]) that the boundedness of
$\|u(t,\cdot)\|_{H^{k+1}({\mathbb{R}}^{d})}$ follows from that of $\|{\bf
u}(t,\cdot)|_{H^{k}({\mathbb{R}}^{d}\colon{\mathbb{C}}^{(Nd+1)})}$ of
${\textbf{u}}=\begin{pmatrix}u\\\ \nabla_{x}u\end{pmatrix}$ which satisfies

$i{\partial}_{t}{\textbf{u}}=H_{0}\textbf{u}+\kappa\begin{pmatrix}V(t,x)&{\textbf{0}}\\\
{\nabla_{x}}V(t,x)&V(t,x){\textbf{1}}\end{pmatrix}{\textbf{u}}\,.$ (1.6)

In §2 we recall the results of [19] which imply that (1.3) generates a unique
propagator $\\{U(t,s,\kappa)\\}$ on ${\mathscr{H}}$. Then, in §3, following
Howland [11], we introduce the extended phase space
${\mathscr{K}}=L^{2}({\mathbb{R}})\otimes{\mathscr{H}}$ and the strongly
continuous one parameter group of bounded operators
$\\{{\mathcal{U}}(\sigma)\colon\sigma\in{\mathbb{R}}\\}$ on ${\mathscr{K}}$ by
$({\mathcal{U}}(s)u)(t)=U(t,t-s)u(t-s)$. Let
$({\mathcal{U}}_{0}(\sigma)u)(t)=e^{-i{\sigma}H_{0}}u(t-s)$. We then state
Theorem 4.1 in §4 that strong limit
${\mathcal{W}}_{+}=\lim_{\sigma\to\infty}e^{i\sigma{\mathcal{K}}}e^{-i\sigma{\mathcal{K}}_{0}}$
exists in ${\mathscr{K}}$, it is an isomorphism of ${\mathscr{K}}$ and
satisfies the intertwing property
$e^{-i\sigma{\mathcal{K}}}={\mathcal{W}}_{+}e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{W}}_{+}^{-1}$.
Postponing the proof of Theorem 4.1 to §6, we show also in §4 that Theorem 4.1
implies that the limit in ${\mathcal{H}}$

$\lim_{t\to\infty}U(s,t,\kappa)e^{-i(t-s)H_{0}}=W_{+}(s)$

exists, it is an isomorphism of ${\mathscr{H}}$ and is uniformly bounded for
$s\in{\mathbb{R}}$ along with the inverse $W_{+}(s)^{-1}$:

$\|W_{+}(s)\|_{{\textbf{B}}({\mathscr{H}})}\leq
C,\quad\|W_{+}(s)^{-1}\|_{{\textbf{B}}({\mathscr{H}})}\leq C,\quad
s\in{\mathbb{R}}$ (1.7)

and that it satisfies the intertwing property:

$U(t,s)=W_{+}(t)e^{-i(t-s)H_{0}}W_{+}(s)^{-1},\quad t,s\in{\mathbb{R}}.$ (1.8)

We prove Theorem 1.1 in §5. The case $m=0$ is evident from (1.7) and (1.8). If
we assume that Theorem 1.1 holds for $m=1,\dots,k$, then the ”potential” of
(1.6) satisfies the condition of Theorem 1.1 for $m=k$ as $M((Nd+1)n)$-valued
function. Hence,
$\|{\textbf{u}}(t)\|_{H^{k}({\mathbb{R}}^{Nd}:{\mathbb{C}}^{(Nd+1)n)})}$ is
bounded for $t\in{\mathbb{R}}$ by the induction hypothesis. and
$\|u(t)\|_{H^{k+1}(R^{Nd}:{\mathbb{C}}^{n})}\leq C$. The proof of Theorem 4.2
is given in §6 by adapting Ioirio-O’Carrol’s argument ([12]) to the extended
phase space and by applying Kato’s theory of smooth perturbations ([13]).
Thus, the method employed here are rather old theory of scattering and it is
expected that its modern theory can produce more refined result.

##  2\. Existence and regularity of propagators

We begin with recalling the result of [19] on the existence and the regularity
of the propagator for (1.3) in the form modified for our purpose. We use some
$N$-body notation due to Agmon [2]: $X$ is the space ${\mathbb{R}}^{Nd}$ with
the so called mass inner product

$(x,y)_{m}\colon=\sum_{j=1}^{N}2m_{j}(x_{j},y_{j})_{{\mathbb{R}}^{d}},\quad
x=(x_{1},\dots,x_{N}),\ y=(y_{1},\dots,y_{N}),$

then $\sum_{j=1}^{N}(2m_{j})^{-1}\Delta_{j}=\Delta_{X}$; for the pair
$\\{j,k\\}$, $1\leq j,k\leq N$, we set

$\displaystyle X_{jk}^{out}=\\{x\in X\colon x_{j}=x_{k}=0\\},\quad
X_{jk}^{in}=X\ominus X_{jk}^{out}\simeq{\mathbb{R}}^{2d},$ $\displaystyle
X_{jk}^{c}=\\{x\in X_{jk}^{in}\colon
x_{j}=x_{k}\\}\simeq{\mathbb{R}}^{d},\quad X_{jk}^{r}=X_{jk}^{in}\ominus
X_{jk}^{c}\simeq{\mathbb{R}}^{d}.$

Then, $X=X_{jk}^{out}\oplus X^{in}_{jk}$ and $X^{in}_{jk}=X_{jk}^{c}\oplus
X_{jk}^{r}$. We let

$X_{jk}=X_{jk}^{out}\oplus X_{jk}^{c}$ (2.1)

and the corresponding orthogonal decomposition of $x\in X$ be

$x=x^{jk}\oplus x_{jk}^{c}\oplus x_{jk}^{out}\in X_{jk}^{r}\oplus
X^{c}_{jk}\oplus X_{jk}^{out};\quad x_{jk}=\colon x_{jk}^{c}\oplus
x_{jk}^{out}\in X_{jk}.$

This orthogonal decomposition leads to

$\displaystyle{\mathscr{H}}=L^{2}(X_{jk}^{r}\colon{\mathscr{H}}_{jk}),\quad{\mathscr{H}}_{jk}\colon=L^{2}(X_{jk}\colon{\mathbb{C}}^{n}).$
(2.2)
$\displaystyle\Delta_{x}=\Delta_{x^{jk}}\otimes\textbf{1}_{{\mathscr{H}}_{jk}}+\textbf{1}_{L^{2}(X_{jk}^{r})}\otimes\Delta_{x_{jk}}.$
(2.3)

We denote $\Delta_{jk}=\Delta_{x^{jk}}$. It is easy to check that

$\displaystyle x^{12}$
$\displaystyle=\left(x^{12}_{1},x^{12}_{2},0,\dots,0\right),\ \
x^{12}_{1}/m_{2}=-x^{12}_{2}/m_{1}=(x_{1}-x_{2})/(m_{1}+m_{2}),$
$\displaystyle x_{12}^{c}$
$\displaystyle=\left(x_{12}^{c},x_{21}^{c},0,\dots,0\right),\quad
x_{12}^{c}=(m_{1}x_{1}+m_{2}x_{2})/(m_{1}+m_{2})$

and $x_{12}^{out}=(0,0,x_{3},\dots,x_{N})$; we have similar identities for
$x^{jk}$, $x_{jk}^{c}$ and $x_{jk}^{out}$ and $V_{jk}(t,x_{j}-x_{k})$ is a
function of $(t,x^{jk})$. Abusing notation, we often write $V_{jk}(t,x^{jk})$
for $V_{jk}(t,x_{j}-x_{k})$.

######  Definition 2.1.

We say a strongly continuous family $\\{U(t,s,\kappa)\colon
s,t\in{\mathbb{R}}\\}$ of bounded operators on
${\mathscr{H}}=L^{2}(X:{\mathbb{C}}^{n})$ is the propagator for (1.3) if it
satisfies the Champan-Kolmogorov equation:
$U(t,s,\kappa)U(s,r,\kappa)=U(t,r,\kappa)$,
$U(t,t,\kappa)={\textbf{1}}_{{\mathscr{H}}}$ for $t,s,r\in{\mathbb{R}}$ and,
if $u(t,x)=(U(t,s,\kappa){\varphi})(x)$, ${\varphi}\in{\mathscr{H}}$, is a
solution of (1.1) such that $u(s,x)={\varphi}(x)$.

For a given $d/2\leq p\leq\infty$ and compact intervals $I$, we let

${\mathcal{X}}^{p}(I)=\bigcap\limits_{1\leq
j<k\leq\infty}L^{\theta}(I,L^{\ell}(X_{jk}^{r},{\mathscr{H}}_{jk}))\,;$ (2.4)

where $\theta={4p}/{d}(\geq 2)$ and ${\ell}=2p/(p-1)(\geq 2d/(d-2))$;
${\mathcal{X}}^{p}_{\textrm{loc}}$ denotes the space of $f$ such that
$f\in{\mathcal{X}}^{p}(I)$ for compact intervals. We remark that the
uniqueness of the solution in the following theorem is under the condition
$u(t,\cdot)\in
C({\mathbb{R}},{\mathscr{H}})\cap{\mathcal{X}}^{p}_{\textrm{loc}}$.

######  Theorem 2.2.

Let $\\{V_{jk}(t,y)\\}_{1\leq j<k\leq N}$ be $M(n)$-valued functions such that
$V_{jk}(t,y)\in
C({\mathbb{R}},L^{p}({\mathbb{R}}^{d}))+C({\mathbb{R}},L^{\infty}({\mathbb{R}}^{d}))$
for a $d/2\leq p\leq\infty$ and

$\sup_{t\in{\mathbb{R}}}(\|V_{jk}(t)\|_{L^{p}({\mathbb{R}}^{d})}+\|V_{jk}(t)\|_{L^{\infty}({\mathbb{R}}^{d})})\leq
C<\infty.$ (2.5)

Then, Eqn. (1.3) uniquely generates a propagator $U(t,s,\kappa)$ such that for
evey ${\varphi}\in{\mathscr{H}}$ $u(t,x)\colon=(U(t,s,\kappa){\varphi})(x)\in
C({\mathbb{R}},{\mathscr{H}})\cap{\mathcal{X}}^{p}_{\textrm{loc}}$. We have
$u(t,\cdot)\in C^{1}({\mathbb{R}},H^{-2}(X:{\mathbb{C}}^{n}))$ and (1.3) is
satisfied in $H^{-2}(X,{\mathbb{C}}^{n})$. Moreover:

  1. (1)

For any $a>0$, $\sup_{t,s\in{\mathbb{R}}\colon|t-s|\leq
a}\|U(t,s,\kappa)\|_{{\textbf{B}}({\mathscr{H}})}=C_{a}<\infty$ and

$\sup_{s\in{\mathbb{R}}}\|u(t)\|_{{\mathcal{X}}^{p}([s-a,s+a])}\leq
C_{a}\|{\varphi}\|_{{\mathscr{H}}}\,.$ (2.6)

  2. (2)

$U(t,s,\kappa)$ is unitary if all $V_{jk}(t,y)$ are Hermitian and
$\kappa\in{\mathbb{R}}$.

The second theorem is on the regularity of the solution obtained in Theorem
2.2.

######  Theorem 2.3.

Suppose $\\{V_{jk}(t,y)\\}_{1\leq j<k\leq N}$ satisfy (2.5) with
$\tilde{p}=\max(2,p)$ replacing $p$ and, in addition,

$\displaystyle\dot{V}_{jk}(t,y)\in
L^{b}_{\textrm{loc}}({\mathbb{R}},L^{q}({\mathbb{R}}^{d}))+L^{1}_{\textrm{loc}}({\mathbb{R}},L^{\infty}({\mathbb{R}}^{d})),$
(2.7) $\displaystyle 1/b=1-d/4p;\ 1/q=\left\\{\begin{array}[]{l}1/2+1/2p\
\mbox{if}\ d=3,\\\ 2/d+1/2p\ \mbox{if}\ d\geq 4.\end{array}\right.$ (2.10)

Then, for ${\varphi}\in H^{2}(X\colon{\mathbb{C}}^{n})$,
$u(t,x)=U(t,s,\kappa){\varphi}$ of Theorem 2.2 satisfies

$u(t,x)\in C^{1}({\mathbb{R}},{\mathscr{H}})\cap
C({\mathbb{R}},H^{2}(X:{\mathbb{C}}^{n})),\quad\dot{u}(t,x)\in{\mathcal{X}}^{{p}}_{\textrm{loc}}.$
(2.11)

We remark that $\tilde{p}=p$ when $d\geq 4$; if $d=3$ and $p<2$ and if
$V_{jk}(t,y)$ satisfies (2.5) with $\tilde{p}=\max(2,p)$ replacing $p$, then
it also satisfies (2.5) with the orginal $p$.

Theorems 2.2 and 2.3 except statement (1) of Theorem 2.2 are stated and proved
in [19] for the case $n=1$ with real valued $V_{jk}(t,y)$ and, with
$\Sigma(2)$ and $\Sigma(-2)$ in place of Sobolev spaces $H^{2}(X)$ and
$H^{-2}(X)$ respectively, where
$\Sigma(2)=\\{u\colon\|u\|_{\Sigma(2)}^{2}=\sum_{|\alpha|+|\beta|\leq
2}\|x^{\alpha}{\partial}^{\beta}u\|_{{\mathscr{H}}}^{2}<\infty\\}$ is domain
of the harmonic oscillator $-\Delta+x^{2}$ and $\Sigma(-2)$ is its dual space.
However, the extension to $n\geq 2$ with matrix-valued $V_{jk}(t,y)$ is
obvious; if external electro-magnetic fields are absent, then the propagator
$e^{-it\Delta}$ for the independent particles has Sobolev spaces
$H^{m}(X:{\mathbb{C}}^{n})$ as invariant subspaces and, $\Sigma(2)$ and
$\Sigma(-2)$ may be replaced by $H^{2}(X)$ and $H^{-2}(X)$ respectively.
Statement (1) of Theorem 2.2 is also evident since assumption (2.5) is
translation invariant with respect to $t\in{\mathbb{R}}$. We omit the proof of
Theorems 2.2 and 2.3,leaving for the readers to check the details.

If ${\varphi}\in H^{2}(X,{\mathbb{C}}^{n})$ and
$u(t,x)=(U(t,s,\kappa){\varphi})(x)$, then it is evident that
$\textbf{u}(t,x)=\begin{pmatrix}u(t,x)\\\ \nabla_{x}u(t,x)\end{pmatrix}\in
C^{(1+Nd)n}$ satisfies (1.6). The next theorem implies that $\textbf{u}(t,x)$
is a unique solution of (1.6). For shortening formulas we write
${\mathscr{H}}^{1}=H^{1}(X\colon{\mathbb{C}}^{n(1+Nd)})$.

######  Theorem 2.4.

Let $\\{V_{jk}(t,y)\\}$ and $\\{\nabla_{y}V_{jk}(t,y)\\}$ satisfy (2.5) with
$p$ being replaced by $\tilde{p}=\max(2,p)$ and ${\varphi}\in
H^{2}(X,{\mathbb{C}}^{n})$. Then, $\textbf{u}(t,x)$ satisfies (1.6). If
$\\{V_{jk}(t,y)\\}$ in addition satisfy (2.7), then u is a unique solution of
(1.6) with the initial condition
${\textbf{u}}(s,x)=\begin{pmatrix}{\varphi}(x)\\\
\nabla_{x}{\varphi}(x)\end{pmatrix}$ and ${\textbf{u}}\in
C({\mathbb{R}},{\mathscr{H}}^{1})(\subset
C({\mathbb{R}},{\mathcal{H}})\cap{\mathcal{X}}^{p}_{\textrm{loc}})$.

###### Proof.

If ${\varphi}\in H^{2}(X,{\mathbb{C}}^{n})$, then Theorem 2.3 and the Sobolev
embedding theorem imply that ${\textbf{u}}\in
C({\mathbb{R}},H^{1}({\mathbb{R}}^{d}))\subset
C(I,{\mathcal{H}})\cap{\mathcal{X}}^{p}_{\textrm{loc}}$. Then, by Theorem 2.2,
${\textbf{u}}(t,x)$ is the unique solution of (1.6) which satisfies the
condition of the theorem. ∎

##  3\. Howland scheme

For proving Theorem 1.1 we use the scattering theory for time dependent
potentials. Following Howland [11], we introduce the extended phase space by
${\mathscr{K}}=L^{2}({\mathbb{R}}^{1},{\mathscr{H}})=L^{2}({\mathbb{R}}^{1})\otimes{\mathscr{H}}$,
where ${\mathscr{H}}=L^{2}(X\colon{\mathbb{C}}^{n})$ in what follows, and “the
free propagator” on ${\mathscr{K}}$ by

$({\mathcal{U}}_{0}(\sigma)u)(t)=e^{-i\sigma H_{0}}u(t-\sigma),\quad
t\in{\mathbb{R}}\,,$

where $H_{0}=-(\Delta_{X})\otimes{\textbf{1}}_{{\mathbb{C}}^{n}}$. It is easy
to see that ${\mathcal{U}}_{0}(\sigma)$ is the unitary group generated by the
selfajoint operator ${\mathcal{K}}_{0}$ on ${\mathscr{K}}$:

${\mathcal{K}}_{0}=(-i{\partial}_{t})\otimes{\textbf{1}}_{{\mathscr{H}}}+{\textbf{1}}_{L^{2}({\mathbb{R}}^{1})}\otimes
H_{0}.$ (3.1)

When condition (1) of Theorem 1.1 is satisfied for $m=0$, then
$\\{V_{jk}(t,y)\\}$ satisfy (2.5) and eqn. (1.3) uniquely generates a
propagator $\\{U(t,s,\kappa)\\}$. We define a one-parameter family of bounded
of operators $\\{{\mathcal{U}}(\sigma,\kappa)\colon\sigma\in{\mathbb{R}}\\}$
on ${\mathscr{K}}$ by

$({\mathcal{U}}(\sigma,\kappa)u)(t)=U(t,t-\sigma,\kappa)u(t-\sigma,\kappa),\quad
t\in{\mathbb{R}}\,.$ (3.2)

In what follows untill the last section, assuming $|\kappa|<\kappa_{0}$ for a
sufficiently small $\kappa_{0}$, we omit the coupling constant $\kappa$ from
various quantities

######  Lemma 3.1 ([11]).

The family $\\{{\mathcal{U}}(\sigma)\colon\sigma\in{\mathbb{R}}\\}$ is a
strongly continuous one-parameter group of bounded operators on
${\mathscr{K}}$. For constants $C\geq 1$ and $M\geq 0$ it satisfies

$\|{\mathcal{U}}(\sigma)\|_{{\textbf{B}}({\mathscr{K}})}\leq
Ce^{|\sigma|M},\quad\sigma\in{\mathbb{R}}.$ (3.3)

There exits a unique closed operator ${\mathcal{K}}$ such that

${\mathcal{U}}(\sigma)=e^{-i\sigma{\mathcal{K}}},\quad-\infty<\sigma<\infty,$
(3.4)

the spectrum of ${\mathcal{K}}$ lies in the strip
$\\{\zeta\colon|\Im\zeta|\leq M\\}$ and

$({\mathcal{K}}-\zeta)^{-1}=\pm i\int_{0}^{\infty}e^{\mp
i({\mathcal{K}}-\zeta)\sigma}d\sigma,\quad\pm\Im\zeta>M.$ (3.5)

######  Remark 3.2.

It will be proved in the next subsection that $M=0$ and
“${\mathcal{K}}={\mathcal{K}}_{0}+V$”.

###### Proof.

The Chapman-Kolmogorov identity implies
${\mathcal{U}}(\sigma){\mathcal{U}}(\tau)={\mathcal{U}}(\sigma+\tau)$ for
$\sigma,\tau\in{\mathbb{R}}$. By Theorem 2.2 (1) we have for an arbitrary
$\sigma_{0}$ that

$\|{\mathcal{U}}(\sigma)u\|_{{\mathscr{K}}}^{2}=\int_{{\mathbb{R}}}\|U(t,t-\sigma)u(t-\sigma)\|_{{\mathscr{H}}}^{2}dt\leq
C_{\sigma_{0}}\|u\|_{{\mathscr{K}}}^{2},\quad|\sigma|<\sigma_{0}.$

It follows that $\\{{\mathcal{U}}(\sigma)\colon\sigma\in{\mathbb{R}}\\}$
defines a one-parameter group of bounded operators on ${\mathscr{K}}$ which is
locally uniformly bounded. Let $u\in C_{0}({\mathbb{R}},{\mathscr{H}})$ be
supported by the interval $[a,b]\Subset{\mathbb{R}}$. Then, for
$|\sigma|\leq\sigma_{0}$,
$\|U(t,t-\sigma)u(t-\sigma)-u(t)\|_{{\mathscr{H}}}\in
C_{0}([a-\sigma_{0},b+\sigma_{0}])$ and it converges to $0$ everywhere for
$t\in{\mathbb{R}}$ as $\sigma\to 0$. Hence

$\|{\mathcal{U}}(\sigma)u-u\|_{{\mathscr{K}}}^{2}=\int_{{\mathbb{R}}}\|U(t,t-\sigma)u(t-\sigma)-u(t)\|_{{\mathscr{H}}}^{2}dt\to
0\quad(\sigma\to 0).$

Since $C_{0}({\mathbb{R}},{\mathscr{H}})$ is dense in ${\mathscr{K}}$, this
implies ${\mathcal{U}}(\sigma)$ is strongly continuous and
$\\{{\mathcal{U}}(\sigma)\\}$ is a $C_{0}$-group of bounded operators. Then,
the rest is well known, see e.g. [20]. ∎

##  4\. Wave operators

In [4], the boundedness of $\|u(t)\|_{H^{s}(X)}$ for the case $N=2$ is proved
by using the local smoothing property and the dispersive estimate for (1.1).
Here we shall prove it for $N=2,3,\dots$ via the following Theorem 4.1 on wave
operators of scattering theory which will be proved in §6.

######  Theorem 4.1.

Let $n=1,2,\dots$. Suppose that $\\{V_{jk}(t,y)\\}_{1\leq j<k\leq N}\subset
M(n)$ are factorized as $V_{jk}(t,y)=B_{jk}^{\ast}(t,y)A_{jk}(t,y)$ by
$A_{jk}(t,y),B_{jk}(t,y)\in M(n)$ which satisfy condition (1) of Theorem 1.1
for $m=0$. Then there exists a $\kappa_{0}$ such that, for all
$|\kappa|<\kappa_{0}$, the following statments are satisfied:

  1. (1)

Both of the following strong limits in ${\mathscr{K}}$ exist:

$\displaystyle{\mathcal{W}}_{\pm}(\kappa)=\lim_{\sigma\to\pm\infty}{\mathcal{U}}(-\sigma,\kappa){\mathcal{U}}_{0}(\sigma),$
(4.1)
$\displaystyle{\mathcal{Z}}_{\pm}(\kappa)=\lim_{\sigma\to\pm\infty}{\mathcal{U}}_{0}(-\sigma){\mathcal{U}}(\sigma,\kappa)\,.$
(4.2)

They satisfy the following identities and are isomorphisms in ${\mathscr{K}}$:

${\mathcal{W}}_{\pm}(\kappa){\mathcal{Z}}_{\pm}(\kappa)={\mathcal{Z}}_{\pm}(\kappa){\mathcal{W}}_{\pm}(\kappa)={\textbf{1}}_{{\mathscr{K}}}\,.$
(4.3)

  2. (2)

The operators ${\mathcal{W}}_{\pm}(\kappa)$ satisfy the intertwing property:

${\mathcal{U}}(\sigma,\kappa)={\mathcal{W}}_{\pm}(\kappa){\mathcal{U}}_{0}(\sigma){\mathcal{W}}_{\pm}(\kappa)^{-1}.$
(4.4)

We postpone the proof of Theorem 4.1 and proceed to proving Theorem 1.1,
admitting Theorem 4.1 has already been proved. We first prove that Theorem 4.1
implies the existence and the completeness of the wave operators for (1.3).

######  Theorem 4.2.

Let $n=1,2,\dots$. Suppose the condition of Theorem 4.1 is satisfied and
$|\kappa|$ is sufficiently small. Then, the following statements are
satisfied:

  1. (1)

Both of the strong limits

$\displaystyle
W_{\pm}(s,\kappa)=\lim_{t\to\pm\infty}U(s,t,\kappa)e^{-i(t-s)H_{0}},$ (4.5)
$\displaystyle
Z_{\pm}(s,\kappa)=\lim_{t\to\pm\infty}e^{i(t-s)H_{0}}U(t,s,\kappa)$ (4.6)

exist in ${\textbf{B}}({\mathscr{H}})$ for all $s\in{\mathbb{R}}$. They
satisfy

$W_{\pm}(s,\kappa)Z_{\pm}(s,\kappa)=Z_{\pm}(s,\kappa)W_{\pm}(s,\kappa)={\textbf{1}}_{{\mathscr{H}}}$
(4.7)

and are isomorphisms in ${\mathscr{H}}$.

  2. (2)

There exists an $s$-independent constant $C$ such that

$\|W_{\pm}(s,\kappa)\|_{{\textbf{B}}({\mathscr{H}})}\leq
C,\quad\|Z_{\pm}(s,\kappa)\|_{{\textbf{B}}({\mathscr{H}})}\leq C.$ (4.8)

  3. (3)

The wave operators satisfy the intertwing property:

$U(t,s,\kappa)=W_{\pm}(t,\kappa)e^{-i(t-s)H_{0}}Z_{\pm}(s,\kappa)\,.$ (4.9)

######  Remark 4.3.

Statement (1) of Theorem 4.2 implies that for any ${\varphi}\in{\mathscr{H}}$,
the solution $U(t,s,\kappa){\varphi}$ becomes asymptotically free as
$t\to\pm\infty$:

$\lim_{t\to\pm\infty}\|U(t,s,\kappa){\varphi}-e^{-i(t-s)H_{0}}{\varphi}_{\pm}\|_{{\mathscr{H}}}=0,\quad{\varphi}_{\pm}=Z_{\pm}(s,\kappa){\varphi}.$

###### Proof.

We prove the $+$ case only. The proof for the other case is similar. Theorem
4.1 implies that for any $u\in{\mathscr{K}}$ the limit

$\lim_{\sigma\to\infty}{\mathcal{U}}(-\sigma){\mathcal{U}}_{0}(\sigma)u(t)=\lim_{\sigma\to\infty}U(t,t+\sigma)U_{0}(t+\sigma,t)u(t)=({\mathcal{W}}_{+}u)(t)$
(4.10)

exists in ${\mathscr{K}}$. Let $u_{0}\in{\mathscr{H}}$, $a>0$ be arbitrary and
set $u(t)=U_{0}(t,0)u_{0}$ for $-a\leq t\leq a$ and $u(t)=0$ for $|t|>a$.
Then, $u\in{\mathscr{K}}$ and, on substituting this $u(t)$ in (4.10), we
obtain that as $\sigma\to\infty$

$\int_{-a}^{a}\|U(t,\sigma)U_{0}(\sigma,0)u_{0}-({\mathcal{W}}_{+}u)(t)\|_{\mathscr{H}}^{2}dt\to
0\quad(\sigma\to\infty).$

Since $\|U(0,t)\|_{{\textbf{B}}({\mathscr{H}})}\leq C$ for $t\in[-a,a]$, it
follows that

$\lim_{\sigma\to\infty}\int_{-a}^{a}\|U(0,\sigma)U_{0}(\sigma,0)u_{0}-U(0,t)({\mathcal{W}}_{+}u)(t)\|_{\mathscr{H}}^{2}=0\,.$

Then Schwarz’s inequality implies that, as $\sigma\to\infty$ ,

$\Big{\|}\int_{-a}^{a}(U(0,\sigma)U_{0}(\sigma,0)u_{0}-U(0,t)({\mathcal{W}}_{+}u)(t))dt\Big{\|}_{{\mathscr{H}}}\\\
\leq\sqrt{2a}\Big{(}\int_{-a}^{a}\|U(0,\sigma)U_{0}(\sigma,0)u_{0}-U(0,t)({\mathcal{W}}_{+}u)(t)\|_{{\mathscr{H}}}^{2}dt\Big{)}^{\frac{1}{2}}\to
0.$

Thus, $U(0,\sigma)U_{0}(\sigma,0)u_{0}$ converges in ${\mathscr{H}}$ as
$\sigma\to\infty$. Since $u_{0}\in{\mathscr{H}}$ is arbitrary, this implies
that for any $t$, the strong limit of (4.5):

$W_{+}(t)=\lim_{\sigma\to\infty}U(t,\sigma)U_{0}(\sigma,t)=U(t,0)(\lim_{\sigma\to\infty}U(0,\sigma)U_{0}(\sigma,0))U_{0}(0,t)=$

exists in ${\mathscr{H}}$ for any $t\in{\mathbb{R}}$ and (4.10) implies

$({\mathcal{W}}_{+}u)(t)=W_{+}(t)u(t),\quad a.e.\ t\in{\mathbb{R}}.$ (4.11)

The proof for $Z_{+}(s)$ is similar. They are bounded operators in
${\mathscr{H}}$ as they are strong limits of bounded operators. We then have

$W_{+}(s)Z_{+}(s)=\lim_{t\to\infty}U(s,t,\kappa)e^{-i(t-s)H_{0}}e^{i(t-s)H_{0}}U(t,s,\kappa)=\textbf{1}$

and, likewise, $Z_{+}(s)W_{+}(s)=\textbf{1}$, which imply the identity (4.7)
and $W_{+}(s)$ and $Z_{+}(s)$ are isomorphisms in ${\mathscr{H}}$.

(2) The Chappman-Kolmogorov identity and the strong convergence imply

$W_{+}(s)u_{0}=U(s,t)W_{+}(t)U_{0}(t,s)u_{0}$

for any $t,s\in{\mathbb{R}}$ and any $u_{0}\in{\mathscr{H}}$. Fix
$s\in{\mathbb{R}}$ and $u_{0}\in{\mathscr{H}}$ arbitary and define
$u(t)=U_{0}(t,s)u_{0}$ for $|t-s|\leq 1$ and $u(t)=0$ for $|t-s|>1$. Then,
since $\|U(t,s)\|_{{\textbf{B}}({\mathscr{H}})}\leq C$ for $|t-s|\leq 1$ and
$\|U_{0}(t,s)\|_{{\textbf{B}}({\mathscr{H}})}=1$,

$\|W_{+}(s)u_{0}\|_{{\mathscr{H}}}\leq\frac{1}{2}\int_{s-1}^{s+1}\|U(s,t)W_{+}(t)u(t)\|_{{\mathscr{H}}}dt\leq\frac{C}{\sqrt{2}}\left(\int_{s-1}^{s+1}\|W_{+}(t)u(t)\|_{{\mathscr{H}}}^{2}dt\right)^{\frac{1}{2}}$

and, by using (4.11), we estimate the right side by

$\frac{C\|{\mathcal{W}}_{+}\|_{{\textbf{B}}({\mathscr{K}})}}{\sqrt{2}}\left(\int_{s-1}^{s+1}\|u(t)\|_{{\mathscr{H}}}^{2}dt\right)^{\frac{1}{2}}\leq
C\|{\mathcal{W}}_{+}\|_{{\textbf{B}}({\mathscr{K}})}\|u_{0}\|\,.$

Here $C$ is independent of $s\in{\mathbb{R}}$ and hence,
$\|W_{+}(s)\|_{{\textbf{B}}({\mathscr{H}})}$ is uniformly bounded for
$s\in{\mathbb{R}}$. Likewise we obtain
$\|Z_{+}(s)\|_{{\textbf{B}}({\mathscr{H}})}\leq C$.

(3) The standard argument for the selfadjoint case applies for (4.9). We omit
the details. ∎

##  5\. Proof of Theorem 1.1

We prove Theorem 1.1 by induction on $m$ admitting Theorem 4.2.

###  Proof of Theorem 1.1 for the case $m=0$ and $n=1,2,\dots$

. Since $\\{W_{\pm}(t)\colon t\in{\mathbb{R}}\\}$ and $\\{Z_{\pm}(s)\colon
s\in{\mathbb{R}}\\}$ are uniformly bounded in ${\mathscr{H}}$ and
$e^{-itH_{0}}$ is unitary for $t\in{\mathbb{R}}$, we have from (4.9) that for
${\varphi}\in{\mathscr{H}}$

$\|U(t,s){\varphi}\|_{{\mathscr{H}}}=\|W_{\pm}(t)e^{-i(t-s)H_{0}}Z_{\pm}(s){\varphi}\|_{{\mathscr{H}}}\leq
C\|{\varphi}\|_{{\mathscr{H}}}$

for a constant $C>0$ independent of $(t,s)$. This proves Theorem 1.1 for the
case $m=0$.

###  Proof of Theorem 1.1 for general $m=1,2,\dots,$

We assume that Theorem 1.1 has already been proved for $m=0,\dots,k$ and for
all $n=1,2,\dots$ and suppose $m=k+1$. We prove

$\|U(t,0){\varphi}\|_{H^{k+1}(X\colon{\mathbb{C}}^{n})}\leq
C_{k+1}\|{\varphi}\|_{H^{k+1}(X\colon{\mathbb{C}}^{n})},\ n=1,2,\dots.$ (5.1)

For shortening formulas we write
${\mathscr{H}}^{s}=H^{s}(X\colon{\mathbb{C}}^{n(Nd+1)})$ and $U(t)$ for
$U(t,0)$. It suffices (5.1) for ${\varphi}\in
H^{k+2}(X\colon{\mathbb{C}}^{n})$ which is dense in
$H^{k+1}(X\colon{\mathbb{C}}^{n})$. Then, since $k+2\geq 2$, Theorem 2.3
implies

${\textbf{u}}(t)\colon=\begin{pmatrix}U(t){\varphi}\\\
\nabla_{x}U(t){\varphi}\end{pmatrix}\in C({\mathbb{R}},{\mathscr{H}}^{1})\cap
C({\mathbb{R}},{\mathscr{H}}^{0})\subset{\mathscr{X}}^{p}_{\textrm{loc}}$

is the unique solution of the initial value problem for the Schrödinger
equation (1.6):

$i{\partial}_{t}{\textbf{u}}=-\Delta_{x}{\textbf{u}}+\kappa\sum_{1\leq j<k\leq
N}\begin{pmatrix}V_{jk}(t)&{\textbf{0}}\\\
{\nabla}V_{jk}(t)&V_{jk}(t){\textbf{1}}\end{pmatrix}{\textbf{u}},\
{\textbf{u}}(0)\colon=\begin{pmatrix}{\varphi}\\\
\nabla_{x}{\varphi}\end{pmatrix}$ (5.2)

and, the “potential” in (5.2) may be factorized as

$\begin{pmatrix}V_{jk}(t)&{\textbf{0}}\\\
{\nabla}V_{jk}(t)&V_{jk}(t){\textbf{1}}\end{pmatrix}=\begin{pmatrix}B_{jk}(t)^{\ast}&({\nabla}B_{jk}(t))^{\ast}\\\
{\textbf{0}}&B_{jk}(t)^{\ast}{\textbf{1}}\end{pmatrix}^{\ast}\begin{pmatrix}A_{jk}(t)&{\textbf{0}}\\\
{\nabla}A_{jk}(t)&A_{jk}(t){\textbf{1}}\end{pmatrix}$

and the factors

$A_{jk}^{\prime}(t,y)=\begin{pmatrix}A_{jk}(t,y)&{\textbf{0}}\\\
{\nabla}A_{jk}(t,y)&A_{jk}(t,y){\textbf{1}}\end{pmatrix},\
B_{jk}^{\prime}(t,y)=\begin{pmatrix}B_{jk}(t,y)^{\ast}&({\nabla}B_{jk}(t,y))^{\ast}\\\
{\textbf{0}}&B_{jk}(t,y)^{\ast}{\textbf{1}}\end{pmatrix}$

satisfy the conditions (2.5) and (2.7) with for $m=k$ as $M(n(Nd+1))$-valued
functions. Hence, the induction hypothese implies

$\sup_{t\in{\mathbb{R}}}\|{\textbf{u}}(t)\|_{{\mathscr{H}}^{k}}\leq
C\|{\textbf{u}}(0)\|_{{\mathscr{H}}^{k}},$

which implies (5.1) and Theorem 1.1 is proved for all integer s $k\geq 0$. ∎

##  6\. Proof of Theorem 4.1

We apply Kato’s theory of smooth perturbation ([13]). Let
$\tilde{{\mathscr{H}}}$ and $\tilde{{\mathscr{H}}}^{\prime}$ be separable
Hilbert spaces (possibly
$\tilde{{\mathscr{H}}}=\tilde{{\mathscr{H}}}^{\prime}$), $T$ a selfadjoint
operator on $\tilde{{\mathscr{H}}}$ and $A$ a closed operator from
$\tilde{{\mathscr{H}}}$ to $\tilde{{\mathscr{H}}}^{\prime}$. $A$ is said to be
$T$-smooth if $A$ is $T$-bounded and for any $u\in\tilde{{\mathscr{H}}}$

$\sup_{\varepsilon>0}\int_{{\mathbb{R}}}\|A(T-\lambda\pm
i\varepsilon)^{-1}u\|_{\tilde{{\mathscr{H}}}^{\prime}}^{2}d\lambda=\frac{1}{2\pi}\int_{0}^{\infty}\|Ae^{\pm
itT}u\|_{\tilde{{\mathscr{H}}}^{\prime}}^{2}dt\leq C\|u\|^{2}..$ (6.1)

Recall ${\mathcal{K}}_{0}$ is the selfadjoint operator on
${\mathscr{K}}=L^{2}({\mathbb{R}})\otimes{\mathscr{H}}$ defined by (3.1). We
have the obvious identity:

$({\mathcal{K}}_{0}-\zeta)^{-1}=\pm
i\int_{0}^{\infty}e^{\mp({\mathcal{K}}_{0}-\zeta)\sigma}d\sigma,\quad\zeta\in{\mathbb{C}}^{\pm}.$
(6.2)

Let $A_{jk}(t)$ and $B_{jk}(t)$ be the multiplication operators on
${\mathscr{H}}$ with $A_{jk}(t,x^{jk})$ and $B_{jk}(t,x^{jk})$ respectively
and ${\mathcal{A}}_{jk}$ and ${\mathcal{B}}_{jk}$ be multiplication operators
on ${\mathscr{K}}$ defined by

${\mathcal{A}}_{jk}u(t,x)=A_{jk}(t)u(t,x),\quad{\mathcal{B}}_{jk}u(t,x)=B_{jk}(t)u(t,x)\,.$

######  Lemma 6.1.

Suppose that the conditions of Theorem 4.1 are satisfied. Then
${\mathcal{A}}_{jk}$ and ${\mathcal{B}}_{jk}$ are ${\mathcal{K}}_{0}$-smooth.

###### Proof.

We prove the lemma for ${\mathcal{A}}_{jk}$. The proof for
${\mathcal{B}}_{jk}$ is similar. We write ${\mathcal{S}}$ for
${\mathcal{S}}({\mathbb{R}}^{1}\times X\colon{\mathbb{C}}^{n})$ for
simplicity. For $u\in{\mathcal{S}}$, it is easy to see that
$(e^{-i\sigma{\mathcal{K}}_{0}}u)(t,\cdot)=e^{-i\sigma{H}_{0}}u(t-\sigma,\cdot)\in{\mathcal{S}}$
and ${\mathcal{A}}_{jk}e^{-i\sigma{{\mathcal{K}}_{0}}}u\in{\mathscr{K}}$.
Then, changing the variable $t$ to $t+\sigma$ and changing the order of
integrations, we have

$\int_{{\mathbb{R}}}\|{\mathcal{A}}_{jk}e^{-i\sigma{{\mathcal{K}}_{0}}}u\|_{{\mathscr{K}}}^{2}d\sigma=\int_{{\mathbb{R}}}\left(\int_{{\mathbb{R}}}\|A_{jk}(t+\sigma)e^{-i\sigma{H_{0}}}u(t)\|_{{\mathscr{H}}}^{2}d\sigma\right)dt.$
(6.3)

Since $e^{-i{\sigma}H_{0}}=e^{i{\sigma}\Delta_{x^{jk}}}\otimes
e^{i{\sigma}\Delta_{x_{jk}}}$ and since $e^{i{\sigma}\Delta_{x_{jk}}}$ is
unitary and commutes with $A_{jk}(t,x^{jk})$, Hölder’s inequality with respect
to the variable $x^{jk}$ implies

$\displaystyle\|A_{jk}(t+\sigma)e^{-i\sigma{H_{0}}}u(t)\|_{{\mathscr{H}}}^{2}=\int_{X_{jk}}\|A_{jk}(t+\sigma)e^{i\sigma\Delta_{x^{jk}}}u(t,x^{jk},x_{jk})\|_{L^{2}(X_{jk}^{r})}^{2}dx_{jk}$
$\displaystyle\leq\sup_{t\in{\mathbb{R}}}\|A_{jk}(t)\|_{L^{d}}^{2}\int_{X_{jk}^{c}}\|e^{i\sigma\Delta_{x_{jk}}}u(t,x^{jk},{x_{jk}})\|_{L^{\frac{2d}{d-2}}(X_{jk}^{r})}^{2}dx_{jk}$
(6.4)

We denote $C_{jk}=\sup_{t\in{\mathbb{R}}}\|A_{jk}(t)\|_{L^{d}}^{2}$ and
integrate both sides of (6.4) by $d\sigma$. Changing the order of integrations
and applying the end point Strichartz inequality ([14]):

$\int_{{\mathbb{R}}}\|e^{i\sigma\Delta_{x^{jk}}}u(t,x^{jk},x_{jk})\|_{L^{\frac{2d}{d-2}}(X_{jk}^{r})}^{2}d\sigma\leq
C_{KT}\|u(t,x^{jk},x_{jk})\|_{L^{2}(X_{jk}^{r})}^{2},$

we have that there exists a constant $C$ independent of $u$ and $t$ such that

$\int_{0}^{\infty}\|A_{jk}(t+\sigma)e^{-i\sigma{H_{0}}}u(t)\|_{{\mathscr{H}}}^{2}d\sigma\leq
C\|u(t)\|_{{\mathscr{H}}}^{2}.\quad u\in{\mathcal{S}}.$ (6.5)

On substituting the result into (6.3) we obtain

$\int_{{\mathbb{R}}}\|{\mathcal{A}}_{jk}e^{-i\sigma{{\mathcal{K}}_{0}}}u\|_{{\mathscr{K}}}^{2}d\sigma\leq
C\|u\|_{{\mathscr{K}}}^{2},\ \ u\in{\mathcal{S}}.$ (6.6)

It follows by Schwarz’ equality that for $\varepsilon>0$

$\int_{0}^{\infty}e^{-\sigma\varepsilon}\|{\mathcal{A}}_{jk}e^{-i\sigma{\mathcal{K}}_{0}}u\|_{{\mathscr{K}}}d\sigma\leq
C(2\varepsilon)^{-\frac{1}{2}}\|u\|_{{\mathscr{K}}}.$ (6.7)

Then, by applying Minkowski’s inequality, we obtain from (6.7) that

$\|{\mathcal{A}}_{jk}({\mathcal{K}}_{0}-\lambda-i\varepsilon)^{-1}u\|_{{\mathscr{K}}}=\left\|\int_{0}^{\infty}e^{i\sigma\lambda}e^{-\sigma\varepsilon}{\mathcal{A}}_{jk}e^{-i\sigma{\mathcal{K}}_{0}}ud\sigma\right\|_{{\mathscr{K}}}\leq
C\varepsilon^{-\frac{1}{2}}\|u\|_{{\mathscr{K}}}.$ (6.8)

Since ${\mathcal{A}}_{jk}$ is closed and since ${\mathcal{S}}$ is dense in
${\mathscr{K}}$, both of (6.6) and (6.8) extend to all $u\in{\mathscr{K}}$ and
${\mathcal{A}}_{jk}$ is ${\mathcal{K}}_{0}$-smooth. ∎

Let
${\mathscr{K}}_{N}={\mathscr{K}}\otimes{\mathbb{C}}^{N(N-1)/2}={\mathscr{K}}^{\oplus(N(N-1)/2)}$
and ${\mathcal{A}}$ and ${\mathcal{B}}$ be the multiplication operators from
${\mathscr{K}}$ to ${\mathscr{K}}_{N}$ defined by

${\mathcal{A}}u=\oplus_{j<k}{\mathcal{A}}_{jk}u,\quad{\mathcal{B}}u=\oplus_{j<k}{\mathcal{B}}_{jk}u$

respectively so that the multiplication operator with $V(t,x)$ on
${\mathscr{K}}$ is equal to ${\mathcal{B}}^{\ast}{\mathcal{A}}$. Denote
${\mathcal{R}}_{0}(\zeta)=({\mathcal{K}}_{0}-\zeta)^{-1}$.

######  Lemma 6.2.

  1. (1)

The operators ${\mathcal{A}}$ and ${\mathcal{B}}$ are
${\mathcal{K}}_{0}$-smooth.

  2. (2)

There exists a constant $C$ independent of
$\zeta\in{\mathbb{C}}^{\pm}=\\{\pm\Im z>0\\}$ such that

$\|{\mathcal{A}}{\mathcal{R}}_{0}(\zeta){\mathcal{B}}^{\ast}u\|_{{\mathscr{K}}_{N}}\leq
C\|u\|_{{\mathscr{K}}_{N}},\quad u\in D({\mathcal{B}}^{\ast}).$ (6.9)

###### Proof.

Statement (1) is obvious by Lemma 6.1 and we prove (2). The following is an
adaptation of Iorio-O’Carrol’s argument ([12]) to the Howland scheme. It
suffices to show that

$\sup_{\pm\Im\zeta>0}\|{\mathcal{A}}_{jk}{\mathcal{R}}_{0}(\zeta){\mathcal{B}}_{lm}^{\ast}u\|_{{\mathscr{K}}}\leq
C\|u\|_{{\mathscr{K}}},\quad j<k,l<m.$ (6.10)

We prove the $+$ case. The proof for the other case is similar. For
$u,v\in{\mathcal{S}}$, we have as in (6.8) that

$|({\mathcal{R}}_{0}(\zeta){\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}|\leq\int_{0}^{\infty}|(e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}|d\sigma.$
(6.11)

1) Let first $\\{j,k\\}\cap\\{l,m\\}=\emptyset$. Define $X_{jklm}$ by
$X=X_{jk}^{r}\oplus X_{lm}^{r}\oplus X_{jklm}$ and let $x=x^{jk}\oplus
x^{lm}\oplus x^{\prime}$ be the corresponding orthogonal decomposition. Then

$H_{0}=-\Delta_{jk}-\Delta_{lm}-\Delta^{\prime},\ \
\Delta_{jk}\colon=\Delta_{x^{jk}},\ \Delta_{lm}\colon=\Delta_{x^{lm}},\
\Delta^{\prime}\colon=\Delta_{x^{\prime}}.$

Since $e^{i\sigma(\Delta_{jk}+\Delta^{\prime})}$ and $e^{i\sigma\Delta_{lm}}$
commute with the multiplications $B_{lm}^{\ast}(t)$ and $A_{jk}^{\ast}(t)$
respectively,
$(e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}$
is equal to

$\displaystyle\int_{{\mathbb{R}}}(e^{i\sigma\left(\Delta_{jk}+\Delta_{lm}+\Delta^{\prime}\right)}B_{lm}^{\ast}(t-\sigma)u(t-\sigma),A_{jk}^{\ast}(t)v(t))_{{\mathscr{H}}}dt$
$\displaystyle=\int_{{\mathbb{R}}}(A_{jk}(t)e^{i\sigma\left(\Delta_{jk}+\Delta^{\prime}\right)}u(t-\sigma),B_{lm}(t-\sigma)e^{-i\sigma\Delta_{lm}}v(t))_{{\mathscr{H}}}dt.$
(6.12)

Then, since $e^{i\sigma\Delta^{\prime}}$ commutes with $A_{jk}(t)$ and is
unitary, Schwarz’s inequality implies

$\displaystyle|(e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}|$
$\displaystyle\leq\left(\int_{{\mathbb{R}}}\|A_{jk}(t)e^{i\sigma\Delta_{jk}}u(t-\sigma)\|_{{\mathscr{H}}}^{2}dt\right)^{\frac{1}{2}}$
$\displaystyle\times\left(\int_{{\mathbb{R}}}\|B_{lm}(t-\sigma)e^{i\sigma\Delta_{lm}}v(t)\|_{{\mathscr{H}}}^{2}dt\right)^{\frac{1}{2}}.$
(6.13)

Change variable $t$ by $t+\sigma$ in the first factor, integrate both sides of
(6.13) by $d\sigma$, apply Schwarz’s inequality once more and change the order
of integrartions. We obtain

$\displaystyle\int_{0}^{\infty}|(e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}|d\sigma\leq\left(\int_{{\mathbb{R}}}\left(\int_{0}^{\infty}\|A_{jk}(t+\sigma)e^{i\sigma\Delta_{jk}}u(t)\|_{{\mathscr{H}}}^{2}d\sigma\right)dt\right)^{\frac{1}{2}}$
$\displaystyle\hskip
56.9055pt\times\left(\int_{{\mathbb{R}}}\left(\int_{0}^{\infty}\|B_{lm}(t-\sigma)e^{i\sigma\Delta_{lm}}v(t)\|_{{\mathscr{H}}}^{2}d\sigma\right)dt\right)^{\frac{1}{2}}$
(6.14)

Then, repeating the argument used for (6.7), we obtain by using Hölder’s, the
end point Strichartz inequality and the density of ${\mathcal{S}}$ in
${\mathscr{K}}$ that

${\textrm{(}\ref{eqn:23})}\leq
C\sup_{t\in{\mathbb{R}}}\|A_{jk}(t,\cdot)\|_{L^{d}}\sup_{t\in{\mathbb{R}}}\|B_{lm}(t,\cdot)\|_{L^{d}}\|u\|_{{\mathscr{K}}}\|v\|_{{\mathscr{K}}}\,,\
u,v\in{\mathscr{K}}.$ (6.15)

It follows from (6.11) and (6.15) that there exists a constant $C>0$
independent of $u,v\in{\mathscr{K}}$ and $\zeta\in{\mathbb{C}}^{+}$ such that

$|({\mathcal{R}}_{0}(\zeta){\mathcal{B}}_{lm}^{\ast}u,{\mathcal{A}}_{jk}^{\ast}v)_{{\mathscr{K}}}|\leq
C\|u\|_{{\mathscr{K}}}\|v\|_{{\mathscr{K}}}$

and we obtain the desired (6.10) for the case
$\\{j,k\\}\cap\\{l,m\\}=\emptyset$.

2) Let next $\\{j,k\\}=\\{l,m\\}$. Let $u\in{\mathcal{S}}$ and
$\Delta_{jk}=\Delta_{x^{jk}}$. We sandwitch (6.2) by ${\mathcal{A}}_{jk}$ and
${\mathcal{B}}_{jk}$ and apply Minowski’s inequlaity. We obtain for any
$\zeta\in{\mathbb{C}}^{+}$ that

$\displaystyle\|{\mathcal{A}}_{jk}{\mathcal{R}}_{0}(\zeta){\mathcal{B}}_{jk}^{\ast}u\|_{{\mathscr{K}}}\leq\int_{0}^{\infty}\|{\mathcal{A}}_{jk}e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{jk}^{\ast}u\|_{{\mathscr{K}}}d\sigma$
$\displaystyle\leq\int_{0}^{\infty}\left(\int_{{\mathbb{R}}}\|A_{jk}(t)e^{i\sigma\Delta_{jk}}B_{jk}^{\ast}(t-\sigma)u(t-\sigma)\|_{L^{2}(X_{jk}^{r})\otimes
L^{2}(X_{jk})}^{2}dt\right)^{\frac{1}{2}}d\sigma.$ (6.16)

The well-known $L^{p}$-$L^{q}$ estimates for $e^{i\sigma\Delta_{jk}}$ on
$L^{2}(X_{jk}^{r})$ and Hölder’s inequality imply that the integrand of the
inner integral of (6.16) is bounded for any $\rho\geq 2$ by

$C_{r}(\sup_{t\in{\mathbb{R}}}\|A_{jk}(t,\cdot)\|_{L^{\rho}(X_{jk}^{r})}\sup_{t\in{\mathbb{R}}}\|B_{jk}(t,\cdot)\|_{L^{\rho}(X_{jk}^{r})})|\sigma|^{-\frac{d}{\rho}}\|u(t-\sigma)\|_{{\mathscr{H}}}^{2}.$
(6.17)

Since $A_{jk}(t,\cdot)$ and $B_{jk}(t,\cdot)$ are $(L^{p}\cap
L^{q})(X_{jk}^{r})$-valued bounded function of $t\in{\mathbb{R}}$ for $p,q$
such that $2\leq p<d<q\leq\infty$, we have by applying (6.17) for $\rho=p$ and
$\rho=q$ that

${\textrm{(}\ref{eqn:3-3})}\leq
C\int_{0}^{\infty}\min_{r=p,q}|\sigma|^{-\frac{d}{r}}\left(\int_{{\mathbb{R}}}\|u(t-\sigma)\|^{2}_{{\mathscr{H}}}dt\right)^{\frac{1}{2}}d\sigma\leq
C\|u\|_{{\mathscr{K}}}.$

Hence (6.10) holds also for $\\{j,k\\}=\\{l,m\\}$.

3) Finally let $|\\{j,k\\}\cap\\{l,m\\}|=1$. We may assume $N=3$ and
$\\{j,k\\}=\\{1,2\\}$ and $\\{l,m\\}=\\{2,3\\}$ without losing generality. We
use Jacobi-coordinates for three particle:

$r=x_{2}-x_{1},\ \ y=x_{3}-\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}},\ \
x_{cm}=\frac{m_{1}x_{1}+m_{2}x_{2}+m_{3}x_{3}m}{m_{1}+m_{2}+m_{3}}.$

Let $\mu$, $\nu$ be reduced masses and $m$ the total mass:

$\mu=m_{1}m_{2}/(m_{1}+m_{2}),\ \ \nu=(m_{1}+m_{2})m_{3}/(m_{1}+m_{2}+m_{3},\
\ m=m_{1}+m_{2}+m_{3}.$

Then, $x_{3}-x_{2}=y-ar$, $a=m_{1}r/(m_{1}+m_{2})$, $dx=Cdrdydx_{cm}$ and

$\Delta_{x}=\frac{1}{2\mu}\Delta_{r}+\frac{1}{2\nu}\Delta_{y}+\frac{1}{2m}\Delta_{x_{cm}}.$

It follows, by writing $u(x)=u(r,y,x_{cm})$, that

$({\mathcal{A}}_{12}e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{23}^{\ast}u)(t,r,y,x_{cm})\\\
=A_{12}(t,r)e^{i\sigma\left(\frac{\Delta_{r}}{2\mu}+\frac{\Delta_{y}}{2\nu}+\frac{\Delta_{x_{cm}}}{2m}\right)}B_{23}^{\ast}(t-\sigma,ar+y)u(t-\sigma,r,y,x_{cm})\,.$
(6.18)

Since $e^{i\sigma(\Delta_{y}/2\nu+\Delta_{x_{cm}}/2m)}$ commutes with
$A_{12}(t,r)$ and is unitary on ${\mathscr{H}}$, (6.18) implies that
$\|{\mathcal{A}}_{12}e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{23}^{\ast}u\|_{{\mathscr{K}}}^{2}$
is equal to

$\int_{X_{c}\times{\mathbb{R}}}\|A_{12}(t,r)e^{i\sigma\Delta_{r}/2\mu}B_{23}^{\ast}(t-\sigma,ar+y)u(t-\sigma,r,y,x_{cm})\|_{L^{2}(dr)}^{2}dydx_{cm}dt.$

We fix $(t,y,x_{cm})$, apply $L^{p}$-$L^{q}$ estimates for
$e^{i\sigma\Delta_{r}/2\mu}$ and Hörder’s inequality on $L^{2}(dr)$ as in the
case 2) and estimate the integrand by

$C\min_{\rho\in\\{p,q\\}}\\{\|A_{12}(t)\|_{\rho}\|B_{23}^{\ast}(t-\sigma)\|_{\rho}|\sigma|^{-d/\rho}\\}\|u(t-s,r,y,x_{cm})\|_{L^{2}(dr)}.$

This yields that
$\|{\mathcal{A}}_{jk}e^{-i\sigma{\mathcal{K}}_{0}}{\mathcal{B}}_{kl}^{\ast}u\|_{{\mathscr{K}}}\leq
C\min_{r\in\\{p,q\\}}|\sigma|^{-d/\rho}\|u\|_{{\mathscr{K}}}$ and we obtain as
previously the desired estimae (6.10) for this case. This completes the proof
of Lemma 6.2. ∎

It follows from Lemma 6.2 that Theorems 1.5 ans 3.9 of [13] may be applied to
the triplet $({\mathcal{A}},{\mathcal{B}},{\mathcal{K}}_{0})$ on
${\mathscr{K}}$ and we obtain the following theorem.

######  Theorem 6.3 (Kato([13])).

There exists a constant $\kappa_{0}>0$ such that for $|\kappa|<\kappa_{0}$
there is a unique closed operator ${\tilde{\mathcal{K}}}(\kappa)$ on
${\mathscr{K}}$ which satisfies following such that (a) to (d).
${\tilde{\mathcal{K}}}(\kappa)$ is uniquely determined by (a) and (b).

  1. (a)

${\mathcal{A}}$ is ${\tilde{\mathcal{K}}}(\kappa)$-smooth and ${\mathcal{B}}$
is ${\tilde{\mathcal{K}}}^{\ast}(\kappa)$-smooth.

  2. (b)

The resolvent
$\tilde{{\mathcal{R}}}(\zeta,\kappa)=({\tilde{\mathcal{K}}}(\kappa)-\zeta)^{-1}$
exists for all $\zeta\in{\mathbb{C}}\setminus{\mathbb{R}}$ and itsatisfies

$\tilde{{\mathcal{R}}}(\zeta,\kappa)-{{\mathcal{R}}}_{0}(\zeta)=-\kappa[{\mathcal{R}}_{0}(\zeta){\mathcal{B}}^{\ast}]{\mathcal{A}}\tilde{{\mathcal{R}}}(\zeta,\kappa)=-\kappa[{\mathcal{R}}(\zeta,\kappa){{\mathcal{B}}}^{\ast}]{{\mathcal{A}}}{\mathcal{R}}_{0}(\zeta)$
(6.19)

where $[{\mathcal{R}}_{0}(\zeta){{\mathcal{B}}}^{\ast}]$ is the closure of
${\mathcal{R}}_{0}(\zeta){{\mathcal{B}}}^{\ast}$ and similarly for
$[{\mathcal{R}}(\zeta,\kappa){{\mathcal{B}}}^{\ast}]$.

Moreover, following statements are satisfied:

  1. (c)

The equations

$({\mathcal{W}}_{\pm}(k)u,v)=(u,v)\mp\lim_{\varepsilon\to
0}\frac{1}{2{\pi}i}\int_{{\mathbb{R}}}({\mathcal{A}}{\mathcal{R}}_{0}(\lambda\pm
i\varepsilon)u,{\mathcal{B}}\tilde{{\mathcal{R}}}(\lambda\mp
i\varepsilon,\kappa)^{\ast}v)d\lambda$

define bounded operators in ${\mathscr{K}}$ which have bounded inverses and
${\tilde{\mathcal{K}}}(\kappa)$ and ${\mathcal{K}}_{0}$ are similar to each
other via ${\mathcal{W}}_{\pm}(\kappa)$ and
${\mathcal{W}}_{\pm}(\kappa)^{-1}$:

${\tilde{\mathcal{K}}}(\kappa)={\mathcal{W}}_{\pm}(\kappa){\mathcal{K}}_{0}{\mathcal{W}}_{\pm}(\kappa)^{-1}.$
(6.20)

  2. (d)

${\tilde{\mathcal{K}}}(\kappa)$ generates a strongly continuous group of
uniformly bounded operators $e^{-i\sigma{\tilde{\mathcal{K}}}(\kappa)}$ in
${\mathscr{K}}$:
$\|e^{-i\sigma{\tilde{\mathcal{K}}}(\kappa)}\|_{{\textbf{B}}({\mathscr{K}})}\leq
M$ and

${\mathcal{W}}_{\pm}(\kappa)=\lim_{\sigma\to\pm\infty}e^{i\sigma{\tilde{\mathcal{K}}}(\kappa)}e^{-i\sigma{\mathcal{K}}_{0}},\quad{\mathcal{W}}_{\pm}(\kappa)^{-1}=\lim_{\sigma\to\pm\infty}e^{i\sigma{\mathcal{K}}_{0}}e^{-i\sigma{\tilde{\mathcal{K}}}(\kappa)}\,.$
(6.21)

In view of Theorem 6.3, the next lemma completes the proof of Theorem 4.1.
Recall that ${\mathcal{K}}(\kappa)$ is the generator of
${\mathcal{U}}(\sigma,\kappa)$,see (3.4).

######  Lemma 6.4.

We have the equality ${\mathcal{K}}(\kappa)={\tilde{\mathcal{K}}}(\kappa)$,
${\tilde{\mathcal{K}}}(\kappa)$ being as in Theorem 6.3.

###### Proof.

Let $M$ be as in (3.3) and ${\tilde{M}}=M+1$ and
${\mathcal{R}}(z,\kappa)=({\mathcal{K}}(\kappa)-\zeta)^{-1}$ for
$|\Im\zeta|>{\tilde{M}}$ (see (3.5)). It suffices to show

${{\mathcal{R}}}(\zeta,\kappa)-{\mathcal{R}}_{0}(\zeta)=-\kappa[{\mathcal{R}}_{0}(\zeta){\mathcal{B}}^{\ast}]{{\mathcal{A}}}{{\mathcal{R}}}(\zeta,\kappa),\
\ |\Im\zeta|>{\tilde{M}}.$ (6.22)

Indeed (6.22) implies
${\mathcal{A}}{{\mathcal{R}}}(\zeta,\kappa)-{\mathcal{A}}{\mathcal{R}}_{0}(\zeta)=-\kappa{\mathcal{A}}[{\mathcal{R}}_{0}(\zeta){\mathcal{B}}^{\ast}]{{\mathcal{A}}}{{\mathcal{R}}}(\zeta,\kappa)$
by virtue of Lemmas 6.1 and 6.2 and

${{\mathcal{R}}}(\zeta,\kappa)={{\mathcal{R}}}_{0}(\zeta,\kappa)-\kappa[{\mathcal{R}}_{0}(\zeta){\mathcal{B}}^{\ast}](1+\kappa[{\mathcal{A}}{{\mathcal{R}}}_{0}(\zeta){{\mathcal{B}}}^{\ast}])^{-1}{{\mathcal{A}}}{\mathcal{R}}_{0}(\zeta)\,.$
(6.23)

Hence ${{\mathcal{R}}}(\zeta,\kappa)=\tilde{{\mathcal{R}}}(\zeta,\kappa)$ for
$|\Im\zeta|>{\tilde{M}}$ and ${\mathcal{K}}={\tilde{\mathcal{K}}}$.

We now prove (6.22). We prove it for $\Im\zeta>{\tilde{M}}$. The proof for
$\Im\zeta<-{\tilde{M}}$ is similar. In what follows we omit the coupling
constant $\kappa$. We first show that

$\int_{0}^{\infty}e^{-{\tilde{M}}\sigma}\|{\mathcal{A}}{\mathcal{U}}(\sigma)u\|_{{\mathscr{K}}_{N}}d\sigma\leq
C\|u\|_{{\mathscr{K}}}.$ (6.24)

Let ${\mathcal{X}}_{jk}=L^{\frac{2d}{d-2}}(X_{jk}^{r},{\mathscr{H}}_{jk})$ and
$C_{d}=\sum_{j<k}\sup_{t\in{\mathbb{R}}}\|A_{jk}(t,\cdot)\|_{L^{d}(X_{jk}^{r})}$.
Then, by Hölder’s inequality

$\displaystyle\int_{0}^{\infty}\|{\mathcal{A}}{\mathcal{U}}(\sigma)u\|_{{\mathscr{K}}_{N}}^{2}e^{-2{\tilde{M}}\sigma}d\sigma$
$\displaystyle=\sum_{j<k}\int_{{\mathbb{R}}}\left(\int_{0}^{\infty}e^{-2{\tilde{M}}\sigma}\|A_{jk}(t+\sigma)U(t+\sigma,t)u(t)\|_{{\mathscr{H}}}^{2}d\sigma\right)dt$
$\displaystyle\leq
C_{d}^{2}\int_{{\mathbb{R}}}\left(\int_{0}^{\infty}e^{-2{\tilde{M}}\sigma}\|U(t+\sigma,t)u(t)\|_{{\mathcal{X}}_{jk}}^{2}d\sigma\right)dt\,.$

The inner $d\sigma$-integral is equal to

$\displaystyle\sum_{n=0}^{\infty}\int_{0}^{1}e^{-2{\tilde{M}}(\sigma+n)}\|U(t+\sigma+n,t)u(t)\|_{{\mathcal{X}}_{jk}}^{2}d\sigma$
$\displaystyle=\sum_{n=0}^{\infty}\int_{0}^{1}e^{-2{\tilde{M}}(\sigma+n)}\|U(t+\sigma+n,t+n)U(t+n,t)u(t)\|_{{\mathcal{X}}_{jk}}^{2}d\sigma.$
(6.25)

The time local Strichartz estimate (2.6) with $p=d/2$ implies that

$\sup_{t\in{\mathbb{R}},n=0,1,\dots}\int_{0}^{1}\|U(t+\sigma+n,t+n,\kappa){\varphi}\|_{{\mathcal{X}}_{jk}}^{2}d\sigma\leq
C\|{\varphi}\|_{{\mathscr{H}}}^{2}$

and, the right of (6.25) is bounded by

$C\sum_{n=0}^{\infty}e^{-2{\tilde{M}}{n}}\|U(t+n,t)u(t)\|_{{\mathscr{H}}}^{2}.$

with a constant independent of $t$. Combining these estimates yields

$\displaystyle\int_{0}^{\infty}\|{\mathcal{A}}{\mathcal{U}}(\sigma)u\|_{{\mathscr{K}}_{N}}^{2}e^{-2{\tilde{M}}\sigma}d\sigma\leq
CC_{d}^{2}\sum_{n=0}^{\infty}\int_{{\mathbb{R}}}e^{-2{\tilde{M}}{n}}\|U(t+n,t)u(t)\|_{{\mathscr{H}}}^{2}dt$
$\displaystyle\hskip
56.9055pt=CC_{d}^{2}\sum_{n=0}^{\infty}e^{-2{\tilde{M}}{n}}\|{\mathcal{U}}(n)u\|_{{\mathscr{K}}}^{2}\leq
C\sum_{n=0}^{\infty}e^{-2n}\|u\|_{{\mathscr{K}}}^{2}=C\|u\|_{{\mathscr{K}}}^{2}.$

We likewise have the corresponding estimate for the integral on
$\sigma\in(-\infty,0)$. Thus, the Schwarz’ inequality implies (6.24) and

$\int_{0}^{\infty}e^{\mp{i\zeta\sigma}}{\mathcal{A}}\,{\mathcal{U}}(\pm\sigma)ud\sigma={\mathcal{A}}{\mathcal{R}}(\zeta)u,\quad
u\in{\mathscr{K}}$ (6.26)

is uniformly bounded for $|\Im\zeta|>{\tilde{M}}$:

$\|{\mathcal{A}}{\mathcal{R}}(\zeta)u\|_{{\mathscr{K}}}\leq
C\|u\|_{{\mathscr{K}}},\quad|\Im\zeta|>{\tilde{M}}.$ (6.27)

As a function of $t\in{\mathbb{R}}$, $U(t,s){\varphi}\in
C({\mathbb{R}},{\mathscr{H}})\cap{\mathcal{X}}_{\textrm{loc}}^{d/2}$ and
${\mathcal{A}}$ and ${\mathcal{B}}$ map
$C({\mathbb{R}},{\mathscr{H}})\cap{\mathcal{X}}_{\textrm{loc}}^{d/2}$ into
$L^{2}_{\textrm{loc}}({\mathbb{R}},{\mathscr{H}})$ and the contrction of the
solution ([19]) implies the Duhamel formula

$U(t,s){\varphi}=e^{-i(t-s)H_{0}}\psi-i\int_{s}^{t}e^{-i(t-r)H_{0}}B^{\ast}(r)A(r)U(r,s){\varphi}dr.$
(6.28)

Let $u,v\in{\mathcal{S}}$. Substituting $t-\sigma$ for $s$ and $u(t-\sigma)$
for ${\varphi}$ in (6.28) yields

$U(t,t-\sigma)u(t-\sigma)=e^{-i{\sigma}H_{0}}u(t-\sigma)\\\
-i\int_{0}^{\sigma}e^{-i(\sigma-\tau)H_{0}}B^{\ast}(t-\sigma+\tau)A(t-\sigma+\tau)U(t-\sigma+\tau,t-\sigma)u(t-\sigma)d\tau,$

which implies that

$({\mathcal{U}}(\sigma)u)(t)=({\mathcal{U}}_{0}(\sigma)u)(t)-i\int_{0}^{\sigma}({\mathcal{U}}_{0}(\sigma-\tau){\mathcal{B}}^{\ast}{\mathcal{A}}{\mathcal{U}}(\tau)u)(t)d\tau,\
\ t\in{\mathbb{R}}.$

Taking the inner products with $v\in{\mathcal{S}}$ in both sides produces

$({\mathcal{U}}(\sigma)u,v)_{{\mathscr{K}}}=({\mathcal{U}}_{0}(\sigma)u,v)_{{\mathscr{K}}}-i\int_{0}^{\sigma}({\mathcal{A}}{\mathcal{U}}(\tau)u,{\mathcal{B}}{\mathcal{U}}_{0}(\tau-\sigma)v)_{{\mathscr{K}}}d\tau\,.$

Multiply both sides by $ie^{i\sigma\zeta}$ with $\Im\zeta>{\tilde{M}}$ and
integrate by $d\sigma$ over $[0,\infty)$. We obtain by changing the order of
integrations that

$({\mathcal{R}}(\zeta)u,v)_{{\mathscr{K}}}=({\mathcal{R}}_{0}(\zeta)u,v)_{{\mathscr{K}}}-({\mathcal{A}}{\mathcal{R}}(\zeta)u,{\mathcal{B}}{\mathcal{R}}_{0}(\overline{\zeta})v)_{{\mathscr{K}}}.$

Restoring the coupling constant, we obtain (6.22): for $\Im z>{\tilde{M}}$.
This completes the proof. ∎

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