[1]Jaume Carbonell

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,France

2]Lebedev Physical Institute, Leninsky prospect 53, 119991 Moscow, Russia

# Abnormal solutions of Bethe–Salpeter equation with massless and massive
exchanges

<EMAIL_ADDRESS>Vladimir A. Karmanov<EMAIL_ADDRESS>Ekaterina A. Kupriyanova<EMAIL_ADDRESS>Hagop Sazdjian
<EMAIL_ADDRESS>*  [

###### Abstract

We summarize the main properties of the so called ”abnormal solutions” of the
Wick–Cutkosky model, i.e. two massive scalar particles interacting via
massless scalar exchange (”photons”), within the Bethe–Salpeter equation.
These solutions do not exist in the non-relativistic limit, in spite of having
very small binding energies. They present a genuine many-body character
dominated by photons, with a norm of the valence constituent wave function
(two-body norm) that vanishes in the limit of zero binding energy.

We present new results concerning the massive-exchange case, in particular
determine under which conditions is it possible to obtain such peculiar
solutions without spoiling the model by tachyonic states ($M^{2}<0$).

###### keywords:

Bethe–Salpeter equation, Wick–Cutkosky model, Abnormal solutions, Hybrid
states

##  1 Introduction

Lorentz invariance of a physical theory does not only manifest itself when the
velocities of the particles are comparable to the speed of light or when their
momenta are comparable to the constituent rest masses. This fundamental
symmetry of nature can also have dynamical consequences in the low-energy
limit and can induce quantitative and qualitative differences with respect to
a non-relativistic description.

One could expect, indeed, that when describing a bound state of two particles
with a very small binding energy, and involving momenta smaller than the
constituent masses, both approaches, a relativistic and a non-relativistic
one, would lead to very similar results. However this is not always the case.

For instance, when considering the zero-binding-energy limit of the Light-
Front [1] and of the Bethe–Salpeter (BS) [2, 3] equations in a
$\varphi^{2}\chi$ scalar theory, it is found [4] that the results of these
covariant theories are very close to each other, but differ from the results
of the non-relativistic Schrödinger equation when the mass $\mu$ of the
exchange particle is non-zero.

More spectacular is the fact that there are families of low-energy solutions
that exist in a relativistic theory while they are totally absent in its non
relativistic limit. This happens within the BS equation when considering two
scalar particles interacting by a massless ($\mu$=0) scalar exchange.

The properties of such states, first discovered by Wick [5] and Cutkosky [6]
in the $\mu$=0 case, and since denoted ”abnormal solutions”, is the main
subject of the present contribution, with special emphasis on its eventual
persistence in the massive ($\mu>0$) case.

We present in Section 2 a brief summary of the BS equation for the scalar
model. Section 3 is aimed to describe the Cutkosky solution for the massless-
exchange case and the properties of normal and abnormal states. Section 4
contains selected results of the massive-exchange case. Concluding remarks
follow in 5.

##  2 The Bethe–Salpeter equation

The BS equation deals with a well-defined object form the Quantum Filed Theory
point of view: the matrix element of the T-product of the Heisenberg operators
taken between the vacuum and the bound state [7]

$\Phi(x_{1},x_{2},P)=<0\mid T\\{\phi(x_{1})\phi(x_{2})\\}\mid P>$ (1)

Its Fourier transform $\Phi(k,P)$

$\Phi(x_{1},x_{2},P)=\int{dp_{1}\over(2\pi)^{4}}{dp_{2}\over(2\pi)^{4}}\Phi(p_{1},p_{2})\;e^{-iPx}\;e^{-ikx}=\;e^{-iPx}\int{dk\over(2\pi)^{4}}\Phi(k,P)\;e^{-ikx}$

written in terms of the total $P=p_{1}+p_{2}$ and relative $2k=p_{1}-p_{2}$
momenta, obeys the equation

$\Phi(k,P)=S_{1}(k,P)\;S_{2}(k,P)\;\int{d^{4}k^{\prime}\over(2\pi)^{4}}\;iK(k,k^{\prime};P)\;\Phi(k^{\prime},P)$
(2)

where

$\displaystyle S_{1}(k,P)$ $\displaystyle=$
$\displaystyle\frac{i}{\left({P\over 2}+k\right)^{2}-m^{2}+i\epsilon}$
$\displaystyle S_{2}(k,P)$ $\displaystyle=$
$\displaystyle\frac{i}{\left({P\over 2}-k\right)^{2}-m^{2}+i\epsilon}$

are the free particle propagators in the case of two equal masses and $iK$ is
the interaction kernel. If $iK$ contained all irreducible graphs of a
Lagrangian density, the solution of (2) would be equivalent to the solution of
the full QFT problem. This kind of ”mantra” is however a wishful thinking, not
only because nobody knows how to construct such a kernel, but, would it be the
case, the corresponding integral equation would not be integrable. One is then
limited to use very simple reductions that keep only a vague flavour of the
underlying Lagrangian theory. In the simplest case of two scalar particles of
mass $m$ interacting via a massive scalar exchange $\mu$ the ladder kernel
reads

$iK(k,k^{\prime})=-{g^{2}\over(k-k^{\prime})^{2}-\mu^{2}+i\epsilon}\quad\Longrightarrow\quad
V(r)=-{g^{2}\over 4\pi}\;{e^{-\mu r}\over r}$ (3)

In the non-relativistic limit it leads to the Yukawa potential $V(r)$.

Equation (2) is an implicit eigenvalue equation with repect to the total mass
squared of the system, $M^{2}\equiv P^{2}$, which in this ladder
approximation, appears only in the free propagators.

Figure 1: Feyman graph representing the BS amplitude in momentum space.

The BS amplitude $\Phi(k,P)$ is quite a nasty mathematical object, plagued
with singularities, as a function of the arguments, which represents the
Feynman amplitude depicted in Fig. 1 and which has not an easy interpretation
in terms of wave functions. These singularities motivated G. Wick [5] to
change the original formulation in Minkowski space into the Euclidean metric,
introducing the, since then famous, Wick rotation in the time-like component:
$k_{0}$=$ik_{4}$ such that $k_{M}=(k_{0},k_{1},k_{2},k_{3})\to
k_{E}=(k_{1},k_{2},k_{3},k_{4}=-ik_{0})$ and
$k_{M}^{2}$=$k_{0}^{2}-k_{1}^{2}-k_{2}^{2}-k_{3}^{2}$=$-k_{E}^{2}$=$-(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+k_{4}^{2})$.
This change of metric paved the way for obtaining the first solutions of BS
equation with kernel (3) with $\mu$=0. Results were published by Wick himself
in the quoted reference [5] and in a much more complete way by Cutkosky in the
subsequent paper of the same journal [6]. They constitute the so-called
Wick–Cutkosky (W-C) model, although we will abusively use the same
denomination for the $\mu\neq$0 case.

For the S-wave and the W-C model, the equation reads:

$\left[(k^{2}+k^{2}_{4}+m^{2}-\frac{1}{4}M^{2})^{2}+M^{2}k^{2}_{4}\right]\Phi_{E}(k_{4},k)=\int_{0}^{\infty}dk^{\prime}\int_{-\infty}^{\infty}dk^{\prime}_{4}V^{E}_{0}(k_{4},k;k^{\prime}_{4},k^{\prime})\Phi_{E}(k^{\prime}_{4},k^{\prime})$
(4)

where the S-wave Euclidean kernel

$\displaystyle V^{E}_{0}(k_{4},k;k^{\prime}_{4},k^{\prime})=\frac{\alpha
m^{2}k^{\prime}}{\pi^{2}k}\;\log\frac{[(k_{4}-k^{\prime}_{4})^{2}+(k+k^{\prime})^{2}+\mu^{2}]}{[(k_{4}-k^{\prime}_{4})^{2}+(k-k^{\prime})^{2}+\mu^{2}]}.$
(5)

is smooth everywhere for $\mu\neq 0$ and presents only logarithmic
singularities in the diagonal
$\\{k^{\prime}=k\\}\times\\{k^{\prime}_{4}=k_{4}\\}$ for $\mu=0$. Here,
$\alpha=g^{2}/(16\pi m^{2})$. Nowadays there exist several methods to solve
accurately the BS equation in Euclidean space for a bound state problem. They
apply to a large variety of kernels with bosons and fermions and even beyond
the ladder approximation. A much more precarious situation is however observed
in the scattering problem.

Other methods have been developed aiming to obtain a Minkowski solution of the
same equation. They are based on an integral representation of the BS
amplitude which collects all the singularities in an analytic term and deals
with a regular weight function that obeys a modified BS equation. They are
also much better adapted to understand the abnormal states for the massless as
well as for the massive exchange cases. These ”Minkowski-space” methods are
widely inspired by the one developed by Cutkosky in his solution of the
massless W-C model, that will be briefly summarized in the coming section.

##  3 Cutkosky’s solution for the massless case

In his first publication [6], Cutkosky searched for the solution of (2) with
the interaction kernel (3) in the form of an integral representation:

$\Phi^{LM}_{n}(k,P)=\sum_{\nu=0}^{n-L-1}\;\int_{-1}^{+1}dz\;g_{nL}^{\nu}(z)\;\frac{Y_{LM}(\hat{k})}{\left[k^{2}+(k\cdot
P)\;z-\left(m^{2}-{P^{2}\over 4}\right)+i\epsilon\right]^{2+n-\nu}}$ (6)

in terms of unknown “weight functions” $g_{n}^{\nu}(z)$.

For the S-wave, which is the only case we are going to consider here, and
disregarding a global normalization factor, (6) turns into

$\Phi_{n}(k,P)=\sum_{\nu=0}^{n-1}\;\int_{-1}^{+1}dz\;\frac{g_{n}^{\nu}(z)}{\left[k^{2}+(k\cdot
P)\;z-\left(m^{2}-{P^{2}\over 4}\right)+i\epsilon\right]^{2+n-\nu}}$ (7)

The solution $\Phi_{n}$, where $n=1,2,3,\ldots$ , which plays the role of the
principal quantum number in the non-relativistic Coulomb problem, is obtained
as a superposition of $n$ components $g_{n}^{\nu}(z)$ labelled by
$\nu=0,1,\ldots,n-1$.

Cutkosky obtained a coupled set of integral equations for the weigh functions
$g_{n}^{\nu}(z)$ in the – highly inspired but useless – form111Note that there
was a misprint in the integration limits over t in the original publication
[6].

$\displaystyle g_{n}^{\nu}(z)$ $\displaystyle=$ $\displaystyle{\lambda\over
2}\sum_{\nu^{\prime}=0}^{\nu}\frac{(n-\nu+1)!\;(n-\nu^{\prime}-1)!}{(n-\nu^{\prime}+1)!\;(n-\nu-1)!}\;\int_{-1}^{+1}dt\;\int_{0}^{1}dx\;x(1-x)^{n-\nu-1}$
(8) $\displaystyle\times$
$\displaystyle\;\int_{-1}^{+1}dz^{\prime}\frac{\delta[z-xt-(1-x)z^{\prime}]}{(1-\eta^{2}+\eta^{2}z^{\prime
2})^{\nu-\nu^{\prime}+1}}\;\;g_{n}^{\nu^{\prime}}(z^{\prime})$ (9)

where

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle{g^{2}\over
16\pi^{2}m^{2}}={\alpha\over\pi}$ (10) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle{M\over 2m}=1-{B\over 2m}$ (11)

After simplifying the coefficients and introducing

$Q(z)=1-\eta^{2}(1-z^{2})$ (12)

one is left with the, still useless, system of integral equations

$\displaystyle g_{n}^{k}(z)$ $\displaystyle=$ $\displaystyle{\lambda\over
2}\sum_{k^{\prime}=0}^{k}\frac{(n-k+1)(n-k)}{(n-k^{\prime}+1)(n-k^{\prime})}$
(13) $\displaystyle\times$
$\displaystyle\int_{-1}^{+1}dt\int_{0}^{1}dx\;x(1-x)^{n-k-1}\;\int_{-1}^{+1}dz^{\prime}\frac{\delta[z-xt-(1-x)z^{\prime}]}{Q(z^{\prime})^{k-k^{\prime}+1}}\;g_{n}^{k^{\prime}}(z^{\prime})$
(14)

The integration over $dt$ can be performed, using the $\delta$ function

  1. 1.

Since $x>0$

$\int_{-1}^{+1}dt\;\delta[z-xt-(1-x)z^{\prime}]=\int_{-1}^{+1}dt\;\delta[xt-(z-(1-x)z^{\prime})]={1\over
x}\int_{-1}^{+1}dt\;\delta\left[t-{z-(1-x)z^{\prime}\over x}\right]$

which takes the form

$\int_{-1}^{+1}dt\;\delta(t-t_{0})\qquad
t_{0}(x,z,z^{\prime})={z-(1-x)z^{\prime}\over x}$ (15)

  2. 2.

Expression (15) vanishes except if $t_{0}\in[-1,+1]$ that is if the following
two conditions are satisfied

$-1<\frac{z-(1-x)z^{\prime}}{x}<+1$

rewritten as

$-1<{z-z^{\prime}\over x}+z^{\prime}<+1$ (16)

To fulfill these conditions we must distinguish the sign of $z-z^{\prime}$

     * •

If $z-z^{\prime}>0$,

\- the left part of (16) is automatically fulfilled 222since
${z-z^{\prime}\over x}>z-z^{\prime}$ and so ${z-z^{\prime}\over
x}+z^{\prime}>z>-1$

\- the right part requires the condition

$x>{z-z^{\prime}\over 1-z^{\prime}}$

and so

$\displaystyle\int dt\;\delta(t-t_{0})\int_{0}^{1}dx\;x(1-x)^{n-k-1}$
$\displaystyle=$ $\displaystyle\int_{{ z-z^{\prime} \over
1-z^{\prime}}}^{1}dx\;(1-x)^{n-k-1}$ (17) $\displaystyle=$
$\displaystyle-{1\over n-k}\left[(1-x)^{n-k}\right]_{ { z-z^{\prime} \over
1-z^{\prime}}}^{1}$ (18) $\displaystyle=$ $\displaystyle{1\over
n-k}\left({1-z\over 1-z^{\prime}}\right)^{n-k}$ (19)

     * •

If $z-z^{\prime}<0$,

\- the right part of (16) is automatically fulfilled since
$z^{\prime}+{z-z^{\prime}\over x}<z^{\prime}<+1$

\- the left part requires

$x>{z^{\prime}-z\over 1+z^{\prime}}$ $\displaystyle\int dt\delta(
t-t_{0})\int_{0}^{1}dx\;x(1-x)^{n-k-1}$ $\displaystyle=$ $\displaystyle\int_{{
z^{\prime}-z \over 1+z^{\prime}}}^{1}dx\;(1-x)^{n-k-1}$ (20) $\displaystyle=$
$\displaystyle-{1\over n-k}\left[(1-x)^{n-k}\right]_{ { z^{\prime}-z \over
1+z^{\prime}}}^{1}$ (21) $\displaystyle=$ $\displaystyle{1\over
n-k}\left({1+z\over 1+z^{\prime}}\right)^{n-k}$ (22)

Gathering (19) and (22) we obtain

$\int dt\delta( t-t_{0})\int_{0}^{1}dx\;x\;(1-x)^{n-k-1}={1\over
n-k}R^{n-k}(z,z^{\prime}) $

where we have introduced the kernel

$R(z,z^{\prime})=\theta(z-z^{\prime})\left({1-z\over
1-z^{\prime}}\right)+\theta(z^{\prime}-z)\left({1+z\over
1+z^{\prime}}\right)=\left\\{\begin{array}[]{lcl}{1-z\over
1-z^{\prime}},&\mbox{if}&\quad z^{\prime}<z\cr{1+z\over
1+z^{\prime}},&\mbox{if}&\quad z^{\prime}>z\end{array}\right.$ (23)

By inserting this expression in (13), one is finally left with the following
triangular system of coupled one-dimensional integral equations, useful for
numerical calculations:

$g_{n}^{\nu}(z)={\lambda\over
2}\sum_{\nu^{\prime}=0}^{\nu}c_{n}^{\nu\nu^{\prime}}\;\int_{-1}^{+1}dz^{\prime}\;{R^{n-\nu}(z,z^{\prime})\over
Q^{\nu-\nu^{\prime}+1}(z^{\prime})}\;g_{n}^{\nu^{\prime}}(z^{\prime})\qquad\nu=0,1,...n-1$
(24)

with coefficients

$c_{n}^{\nu\nu^{\prime}}=\frac{(n-\nu+1)}{(n-\nu^{\prime}+1)(n-\nu^{\prime})}$

The kernels $R(z,z^{\prime})$ and $Q(z)$, defined respectively in (23) and
(12), are displayed in Fig. 2. $R(z,z^{\prime})$ is continuous in both
arguments but has a cusp at $z^{\prime}=z$, while $1/Q(z)$ is peaked around
$z=0$. This peak becomes increasingly sharp when the binding energy tends to
zero. Since both kernels appear in equation (24) to some integer power, when
this power is large they become quasi-singular around the critical points
($z^{\prime}=z$ and $z=0$), and make the calculations difficult.

Figure 2: Left panel kernel R(z,z’) defined in (23) and right panel kernel $Q$
defined in (12) for different values of $\eta$.

For $n=1$ there is a single equation determining the unique component of
$\Phi_{1}$

$g_{1}(z)={\lambda\over 2}\;\int_{-1}^{+1}dz^{\prime}\;{R(z,z^{\prime})\over
Q(z^{\prime})}\;g_{1}(z^{\prime})$ (25)

For $n=2$, the solution $\Phi_{2}$ is determined by two components,
$g_{2}^{0}$ and $g_{2}^{1}$ , satisfying

$\displaystyle g_{2}^{0}(z)$ $\displaystyle=$ $\displaystyle{\lambda\over
2}\;\;\;{1\over 2}\int_{-1}^{+1}dz^{\prime}\;{R^{2}(z,z^{\prime})\over
Q(z^{\prime})}\;g_{2}^{0}(z^{\prime})$ $\displaystyle g_{2}^{1}(z)$
$\displaystyle=$ $\displaystyle{\lambda\over 2}\left[{1\over
3}\;\int_{-1}^{+1}dz^{\prime}\;{R(z,z^{\prime})\over
Q^{2}(z^{\prime})}\;g_{2}^{0}(z^{\prime})+\int_{-1}^{+1}dz^{\prime}\;{R(z,z^{\prime})\over
Q(z^{\prime})}\;g_{2}^{1}(z^{\prime})\right]$

For $n=3$ the solution is given by

$\displaystyle g_{3}^{0}(z)$ $\displaystyle=$ $\displaystyle{\lambda\over
2}\;\;\;{1\over 3}\int_{-1}^{+1}dz^{\prime}\;{R^{3}(z,z^{\prime})\over
Q(z^{\prime})}\;g_{3}^{0}(z^{\prime})$ $\displaystyle g_{3}^{1}(z)$
$\displaystyle=$ $\displaystyle{\lambda\over
2}\left[c_{3}^{10}\;\int_{-1}^{+1}dz^{\prime}\;{R^{2}(z,z^{\prime})\over
Q^{2}(z^{\prime})}\;g_{3}^{0}(z^{\prime})+c_{3}^{11}\int_{-1}^{+1}dz^{\prime}\;{R^{2}(z,z^{\prime})\over
Q(z^{\prime})}\;g_{3}^{1}(z^{\prime})\right]$ $\displaystyle g_{3}^{2}(z)$
$\displaystyle=$ $\displaystyle{\lambda\over
2}\left[c_{3}^{20}\;\int_{-1}^{+1}dz^{\prime}\;{R(z,z^{\prime})\over
Q^{3}(z^{\prime})}\;g_{3}^{0}(z^{\prime})+c_{3}^{21}\int_{-1}^{+1}dz^{\prime}\;{R(z,z^{\prime})\over
Q^{2}(z^{\prime})}\;g_{3}^{1}(z^{\prime})+c_{3}^{22}\int_{-1}^{+1}dz^{\prime}\;{R^{2}(z,z^{\prime})\over
Q(z^{\prime})}\;g_{3}^{2}(z^{\prime})\right]$

It is worth noticing that, for each $n$, the component $g_{n}^{0}$ decouples
from the rest. It is determined by a single equation

$g_{n}^{0}(z)={\lambda\over
2n}\;\int_{-1}^{+1}dz^{\prime}\;{R^{n}(z,z^{\prime})\over
Q(z^{\prime})}\;g_{n}^{0}(z^{\prime})$ (26)

and provides the full spectrum of the W-C model. The rest of the components
are needed to reconstruct the full BS amplitude and other observables, like,
e.g., form factors.

Since (26) is homogeneous, the norm of $g_{n}^{0}$ is not fixed by (26), and
can be only determined by normalizing $\Phi_{n}$ in (7). On the contrary,
there is no choice in the norm of the components $g_{n}^{\nu>0}$, since they
obey an inhomogeneous equation with a source term proportional to $g_{n}^{0}$.

There is an equivalent formulation of (24) in differential form. For instance,
eq. (26) providing $g_{n}^{0}$ for any $n$, is equivalent to the differential
equation

${g^{\prime\prime}}_{n}^{0}(z)+{2(n-1)z\over
1-z^{2}}\;{g^{\prime}}_{n}^{0}(z)-{n(n-1)\over
1-z^{2}}\;g_{n}^{0}(z)+{\lambda\over(1-z^{2})Q(z,\eta)}\;g_{n}^{0}(z)=0$ (27)

with the boundary conditions $g_{n}^{0}(\pm 1)=0$.

For n=2, and after having obtained $g_{2}^{0}$ by (27), the component
$g_{2}^{1}$ is a solution of the inhomogeneous equation

${g^{\prime\prime}}_{2}^{1}+{\alpha\over\pi}{1\over(1-z^{2})Q(z,\eta)}\;g_{2}^{1}=-{\alpha\over
3\pi}\;{1\over(1-z^{2})Q^{2}(z,\eta)}\;g_{2}^{0}$ (28)

with $g_{2}^{0}$ as a source term, and so forth for the other $n$’s.

###  3.1 Normal and abnormal states

The W-C model has an infinite family of solutions ($\Phi_{n},M_{n}$), which,
for small values of the coupling constant $\alpha$, are ”logarithmically
tangent” to the non-relativistic Coulomb bound-state spectrum [8]:

$B_{n}(\alpha)={m\alpha^{2}\over
4n^{2}}\;\left[1+{4\over\pi}\alpha\ln\alpha+o(\alpha^{2})\right]$ (29)

where $M_{n}=2m-B_{n}$.

For a given $n$, the function $g_{n}^{0}(z)$ satisfies the homogeneous
equation (27), in which $n$ plays role of a parameter. Such equations usually
have a whole spectrum of solutions. This just takes place in the equation
(27). Therefore, for each value of $n$=1,2,…, there exists an additional
series of eigenvalues labeled by a new quantum number $\kappa=0,1,\ldots$,
which is a consequence of the symmetry of the 4D Coulomb problem. The
components of the corresponding eigenstates $g^{\nu}_{n\kappa}$ have a well
defined parity determined by $\kappa$.

$g_{n\kappa}^{\nu}(-z)=(-)^{\kappa}g^{\nu}_{n\kappa}(z)$

The subset $\kappa=0$ corresponds, for small values of $\alpha$, to the
standard Balmer series (29), that is, to the ”normal” non-relativistic
solutions. The other states, with $\kappa\neq 0$ , do not have a counterpart
in the non-relativistic theory and for this reason were named by Wick
”abnormal states”. It is worth mentioning that the states with odd values of
$\kappa$ have vanishing contributions to the S-matrix [9]. So the first
abnormal state with dynamical content is $\kappa=2$.

We have displayed in Fig. 3 the energies of the lower states as a function of
the coupling constant $\lambda={\alpha\over\pi}$ in a log-log scale. One sees
clearly that two different families of states arise: in solid black lines is
the ensemble of normal states ($\kappa=0$) with different values of the
principal quantum number $n$. They have an accumulation point in
($\lambda,B$)=(0,0). We have included in black dashed line the non-
relativistic limit for the ground state. Colored lines correspond to the
ground state ($n=1$) of the abnormal solutions: $\kappa=1$ in red, $\kappa=2$
in green,…. This family is totally decoupled, in its $\lambda(B)$ trajectory,
from the first one, though the normal and abnormal series intersect. As seen
in Fig. 3, between two excited normal states there are the abnormal ones. The
abnormal series has an accumulation point at ($\lambda,B$)=(1/4,0). One can
show [5, 6] that, for small values of B, the energies of the abnormal states
behave as

$\lambda(B)\approx{1\over 4}+\frac{4\pi^{2}(\kappa-1)^{2}}{\ln^{2}{B\over m}}$
(30)

independently of the main quantum number $n$. Notice from Fig. 3 that the
asymptotic value $\lambda_{min}=1/4$, independent of $\kappa$, is reached very
slowly when $B$ tends to zero. For instance, for $\kappa$=2 and B=10-6, the
value of $\lambda$ is still a factor two larger than $1/4$.

Figure 3: Solid black lines correspond to binding energies of the first
(n=1,2,…) normal states ($\kappa=0$) as a function of the coupling constant
$\alpha$. Abnormals states, limited to ground n=1, are in colored lines:
$\kappa=1$ in red, $\kappa=2$ in green,… .

The fact that this horizontal asymptote is reached always from above means
that the abnormal solutions exist only for large values of the coupling
constant: $\lambda\geq{1\over 4}$ or $\alpha\geq{\pi\over 4}$. This is the
meaning of the horizontal bold brown line at $\lambda=\lambda_{min}=1/4$.

The existence of a minimal coupling constant to bind a two-body system is
typical of a Yukawa-like massive-exchange potential. From this point of view,
the ground abnormal state ($\kappa=2$) behaves as if it was created by the
exchange of a ”massive photon” with a non zero effective mass $\mu_{eff}$. One
can get an evaluation of its value by means of the solution of the non-
relativistic Yukawa model [31]. One can see there that the minimal coupling
constant for the appearance of the first bound state in the dimensionless
problem is $G_{0}\approx 1.680$. It is related to the minimal coupling
constant of the dimensionful problem by $\alpha_{min}={\mu\over m}G_{0}$. For
all abnormal solutions one has $\alpha_{min}=\pi/4$, which gives
$\mu_{eff}/m=0.47$.

On another hand, it turns out that for values of $\lambda>2$ the ground state
of the model, i.e., the normal state with $n$=1 and $\kappa$=0, has $M^{2}<0$.
If we want to restrict ourselves to non-tachyonic states, the coupling
constant must be limited to $\lambda\leq\lambda_{max}=2$. This is the reason
for the horizontal brown line at $\lambda=\lambda_{max}$ in Fig. 3. Since the
first abnormal state is $\kappa$=2 and it crosses the line
$\lambda=\lambda_{max}$ at $B\approx 0.00903$, this means that all abnormal
states, although requiring large values of the coupling constants to exist,
concern very low-energy solutions.

It is quite a paradoxical situation for a relativistic theory to predict a
series of new low-energy states. We would like to mention here that the
existence of such states, would they be obtained with a simplified kernel,
require the full covariance of the theory. If the excitation in the time-like
degrees of freedom are frozen, the abnormal states disappear [10].

###  3.2 Characterization of the abnormal states

In a series of papers [11, 12, 13], summarized in [14], we have studied the
properties of the abnormal states in the W-C model, with the main aim of
giving an intrinsic characterization of such states, rather than a relative
position in the full spectrum of the model.

We have extensively compared the elastic and inelastic form factors of normal
versus abnormal states. The most striking difference lies in the two-body
contents of the corresponding state vectors. Indeed, in the Light-Front
approach, the state vector $\mid P\rangle$ appearing in the definition of the
BS amplitude (1) is a QFT state involving many-body components (Fock
expansion):

$\mid P\rangle=\sum_{n\geq 2}^{\infty}\Psi_{n}(k_{1},k_{2},\ldots,k_{,})\mid
n\rangle\qquad\mid
n\rangle=a^{\dagger}_{k_{1}}a^{\dagger}_{k_{2}},\ldots,a^{\dagger}_{k_{n-2}}b^{\dagger}_{q_{1}}b^{\dagger}_{q_{2}}$

where the operators $b^{\dagger}$ and $a^{\dagger}$ are the creation operators
of the constituent massive particles and of the exchanged particle,
respectively, and where $\Psi_{n}$ is, by definition, the n-body wave
function.

The total norm of a state vector results from adding the partial norms of the
corresponding 2-, 3- and many-body components

$\langle P\mid
P\rangle=1=\int\Psi_{2}^{2}+\int\Psi_{3}^{2}+\int\Psi_{4}^{2}+\ldots=N_{2}+N_{3}+N_{4}+\ldots$

Having normalized the BS amplitude $\Phi$ by the condition $F$(0)=1, where
$F(Q)$ is the elastic electric form factor of the full bound system, we have
obtained the two-body wave function by projecting on the Light-Front plane,
$\omega\cdot x$=0 with $\omega^{2}=0$, the BS amplitude

$\psi_{2}(k_{1},k_{2},P,\omega)=\frac{(\omega\cdot k_{1})(\omega\cdot
k_{2})}{\pi(\omega\cdot
P)}\int_{-\infty}^{+\infty}\Phi(k+\beta\omega,P)\;d\beta$

and by that, the two-body norm

$N_{2}=\frac{1}{(2\pi)^{3}}\int\psi_{2}^{2}(k_{\perp},x)\frac{d^{2}k_{\perp}dx}{2x(1-x)}.$

We have displayed in Fig. 4 the two-body norm $N_{2}$ of the first two normal
(left panel) and abnormal (right panel) states as functions of their binding
energy $B$. As one can see, an essential difference appears between the two-
body contents of a normal and an abnormal state. In the limit of small binding
energies, the two-body norm $N_{2}$ of a normal state tends to 1 (left panel)
, indicating that it is described by a 2-body. valence wave function. On the
contrary, the abnormal states has a two-body norm that, in this $B\to 0$
limit, tends to 0, indicating a genuine many body character of such states.
This difference holds for the ground as well as for the first excited states.

This is an intrinsic difference between such kind of states, related to only
their very internal structure and independent of their energy, and was
probably the most striking result of our previous work [14].

It is worth mentioning here some kind of skepticism expressed by Wick himself,
when commenting on the existence of such states at the very end of his seminal
article [5]. He wrote ”About the possible existence of these abnormal
solutions, we shall not try to speculate. Since they occur only for finite
values of $\lambda$ ($\lambda\geq 1/4$) it will be unwise to assume that they
are a property of the complete BS equation. Certainly the ladder approximation
cannot be trusted to such extent”. To our knowledge, this judicious remark
still remains an open question. However the many-body character of the
abnormal states that we have put in evidence, seems to provide an argument in
favour of their existence. It is hard to imagine a possible mechanism with
which the non-ladder many-body contributions could inhibit the construction of
such collective many-body states.

Figure 4: $N_{2}$-dependence on the binding energy $B/m$ for the ground
($n=1$) and first excited ($n=2$) states. On left panel for the normal states
($\kappa=0$) and on right panel for the abnormal ones ($\kappa=2$).

So far we have considered the case of equal constituent masses. If the Coulomb
field would be provided by a heavy ion, interacting with an electron, we would
deal with strongly unequal constituent masses. The existence of abnormal
states in this case was studied in [6, 15]. It turns out that in a system with
such different masses the abnormal states still exist. Moreover, the effect of
unequal masses is attractive. The balance between the exchanged photons and
the massive constituents is little sensitive to the mass ratio, and so the
many-photon component still predominates.

It remains to see whether the abnormal states survive when the exchange mass
differs from zero. This will be the content of our forthcoming publication
[29] . The first results are summarized in the next section.

##  4 Solutions for the massive-exchange case

The solution of the massive-exchange case can be obtained by directly solving
the BS equation in Euclidean space (4). There are however other alternatives
which are inspired by the Cutkosky solution of the massless-exchange case,
previously discussed. They are also based on a, now two-dimensional, integral
representation of the BS amplitude that is due to Nakanishi [16] and that
reads

$\displaystyle\Phi(k,p)$ $\displaystyle=$
$\displaystyle-i\int_{-1}^{1}dz^{\prime}\int_{0}^{\infty}d\gamma\frac{g(\gamma,z)}{\left[\gamma+m^{2}-\frac{1}{4}M^{2}-k^{2}-p\cdot
k\;z-i\epsilon\right]^{3}}.$ (31)

Once inserted in the BS equation, one can obtain an integral equation for the
bi-dimensional spectral function $g(\gamma,z)$. This approach was first
suggested by Kusaka and Williams [17, 18] with the aim of obtaining the BS
solutions in Minkowski space, and has taken different forms during the almost
thirty years of sustained developments [19, 20, 21, 22, 23, 24, 25, 26, 27].

Our last formulation [26] is based on a combined use of the Nakanishi
representation, the Light-Front projection of the BS amplitude and the
Stieltjes transform. The (bound state) BS equation for the weight function $g$
takes the standard form

$g(\gamma,z)=\int_{0}^{\infty}d\gamma^{\prime}\int_{-1}^{1}dz^{\prime}\;N(\gamma,z;\gamma^{\prime},z^{\prime})\;g(\gamma^{\prime},z^{\prime})$
(32)

Several, a priori equivalent, forms of the kernel $N$ corresponding to the W-C
model can be found in [22, 26, 28].

Although the spectral function $g(\gamma,z)$ is smooth, the kernel $N$ has
several moving singularities in both integration variables
$\gamma^{\prime},z^{\prime}$. They are of course integrable but must be
treated with some care to avoid spurious structures. The detailed analysis of
these singularities depends on the particular form of the kernel and will be
given in [29].

Figure 5: Upper panel: The Euclidean BS amplitude for $\mu=0.15$ and $B=0.2$.
On the left, as a function of $k_{4}$ for selected values of $k$, and on the
right, as a function of $k$ for selected values of $k_{4}$. Lower panel: the
same state in terms of the Nakanishi weight function $g(\gamma,z)$

We display in Fig. 5 two representation of the same state, one (upper part)
obtained with the BS solution in Euclidean space (4) and the other one (lower
part) with the solution of eq. (32). They correspond to $\mu=0.15$, a binding
energy $B=0.2$, with the coupling constant $\alpha=2.10$. If the Euclidean
solution in the upper part of the figure is a very smooth function of both
arguments, the $\gamma$-dependence of the Nakanishi weight function $g$ is
non-trivial. We can show analytically [29] that $g$ is constant in a
triangular domain $\Delta$ of the ($\gamma,z$) plane, illustrated in Fig. 6,
presents a cusp on its border and evolves continuously outside. The analytic
expression of this domain is $\Delta=\\{(\gamma,z)\in R^{2}:z\in[-1,+1]\;{\rm
and}\;\gamma=\gamma_{0}(z)\\}$ with $\gamma_{0}(z)$ given by

$\gamma_{0}(z)=(1-|z|){M^{2}\over 4m^{2}}\;\left({\mu^{2}\over
m^{2}}+4{\mu\over M}\sqrt{1-{M^{2}\over 4m^{2}}}\sqrt{1-{\mu^{2}\over
4m^{2}}}\right)$ (33)

This peculiar behaviour was missed in our first publications [19, 20] because
of some numerical instabilities in the left-hand side kernel333We were solving
at that time a generalized eigenvalue equation, formally writen as
$V_{L}\;g=V_{R}\;g$ and was only vaguely suggested in [22] due to an unadapted
basis set used in the numerical solution. It was however well reproduced in
Figs. 2 and 3 of [17]. It seems to be also well reproduced in Fig. 5 of Ref.
[27], although in this work the domain of constant $g$ is half a circle rather
than a triangle, maybe due to the used logarithmic scale or to some change of
variable. Notice also that the negative part of $g$, visible in Fig. 5 for
$\gamma\sim 0.2$, is totally absent in reference [17] and not clearly seen in
[27]. On the contrary the results from [18] are not understandable in terms of
the previous analysis.

Figure 6: Domain $\Delta$ of the ($\gamma,z$) where the solution $g(\gamma,z)$
is constant.

When $\mu\neq 0$ the symmetry of the 4D Coulomb problem is lost, as well as
the $\kappa$ quantum number identifying the abnormal states in the $\mu$=0
case. The solutions of the BS equation are then labeled by a single quantum
number $n$ which tells us nothing about the normal or abnormal character of
the state. On another hand, the level ordering of the abnormal states in the
$\mu$=0 case depends critically on the binding energy $B$ of the state. As one
can see in Fig. 3, for B=0.1, the first abnormal state ($\kappa$=2, $n$=1) (in
green) is the 6th excitation in the $\alpha$ spectrum, while for B=0.01 it
corresponds to the 11th… and for B=0 there is an infinity of normal states
below the first abnormal one. So, even for $\mu$=0, just by computing the
$\alpha$ spectrum for a given $B$, there is no way to identify the abnormal
state without study its wave function: there is an infinity of level crossing
among normal and abnormal states when $B$ is changed.

In a recent work we have extensively studied the survival of the abnormal
states in the massive case. The method and the results will be detailed in a
forthcoming publication [29]. We present in what follows a summary of the main
results concerning the very existence of such states.

The procedure is based on tracing the trajectories of a well-identified
abnormal state for $\mu$=0 as a function of $\mu$, and on determining in this
way the ensemble of parameters ($\alpha,\mu$) allowing the existence of the
abnormal states, as well as their binding energies $B(\alpha,\mu)$. We will
restrict here to the non-tachyonic domain of the coupling constant:
$1/4<\lambda={\alpha\over\pi}<2$.

An illustrative example is given in the left panel of Fig. 7 for the ground
abnormal state ($n$=1, $\kappa$=2) with binding energy $B$=0.007. This state
(in blue solid line) corresponds to the 12th excitation at $\mu$=0. When $\mu$
is increased, the corresponding value of $\alpha$ increases until it reaches
the maximum allowed value of the coupling constant $\alpha/\pi=2$. This
determines, the maximum allowed exchanged mass $\mu$ for this state:
$\mu_{max}\approx$0.0030. This result constitutes the first evidence, a
numerical proof of existence, of abnormal states in the W-C model with non-
zero $\mu$. Such a possibility was considered in Ref. [30], in relation with
its eventual contribution to the S-matrix.

By repeating this study for several values of $B$, one can determine
$\mu_{max}(B)$, that is the maximum value of $\mu$ compatible with a non-
tachyonic ground state solution ($M^{2}>0$) as a function of its binding
energy $B$. It is worth noticing here that for $\mu>0$ the non-tachyonic
condition is not exactly given by $\alpha=2\pi$ but by a a slightly larger
value $\alpha_{max}(\mu)>2\pi$ due to the short-range character of the $\mu>0$
interaction. Since the involved values of $\mu$ are very small, one can take
for practical purposes $\alpha_{max}(\mu)=2\pi$. The $\mu_{max}(B)$ dependence
is the essential ingredient in our study and it is displayed in the right
panel of Fig. 7. The maximum allowed value of the exchanged mass is reached
for $B=0$ and is $\mu_{max}(0)=0.087$ (in constituent units).

Figure 7: Left panel: dependence of the $\alpha$-spectrum as a function of
$\mu$, for B=0.007. The horizontal dashed line corresponds
$\alpha_{max}(\mu)\approx 2\pi$. The trajectory $\alpha(\mu)$ – corresponding
to the ground abnormal state (in blue solid line) – remains smaller than
$\alpha_{max}(\mu)$ until $\mu\approx$0.0030. The ground abnormal state, which
at $\mu$=0 is the 12th excitation of $\alpha$ spectrum, displays a level
crossing at $\mu\approx$0.0034 and becomes the 11th excited state. Right
panel: by repeating the study for several values of B one determines the
domain $\mu_{max}(B)$.

We have repeated this study for several values of the binding energy and
obtained the results of Fig. 8. Our technology does not allow us to go below
B=10-5, essentially due to the difficulty of accurately computing higher
excited states. For $B$=10-6 and $B$=0 we have just inserted the values for
$\mu$=0. On the other hand, it is worth noticing that the $\mu=0$ limit of the
W-C model is non-analytic and highly singular. This is manifested already by
the fact that the two-dimensional weigh function $g(\gamma,z)$ generates in
this limit a $\delta(\gamma)$ function and there remains only the
$z$-dependence. When computing the solutions for very small values of $\mu$ we
are faced to this embarrassing vicinity. Let us also mention that the slope at
$\mu=0$ of the $\alpha(\mu,B)$ represented at B=0 is infinite.

Figure 8: The parameter window ($\alpha,\mu$) of the ground abnormal state,
for several values of B. For $B<10^{-5}$ the ground abnormal state is too high
in the excitation spectrum to be numerically accessible: we give only the
values of the coupling constant corresponding to $\mu=0$.

The curves displayed in Fig. 8 delineate the parameter domain of the W-C
model, for which the abnormal states exist. It constitutes the main result of
our work. They allow us to draw two conclusions. The first one is that
abnormal states exist for $\mu>0$. The second one is that, due to stability
reasons of the theory, their binding energy is smaller than $B/m\approx
0.00903$ and the exchanged mass is limited to very small values of $\mu$,
$\mu/m<0.0087$.

##  5 Conclusion

We have reviewed the main properties of the “abnormal solutions” of the
Bethe–Salpeter equation with the Wick–Cutkosky model, i.e., scalar particles
with mass $m$ interacting via massless ($\mu$=0) exchange, as wells as its
extension to the massive-exchange case.

These are low-energy ($B/m<0.009$) solutions that exist in this particular
relativistic approach, but are absent in the non-relativistic limit (Coulomb
problem). Their position in the full spectrum of the model is totally
decoupled from the normal solutions, which tend to the standard Coulomb
states, and require, even in the zero binding limit, a minimal value of the
coupling constant, $\alpha_{min}=\pi/4$, to exist. From this point of view the
abnormal solutions behave as if they were created by an effective ”massive
photon”. In the $\mu=0$ case, this is equivalent to a ”massive photon” with an
effective mass of $\mu_{eff}/m\approx 0.47$. For the $\mu>0$ case, the value
of $\mu_{eff}$ is roughly the same than for the massless case since the values
of $\alpha_{min}$ are practically unchanged with $\mu$ (See left panel of Fig
7).

Discovered by Cutkosky [6] soon after the formulation of the Bethe–Salpeter
equation and its first solution in Euclidean space by Wick [5], we have given
them [14] an intrinsic characterization in terms of the small two-body norm of
their valence wave function, which vanishes in the $B\to 0$ limit. This
confers to them a genuine many body status.

We have presented new results concerning the massive-exchange case, where we
have obtained the ensemble of parameters of the model, in particular the
values of the exchanged mass $\mu$, that allow the existence of such peculiar
solutions without spoiling the model by tachyonic states ($M^{2}<0$).

As our previous analysis shows, the reason for the existence of abnormal
states, dominated by multi-photon exchange, is the strong electrical field
between constituents and therefore it can be hardly affected by the eventual
spin degrees of freedom which were not included in our consideration. The
experimental creation and observation of these systems do not seem to be an
easy task, but they would be of great interest.

Acknowledgments J.C. thanks the financial support from FAPESP (Fundação de
Amparo à Pesquisa do Estado de São Paulo) grant 2022/10580-3. H.S.
acknowledges financial support from the EU research and innovation programme
Horizon 2020, under Grant agreement No. 824093.

## References

  * [1] J. Carbonell, B. Desplanques, V.A. Karmanov, J.-F. Mathiot, _Explicitly covariant light front dynamics and relativistic few-body systems_ , Phys. Rep. 300, 215 (1998)
  * [2] H.A. Bethe, E.E. Salpeter, _A relativistic equation for Bound State Problems_ , Phys. Rev. 82, 309 (1951)
  * [3] E.E. Salpeter, H. Bethe, _A Relativistic equation for bound-state problems_ , Phys. Rev. 84, 1232 (1951).
  * [4] M. Mangin-Brinet, J. Carbonell, _Solutions of the Wick-Cutkosky model in the Light Front Dynamics_ , Phys. Lett. B 474, 237(2000)
  * [5] G.C. Wick, _Properties of Bethe-Salpeter wave functions_ , Phys. Rev. 96, 1124 (1954).
  * [6] R.E. Cutkosky, _Solutions of a Bethe-Salpeter equation_ , Phys. Rev. 96, 1135 (1954).
  * [7] M. Gell-Mann, F. Low, Bound states in quantum field theory, Phys. Rev. 84 (1951) 350
  * [8] G. Feldman, T. Fulton, J. Townsend, Wick equation, the infinite-momentum frame, and perturbation theory, Phys. Rev. D 7, 1814 (1973).
  * [9] M. Ciafaloni, P. Menotti, _Operator analysis of the Bethe-Salpeter equation_ , Phys. Rev. 140, B929 (1965).
  * [10] H. Jallouli, H. Sazdjian, There are no abnormal solutions of the Bethe-Salpeter equation in the static model. J. Phys. G 22, 1119 (1996).
  * [11] V. A. Karmanov, J. Carbonell, H. Sazdjian, _Structure and EM form factors of purely relativistic systems_ , PoS(LC2019)050; arXiv:2001.00401.
  * [12] V. A. Karmanov, J. Carbonell, H. Sazdjian, _Bound states of purely relativistic nature_ , EPJ Web Conf. 204 (2019) 01014
  * [13] V. A. Karmanov, J. Carbonell, H. Sazdjian, _Bound states of relativistic nature_ , Contribution to NTSE 2018, 1903.02892 [hep-ph]
  * [14] J. Carbonell, V.A. Karmanov, H. Sazdjian, _Hybrid nature of the abnormal solutions of the Bethe-Salpeter equation in the Wick-Cutkosky model_ , Eur. Phys. J. C (2021) 81:50
  * [15] V.A. Karmanov, _Abnormal states with unequal constituent masses_ , Eur. Phys. J. C (2024) 84:58
  * [16] N. Nakanishi, Partial-wave Bethe–Salpeter equation, Phys. Rev. 130, 1230 (1963);General Survey of the Theory of the Bethe-Salpeter Equation, Prog. Theor. Phys. Suppl. 43, 1 (1969); Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971).
  * [17] K. Kusaka and A.G. Williams, _Solving the Bethe-Salpeter equation for scalar theories in Minkowski space_ , Phys. Rev. D 51 (1995) 7026;
  * [18] K. Kusaka, K. Simpson and A.G. Williams, _Solving the Bethe-Salpeter equation for bound states of scalar theories in Minkowski space_ , Phys. Rev. D 56 (1997) 5071.
  * [19] V.A. Karmanov and J. Carbonell, _Solving Bethe-Salpeter equation in Minkowski space_ , Eur. Phys. J. A 27 (2006) 1.
  * [20] J. Carbonell, V.A. Karmanov, _Cross-ladder effects in Bethe-Salpeter and light-front equations_ , Eur. Phys. J. A 27 (2006) 11.
  * [21] T. Frederico, G. Salmè and M. Viviani, _Two-body scattering states in Minkowski space and the Nakanishi integral representation onto the null plane_ , Phy. Rev. D 85 036009 (2012)
  * [22] T. Frederico, G. Salmè and M. Viviani, _Quantitative studies of the homogeneous Bethe-Salpeter equation in Minkowski space_ , Phy. Rev. D 89 (2014) 016010.
  * [23] T. Frederico, G. Salmè and M. Viviani, _Solving the inhomogeneous Bethe-Salpeter Equation in Minkowski space: the zero-energy limit_ , Eur. Phys. J. C 75 (2015) 398.
  * [24] W. de Paula, T. Frederico, G. Salmè, M. Viviani, _Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space_ , Phys.Rev. D 94 (2016) 071901.
  * [25] C. Gutierrez, V. Gigante, T. Frederico, G. Salmè, M. Viviani, L. Tomio, _Bethe?Salpeter bound-state structure in Minkowski space_ , Phys. Lett. B 759 (2016) 131.
  * [26] J. Carbonell, T. Frederico, V.A. Karmanov, Bound state equation for the Nakanishi weight function, Phys. Lett. B, 769, 418 (2017); arXiv:1704.04160.
  * [27] Shaoyang Jia, _Direct solution of Minkowski-space Bethe-Salpeter equation in the massive Wick-Cutkosky model_ , Phys. Rev. D 109, 036020 (2024)
  * [28] V.A. Karmanov, _New form of kernel in equation for Nakanishi function_ , Phys. Rev. D 104, 056012 (2021)
  * [29] J. Carbonell, V.A. Karmanov, E.A. Kupriyanova and H. Sazdjian, _Abnormal solutions of the Bethe-Salpeter equation: the massive exchange case_ , to be published
  * [30] S. Naito, _S-Matrix and Abnormal Solutions of the Bethe-Salpeter Equation_ , Prog. Theor. Phys. 40, 628 (1968)
  * [31] J. Carbonell · F. de Soto · V. A. Karmanov, _Three Different Approaches to the Same Interaction: The Yukawa Model in Nuclear Physics_ , Few-Body Syst (2013) 54:2255-2269


# TIME-DEPENDENT STABLE OPERATORS

###### Abstract.

We prove that every invertible generalized hyperbolic operator on a Banach
space is time-dependent stable.

###### Key words and phrases:

Hyperbolicity; Generalized hyperbolicity; Structural stability; Time-dependent
stability; Linear operators; Banach spaces.

###### 2020 Mathematics Subject Classification:

Primary 37C20, 37B25; Secondary 47A16.

Nilson C. Bernardes Jr.

Institut Universitari de Matemàtica Pura i Aplicada

Universitat Politècnica de València

Camí de Vera S/N, Edifici 8E, Acces F, 4a Planta, València, 46022, Spain

and

Departamento de Matemática Aplicada, Instituto de Matemática

Universidade Federal do Rio de Janeiro, Caixa Postal 68530

Rio de Janeiro, RJ, 21941-909, Brazil

e-mail<EMAIL_ADDRESS>
##  1\. Introduction

Structural stability is one of the most important concepts in the modern
theory of dynamical systems and differential equations. It was originally
introduced by A. Andronov and L. Pontrjagin [1] for a certain class of
differentiable flows on the plane. Later, variations of the original concept
were introduced and investigated in different contexts. In the area of
differentiable dynamics, a diffeomorphism $f:M\to M$ on a compact manifold is
said to be structurally stable if every diffeomorphism $g:M\to M$ that is
sufficiently close to $f$ in the $C^{1}$-topology is topologically conjugate
to $f$, that is, there exists a homeomorphism $h:M\to M$ such that $g=h\circ
f\circ h^{-1}$. The importance of this concept lies in the fact that models of
physical systems are only approximations of the true systems and it is
important to know whether the qualitative behavior of the approximation and
the true dynamical system are the same. We refer the reader to [12, 16, 17]
for nice expositions about structural stability.

J. Franks [8] observed that, in most physical situations, it seems likely that
the true dynamical system is not really autonomous but is instead to a certain
extend time dependent. So, he proposed the following concept of stability: A
diffeomorphim $f:M\to M$ on a compact manifold is said to be time-dependent
stable if there is a neighborhood $\mathcal{N}$ of $f$ in the $C^{1}$-topology
such that $g_{1}\circ\cdots\circ g_{p}$ is topologically conjugate to $f^{p}$
whenever $g_{1},\ldots,g_{p}\in\mathcal{N}$ (an important point here is that
$\mathcal{N}$ is independent of $p$). It is clear that this concept is
stronger than structural stability.

In the present note we are interested in the dynamics of invertible operators
on Banach spaces. Our main motivation comes from the following classical
result.

Hartman’s theorem. Every invertible hyperbolic operator on a Banach space is
structurally stable.

The name “Hartman’s theorem” for the above result was coined by C. Pugh [15].
Actually, P. Hartman [11] established it for operators on finite-dimensional
euclidean spaces. The extension to arbitrary Banach spaces was independently
obtained by J. Palis [14] and C. Pugh [15] (both were motivated by an argument
due to J. Moser [13]). The above theorem is the major tool for the proof of
the Grobman-Hartman theorem in Banach spaces as established in [14, 15].

A basic question is whether or not the converse of Hartman’s theorem is always
true. It was soon realized that the answer is positive in the finite-
dimensional setting (see J. Robbin [16]), but the full question remained open
for more than 50 years. This question was finally answered in the negative in
the recent paper [4] of the author with A. Messaoudi, where the first examples
of structurally stable operators that are not hyperbolic were obtained. A
little later, the same authors proved that every invertible generalized
hyperbolic operator on a Banach space is structurally stable [3, Theorem 1],
which implied a generalized Grobman-Hartman theorem [3, Theorem 3].

In this note we address another basic question related to Hartman’s theorem,
namely:

> Is every invertible hyperbolic operator on a Banach space time-dependent
> stable?

Our goal is to give a positive answer to this question. Actually, we will
establish the following more general result.

######  Theorem 1.

Every invertible generalized hyperbolic operator on a Banach space is time-
dependent stable.

In fact, this theorem will be derived from an even more general result
(Theorem 4).

The note is organized as follows. In Section 2 we present all the necessary
preliminaries for the present work. In Section 3 we prove a refinement of
Theorem 1, namely Theorem 4, in which we also analyze the problem of
uniqueness of the homeomorphism that establishes the conjugation and the
possibility of choosing this homeomorphism in such a way that we have
continuous dependence with respect to the pertubation. Some additional remarks
are also presented at the end of the section.

##  2\. Preliminaries

Throughout $\mathbb{K}$ denotes either the field $\mathbb{R}$ of real numbers
or the field $\mathbb{C}$ of complex numbers, $\mathbb{N}$ denotes the set of
all positive integers and $\mathbb{N}_{0}\\!:=\mathbb{N}\cup\\{0\\}$.

All Banach spaces are assumed to be over $\mathbb{K}$, unless otherwise
specified.

Given a continuous linear map $T$ from a Banach space $X$ into a Banach space
$Y$, recall that the norm of $T$ is the non-negative real number given by

$\|T\|=\sup_{\|x\|\leq 1}\|T(x)\|.$

This notion will play an important role in the next section. If $F:X\to Y$ is
any map, we define

$\|F\|_{\infty}=\sup_{x\in X}\|F(x)\|\ \ \text{ and }\ \
\operatorname{Lip}(F)=\sup_{x\neq
x^{\prime}}\frac{\|F(x)-F(x^{\prime})\|}{\|x-x^{\prime}\|}\,\cdot$

We say that $F$ is a bounded map if $\|F\|_{\infty}<\infty$ and it is a
Lipschitz map if $\operatorname{Lip}(F)<\infty$.

A continuous linear map from a Banach space $X$ into itself will be called
simply an operator on $X$. We denote by $I_{X}$ the identity operator on $X$.
In this note we are interested in the dynamics of invertible operators.

Recall that an invertible operator $T$ on a Banach space $X$ is said to be
hyperbolic if its spectrum $\sigma(T)$ does not intersect the unit circle in
the complex plane. It is well known that $T$ is hyperbolic if and only if
there are an equivalent norm $\|\cdot\|$ on $X$ and a direct sum decomposition

$X=X_{s}\oplus X_{u},\ T=T_{s}\oplus T_{u}\ \ \ (\text{the {\em hyperbolic
splitting} of }T),$

where $X_{s}$ and $X_{u}$ are closed $T$-invariant subspaces of $X$ (the
stable and the unstable subspaces for $T$), $T_{s}=T|_{X_{s}}$ is a proper
contraction (i.e., $\|T_{s}\|<1$) and $T_{u}=T|_{X_{u}}$ is a proper dilation
(i.e., $\|T_{u}^{-1}\|<1$).

Recall that an invertible operator $T$ on a Banach space $X$ is said to be
generalized hyperbolic if there is a direct sum decomposition

$X=M\oplus N,$

where $M$ and $N$ are closed subspaces of $X$ with the following properties:

  * (GH1)

$T(M)\subset M$ and $\sigma(T|_{M})\subset\mathbb{D}$,

  * (GH2)

$T^{-1}(N)\subset N$ and $\sigma(T^{-1}|_{N})\subset\mathbb{D}$,

where $\mathbb{D}$ denotes the open unit disc in the complex plane. This class
of operators appeared in the statement of [2, Theorem A], which was the major
tool to prove the existence of operators that have the shadowing property but
are not hyperbolic [2, Theorem B]. However, the terminology “generalized
hyperbolic” was introduced in [6], where additional dynamical properties of
these operators were investigated (see also [3, 5]).

In the definition of the concept of structural stability, it is usual to
consider pertubations that are small in the sense of the $C^{1}$-topology.
However, in certain situations, some authors allow more general pertubations,
namely: Lipschitz pertubations with small sup norm and small Lipschitz
constant (see [15], for instance). As in [3, 4], we adopt the following
definition of structural stability in the setting of linear dynamics.

######  Definition 2.

An invertible operator $T$ on a Banach space $X$ is said to be structurally
stable if there exists $\varepsilon>0$ such that

$T+L\text{ is topologically conjugate to }T$

whenever $L:X\to X$ is a Lipschitz map with $\|L\|_{\infty}<\varepsilon$ and
$\operatorname{Lip}(L)<\varepsilon$.

Therefore, it is natural to adopt the following definition of time-dependent
stability in the present context.

######  Definition 3.

An invertible operator $T$ on a Banach space $X$ is said to be time-dependent
stable if there exists $\varepsilon>0$ such that

$(T+L_{1})\circ\cdots\circ(T+L_{p})\text{ is topologically conjugate to
}T^{p}$

whenever $p\in\mathbb{N}$ and $L_{1},\ldots,L_{p}$ are Lipschitz maps from $X$
into itself with $\|L_{j}\|_{\infty}<\varepsilon$ and
$\operatorname{Lip}(L_{j})<\varepsilon$ for all $j\in\\{1,\ldots,p\\}$.

##  3\. Main result

Let us now state and prove our main result, which clearly implies Theorem 1.

######  Theorem 4.

For any invertible generalized hyperbolic operator $T$ on any Banach space
$X$, the following properties hold:

  * (A)

There exists $\varepsilon>0$ such that for any $p\in\mathbb{N}$ and any
bounded Lipschitz maps $L_{j}:X\to X$, $j=1,\ldots,p$, with

$\operatorname{Lip}(L_{j})<\varepsilon\ \text{ for all }j\in\\{1,\ldots,p\\},$

there exists a homeomorphism $h:X\to X$ satisfying

$(T+L_{1})\circ\cdots\circ(T+L_{p})=h\circ T^{p}\circ h^{-1}\ \text{ and }\ \
\|h-I_{X}\|_{\infty}<\infty.$

  * (B)

Given $\delta>0$, there exists $\varepsilon>0$ such that for any
$p\in\mathbb{N}$ and any Lipschitz maps $L_{j}:X\to X$, $j=1,\ldots,p$, with

$\|L_{j}\|_{\infty}<\varepsilon\ \text{ and }\
\operatorname{Lip}(L_{j})<\varepsilon\ \text{ for all }j\in\\{1,\ldots,p\\},$

there exists a homeomorphism $h:X\to X$ satisfying

$(T+L_{1})\circ\cdots\circ(T+L_{p})=h\circ T^{p}\circ h^{-1}\ \text{ and }\ \
\|h-I_{X}\|_{\infty}<\delta.$

Moreover, the homeomorphism $h$ given in (A) (or (B)) is unique if and only if
the operator $T$ is hyperbolic. Nevertheless, it is always possible to choose
such a homeomorphism $h_{\mathcal{L}}$ for each $p$-tuple
$\mathcal{L}=(L_{1},\ldots,L_{p})$ of maps as in (A) (resp. (B)) in such a way
that the correspondence $\mathcal{L}\to h_{\mathcal{L}}$ is uniformly
continuous in the following sense: given $\gamma>0$, there exists $\eta>0$
such that

$\|h_{\mathcal{L}}-h_{\mathcal{L}^{\prime}}\|_{\infty}<\gamma$

whenever $\mathcal{L}=(L_{1},\ldots,L_{p})$ and
$\mathcal{L}^{\prime}=(L^{\prime}_{1},\ldots,L^{\prime}_{p})$ are $p$-tuples
of maps as in (A) (resp. (B)) with

$\|L_{j}-L^{\prime}_{j}\|_{\infty}<\eta\ \text{ for all
}j\in\\{1,\ldots,p\\}.$

Note that the assumption that the $L_{j}$:s are small in the sup norm is not
necessary to guarantee that $(T+L_{1})\circ\cdots\circ(T+L_{p})$ is
topologically conjugate to $T^{p}$, the important assumption is that the
$L_{j}$:s are bounded maps with small Lipschitz constants. However, by adding
the hypothesis that the $L_{j}$:s are small in the sup norm, we can ensure
that the homeomorphism that establishes the conjugation can be chosen to be
uniformly close to the identity map.

###### Proof.

Fix an invertible generalized hyperbolic operator $T$ on a Banach space $X$.
Let

(1) $X=M\oplus N$

be a direct sum decomposition given by the definition of generalized
hyperbolicity and consider the closed subspace of $X$ given by

$Y=M+T^{-1}(N).$

By the spectral radius formula, there exist $a\geq 1$ and $t\in(0,1)$ such
that

(2) $\|T^{n}(y)\|\leq a\,t^{n}\|y\|\ \text{ and }\ \|T^{-n}(z)\|\leq
a\,t^{n}\|z\|\ \text{ whenever }n\in\mathbb{N}_{0},y\in M,z\in N.$

Let $P_{M}:X\to M$ and $P_{N}:X\to N$ denote the canonical projections
associated to the direct sum decomposition (1) and define

(3) $b=\max\\{\|P_{M}\|,\|P_{N}\|\\}.$

Choose $\delta\in(0,1)$. We shall prove that the number

$\varepsilon=\min\Big{\\{}\frac{1-t}{a\,b\,(1+t)},\frac{1}{\|T^{-1}\|}\Big{\\}}\,\delta$

satisfies properties (A) and (B). Indeed, fix $p\in\mathbb{N}$ and a $p$-tuple
$\mathcal{L}=(L_{0},\ldots,L_{p-1})$ of bounded Lipschitz maps $L_{j}:X\to X$
such that

$\operatorname{Lip}(L_{j})<\varepsilon\ \text{ for all
}j\in\\{0,\ldots,p-1\\}.$

Let $S_{j}=T+L_{j}$ for each $j$. Since
$\operatorname{Lip}(L_{j})<\|T^{-1}\|^{-1}$, we have that $S_{j}$ is a
homeomorphism. We shall prove that $S_{p-1}\circ\cdots\circ S_{0}$ is
topologically conjugate to $T^{p}$. For this purpose, let
$\mathbb{Z}_{p}=\\{0,\ldots,p-1\\}$ endowed with the operation $+_{p}$ of
addition mod $p$ and with the trivial metric $d_{0}$. Consider the product
$\widetilde{X}=X\times\mathbb{Z}_{p}$ endowed with the metric

$d((x,j),(x^{\prime},j^{\prime}))=\max\\{\|x-x^{\prime}\|,d_{0}(j,j^{\prime})\\}.$

For $(x,j),(x^{\prime},j)\in\widetilde{X}$ and $\lambda\in\mathbb{K}$, we
define

$(x,j)\pm(x^{\prime},j)=(x\pm x^{\prime},j),\ \ \ \lambda(x,j)=(\lambda x,j)\
\ \text{ and }\ \ |(x,j)|=\|x\|.$

Note that the above operations of “addition” and “subtraction” on
$\widetilde{X}$ are defined only for pairs of points with the same second
coordinate. Moreover, we define

$\sum_{k=0}^{\infty}(x_{k},j)=\Big{(}\sum_{k=0}^{\infty}x_{k}\,,j\Big{)},$

provided $\sum_{k=0}^{\infty}x_{k}$ is a convergent series in $X$. We will
need the function spaces defined below:

  * •

$\mathcal{F}$ is the set of all continuous bounded maps
$F:\widetilde{X}\to\widetilde{X}$ such that

$F(X\times\\{j\\})\subset Y\times\\{j\\}\ \text{ for all }j\in\mathbb{Z}_{p}.$

  * •

$\mathcal{G}$ is the set of all continuous bounded maps
$F:\widetilde{X}\to\widetilde{X}$ such that

$F(X\times\\{j\\})\subset X\times\\{j+_{p}1\\}\ \text{ for all
}j\in\mathbb{Z}_{p}.$

We recall that $F:\widetilde{X}\to\widetilde{X}$ is a bounded map if its range
$F(\widetilde{X})$ is a bounded set in $\widetilde{X}$, i.e., a set with
finite diameter. Although $\widetilde{X}$ is not a vector space and $|\cdot|$
is not a norm, both $\mathcal{F}$ and $\mathcal{G}$ are Banach spaces when
endowed with the algebraic operations defined pointwise

$(F_{1}+F_{2})(x,j)=F_{1}(x,j)+F_{2}(x,j)\ \ \text{ and }\ \ (\lambda
F)(x,j)=\lambda F(x,j),$

and the supremum norm

$|F|_{\infty}=\sup_{(x,j)\in\widetilde{X}}|F(x,j)|.$

Note that the zero vector of $\mathcal{F}$ (resp. $\mathcal{G}$) is the map
given by

$O_{\mathcal{F}}(x,j)=(0,j)\ \ \ \ (\text{resp.}\
O_{\mathcal{G}}(x,j)=(0,j+_{p}1)).$

Let $Q_{M}:\widetilde{X}\to M\times\mathbb{Z}_{p}$ and $Q_{N}:\widetilde{X}\to
N\times\mathbb{Z}_{p}$ be given by

$Q_{M}(x,j)=(P_{M}(x),j)\ \ \text{ and }\ \ Q_{N}(x,j)=(P_{N}(x),j).$

Consider the homeomorphisms $\widetilde{T}:\widetilde{X}\to\widetilde{X}$ and
$S:\widetilde{X}\to\widetilde{X}$ defined by

$\widetilde{T}(x,j)=(T(x),j+_{p}1)\ \ \text{ and }\ \
S(x,j)=(S_{j}(x),j+_{p}1).$

Given a homeomorphism $R:\widetilde{X}\to\widetilde{X}$ with
$R(X\times\\{j\\})=X\times\\{j+_{p}1\\}$ for all $j$, we consider the map

(4) $\Psi:F\in\mathcal{F}\mapsto F\circ R-\widetilde{T}\circ F\in\mathcal{G}.$

It is easy to check that $\Psi$ is linear and continuous.

We claim that $\Psi$ is injective. Indeed, let $F\in\mathcal{F}$ with
$\Psi(F)=O_{\mathcal{G}}$. We have to show that $F=O_{\mathcal{F}}$. The
equality $\Psi(F)=O_{\mathcal{G}}$ gives

(5) $F\circ R^{n}=\widetilde{T}^{n}\circ F\ \ \text{ for all }n\in\mathbb{N}.$

Hence, for every $(x,j)\in\widetilde{X}$ and $n\in\mathbb{N}$,

(6) $F(x,j)=\widetilde{T}^{n}(F(R^{-n}(x,j)))=\alpha_{n}(x,j)+\beta_{n}(x,j),$

where

$\alpha_{n}(x,j)=\widetilde{T}^{n}(Q_{M}(F(R^{-n}(x,j))))\ \text{ and }\
\beta_{n}(x,j)=\widetilde{T}^{n}(Q_{N}(F(R^{-n}(x,j)))).$

It is clear that, for each $n\in\mathbb{N}$,

(7) $\alpha_{n}(x,j)\in M\times\mathbb{Z}_{p}\ \ \text{ for all
}(x,j)\in\widetilde{X}.$

Let us show that, for each $n\in\mathbb{N}$,

(8) $\beta_{n}(x,j)\in T^{-1}(N)\times\mathbb{Z}_{p}\ \ \text{ for all
}(x,j)\in\widetilde{X}.$

Since $F(R^{-1}(x,j))\in Y\times\mathbb{Z}_{p}$, we must have
$Q_{N}(F(R^{-1}(x,j)))\in T^{-1}(N)\times\mathbb{Z}_{p}$, and so

$\beta_{1}(x,j)=\widetilde{T}(Q_{N}(F(R^{-1}(x,j))))\in
N\times\mathbb{Z}_{p}.$

Hence, since $\alpha_{1}(x,j)\in M\times\mathbb{Z}_{p}$ and
$\alpha_{1}(x,j)+\beta_{1}(x,j)=F(x,j)\in Y\times\mathbb{Z}_{p}$, we see that
(8) holds for $n=1$. Assume that (8) holds for a certain $n\in\mathbb{N}$.
Then,

$\beta_{n+1}(x,j)=\widetilde{T}(\beta_{n}(R^{-1}(x,j)))\in
N\times\mathbb{Z}_{p}.$

As before, we conclude that (8) holds with $n+1$ in the place of $n$. By
induction, (8) holds for all $n\in\mathbb{N}$. By (6), (7) and (8),

$Q_{M}(F(x,j))=\alpha_{n}(x,j)=\widetilde{T}^{n}(Q_{M}(F(R^{-n}(x,j))))\ \
\text{ for all }n\in\mathbb{N}.$

Thus, by (2) and (3),

$|Q_{M}(F(x,j))|\leq a\,b\,t^{n}|F|_{\infty}\ \ \text{ for all
}n\in\mathbb{N},$

implying that

(9) $Q_{M}(F(x,j))=(0,j)\ \ \text{ for all }(x,j)\in\widetilde{X}.$

Now, by (5), for every $(x,j)\in\widetilde{X}$ and $n\in\mathbb{N}$,

(10)
$F(x,j)=\widetilde{T}^{-n}(F(R^{n}(x,j)))=\varphi_{n}(x,j)+\psi_{n}(x,j),$

where

$\varphi_{n}(x,j)=\widetilde{T}^{-n}(Q_{M}(F(R^{n}(x,j))))\ \text{ and }\
\psi_{n}(x,j)=\widetilde{T}^{-n}(Q_{N}(F(R^{n}(x,j)))).$

It is clear that, for each $n\in\mathbb{N}$,

(11) $\psi_{n}(x,j)\in T^{-1}(N)\times\mathbb{Z}_{p}\ \ \text{ for all
}(x,j)\in\widetilde{X}.$

Let us show that, for each $n\in\mathbb{N}$,

(12) $\varphi_{n}(x,j)\in M\times\mathbb{Z}_{p}\ \ \text{ for all
}(x,j)\in\widetilde{X}.$

Since

$\displaystyle\widetilde{T}(Q_{N}(\varphi_{1}(x,j)))$
$\displaystyle=\widetilde{T}(\varphi_{1}(x,j))-\widetilde{T}(Q_{M}(\varphi_{1}(x,j)))$
$\displaystyle=Q_{M}(F(R(x,j)))-\widetilde{T}(Q_{M}(\varphi_{1}(x,j)))\in
M\times\mathbb{Z}_{p},$

we have that

(13) $Q_{N}(\varphi_{1}(x,j))\in T^{-1}(M)\times\mathbb{Z}_{p}.$

By (10) with $n=1$, we can write

$F(x,j)=Q_{M}(\varphi_{1}(x,j))+(Q_{N}(\varphi_{1}(x,j))+\psi_{1}(x,j)).$

Since $Q_{M}(\varphi_{1}(x,j))\in M\times\mathbb{Z}_{p}$ and
$Q_{N}(\varphi_{1}(x,j))+\psi_{1}(x,j)\in N\times\mathbb{Z}_{p}$, the fact
that $F(x,j)\in Y\times\mathbb{Z}_{p}$ implies that
$Q_{N}(\varphi_{1}(x,j))+\psi_{1}(x,j)\in T^{-1}(N)\times\mathbb{Z}_{p}$.
Hence, by (11),

(14) $Q_{N}(\varphi_{1}(x,j))\in T^{-1}(N)\times\mathbb{Z}_{p}.$

By (13) and (14), $Q_{N}(\varphi_{1}(x,j))\in\\{0\\}\times\mathbb{Z}_{p}$,
which gives (12) for $n=1$. Assume that (12) holds for a certain
$n\in\mathbb{N}$. Then,

$\displaystyle\widetilde{T}(Q_{N}(\varphi_{n+1}(x,j)))$
$\displaystyle=\widetilde{T}(\varphi_{n+1}(x,j))-\widetilde{T}(Q_{M}(\varphi_{n+1}(x,j)))$
$\displaystyle=\varphi_{n}(R(x,j))-\widetilde{T}(Q_{M}(\varphi_{n+1}(x,j)))\in
M\times\mathbb{Z}_{p},$

and so

$Q_{N}(\varphi_{n+1}(x,j))\in T^{-1}(M)\times\mathbb{Z}_{p}.$

By arguing as above, we obtain
$Q_{N}(\varphi_{n+1}(x,j))\in\\{0\\}\times\mathbb{Z}_{p}$, and so (12) holds
with $n+1$ in the place of $n$. By induction, (12) holds for all
$n\in\mathbb{N}$. By (10), (11) and (12),

$Q_{N}(F(x,j))=\psi_{n}(x,j)=\widetilde{T}^{-n}(Q_{N}(F(R^{n}(x,j))))\ \
\text{ for all }n\in\mathbb{N}.$

Therefore, by (2) and (3),

$|Q_{N}(F(x,j))|\leq a\,b\,t^{n}|F|_{\infty}\ \ \text{ for all
}n\in\mathbb{N},$

which implies that

(15) $Q_{N}(F(x,j))=(0,j)\ \ \text{ for all }(x,j)\in\widetilde{X}.$

By (9) and (15), we conclude that $F=O_{\mathcal{F}}$, as desired.

Let us now prove that $\Psi$ is surjective (hence bijective) and its inverse
is given by

(16)
$\Psi^{-1}(G)(x,j)=\sum_{k=0}^{\infty}\widetilde{T}^{k}(Q_{M}(G(R^{-k-1}(x,j))))-\sum_{k=1}^{\infty}\widetilde{T}^{-k}(Q_{N}(G(R^{k-1}(x,j)))).$

For this purpose, take $G\in\mathcal{G}$. Note that the partial sums of both
series in (16) are well defined. Moreover, by (2) and (3),

(17)
$\sum_{k=0}^{\infty}|\widetilde{T}^{k}(Q_{M}(G(R^{-k-1}(x,j))))|\leq\sum_{k=0}^{\infty}a\,b\,t^{k}|G(R^{-k-1}(x,j))|\leq\frac{a\,b}{1-t}\,|G|_{\infty}$

and

(18)
$\sum_{k=1}^{\infty}|\widetilde{T}^{-k}(Q_{N}(G(R^{k-1}(x,j))))|\leq\sum_{k=1}^{\infty}a\,b\,t^{k}|G(R^{k-1}(x,j))|\leq\frac{a\,b\,t}{1-t}\,|G|_{\infty}.$

This implies that the first (resp. second) series in (16) is well defined and
determines an element of $M\times\\{j\\}$ (resp. $T^{-1}(N)\times\\{j\\}$).
Hence, the difference in (16) is an element of $Y\times\\{j\\}$. Since the
convergence is uniform, we conclude that the right hand side of (16) defines a
map $F\in\mathcal{F}$. Now,

$\displaystyle\Psi(F)(x,j)$ $\displaystyle=F(R(x,j))-\widetilde{T}(F(x,j))$
$\displaystyle=\big{(}Q_{M}(F(R(x,j)))-\widetilde{T}(Q_{M}(F(x,j)))\big{)}$
$\displaystyle\ \ \ \ \ \
+\big{(}Q_{N}(F(R(x,j)))-\widetilde{T}(Q_{N}(F(x,j)))\big{)}$
$\displaystyle=\Big{(}\sum_{k=0}^{\infty}\widetilde{T}^{k}(Q_{M}(G(R^{-k}(x,j))))-\sum_{k=0}^{\infty}\widetilde{T}^{k+1}(Q_{M}(G(R^{-k-1}(x,j))))\Big{)}$
$\displaystyle\ \ \ \ \ \
+\Big{(}-\sum_{k=1}^{\infty}\widetilde{T}^{-k}(Q_{N}(G(R^{k}(x,j))))+\sum_{k=1}^{\infty}\widetilde{T}^{-k+1}(Q_{N}(G(R^{k-1}(x,j))))\Big{)}$
$\displaystyle=Q_{M}(G(x,j))+Q_{N}(G(x,j))$ $\displaystyle=G(x,j),$

showing that $\Psi(F)=G$. Thus, $\Psi$ is bijective and its inverse is given
by (16). Moreover, the estimates in (17) and (18) imply that

(19) $\|\Psi^{-1}\|\leq\frac{a\,b\,(1+t)}{1-t}\,\cdot$

We will need maps of the form (4) for two choices of $R$, namely:
$R=\widetilde{T}$ and $R=S$. In other words, we will consider the bijective
continuous linear maps

$\Psi_{1}:F\in\mathcal{F}\mapsto F\circ\widetilde{T}-\widetilde{T}\circ
F\in\mathcal{G}\ \ \text{ and }\ \ \Psi_{2}:F\in\mathcal{F}\mapsto F\circ
S-\widetilde{T}\circ F\in\mathcal{G}.$

Let $\overline{\mathcal{L}}\in\mathcal{G}$ be defined by

$\overline{\mathcal{L}}(x,j)=(L_{j}(x),j+_{p}1).$

We define the maps $\Phi_{1}:\mathcal{F}\to\mathcal{G}$ and
$\Phi_{2}:\mathcal{F}\to\mathcal{G}$ by

$\Phi_{1}(F)(x,j)=\overline{\mathcal{L}}((x,j)+F(x,j))\ \ \text{ and }\ \
\Phi_{2}(F)(x,j)=\Phi_{1}(F)(x,j)-\overline{\mathcal{L}}(x,j).$

We claim that $\Phi_{1}$ and $\Phi_{2}$ are Lipschitz maps with

(20)
$\operatorname{Lip}(\Phi_{1})=\operatorname{Lip}(\Phi_{2})\leq\varepsilon.$

Indeed, take $F,G\in\mathcal{F}$ with $F(x,j)=(f_{j}(x),j)$ and
$G(x,j)=(g_{j}(x),j)$. Since

$\displaystyle|\overline{\mathcal{L}}((x,j)+F(x,j))-\overline{\mathcal{L}}(($
$\displaystyle x,j)+G(x,j))|$
$\displaystyle=|(L_{j}(x+f_{j}(x)),j+_{p}1)-(L_{j}(x+g_{j}(x)),j+_{p}1)|$
$\displaystyle=\|L_{j}(x+f_{j}(x))-L_{j}(x+g_{j}(x))\|$
$\displaystyle\leq\operatorname{Lip}(L_{j})\|f_{j}(x)-g_{j}(x)\|$
$\displaystyle\leq\varepsilon\,|F-G|_{\infty},$

we obtain $\operatorname{Lip}(\Phi_{1})\leq\varepsilon$. It is clear that
$\operatorname{Lip}(\Phi_{2})=\operatorname{Lip}(\Phi_{1})$.

By (19) and (20), the map
$\Psi_{1}^{-1}\circ\Phi_{1}:\mathcal{F}\to\mathcal{F}$ satisfies

(21)
$\operatorname{Lip}(\Psi_{1}^{-1}\circ\Phi_{1})\leq\|\Psi_{1}^{-1}\|\operatorname{Lip}(\Phi_{1})\leq\frac{a\,b\,(1+t)}{1-t}\,\varepsilon\leq\delta<1.$

Hence, by Banach’s fixed point theorem, $\Psi_{1}^{-1}\circ\Phi_{1}$ has a
unique fixed point $U$ in $\mathcal{F}$. Write $U(x,j)=(u_{j}(x),j)$ and let
$H:\widetilde{X}\to\widetilde{X}$ be the continuous map defined by

$H(x,j)=(x+u_{j}(x),j).$

We claim that

(22) $S\circ H=H\circ\widetilde{T}.$

Indeed, since $U$ is a fixed point of $\Psi_{1}^{-1}\circ\Phi_{1}$,

$U\circ\widetilde{T}-\widetilde{T}\circ U=\Psi_{1}(U)=\Phi_{1}(U).$

Hence,

$(u_{j+_{p}1}(T(x)),j+_{p}1)-(T(u_{j}(x)),j+_{p}1)=(L_{j}(x+u_{j}(x)),j+_{p}1),$

and so

$u_{j+_{p}1}(T(x))=T(u_{j}(x))+L_{j}(x+u_{j}(x)).$

Therefore,

$\displaystyle(S\circ H)(x,j)$ $\displaystyle=(S_{j}(x+u_{j}(x)),j+_{p}1)$
$\displaystyle=(T(x)+T(u_{j}(x))+L_{j}(x+u_{j}(x)),j+_{p}1)$
$\displaystyle=(T(x)+u_{j+_{p}1}(T(x)),j+_{p}1)$
$\displaystyle=(H\circ\widetilde{T})(x,j).$

Now, let $V=-\Psi_{2}^{-1}(\overline{\mathcal{L}})\in\mathcal{F}$. Write
$V(x,j)=(v_{j}(x),j)$ and let $K:\widetilde{X}\to\widetilde{X}$ be the
continuous map defined by

$K(x,j)=(x+v_{j}(x),j).$

We claim that

(23) $K\circ S=\widetilde{T}\circ K.$

Indeed, by the definition of $V$,

$V\circ S-\widetilde{T}\circ V=\Psi_{2}(V)=-\overline{\mathcal{L}}.$

Hence,

$(v_{j+_{p}1}(S_{j}(x)),j+_{p}1)-(T(v_{j}(x)),j+_{p}1)=(-L_{j}(x),j+_{p}1),$

and so

$L_{j}(x)+v_{j+_{p}1}(S_{j}(x))=T(v_{j}(x)).$

Therefore,

$\displaystyle(K\circ S)(x,j)$
$\displaystyle=(S_{j}(x)+v_{j+_{p}1}(S_{j}(x)),j+_{p}1)$
$\displaystyle=(T(x)+L_{j}(x)+v_{j+_{p}1}(S_{j}(x)),j+_{p}1)$
$\displaystyle=(T(x)+T(v_{j}(x)),j+_{p}1)$ $\displaystyle=(\widetilde{T}\circ
K)(x,j).$

Let us now show that

(24) $K(H(x,j))=(x,j)\ \ \text{ for all }(x,j)\in\widetilde{X}.$

Let $W\in\mathcal{F}$ be given by

$W(x,j)=(w_{j}(x),j),\ \text{ where }w_{j}(x)=u_{j}(x)+v_{j}(x+u_{j}(x)).$

Note that

$(K\circ H)(x,j)=(x+w_{j}(x),j).$

By (22) and (23), $K\circ H\circ\widetilde{T}=\widetilde{T}\circ K\circ H$,
that is,

$(T(x)+w_{j+_{p}1}(T(x)),j+_{p}1)=(T(x)+T(w_{j}(x)),j+_{p}1).$

Hence,

$\displaystyle\Psi_{1}(W)(x,j)$
$\displaystyle=W(\widetilde{T}(x,j))-\widetilde{T}(W(x,j))$
$\displaystyle=(w_{j+_{p}1}(T(x)),j+_{p}1)-(T(w_{j}(x)),j+_{p}1)=O_{\mathcal{G}}(x,j).$

Thus, $W=O_{\mathcal{F}}$ and (24) holds.

Let us now prove that

(25) $H(K(x,j))=(x,j)\ \ \text{ for all }(x,j)\in\widetilde{X}.$

Let $Z\in\mathcal{F}$ be given by

$Z(x,j)=(z_{j}(x),j),\ \text{ where }z_{j}(x)=v_{j}(x)+u_{j}(x+v_{j}(x)).$

Note that

$(H\circ K)(x,j)=(x+z_{j}(x),j).$

By (22) and (23), $H\circ K\circ S=S\circ H\circ K$, that is,

$(S_{j}(x)+z_{j+_{p}1}(S_{j}(x)),j+_{p}1)=(S_{j}(x+z_{j}(x)),j+_{p}1).$

This is equivalent to

$(T(x)+L_{j}(x)+z_{j+_{p}1}(S_{j}(x)),j+_{p}1)=(T(x)+T(z_{j}(x))+L_{j}(x+z_{j}(x)),j+_{p}1),$

which gives

$z_{j+_{p}1}(S_{j}(x))-T(z_{j}(x))=L_{j}(x+z_{j}(x))-L_{j}(x).$

Hence,

$\displaystyle\Psi_{2}(Z)(x,j)$
$\displaystyle=Z(S(x,j))-\widetilde{T}(Z(x,j))$
$\displaystyle=(z_{j+_{p}1}(S_{j}(x)),j+_{p}1)-(T(z_{j}(x)),j+_{p}1)$
$\displaystyle=(L_{j}(x+z_{j}(x)),j+_{p}1)-(L_{j}(x),j+_{p}1)$
$\displaystyle=\overline{\mathcal{L}}((x,j)+Z(x,j))-\overline{\mathcal{L}}(x,j)$
$\displaystyle=\Phi_{2}(Z)(x,j).$

Thus, $Z$ is a fixed point of $\Psi_{2}^{-1}\circ\Phi_{2}$. As in (21), the
map $\Psi_{2}^{-1}\circ\Phi_{2}:\mathcal{F}\to\mathcal{F}$ satisfies

$\operatorname{Lip}(\Psi_{2}^{-1}\circ\Phi_{2})\leq\delta<1.$

Hence, $\Psi_{2}^{-1}\circ\Phi_{2}$ has a unique fixed point in $\mathcal{F}$.
Since $(\Psi_{2}^{-1}\circ\Phi_{2})(O_{\mathcal{F}})=O_{\mathcal{F}}$, we
conclude that $Z=O_{\mathcal{F}}$ and (25) holds.

By (24) and (25), $H$ is a homeomorphism with inverse $K$. By definition,

$H(x,0)=(h(x),0),\ \ \text{ where }h(x)=x+u_{0}(x).$

It follows that $h:X\to X$ is a homeomorphism and

$H^{-1}(x,0)=(h^{-1}(x),0).$

By (22), $S=H\circ\widetilde{T}\circ H^{-1}$, which gives

$S^{p}=H\circ\widetilde{T}^{p}\circ H^{-1}.$

In particular,

$((S_{p-1}\circ\cdots\circ
S_{0})(x),0)=S^{p}(x,0)=(H\circ\widetilde{T}^{p}\circ
H^{-1})(x,0)=(h(T^{p}(h^{-1}(x))),0).$

Thus, $h$ establishes a conjugation between $S_{p-1}\circ\cdots\circ S_{0}$
and $T^{p}$. Finally,

(26)
$\|h-I_{X}\|_{\infty}=\|u_{0}\|_{\infty}\leq|U|_{\infty}=|\Psi_{1}^{-1}(\Phi_{1}(U))|_{\infty}\leq\frac{a\,b\,(1+t)}{1-t}\,|\overline{\mathcal{L}}|_{\infty}<\infty.$

Under the additional hypothesis that

$\|L_{j}\|_{\infty}<\varepsilon\text{ for all }j\in\\{0,\ldots,p-1\\},$

we obtain

$\|h-I_{X}\|_{\infty}<\frac{a\,b\,(1+t)}{1-t}\,\varepsilon\leq\delta.$

This completes the proof of (A) and (B).

Let us assume that $T$ is hyperbolic and prove the uniqueness of the
homeomorphism $h$. Let $g:X\to X$ be any homeomorphism satisfying

(27) $S_{p-1}\circ\cdots\circ S_{0}=g\circ T^{p}\circ g^{-1}\ \text{ and }\ \
\|g-I_{X}\|_{\infty}<\infty.$

We define recursively maps $b_{0},\ldots,b_{p-1}$ from $X$ into itself as
follows:

$b_{0}=g-I_{X}\ \ \text{ and }\ \ b_{j}=S_{j-1}\circ(I_{X}+b_{j-1})\circ
T^{-1}-I_{X}\ \text{ for }j\in\\{1,\ldots,p-1\\}.$

Clearly, these maps are continuous. Since $b_{0}$ is bounded and

(28) $b_{j}(x)=T(b_{j-1}(T^{-1}(x)))+L_{j-1}(T^{-1}(x)+b_{j-1}(T^{-1}(x))),$

it follows recursively that all the $b_{j}$:s are bounded. Thus,

$B:(x,j)\in\widetilde{X}\mapsto(b_{j}(x),j)\in\widetilde{X}$

is a continuous bounded map. Since we are assuming that $T$ is hyperbolic,
$Y=X$, and so $B\in\mathcal{F}$. If we prove that

(29) $B\text{ is a fixed point of }\Psi_{1}^{-1}\circ\Phi_{1},$

then the uniqueness of such a fixed point would imply $B=U$ and we would
obtain

$g=I_{X}+b_{0}=I_{X}+u_{0}=h,$

as desired. Given $(x,j)\in\widetilde{X}$, we have to show that

$\Psi_{1}(B)(x,j)=\Phi_{1}(B)(x,j).$

If $j\in\\{0,\ldots,p-2\\}$, then (28) gives

$b_{j+1}(T(x))=T(b_{j}(x))+L_{j}(x+b_{j}(x)),$

and so

$\displaystyle\Psi_{1}(B)(x,j)$
$\displaystyle=B(\widetilde{T}(x,j))-\widetilde{T}(B(x,j))$
$\displaystyle=(b_{j+1}(T(x))-T(b_{j}(x)),j+1)$
$\displaystyle=(L_{j}(x+b_{j}(x)),j+1)$ $\displaystyle=\Phi_{1}(B)(x,j).$

In order to analyze the case $j=p-1$, first note that the recursive definition
of the $b_{j}$:s implies that

$b_{j}=S_{j-1}\circ\cdots\circ S_{0}\circ(I_{X}+b_{0})\circ T^{-j}-I_{X}\
\text{ for all }j\in\\{1,\ldots,p-1\\}.$

In particular,

$I_{X}+b_{p-1}=S_{p-2}\circ\cdots\circ S_{0}\circ g\circ T^{-p+1}.$

Consequently,

$\displaystyle T(b_{p-1}(x))+L_{p-1}(x+b_{p-1}(x))$
$\displaystyle=S_{p-1}(x+b_{p-1}(x))-T(x)$
$\displaystyle=(S_{p-1}\circ\cdots\circ S_{0}\circ g)(T^{-p+1}(x))-T(x)$
$\displaystyle=(g\circ T^{p})(T^{-p+1}(x))-T(x)$ $\displaystyle=g(T(x))-T(x)$
$\displaystyle=b_{0}(T(x)),$

where in the third equality we used the equality in (27). Therefore,

$\displaystyle\Psi_{1}(B)(x,p-1)$
$\displaystyle=B(\widetilde{T}(x,p-1))-\widetilde{T}(B(x,p-1))$
$\displaystyle=(b_{0}(T(x))-T(b_{p-1}(x)),0)$
$\displaystyle=(L_{p-1}(x+b_{p-1}(x)),0)$ $\displaystyle=\Phi_{1}(B)(x,p-1).$

This completes the proof of (29), as it was to be shown.

Now, suppose that the operator $T$ is not hyperbolic. Then, we can take a
nonzero vector $y\in M\cap T(N)$, and so

$z=\sum_{n=-\infty}^{\infty}T^{n}(y)$

defines a nontrivial fixed point of $T$. For each $\lambda\in\mathbb{K}$,

$h_{\lambda}:x\in X\mapsto h(x+\lambda z)\in X$

is a homeomorphism whose inverse is given by

$h_{\lambda}^{-1}:x\in X\mapsto h^{-1}(x)-\lambda z\in X.$

Since $z$ is a fixed point of $T$,

$h_{\lambda}\circ T^{p}\circ h_{\lambda}^{-1}=h\circ T^{p}\circ
h^{-1}=S_{p-1}\circ\cdots\circ S_{0}\ \ \text{ for all }\lambda\in\mathbb{K}.$

Moreover,

$\|h_{\lambda}-I_{X}\|_{\infty}\leq\|h-I_{X}\|_{\infty}+|\lambda|\|z\|.$

Hence, if $\|h-I_{X}\|_{\infty}<\delta$, then we will also have
$\|h_{\lambda}-I_{X}\|_{\infty}<\delta$ whenever $|\lambda|$ is suffciently
small. In any case, we see that $h$ can be replaced by uncountably many other
homeomorphisms.

Finally, let us prove the last assertion of the theorem. For this purpose, we
denote the maps $\Phi_{1}$, $U$ and $h$ by $\Phi_{\mathcal{L},1}$,
$U_{\mathcal{L}}$ and $h_{\mathcal{L}}$, respectively, in order to make it
clear that they actually depend on $\mathcal{L}$. Recall that:

  * (a)

$\Phi_{\mathcal{L},1}:\mathcal{F}\to\mathcal{G}$ is given by
$\Phi_{\mathcal{L},1}(F)(x,j)=\overline{\mathcal{L}}((x,j)+F(x,j))$.

  * (b)

$\Psi_{1}^{-1}\circ\Phi_{\mathcal{L},1}:\mathcal{F}\to\mathcal{F}$ satisfies
$\operatorname{Lip}(\Psi_{1}^{-1}\circ\Phi_{\mathcal{L},1})\leq\delta<1$.

  * (c)

$U_{\mathcal{L}}\in\mathcal{F}$ is the unique fixed point of
$\Psi_{1}^{-1}\circ\Phi_{\mathcal{L},1}$.

  * (d)

$h_{\mathcal{L}}:X\to X$ is the homeomorphism given by
$h_{\mathcal{L}}(x)=x+u_{\mathcal{L},0}(x)$, where
$U_{\mathcal{L}}(x,j)=(u_{\mathcal{L},j}(x),j)$.

Let $\Lambda$ denote the set of all $p$-tuples
$\mathcal{L}=(L_{0},\ldots,L_{p-1})$ of bounded Lipschitz maps $L_{j}:X\to X$
with $\operatorname{Lip}(L_{j})<\varepsilon$ (and
$\|L_{j}\|_{\infty}<\varepsilon$ in the case of item (B)) endowed with the
metric

$D(\mathcal{L},\mathcal{L}^{\prime})=\max_{0\leq
j<p}\|L_{j}-L^{\prime}_{j}\|_{\infty}.$

We want to prove that the correspondence $\mathcal{L}\to h_{\mathcal{L}}$ is
uniformly continuous in the sense described in the statement of the theorem.
Since

(30)
$\|h_{\mathcal{L}}-h_{\mathcal{L}^{\prime}}\|_{\infty}=\|u_{\mathcal{L},0}-u_{\mathcal{L}^{\prime},0}\|_{\infty}\leq|U_{\mathcal{L}}-U_{\mathcal{L}^{\prime}}|_{\infty},$

it is enough to show that the map

$\mathcal{L}\in\Lambda\mapsto U_{\mathcal{L}}\in\mathcal{F}$

is uniformly continuous. In view of (b) and (c), this will follow from the
parametrized version of Banach’s fixed point theorem (see [9], for instance)
as soon as we show that the map

(31)
$f:(F,\mathcal{L})\in\mathcal{F}\times\Lambda\mapsto\Psi_{1}^{-1}(\Phi_{\mathcal{L},1}(F))\in\mathcal{F}$

is uniformly continuous. But this is true because simple computations show
that

(32)
$|f(F,\mathcal{L})-f(F^{\prime},\mathcal{L}^{\prime})|_{\infty}\leq\|\Psi_{1}^{-1}\|(\varepsilon|F-F^{\prime}|_{\infty}+D(\mathcal{L},\mathcal{L}^{\prime})).$

This completes the proof of the theorem. ∎

######  Remark 5.

Actually, the correspondence $\mathcal{L}\to h_{\mathcal{L}}$ is Lipschitz. In
fact, by (31) and (c), we have that

$f(U_{\mathcal{L}},\mathcal{L})=\Psi_{1}^{-1}(\Phi_{\mathcal{L},1}(U_{\mathcal{L}}))=U_{\mathcal{L}}\
\ \text{ for all }\mathcal{L}\in\Lambda.$

In view of (b), it follows from the proof of the parametrized version of
Banach’s fixed point theorem that

$|U_{\mathcal{L}^{\prime}}-U_{\mathcal{L}}|_{\infty}\leq\frac{2}{1-\delta}\,|f(U_{\mathcal{L}^{\prime}},\mathcal{L}^{\prime})-f(U_{\mathcal{L}^{\prime}},\mathcal{L})|_{\infty}.$

Hence, by (19), (30) and (32), we obtain

$\|h_{\mathcal{L}}-h_{\mathcal{L}^{\prime}}\|_{\infty}\leq\frac{2\,a\,b\,(1+t)}{(1-\delta)(1-t)}\,D(\mathcal{L},\mathcal{L}^{\prime}).$

######  Remark 6.

Theorem 4 remains true if we replace “homeomorphism” by “uniform
homeomorphism” throughout its statement. The proof is essentially the same,
but we need to work with the function spaces $\mathcal{F}$ and $\mathcal{G}$
defined as follows:

  * •

$\mathcal{F}$ is the set of all uniformly continuous bounded maps
$F:\widetilde{X}\to\widetilde{X}$ such that

$F(X\times\\{j\\})\subset Y\times\\{j\\}\ \text{ for all }j\in\mathbb{Z}_{p}.$

  * •

$\mathcal{G}$ is the set of all uniformly continuous bounded maps
$F:\widetilde{X}\to\widetilde{X}$ such that

$F(X\times\\{j\\})\subset X\times\\{j+_{p}1\\}\ \text{ for all
}j\in\mathbb{Z}_{p}.$

######  Remark 7.

Recall that a diffeomorphism $f:M\to M$ on a compact manifold is said to be
absolutely structurally stable [7, 10] if there exist a neighborhood
$\mathcal{N}$ of $f$ in the $C^{1}$-topology and a constant $C\in(0,\infty)$
such that each $g\in\mathcal{N}$ is topologically conjugate to $f$ by means of
a homeomorphism $h:M\to M$ satisfying

$\|h-I_{X}\|_{\infty}\leq C\,\|g-f\|_{\infty}.$

By using the notations in the proof of Theorem 4, it follows from (26) that
the constant

$C=\frac{a\,b\,(1+t)}{1-t}$

has the following property: for each
$\mathcal{L}=(L_{0},\ldots,L_{p-1})\in\Lambda$, the homeomorphism
$h_{\mathcal{L}}:X\to X$ conjugating $S_{p-1}\circ\cdots\circ S_{0}$ and
$T^{p}$ (where $S_{j}=T+L_{j}$) satisfies

$\|h_{\mathcal{L}}-I_{X}\|_{\infty}\leq C\max_{0\leq
j<p}\|L_{j}\|_{\infty}=C\max_{0\leq j<p}\|S_{j}-T\|_{\infty}.$

## Acknowledgement

The author is beneficiary of a grant within the framework of the grants for
the retraining, modality María Zambrano, in the Spanish university system
(Spanish Ministry of Universities, financed by the European Union,
NextGenerationEU). The author was also partially supported by CNPq – Project
#308238/2021-4, by CAPES – Finance Code 001, and by
MCIN/AEI/10.13039/501100011033/FEDER, UE, Projects PID2019-105011GB-I00 and
PID2022-139449NB-I00.

## References

  * [1] A. Andronov and L. Pontrjagin, Structurally stable systems, Dokl. Akad. Nauk SSSR 14 (1937), 247–250. (Russian)
  * [2] N. C. Bernardes Jr., P. R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018), no. 1, 796–816.
  * [3] N. C. Bernardes Jr. and A. Messaoudi, A generalized Grobman-Hartman theorem, Proc. Amer. Math. Soc. 148 (2020), no. 10, 4351–4360.
  * [4] N. C. Bernardes Jr. and A. Messaoudi, Shadowing and structural stability for operators, Ergodic Theory Dynam. Systems 41 (2021), no. 4, 961–980.
  * [5] N. C. Bernardes Jr. and A. Peris, On shadowing and chain recurrence in linear dynamics, Adv. Math. 441 (2024), Paper No. 109539, 46 pp.
  * [6] P. R. Cirilo, B. Gollobit and E. R. Pujals, Dynamics of generalized hyperbolic linear operators, Adv. Math. 387 (2021), Paper No. 107830, 37 pp.
  * [7] J. M. Franks, Absolutely structurally stable diffeomorphisms, Proc. Amer. Math. Soc. 37 (1973), 293–296.
  * [8] J. M. Franks, Time dependent stable diffeomorphisms, Invent. Math. 24 (1974), 163–172.
  * [9] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  * [10] J. Guckenheimer, Absolutely $\Omega$-stable diffeomorphisms, Topology 11 (1972), 195–197.
  * [11] P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620.
  * [12] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
  * [13] J. K. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969), 411-440.
  * [14] J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ci. 40 (1968), 263–266.
  * [15] C. C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363–367.
  * [16] J. W. Robbin, Topological conjugacy and structural stability for discrete dynamical systems, Bull. Amer. Math. Soc. 78 (1972), no. 6, 923–952.
  * [17] M. Shub, Global Stability of Dynamical Systems (with the collaboration of A. Fathi and R. Langevin), Springer-Verlag, New York, 1987.