# Berry phase and the Mandel parameter of the non-degenerate parametric
amplifier

J. C. Vegaa, E. Choreñob,
D. Ojeda-Guillénc,111E-mail address<EMAIL_ADDRESS>and R. D. Motad

a Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional,
Ed. 9, U.P. Adolfo López Mateos, Alc. Gustavo A. Madero, C.P. 07738 Ciudad de
México, Mexico.

b Unidad Profesional Interdisciplinaria de Energía y Movilidad, Instituto
Politécnico Nacional, Av. Wilfrido Massieu, U.P. Adolfo López Mateos, Alc.
Gustavo A. Madero, C.P. 07738, Ciudad de México, Mexico.

b Escuela Superior de Cómputo, Instituto Politécnico Nacional, Av. Juan de
Dios Bátiz esq. Av. Miguel Othón de Mendizábal, Col. Lindavista, Alc. Gustavo
A. Madero, C.P. 07738, Ciudad de México, Mexico.

d Escuela Superior de Ingeniería Mecánica y Eléctrica, Unidad Culhuacán,
Instituto Politécnico Nacional, Av. Santa Ana No. 1000, Col. San Francisco
Culhuacán, Alc. Coyoacán, C.P. 04430, Ciudad de México, Mexico.

## Abstract

We study the non-degenerate parametric amplifier problem from an algebraic
approach of the $SU(1,1)$ group. We write the Hamiltonian of this problem in
terms of the boson generators of the $SU(1,1)$ group and the difference
operator. We apply the tilting transformation to our results to exactly solve
this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by
assuming that our Hamiltonian is an explicit function of time we calculate its
Berry phase. Finally we obtain the Mandel $Q-$parameter of the photon numbers
$n_{a}$ and $n_{b}$.

##  1 Introduction

Within quantum optics, the production and study of non-classical states of
light has been relevant in recent years, since they allow us to study
fundamental concepts within quantum mechanics. Some of the most interesting
applications have occurred in interdisciplinary fields such as quantum
computing, quantum communication or quantum metrology [1, 2, 3].

Among the most efficient and simplest methods to produce non-classical states
of light are optical parametric amplifiers, where optical cavities and non-
linear crystals are used. In these amplifiers, one photon of a pump field of
frequency $\omega_{3}$ transforms, via the nonlinear medium, into two photons
called signal and idler, of frequencies $\omega_{1}$ and $\omega_{2}$,
respectively. These output beams have the same frequency
$(\omega_{1}=\omega_{2})$ and polarization in the degenerate case and
different ones $(\omega_{1}\neq\omega_{2})$ in the non-degenerate case [4, 5,
6]. This pairwise production of photons results in the conservation of the
photon-number difference between the signal and idler modes in the absence of
any loss. The high correlation between the signal and idler fields is
responsible for the generation of a squeezed-vacuum state in the output of the
device [7].

The study of parametric amplifiers and their properties is still relevant
today. Some of the most important applications are reported in the Refs. [8,
9, 10, 11, 12, 13, 14].

The aim of this work is to study and exactly solve the non-degenerate
parametric amplifier. We obtain its eigenfunctions and energy spectrum by
using the $su(1,1)$ Lie algebra. Also, we obtain some important properties as
its Berry phase and the Mandel $Q-$parameter of the photon numbers $n_{a}$ and
$n_{b}$.

This work is organized as it follows. In Sec. $2$ we write the Hamiltonian of
the non-degenerate parametric amplifier in terms of a boson realization of the
$su(1,1)$ Lie algebra. Then, we use the $SU(1,1)$ tilting transformation and
the eigenfunctions of the two dimensional harmonic oscillator to obtain its
energy spectrum and eigenfunctions. The Sec. $3$ is dedicated to compute the
Berry phase of the non-degenerate parametric amplifier by supposing that its
Hamiltonian is an explicit function of time. The Mandel $Q-$parameter of this
problem for the photon numbers $n_{a}$ and $n_{b}$ is obtained in Sec. $4$.
Finally, we give some concluding remarks.

##  2 The non-degenerate parametric amplifier

The Hamiltonian which describes the stationary non-degenerate parametric
amplifier is (with $\hbar=1$) [4, 15, 16]

$H=\omega_{1}a^{\dagger}a+\omega_{2}b^{\dagger}b+\lambda
ab+\lambda^{*}a^{\dagger}b^{\dagger},$ (1)

where $\lambda=\chi e^{-i\psi}$ is the coupling constant. Now, we can
introduce the two-mode realization of the $su(1,1)$ Lie algebra (see Appendix)
[17]

$K_{0}=\frac{1}{2}(a^{\dagger}a+b^{\dagger}b+1),\hskip
8.5359ptK_{+}=a^{\dagger}b^{\dagger},\hskip 8.5359ptK_{-}=ab.$ (2)

These operators together with the operator
$J_{0}=\frac{1}{2}(a^{\dagger}a-b^{\dagger}b)$, which belongs to the $SU(2)$
group, allow us to write the Hamiltonian in Eq. (1) as

$H=(\omega_{1}+\omega_{2})\left(K_{0}-\frac{1}{2}\right)+(\omega_{1}-\omega_{2})J_{0}+\chi
e^{-i\psi}K_{-}+\chi e^{i\psi}K_{+}.$ (3)

In order to diagonalize this Hamiltonian we apply the tilting transformation
to the stationary Schrödinger equation $H\Psi=E\Psi$ as it follows [18, 19,
20]

$D^{\dagger}(\xi)HD(\xi)D^{\dagger}(\xi)\Psi=ED^{\dagger}(\xi)\Psi.$ (4)

Now, we can define the tilted Hamiltonian $H^{\prime}$ and its wavefunction
$\Psi^{\prime}$ as $H^{\prime}=D^{\dagger}(\xi)HD(\xi)$ and
$\Psi^{\prime}=D^{\dagger}(\xi)\Psi$, respectively. From the similarity
transformations of Eqs. (84)-(86) we obtain

$\begin{gathered}H^{\prime}=\left[(\omega_{1}+\omega_{2})(2\beta+1)+\chi
e^{-i\psi}\frac{\xi}{\absolutevalue{\xi}}\alpha+\chi
e^{i\psi}\frac{\xi^{*}}{\absolutevalue{\xi}}\alpha\right]K_{0}+\left[(\omega_{1}+\omega_{2})\frac{\alpha\xi}{2\absolutevalue{\xi}}+\chi
e^{-i\psi}\beta\frac{\xi}{\xi^{*}}+\chi e^{i\psi}(\beta+1)\right]K_{+}\\\
+\left[(\omega_{1}+\omega_{2})\frac{\alpha\xi^{*}}{2\absolutevalue{\xi}}+\chi
e^{-i\psi}(\beta+1)+\chi
e^{i\psi}\beta\frac{\xi^{*}}{\xi}\right]K_{-}+(\omega_{1}-\omega_{2})J_{0}-\frac{1}{2}(\omega_{1}+\omega_{2}).\end{gathered}$
(5)

Therefore, the tilted Hamiltonian becomes diagonal if the coefficients of the
operators $K_{\pm}$ vanish. This is achieved by defining the parameters of the
coherent state as

$\theta=\tanh^{-1}{\left(\frac{2\chi}{\omega_{1}+\omega_{2}}\right)},\hskip
8.5359pt\gamma=\psi.$ (6)

If we substitute these parameters into equation (5) we can simplify the tilted
Hamiltonian $H^{\prime}$ to

$H^{\prime}=[(\omega_{1}+\omega_{2})\cosh{(\theta)}-2\chi\sinh{(\theta)}]K_{0}+(\omega_{1}-\omega_{2})J_{0}-\frac{1}{2}(\omega_{1}+\omega_{2}).$
(7)

By using the results

$\cosh{(\theta)}=\frac{\omega_{1}+\omega_{2}}{\sqrt{(\omega_{1}+\omega_{2})^{2}-4\chi^{2}}},\quad\quad\sinh{(\theta)}=\frac{2\chi}{\sqrt{(\omega_{1}+\omega_{2})^{2}-4\chi^{2}}},$
(8)

we obtain that the diagonalized Hamiltonian $H^{\prime}$ becomes

$H^{\prime}=\sqrt{(\omega_{1}+\omega_{2})^{2}-4\chi^{2}}K_{0}+(\omega_{1}-\omega_{2})J_{0}-\frac{1}{2}(\omega_{1}+\omega_{2}).$
(9)

Now, let us notice that $2\omega K_{0}=H_{HO}$, where $H_{HO}$ is the
Hamiltonian of the two-dimensional harmonic oscillator. Therefore, since the
operators $K_{0}$ and $J_{0}$ commute, the eigenfunctions of $H^{\prime}$ are
those of the two-dimensional harmonic oscillator [21]

$\Psi^{\prime}_{N,m}(r,\varphi)=\frac{1}{\sqrt{\pi}}e^{im\varphi}(-1)^{\frac{N-m}{2}}\sqrt{\frac{2\left(\frac{N-m}{2}\right)!}{\left(\frac{N+m}{2}\right)!}}r^{m}L_{\frac{1}{2}(N-m)}^{m}(r^{2})e^{-\frac{1}{2}r^{2}},$
(10)

or

$\Psi^{\prime}_{n_{r},m}(r,\varphi)=\frac{1}{\sqrt{\pi}}e^{im\varphi}(-1)^{n_{r}}\sqrt{\frac{2\left(n_{r}\right)!}{\left(n_{r}+m\right)!}}r^{m}L_{n_{r}}^{m}(r^{2})e^{-\frac{1}{2}r^{2}},$
(11)

where $n_{r}=\frac{1}{2}(N-m)$ is the radial quantum number.

On the other hand, in polar coordinates the creation and annihilation
operators take the form

$\displaystyle a=\frac{e^{-i\varphi}}{2}\left[r+\frac{\partial}{\partial
r}-\frac{i}{r}\frac{\partial}{\partial\varphi}\right],\quad\quad
a^{{\dagger}}=\frac{e^{i\varphi}}{2}\left[r-\frac{\partial}{\partial
r}-\frac{i}{r}\frac{\partial}{\partial\varphi}\right],$ (12) $\displaystyle
b=\frac{e^{i\varphi}}{2}\left[r+\frac{\partial}{\partial
r}+\frac{i}{r}\frac{\partial}{\partial\varphi}\right],\quad\quad
b^{{\dagger}}=\frac{e^{-i\varphi}}{2}\left[r-\frac{\partial}{\partial
r}+\frac{i}{r}\frac{\partial}{\partial\varphi}\right].$ (13)

The action of these operators on the basis $|N,m\rangle$ is given by [22]

$\displaystyle a|N,m\rangle=\sqrt{\frac{1}{2}(N+m)}|N-1,m-1\rangle,\quad
a^{{\dagger}}|N,m\rangle=\sqrt{\frac{1}{2}(N+m)+1}|N+1,m+1\rangle,$ (14)
$\displaystyle b|N,m\rangle=\sqrt{\frac{1}{2}(N-m)}|N-1,m+1\rangle,\quad
b^{{\dagger}}|N,m\rangle=\sqrt{\frac{1}{2}(N-m)+1}|N+1,m-1\rangle.$ (15)

Thus, by substituting equations (12) and (13) into equation (2), we obtain
that the $su(1,1)$ Lie algebra generators $K_{\pm},K_{0}$ and the Casimir
operator $K^{2}$ in polar coordinates are given by [19]

$\begin{gathered}K_{+}=\frac{1}{4}\left[r^{2}-2r\frac{\partial}{\partial
r}-2+\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right],\\\
K_{-}=\frac{1}{4}\left[r^{2}+2r\frac{\partial}{\partial
r}+2+\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right],\\\
K_{0}=\frac{1}{4}\left[r^{2}-\frac{\partial^{2}}{\partial
r^{2}}-\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right],\\\
K^{2}=-\frac{1}{4}\left(1+\frac{\partial^{2}}{\partial\varphi^{2}}\right).\end{gathered}$
(16)

Similarly we can obtain that the operator $J_{0}$ is written in polar
coordinates by

$J_{0}=\frac{i}{2}\frac{\partial}{\partial\varphi}.$ (17)

In order to establish the relationship between the quantum numbers of our
problem $N,m$ and the quantum numbers of the $SU(1,1)$ group $n,k$, we need to
consider the action of the operators $K_{0}$ and $K^{2}$ of equations (77) and
(78) of Appendix on the states $\ket{N,m}$. Thus, from the expression

$K^{2}\ket{N,m}=\frac{1}{4}(m^{2}+1)\ket{N,m}=k(k-1)\ket{N,m},$ (18)

we obtain that the Bargmann $k$ index is $k=\frac{1}{2}(m+1)$. This result of
the Bargmann index and the action of $K_{0}$ on the states $\ket{N,m}$

$K_{0}\ket{N,m}=\frac{1}{2}(N+1)\ket{N,m}.$ (19)

lead us to conclude that $n=\frac{1}{2}(N-m)=n_{r}$. The action of the
operator $J_{0}$ on the states $\ket{N,m}$ can be directly computed from Eq.
(17) or by using the relationship $K^{2}=J_{0}^{2}-\frac{1}{4}$. Hence, we
obtain that

$J_{0}\ket{N,m}=\frac{m}{2}\ket{N,m}.$ (20)

Therefore, by substituting the Eqs. (19) and (20) into Eq. (9) we obtain that
the energy spectrum of the non-degenerate parametric amplifier is given by

$E=\left[\sqrt{(\omega_{1}+\omega_{2})^{2}-4\chi^{2}}\left(\frac{1}{2}(N+1)\right)+\frac{1}{2}(\omega_{1}-\omega_{2})m-\frac{1}{2}(\omega_{1}+\omega_{2})\right],$
(21)

or in terms of the quantum number $n,m$

$E=\left[\sqrt{(\omega_{1}+\omega_{2})^{2}-4\chi^{2}}\left(n+\frac{m}{2}+\frac{1}{2}\right)+\frac{1}{2}m(\omega_{1}-\omega_{2})-\frac{1}{2}(\omega_{1}+\omega_{2})\right].$
(22)

The eigenfunctions of the non-degenerate parametric amplifier are obtained
from the relationship $\Psi=D(\xi)\Psi^{\prime}$ and explicitly are the
Perelomov number coherent states for the two-dimensional harmonic oscillator
[19]

$\displaystyle\Psi_{n,m}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m+1)}}\frac{(-1)^{n}}{\sqrt{\pi}}e^{im\varphi}\frac{(-\zeta^{*})^{n}(1-|\zeta|^{2})^{\frac{m}{2}+\frac{1}{2}}(1+\sigma)^{n}}{(1-\zeta)^{m+1}}$
(23) $\displaystyle\times
e^{-\frac{r^{2}(\zeta+1)}{2(1-\zeta)}}r^{m}L_{n}^{m}\left(\frac{r^{2}\sigma}{(1-\zeta)(1-\sigma)}\right),$

where $\sigma$ is defined as

$\sigma=\frac{1-|\zeta|^{2}}{(1-\zeta)(-\zeta^{*})}.$ (24)

It is important to note that when we consider the isotropic case
$\omega_{1}=\omega_{2}$, the energy spectrum of equations (21) and (22) are
properly reduced to those reported in Refs. [18, 19]. Therefore, we have
exactly solved the anisotropic non-degenerate parametric amplifier problem
using the tilting transformation and the $SU(1,1)$ group theory.

##  3 The Berry phase for the non-degenerate parametric amplifier

In order to compute the Berry phase of the non-degenerate parametric
amplifier, we can assume that our Hamiltonian of Eq. (3) is an explicit
function of time

$H(t)=\Omega(t)\left(K_{0}-\frac{1}{2}\right)+a_{1}(t)K_{+}+a_{2}(t)K_{-}+\Delta\omega(t)J_{0},$
(25)

where $\Omega(t)=\omega_{1}+\omega_{2}$,
$\Delta\omega(t)=\omega_{1}-\omega_{2}$ and $a_{2}(t)=a^{*}_{1}(t)$. Here, the
coefficient $a_{1}(t)$ is explicitly given by

$a_{1}(t)=\chi(t)e^{i\psi(t)},$ (26)

where $\chi(t)$ and $\psi(t)$ are arbitrary real functions of time.

To describe the quantum dynamics of the system, we shall use the Schrödinger
picture (with $\hbar=1$)

$i\frac{d}{dt}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle.$ (27)

The time evolution of the states of our Hamiltonian $H(t)$ can be studied by
using the time-dependent nontrivial invariant Hermitian operator $I(t)$ [23,
24], which satisfies the condition

$i\frac{\partial}{\partial t}I(t)+[I(t),H(t)]=0.$ (28)

By using the $SU(1,1)$ time-dependent displacement operators
$D(\xi)=\exp\\{\xi(t)K_{+}-\xi(t)^{*}K_{-}\\}$, with
$\xi(t)=-\frac{1}{2}\theta(t)e^{-i\gamma(t)}$ and where now $\theta(t)$ and
$\gamma(t)$ are arbitrary real functions of time, we can define the invariant
operator $I(t)$ as (see Ref. [25])

$I(t)=D(\xi(t))K_{0}D^{{\dagger}}(\xi(t)).$ (29)

From the definition of the displacement operator we obtain that
$D^{{\dagger}}(\xi)=D(-\xi)$. Thus, from equation (86) of Appendix we obtain
that the invariant operator $I(t)$ is explicitly

$I(t)=\cosh(\theta)K_{0}+\frac{\sinh(\theta)}{2}e^{-i\gamma}K_{+}+\frac{\sinh(\theta)}{2}e^{i\gamma}K_{-}.$
(30)

Now, if we substitute the time-dependent Hamiltonian (25) and the invariant
operator $I(t)$ of expression (30) into the equation (28), we obtain that the
time-dependent physical parameters $\Omega(t)$, $\Delta\omega(t)$, $\chi(t)$
and $\psi(t)$ are related with the parameters $\theta(t)$ and $\gamma(t)$ as
follows

$\dot{\theta}=-2\chi\sin(\psi+\gamma),\quad\quad(\dot{\gamma}-\Omega)\sinh(\theta)=-2\chi\cosh(\theta)\cos(\psi+\gamma).$
(31)

On the other hand, the transformations of the generators $\\{K_{0},K_{\pm}\\}$
under the time-dependent displacement operators $D(\xi(t))$ are given by

$\displaystyle
D^{{\dagger}}(t)K_{+}D(t)=-e^{i\gamma}\sinh(\theta)K_{0}+\frac{\cosh(\theta)+1}{2}K_{+}+e^{2i\gamma}\frac{\cosh(\theta)-1}{2}K_{-},$
(32) $\displaystyle
D^{{\dagger}}(t)K_{-}D(t)=-e^{-i\gamma}\sinh(\theta)K_{0}+\frac{\cosh(\theta)+1}{2}K_{-}+e^{-2i\gamma}\frac{\cosh(\theta)-1}{2}K_{+},$
(33) $\displaystyle
D^{{\dagger}}(t)K_{0}D(t)=\cosh(\theta)K_{0}-e^{-i\gamma}\frac{\sinh(\theta)}{2}K_{+}-e^{i\gamma}\frac{\sinh(\theta)}{2}K_{-}.$
(34)

In addition, by using of the BCH identity (Eq. (83) of Appendix) the operator
$i\frac{\partial}{\partial t}$ is transformed under the $SU(1,1)$ time-
dependent displacement operators $D(\xi)$ as

$D^{{\dagger}}(t)\left(i\frac{\partial}{\partial
t}\right)D(t)=\dot{\gamma}(\cosh(\theta)-1)K_{0}-\frac{e^{-i\gamma}}{2}\left(\dot{\gamma}\sinh(\theta)+i\dot{\theta}\right)K_{+}-\frac{e^{i\gamma}}{2}\left(\dot{\gamma}\sinh(\theta)-i\dot{\theta}\right)K_{-}.$
(35)

As it is shown in Ref. [24], if the eigenstates of the invariant operator
satisfy the Schrödinger equation its eigenvalues are real. Therefore, from the
Eq. (77) of Appendix and the expression (29) we have

$D(\xi)K_{0}|k,n\rangle=(k+n)D(\xi)|k,n\rangle,\quad\quad
I(t)D(\xi)|k,n\rangle=(k+n)D(\xi)|k,n\rangle.$ (36)

Thus, the eigenvalues of the invariant operator $I(t)$ are $(k+n)$ and its
eigenstates are $D(\xi)|k,n\rangle=|\zeta(t),k,n\rangle$, which are the
$SU(1,1)$ Perelomov number coherent states of Eq. (82), but now these are
functions of time through the parameters

$\zeta(t)=-\tanh\left(\frac{\theta(t)}{2}\right)e^{-i\gamma(t)},\quad\quad\eta(t)=\ln(1-|\zeta(t)|^{2}).$
(37)

Moreover, if the states $|\Psi(t)\rangle$ satisfy the Eq. (27), then they can
be expanded through the states $|\zeta(t),k,n\rangle$ in the form

$|\Psi(t)\rangle=\sum_{n}a_{n}e^{i\alpha_{n}}|\zeta(t),k,n\rangle,$ (38)

where according to Lewis [24] the phase $\alpha_{n}$ is given as

$\alpha_{n}=\int_{0}^{t}dt^{\prime}\langle\lambda,\kappa|i\frac{\partial}{\partial
t^{\prime}}-H(t^{\prime})|\lambda,\kappa\rangle.$ (39)

Here, $|\lambda,\kappa\rangle$ are the eigenstates and $\lambda$ are the
eigenvalues of the invariant operator $I(t)$. For our problem
$|\lambda,\kappa\rangle=|\zeta(t),k,n\rangle=D(\xi)|N,m\rangle$. Therefore,

$\alpha_{n}=\int_{0}^{t}dt^{\prime}\langle
N,m|D^{{\dagger}}(\xi)\left[i\frac{\partial}{\partial
t^{\prime}}-H(t^{\prime})\right]D(\xi)|N,m\rangle.$ (40)

By substituting the equation (35) and the results of expressions (32)-(34)
into equation (40) we obtain that the phase $\alpha_{n}$ in a non-adiabatic
process is given by

$\displaystyle\alpha_{n}$ $\displaystyle=\int_{0}^{t}\langle
N,m|\left[(\dot{\gamma}-\Omega)(\cosh(\theta)-1)+2\chi\cos(\gamma+\psi)\sinh(\theta)-\Omega\right]K_{0}-\Delta\omega
J_{0}+\frac{1}{2}\Omega|N,m\rangle dt^{\prime},$
$\displaystyle=\int_{0}^{t}\left[\frac{1}{2}(N+1)\left[(\dot{\gamma}-\Omega)(\cosh(\theta)-1)+2\chi\cos(\gamma+\psi)\sinh(\theta)-\Omega\right]-\Delta\omega\frac{m}{2}+\frac{1}{2}\Omega\right]dt^{\prime}.$
(41)

In this expression we have used the results of Eqs. (19) and (20). Unlike in a
non-adiabatic process, in an adiabatic process we have that
$\dot{\theta}=\dot{\gamma}=0$ and the equations (31) become

$\psi+\gamma=n\pi,\quad\quad\tanh(\theta)=\frac{2\chi}{\Omega}(-1)^{n}.$ (42)

If we set $n=0$, the above conditions are reduced to the time-dependent
versions of the expressions

$\psi=-\gamma,\quad\quad\tanh(\theta)=\frac{2\chi}{\omega_{1}+\omega_{2}}.$
(43)

We note that these results are very similar to those obtained in Section 2 for
the diagonalization process of the time-independent parametric amplifier
Hamiltonian (see equation (6)). Therefore, in an adiabatic process the phase
of the states $D(\xi)|N,m\rangle$ is reduced to

$\alpha_{n}=-\int_{0}^{t}\left[\frac{1}{2}(N+1)\sqrt{\Omega(t^{\prime})^{2}-4\chi^{2}(t^{\prime})}+\Delta\omega(t^{\prime})\frac{m}{2}-\frac{1}{2}\Omega\right]dt^{\prime}.$
(44)

The second term of equation (40) is known as the dynamical phase and is
defined as

$\dot{\epsilon}_{N}=\langle
N,m|D^{{\dagger}}(\xi)H(t^{\prime})D(\xi)|N,m\rangle,$ (45)

while the first term is known as the Berry phase

$\dot{\Gamma}_{N}=i\langle N,m|D^{{\dagger}}(\xi)\frac{\partial}{\partial
t}D(\xi)|N,m\rangle.$ (46)

Thus, the Berry phase of the states $D(\xi)|N,m\rangle$ is obtained in the
adiabatic limit as follows

$\Gamma_{N}(T)=\frac{1}{2}(N+1)\oint(\cosh(\theta)-1)d\psi,$ (47)

where $T$ denotes the period. It is obvious that in this case the Berry phase
do not depend on an explicit form of $\psi(t)$. Therefore, since $\oint
d\psi=2\pi$ for one period, we finally obtain that the Berry phase for the
non-degenerate parametric amplifier only depends on the time-dependent
parameters in the Hamiltonian, i.e.,

$\Gamma_{N}(T)=\pi\left(N+1\right)\frac{\Omega(t)-\sqrt{\Omega^{2}(t)-4\chi^{2}(t)}}{\sqrt{\Omega^{2}(t)-4\chi^{2}(t)}}.$
(48)

##  4 The Mandel $Q-$parameter of the non-degenerate parametric amplifier

A simple way to gauge the nature of the photon statistics of any states to
calculate the so-called $Q-$parameter [6, 26], which is defined by

$Q=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}}{\langle\hat{n}\rangle}-1,$
(49)

and where

$Q=\left\\{\begin{array}[]{lcc}>0,&super\leavevmode\nobreak\
Poissonian\leavevmode\nobreak\ distribution&\\\ \\\
=0,&Poissonian\leavevmode\nobreak\ distribution\leavevmode\nobreak\
(coherent\leavevmode\nobreak\ state)&\\\ \\\ <0,&sub-Poissonian&\\\ \\\
=-1,&number\leavevmode\nobreak\ state.\end{array}\right.$ (50)

In this Section we will compute the Mandel $Q$-parameter for the eigenstates
$|\Psi\rangle=D(\xi)|\Psi^{\prime}\rangle$ of the non-degenerate parametric
amplifier. Hence, we can obtain the Mandel $Q_{a}$ parameter from the
expression (49) as

$Q_{a}=\frac{\langle n^{2}_{a}\rangle-\langle n_{a}\rangle^{2}}{\langle
n_{a}\rangle}-1.$ (51)

In this expression we have that

$\displaystyle\langle n_{a}^{2}\rangle=\langle(a^{{\dagger}}a)^{2}\rangle$
$\displaystyle=\langle\Psi|(a^{{\dagger}}a)^{2}|\Psi\rangle=\langle\Psi^{\prime}|D^{{\dagger}}(\xi)(a^{{\dagger}}a)^{2}D(\xi)|\Psi^{\prime}\rangle$
$\displaystyle=\langle\Psi^{\prime}|\left[D^{{\dagger}}(\xi)a^{{\dagger}}aD(\xi)\right]^{2}|\Psi^{\prime}\rangle$
(52) $\langle n_{a}\rangle^{2}=\langle
a^{{\dagger}}a\rangle^{2}=\left[\langle\Psi^{\prime}|D^{{\dagger}}(\xi)a^{{\dagger}}aD(\xi)|\Psi^{\prime}\rangle\right]^{2},$
(53)

and

$\langle n_{a}\rangle=\langle
a^{{\dagger}}a\rangle=\langle\Psi^{\prime}|D^{{\dagger}}(\xi)a^{{\dagger}}aD(\xi)|\Psi^{\prime}\rangle.$
(54)

Now, we can express the term $a^{{\dagger}}a$ as a linear combination of the
operators $K_{0}$ and $J_{0}$ as

$a^{{\dagger}}a=K_{0}+J_{0}-\frac{1}{2}.$ (55)

Therefore, since $|\Psi^{\prime}\rangle=|N,m\rangle=|\rangle_{N,m}$, we obtain
the following results

$\langle
n_{a}^{2}\rangle=\left\langle\left[D^{{\dagger}}(\xi)\left(K_{0}+J_{0}-\frac{1}{2}\right)D(\xi)\right]^{2}\right\rangle_{N,m},$
(56) $\langle n_{a}\rangle^{2}=\left[\langle
D^{{\dagger}}(\xi)K_{0}D(\xi)\rangle_{N,m}+\langle
J_{0}\rangle_{N,m}-\frac{1}{2}\right]^{2},$ (57) $\langle n_{a}\rangle=\langle
D^{{\dagger}}(\xi)K_{0}D(\xi)\rangle_{N,m}+\langle
J_{0}\rangle_{N,m}-\frac{1}{2}.$ (58)

By using the $SU(1,1)$ displacement operator $D(\xi)$ of Eq. (79) we arrive to

$\displaystyle D^{{\dagger}}(\xi)a^{{\dagger}}aD(\xi)$
$\displaystyle=D^{{\dagger}}(\xi)\left[K_{0}+J_{0}-\frac{1}{2}\right]D(\xi)=D^{{\dagger}}(\xi)K_{0}D(\xi)+J_{0}-\frac{1}{2},$
$\displaystyle=\cosh(\theta)K_{0}-\frac{e^{-i\gamma}}{2}\sinh(\theta)K_{+}-\frac{e^{i\gamma}}{2}\sinh(\theta)K_{-}+J_{0}-\frac{1}{2}.$
(59)

Thus, by substituting Eq. (4) into Eq. (56) we have

$\langle n^{2}_{a}\rangle=\cosh^{2}(\theta)\langle
K^{2}_{0}\rangle_{N,m}+2\cosh(\theta)\langle
J_{0}K_{0}-K_{0}/2\rangle_{N,m}+\frac{\sinh^{2}(\theta)}{4}\left[\langle
K_{+}K_{-}\rangle_{N,m}+\langle
K_{-}K_{+}\rangle_{N,m}\right]+\langle\left(J_{0}-1/2\right)^{2}\rangle_{N,m}.$
(60)

Similarly, from Eqs. (4), (57) and (58) it is easily seen that

$\displaystyle\langle n_{a}\rangle^{2}=\cosh^{2}(\theta)\langle
K_{0}\rangle^{2}_{N,m}+\langle J_{0}\rangle^{2}_{N,m}+2\cosh(\theta)\langle
K_{0}\rangle_{N,m}\langle J_{0}\rangle_{N,m}-\cosh(\theta)\langle
K_{0}\rangle_{N,m}-\langle J_{0}\rangle_{N,m}+\frac{1}{4},$ (61)

and

$\displaystyle\langle n_{a}\rangle=\cosh(\theta)\langle
K_{0}\rangle_{N,m}+\langle J_{0}\rangle_{N,m}-\frac{1}{2}.$ (62)

Therefore, from the results of equations (60)-(62), the Mandel $Q_{a}$
parameter can be expressed in terms of the $su(1,1)$ Lie algebra generators as

$Q_{a}=\frac{\sinh^{2}(\theta)}{4}\left[\frac{\langle
K_{+}K_{-}\rangle_{N,m}+\langle K_{-}K_{+}\rangle_{N,m}}{\cosh(\theta)\langle
K_{0}\rangle_{N,m}+\langle J_{0}\rangle_{N,m}-\frac{1}{2}}\right]-1.$ (63)

We can compute the expected values in this expression from Eq. (2), and the
actions of the boson operators $a$ and $b$ on the states $|N,m\rangle$ (see
Eqs. (14) and (15))

$\langle K_{0}\rangle_{N,m}=\frac{N+1}{2},\quad\quad\langle
J_{0}\rangle_{N,m}=\frac{m}{2},\quad\quad\langle
K_{+}K_{-}\rangle_{N,m}=\frac{N^{2}-m^{2}}{4},\quad\quad\langle
K_{-}K_{+}\rangle_{N,m}=\frac{N^{2}-m^{2}}{4}+N+1.$ (64)

Thus, in terms of the quantum numbers $N,m$ it follows that

$Q_{a}=\frac{\sinh^{2}(\theta)}{4}\left[\frac{N^{2}-m^{2}+2N+2}{\cosh(\theta)(N+1)+m-1}\right]-1.$
(65)

Furthermore, by using the results of Eq. (8) with
$\Omega=\omega_{1}+\omega_{2}$, we obtain that the $Q_{a}$-parameter for the
non-degenerate parametric amplifier, the expression (65) can be rewritten as

$Q_{a}=\frac{\chi^{2}}{\sqrt{\Omega^{2}-4\chi^{2}}}\left[\frac{N^{2}-m^{2}+2N+2}{\Omega(N+1)+(m-1)\sqrt{\Omega^{2}-4\chi^{2}}}\right]-1.$
(66)

Analogously, we can compute the Mandel $Q_{b}$ parameter considering that

$b^{{\dagger}}b=K_{0}-J_{0}-\frac{1}{2},$ (67)

and therefore we have the following results for $\langle n^{2}_{b}\rangle$,
$\langle n_{b}\rangle^{2}$ and $\langle n_{b}\rangle$

$\langle n^{2}_{b}\rangle=\cosh^{2}(\theta)\langle
K^{2}_{0}\rangle_{N,m}-2\cosh(\theta)\langle
J_{0}K_{0}+K_{0}/2\rangle_{N,m}+\frac{\sinh^{2}(\theta)}{4}\left[\langle
K_{+}K_{-}\rangle_{N,m}+\langle
K_{-}K_{+}\rangle_{N,m}\right]+\langle\left(J_{0}+1/2\right)^{2}\rangle_{N,m},$
(68) $\displaystyle\langle n_{b}\rangle^{2}=\cosh^{2}(\theta)\langle
K_{0}\rangle^{2}_{N,m}+\langle J_{0}\rangle^{2}_{N,m}-2\cosh(\theta)\langle
K_{0}\rangle_{N,m}\langle J_{0}\rangle_{N,m}-\cosh(\theta)\langle
K_{0}\rangle_{N,m}+\langle J_{0}\rangle_{N,m}+\frac{1}{4},$ (69)
$\displaystyle\langle n_{b}\rangle=\cosh(\theta)\langle
K_{0}\rangle_{N,m}-\langle J_{0}\rangle_{N,m}-\frac{1}{2}.$ (70)

Thus, by substituting these results in the definition of the Mandel parameter
(Eq. (49)) we obtain

$Q_{b}=\frac{\sinh^{2}(\theta)}{4}\left[\frac{\langle
K_{+}K_{-}\rangle_{N,m}+\langle K_{-}K_{+}\rangle_{N,m}}{\cosh(\theta)\langle
K_{0}\rangle_{N,m}-\langle J_{0}\rangle_{N,m}-\frac{1}{2}}\right]-1.$ (71)

Finally by using the relations (8) and (64), we obtain that the Mandel
parameter $Q_{b}$ for the non-degenerate parametric amplifier is explicitly
given by

$Q_{b}=\frac{\chi^{2}}{\sqrt{\Omega^{2}-4\chi^{2}}}\left[\frac{N^{2}-m^{2}+2N+2}{\Omega(N+1)-(m+1)\sqrt{\Omega^{2}-4\chi^{2}}}\right]-1.$
(72)

##  5 Concluding remarks

In this paper we were able to obtain the exact solution of the non-degenerate
parametric amplifier by means of the tilting transformation and the unitary
irreducible representation theory of the $SU(1,1)$ group. This allowed us to
diagonalize our original Hamiltonian and write it in terms of the Hamiltonian
of the two-dimensional harmonic oscillator.

Furthermore, by assuming that our Hamiltonian was explicitly time-dependent
and using the similarity transformations of the operator
$i\frac{\partial}{\partial t}$, we computed the Berry phase of the non-
degenerate parametric amplifier.

Finally, the bosonic realization of the $su(1,1)$ Lie algebra and the
similarity transformations of the displacement operator $D(\xi)$ allowed us to
calculate the Mandel parameters of the photon numbers $n_{a}$ and $n_{b}$.

## Acknowledgments

This work was partially supported by SNII-México, EDI-IPN, SIP-IPN Project
Number $20230633$.

##  6 Appendix.

###  6.1 The $su(1,1)$ Lie algebra and its coherent states

Three operators $K_{\pm},K_{0}$ close the $su(1,1)$ Lie algebra if they
satisfy the commutation relations [27]

$\displaystyle[K_{0},K_{\pm}]=\pm K_{\pm},\quad\quad[K_{-},K_{+}]=2K_{0}.$
(73)

The Casimir operator $K^{2}$, which commutes with these three operators is
given for any irreducible representation of this group by

$K^{2}=K^{2}_{0}-\frac{1}{2}\left(K_{+}K_{-}+K_{-}K_{+}\right).$ (74)

The action of these four operators on the Fock space states
$\\{|k,n\rangle,n=0,1,2,...\\}$ is

$K_{+}|k,n\rangle=\sqrt{(n+1)(2k+n)}|k,n+1\rangle,$ (75)
$K_{-}|k,n\rangle=\sqrt{n(2k+n-1)}|k,n-1\rangle,$ (76)
$K_{0}|k,n\rangle=(k+n)|k,n\rangle,$ (77)
$K^{2}|k,n\rangle=k(k-1)|k,n\rangle.$ (78)

The Bargmann index $k$ determines a representation of the $su(1,1)$ algebra.
The discrete series are those for which $k>0$.

The displacement operator $D(\xi)$ is defined in terms of the creation and
annihilation operators $K_{+},K_{-}$ as

$D(\xi)=\exp(\xi K_{+}-\xi^{*}K_{-}),$ (79)

where $\xi=-\frac{1}{2}\theta e^{-i\gamma}$, $-\infty<\theta<\infty$ and
$0\leq\gamma\leq 2\pi$. The so-called normal form of the displacement operator
is given by

$D(\xi)=\exp(\zeta K_{+})\exp(\eta K_{0})\exp(-\zeta^{*}K_{-}),$ (80)

where $\zeta=-\tanh(\frac{1}{2}\theta)e^{-i\gamma}$ and
$\eta=-2\ln\cosh|\xi|=\ln(1-|\zeta|^{2})$ [28].

The $SU(1,1)$ Perelomov coherent states are defined as the action of the
displacement operator $D(\xi)$ on the lowest normalized state $|k,0\rangle$ as
[29]

$|\zeta\rangle=D(\xi)|k,0\rangle=(1-|\zeta|^{2})^{k}\sum_{n=0}^{\infty}\sqrt{\frac{\Gamma(n+2k)}{n!\Gamma(2k)}}\zeta^{n}|k,n\rangle.$
(81)

The $SU(1,1)$ Perelomov number coherent state $|\zeta,k,n\rangle$ is defined
as the action of the displacement operator $D(\xi)$ on an arbitrary excited
state $|k,n\rangle$ [19, 30]

$\displaystyle|\zeta,k,n\rangle$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}\frac{\zeta^{s}}{s!}\sum_{j=0}^{n}\frac{(-\zeta^{*})^{j}}{j!}e^{\eta(k+n-j)}\frac{\sqrt{\Gamma(2k+n)\Gamma(2k+n-j+s)}}{\Gamma(2k+n-j)}$
(82)
$\displaystyle\times\frac{\sqrt{\Gamma(n+1)\Gamma(n-j+s+1)}}{\Gamma(n-j+1)}|k,n-j+s\rangle.$

The similarity transformations $D^{{\dagger}}(\xi)K_{+}D(\xi)$,
$D^{{\dagger}}(\xi)K_{-}D(\xi)$, and $D^{{\dagger}}(\xi)K_{0}D(\xi)$ of the
$su(1,1)$ Lie algebra generators are computed by using the displacement
operator $D(\xi)$ an the Baker-Campbell-Hausdorff identity

$e^{A}Be^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...$
(83)

These results explicitly are

$D^{{\dagger}}(\xi)K_{+}D(\xi)=\frac{\xi^{*}}{|\xi|}\alpha
K_{0}+\beta\left(K_{+}+\frac{\xi^{*}}{\xi}K_{-}\right)+K_{+},$ (84)
$D^{{\dagger}}(\xi)K_{-}D(\xi)=\frac{\xi}{|\xi|}\alpha
K_{0}+\beta\left(K_{-}+\frac{\xi}{\xi^{*}}K_{+}\right)+K_{-},$ (85)
$D^{{\dagger}}(\xi)K_{0}D(\xi)=(2\beta+1)K_{0}+\frac{\alpha\xi}{2|\xi|}K_{+}+\frac{\alpha\xi^{*}}{2|\xi|}K_{-},$
(86)

where $\alpha=\sinh(2|\xi|)$ and
$\beta=\frac{1}{2}\left[\cosh(2|\xi|)-1\right]$.

A particular realization of the $su(1,1)$ Lie algebra is given by the Jordan-
Schwinger operators [17]

$K_{0}=\frac{1}{2}\left(a^{{\dagger}}a+b^{{\dagger}}b+1\right),\quad
K_{+}=a^{{\dagger}}b^{{\dagger}},\ \quad K_{-}=ba,$ (87)

where the two sets of operators $(a,a^{{\dagger}})$ and $(b,b^{{\dagger}})$
satisfy the bosonic algebra

$[a,a^{{\dagger}}]=[b,b^{{\dagger}}]=1,\quad\quad[a,b^{{\dagger}}]=[a,b]=0.$
(88)

If we introduce the operator $J_{0}$ defined as half the difference of the
number operators of the two oscillators, then $J_{0}$ commutes with all the
generators of this algebra realization. Hence, we have the following
results[31]

$K^{2}=J_{0}^{2}-\frac{1}{4},\quad\quad
J_{0}=\frac{1}{2}\left(a^{{\dagger}}a-b^{{\dagger}}b\right),$
$[J_{0},K_{0}]=[J_{0},K_{+}]=[J_{0},K_{-}]=0.$ (89)

In fact, the operator $J_{0}$ is one of the generators of the $su(2)$ Jordan-
Schwinger Lie algebra realization.

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$\operatorname{RHom}(k_{[a,b[},k_{[c,d[})\simeq\begin{cases}k&\text{for \
}a\leq c<b\leq d,\\\ k[-1]\quad&\text{for \ }c<a\leq d<b,\\\
0&\text{otherwise.}\end{cases}$

###### Proof.

The formula
$k_{[c,d[}\simeq\operatorname{R\mathcal{H}om}(k_{]d,c]},k_{\mathbb{R}})$
together with the adjunction between $\otimes$ and
$\operatorname{R\mathcal{H}om}$ give
$\operatorname{RHom}(k_{[a,b[},k_{[c,d[})\simeq\operatorname{RHom}(k_{I},k_{\mathbb{R}})$,
where $I=[a,b\mathclose{[}\cap\mathopen{]}c,d]$. If $I$ is half closed or
empty, the result is $0$. The two other cases correspond to $I$ open or $I$
closed. ∎

##  Appendix B Decomposition in the completion of $D_{lc}({\mathbb{R}})$

###  B.1. Limits of constructible sheaves

We assume in this section that $N$ is endowed with a real analytic structure
and a real analytic metric $g$. We recall that we work with coefficients in a
field $k$. We denote by $D_{c}(N)$ the subcategory of $D(N)$ of constructible
sheaves. We let $D_{lc}(N)$ be the set of objects which are limits of
constructible sheaves with respect to the $\gamma_{g}$-topology.

######  Lemma B.1.

Let ${\mathcal{F}},{\mathcal{G}}\in D_{lc}(N)$ with compact supports and
$a<b$. Then the morphism ${\rm
Hom}({\mathcal{F}},K_{\varphi}^{a}({\mathcal{G}}))\xrightarrow[]{}{\rm
Hom}({\mathcal{F}},K_{\varphi}^{b}({\mathcal{G}}))$ induced by
$\tau_{a,b}({\mathcal{G}})$ has finite dimensional image.

###### Proof.

We set $c=(b-a)/2$. Applying $K_{\varphi}^{-c}$ we identify ${\rm
Hom}({\mathcal{F}},K_{\varphi}^{b}({\mathcal{G}}))$ with ${\rm
Hom}(K_{\varphi}^{-c}({\mathcal{F}}),K_{\varphi}^{b-c}({\mathcal{G}}))$ and
the morphism of the lemma gets identified with

$\displaystyle f\colon{\rm Hom}({\mathcal{F}},K_{\varphi}^{a}({\mathcal{G}}))$
$\displaystyle\xrightarrow[]{}{\rm
Hom}(K_{\varphi}^{-c}({\mathcal{F}}),K_{\varphi}^{b-c}({\mathcal{G}}))$
$\displaystyle u$ $\displaystyle\mapsto\tau_{a,b-c}({\mathcal{G}})\circ
u\circ\tau_{-c,0}({\mathcal{F}})$

Since ${\mathcal{F}},{\mathcal{G}}\in D_{lc}(N)$, there exist constructible
sheaves ${\mathcal{F}}^{\prime}$, ${\mathcal{G}}^{\prime}$ and morphisms
$K_{\varphi}^{-c}{\mathcal{F}}\xrightarrow[]{u_{1}}{\mathcal{F}}^{\prime}\xrightarrow[]{u_{2}}{\mathcal{F}}$,
$K_{\varphi}^{a}({\mathcal{G}})\xrightarrow[]{v_{1}}{\mathcal{G}}^{\prime}\xrightarrow[]{v_{2}}K_{\varphi}^{b-c}({\mathcal{G}})$
such that $u_{2}\circ u_{1}=\tau_{-c,0}({\mathcal{F}})$, $v_{2}\circ
v_{1}=\tau_{a,b-c}({\mathcal{G}})$. We choose a compact set $Z$ containing the
supports of ${\mathcal{F}}$ and $K_{\varphi}^{-c}{\mathcal{F}}$. Applying the
functor $(-)_{Z}$ we obtain
$K_{\varphi}^{-c}{\mathcal{F}}\xrightarrow[]{(u_{1})_{Z}}{\mathcal{F}}^{\prime}_{Z}\xrightarrow[]{(u_{2})_{Z}}{\mathcal{F}}$
and we still have $(u_{2})_{Z}\circ(u_{1})_{Z}=(u_{2}\circ
u_{1})_{Z}=\tau_{-c,0}({\mathcal{F}})$. So we may assume from the beginning
that ${\mathcal{F}}^{\prime}$ has a compact support. Now $f$ factorizes
through ${\rm Hom}({\mathcal{F}}^{\prime},{\mathcal{G}}^{\prime})$ which is
finite dimensional. ∎

As a consequence, for a given $\varepsilon>0$, the images of ${\rm
Hom}({\mathcal{F}},K_{\varphi}^{\delta}({\mathcal{G}}))$ in ${\rm
Hom}({\mathcal{F}},K_{\varphi}^{\varepsilon}({\mathcal{G}}))$, for
$0<\delta\leq\varepsilon$, stabilize when $\delta\xrightarrow[]{}0$. In other
words the projective system $({\rm
Hom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}})))_{n\in\mathbb{N}}$, with
morphisms induced by $\tau_{1/(n+1),1/n}({\mathcal{G}})$, satisfies the
Mittag-Leffler criterion. Let us introduce the notations, for $\varepsilon>0$,
${\rm Hom}_{\varepsilon}({\mathcal{F}},{\mathcal{G}})=\operatorname{Im}({\rm
Hom}({\mathcal{F}},K_{\varphi}^{-\varepsilon}({\mathcal{G}}))\xrightarrow[]{}{\rm
Hom}({\mathcal{F}},{\mathcal{G}}))$ and ${\rm
Hom}_{(\varepsilon)}({\mathcal{F}},{\mathcal{G}})=\bigcap_{\delta<\varepsilon}{\rm
Hom}_{\delta}({\mathcal{F}},{\mathcal{G}})$. A consequence of Lemma B.3 below
is that ${\rm Hom}_{\varepsilon}({\mathcal{F}},{\mathcal{G}})={\rm
Hom}_{(\varepsilon)}({\mathcal{F}},{\mathcal{G}})$. We will mainly consider

${\rm
Hom}_{(\varepsilon)}({\mathcal{F}},K_{\varphi}^{\varepsilon}({\mathcal{G}}))=\bigcap_{0<\delta\leq\varepsilon}\operatorname{Im}\big{(}{\rm
Hom}({\mathcal{F}},K_{\varphi}^{\delta}({\mathcal{G}}))\xrightarrow[]{}{\rm
Hom}({\mathcal{F}},K_{\varphi}^{\varepsilon}({\mathcal{G}}))\big{)}$

By Lemma B.1 this space has finite dimension and, by construction, the
morphisms ${\rm
Hom}_{(\varepsilon)}({\mathcal{F}},K_{\varphi}^{\varepsilon}({\mathcal{G}}))\xrightarrow[]{}{\rm
Hom}_{(\varepsilon^{\prime})}({\mathcal{F}},K_{\varphi}^{\varepsilon^{\prime}}({\mathcal{G}}))$,
for $0<\varepsilon\leq\varepsilon^{\prime}$, are surjective.

In the next lemma $\operatorname{Ab}$ denotes the category of Abelian groups
and $D(\operatorname{Ab})$ its derived category. The statement is a well known
consequence of the Mittag-Leffler condition; it is for example a variation on
[KS90, Prop. 1.12.4].

######  Lemma B.2.

Let $(A_{n},a_{n})$, $a_{n}\colon A_{n}\xrightarrow[]{}A_{n-1}$, be a
projective system in $D(\operatorname{Ab})$ and define
$\operatorname{holim}_{n}A_{n}$ by the distinguished triangle
$\operatorname{holim}_{n}A_{n}\xrightarrow[]{}\prod_{n}A_{n}\xrightarrow[]{\alpha}\prod_{n}A_{n}\xrightarrow[]{+1}$
where $\alpha={\rm id}-\prod_{n}a_{n}$. We assume that $H^{i}A_{n}$ satisfies
the Mittag-Leffler criterion, for some $i$. Then
$H^{i+1}(\operatorname{holim}_{n}A_{n})\simeq\varprojlim_{n}H^{i+1}A_{n}$.

###### Proof.

In $\operatorname{Ab}$ the product $\prod$ is exact, hence $H^{i}$ and $\prod$
commute and we have a long exact sequence
$\prod_{n}H^{i}A_{n}\xrightarrow[]{H^{i}\alpha}\prod_{n}H^{i}A_{n}\xrightarrow[]{}H^{i+1}(\operatorname{holim}_{n}A_{n})\xrightarrow[]{}\prod_{n}H^{i+1}A_{n}\xrightarrow[]{}\prod_{n}H^{i+1}A_{n}$.
Hence the statement is equivalent to the surjectivity of $H^{i}\alpha$. We
write for short $B_{n}=H^{i}A_{n}$, $b_{n}=H^{i}a_{n}$ and
$B_{n}^{\infty}=\operatorname{im}(B_{m}\xrightarrow[]{}B_{n})$ for $m\gg n$.
We let $b_{n}^{\infty}$, $\bar{b}_{n}$ be the maps induced by $b_{n}$ on
$B_{n}^{\infty}$, $B_{n}/B_{n}^{\infty}$. We have $H^{i}\alpha={\rm id}-\prod
b_{n}$ and the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod
B_{n}^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}-\prod b_{n}^{\infty}}$$\textstyle{\prod
B_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}-\prod b_{n}}$$\textstyle{\prod
B_{n}/B_{n}^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}-\prod\bar{b}_{n}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod
B_{n}^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod
B_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\prod
B_{n}/B_{n}^{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

We see that ${\rm id}-\prod b_{n}^{\infty}$ is surjective because the
$b_{n}^{\infty}$’s are surjective. Let us write $t=\prod\bar{b}_{n}$. By the
Mittag-Leffler criterion, for each $n$, there exists $m\geq n$ such that
$\bar{b}_{n+1}\circ\bar{b}_{n+2}\circ\dots\circ\bar{b}_{m}$ is the zero map.
It follows that $t^{\prime}=\sum_{i=0}^{\infty}t^{i}$ is well-defined. Hence
${\rm id}-t$ is an isomorphism, with inverse $t^{\prime}$. We deduce that
${\rm id}-\prod b_{n}$ is also surjective, as required. ∎

######  Lemma B.3.

Let ${\mathcal{F}},{\mathcal{G}}\in D_{lc}(N)$ with compact supports. We have
natural isomorphisms

${\rm
Hom}({\mathcal{F}},{\mathcal{G}})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\varprojlim_{n}{\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\varprojlim_{n}{\rm
Hom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$

and, for any $n$, the map ${\rm
Hom}({\mathcal{F}},{\mathcal{G}})\xrightarrow[]{}{\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$,
$u\mapsto[u]_{n}:=\tau_{0,1/n}({\mathcal{G}})\circ u$, is surjective.

###### Proof.

By Lemma 6.23 we have
$\operatorname{hocolim}K_{\varphi}^{-1/n}({\mathcal{F}})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}$
and we deduce
$\operatorname{RHom}({\mathcal{F}},{\mathcal{G}})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\operatorname{holim}\operatorname{RHom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$.
Since the projective system of vector spaces
$H^{-1}\operatorname{RHom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))\simeq{\rm
Hom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}})[-1])$ satisfies Mittag-
Leffler, Lemma B.2 gives ${\rm
Hom}({\mathcal{F}},{\mathcal{G}})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\varprojlim_{n}{\rm
Hom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$. The second isomorphism
of the lemma and the last assertion are clear by the definition of ${\rm
Hom}_{(1/n)}$. ∎

######  Lemma B.4.

For ${\mathcal{F}},{\mathcal{G}},\mathcal{H}\in D_{lc}(N)$ with compact
supports, the composition induces a well-defined map, for any $n$,

$\displaystyle\circ_{n}\colon{\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))\times{\rm
Hom}_{(1/n)}({\mathcal{G}},K_{\varphi}^{1/n}(\mathcal{H}))$
$\displaystyle\xrightarrow[]{}{\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}(\mathcal{H}))$
$\displaystyle(\bar{u},\bar{v})$
$\displaystyle\mapsto\bar{v}\circ_{n}\bar{u}:=[v\circ u]_{n},$

where $u,v$ in ${\rm Hom}({\mathcal{F}},{\mathcal{G}})$, ${\rm
Hom}({\mathcal{G}},\mathcal{H})$ satisfy $[u]_{n}=\bar{u}$, $[v]_{n}=\bar{v}$,
and we use the notation $[-]_{n}$ of Lemma B.3. This turns ${\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{F}}))$ into an algebra
which is a finite dimensional quotient of ${\rm
Hom}({\mathcal{F}},{\mathcal{F}})$ and whose unit is
$\tau_{0,1/n}({\mathcal{F}})$ (assuming ${\mathcal{F}}\not=0$ and $n$ is big
enough so that $\tau_{0,1/n}({\mathcal{F}})\not=0$ – see Lemma 6.4).

###### Proof.

It is enough to see that if $[u]_{n}=0$ or $[v]_{n}=0$, then $[v\circ
u]_{n}=0$. For $u$ this follows from the commutative diagram

$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{[u]_{n}}$$\textstyle{{\mathcal{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\scriptstyle{\tau_{0,1/n}({\mathcal{G}})}$$\textstyle{\mathcal{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0,1/n}(\mathcal{H})}$$\textstyle{K_{\varphi}^{1/n}({\mathcal{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K_{\varphi}^{1/n}(v)}$$\textstyle{K_{\varphi}^{1/n}(\mathcal{H}))}$

which shows that $[v\circ u]_{n}=K_{\varphi}^{1/n}(v)\circ[u]_{n}$. The case
$[v]_{n}=0$ is similar. ∎

######  Lemma B.5.

Let ${\mathcal{F}},{\mathcal{G}}\in D_{lc}(N)$ with compact supports such that
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=0$. We set $A_{n}={\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{F}}))$ and let
$A_{n}^{\times}$ be its subset of invertible elements. We also define, using
the notation $\circ_{n}$ of Lemma B.4,

$B_{n}=\big{\\{}(u,v)\in{\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))\times{\rm
Hom}_{(1/n)}({\mathcal{G}},K_{\varphi}^{1/n}({\mathcal{F}}));\\\
v\circ_{n}u=\tau_{0,1/n}({\mathcal{F}})$,
$u\circ_{n}v=\tau_{0,1/n}({\mathcal{G}})\big{\\}}$

Then

  1. (1)

for any $n$ the set $B_{n}$ is non empty,

  2. (2)

for any $n$ and any $(u,v)\in B_{n}$, the maps $\alpha_{u,v}\colon
A^{\times}_{n}\xrightarrow[]{}B_{n}$, $a\mapsto(u\circ_{n}a,a^{-1}\circ_{n}v)$
and $\beta_{u,v}\colon B_{n}\xrightarrow[]{}A^{\times}_{n}$,
$(u^{\prime},v^{\prime})\mapsto v\circ_{n}u^{\prime}$, are mutually inverse
bijections,

  3. (3)

for $m\geq n$ the natural map $B_{m}\xrightarrow[]{}B_{n}$ is surjective.

###### Proof.

(1) By Lemma B.1 there exists $m\geq n$ such that ${\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$ is the image of
${\rm Hom}({\mathcal{F}},K_{\varphi}^{1/m}({\mathcal{G}}))$ in ${\rm
Hom}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$ (and the same with
${\mathcal{F}}$, ${\mathcal{G}}$ switched). Since
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=0$, we can find
$u_{1}\colon{\mathcal{F}}\xrightarrow[]{}K_{\varphi}^{1/2m}({\mathcal{G}})$
and
$v_{1}\colon{\mathcal{G}}\xrightarrow[]{}K_{\varphi}^{1/2m}({\mathcal{F}})$
such that $u_{1}\circ K_{\varphi}^{1/2m}(v_{1})$ and $v_{1}\circ
K_{\varphi}^{1/2m}(u_{1})$ are the morphisms $\tau$. Letting $u$ be the image
of $u_{1}$ in ${\rm
Hom}_{(1/n)}({\mathcal{F}},K_{\varphi}^{1/n}({\mathcal{G}}))$ and defining $v$
in the same way, we then have $(u,v)\in B_{n}$.

(2) We first remark that $\beta_{u,v}(u^{\prime},v^{\prime})$ belongs to
$A^{\prime}_{n}$: indeed the inverse of $v\circ_{n}u^{\prime}$ is
$v^{\prime}\circ_{n}u$. Now
$\beta_{u,v}(\alpha_{u,v}(a))=v\circ_{n}u\circ_{n}a=a$ and
$\alpha_{u,v}(\beta_{u,v}(u^{\prime},v^{\prime}))=(u\circ_{n}v\circ_{n}u^{\prime},v^{\prime}\circ_{n}u\circ_{n}v)=(u^{\prime},v^{\prime})$.

(3) We pick $(u,v)\in B_{m}$ and let $(\bar{u},\bar{v})$ be its image by the
natural map $B_{m}\xrightarrow[]{}B_{n}$. We obtain a commutative diagram

$\textstyle{A^{\prime}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{u,v}}$$\scriptstyle{q}$$\textstyle{B_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\bar{u},\bar{v}}}$$\textstyle{B_{n}}$

where the horizontal maps are bijections and the vertical map $q$ is
surjective by Lemma B.6. The result follows. ∎

######  Lemma B.6.

Let $A$, $B$ be finite dimensional algebras over $k$ and let $f\colon
A\xrightarrow[]{}B$ be a surjective algebra morphism. Then the map induced on
the sets of invertible elements $f^{\times}\colon
A^{\times}\xrightarrow[]{}B^{\times}$ is also surjective.

###### Proof.

We pick $x\in A$ and assume that $f(x)$ is invertible. We have to prove that
there exists $z\in\ker(f)$ such that $x+z$ is invertible. We let
$A^{\prime}=\langle x\rangle$ be the subalgebra of $A$ generated by $x$ and
define similarly $B^{\prime}=\langle f(x)\rangle$. The maps $u\colon
k[X]\xrightarrow[]{}A$, $X\mapsto x$, and $f\circ u$ identify $A^{\prime}$ and
$B^{\prime}$ with quotients of $k[X]$. Since $k[X]$ is principal, we can write
$A^{\prime}=k[X]/\langle PQ\rangle$, with $x=[X]_{A^{\prime}}$, and
$B^{\prime}=k[X]/\langle Q\rangle$, with $f(x)=[X]_{B^{\prime}}$.

Since $f(x)$ is invertible in $B$, it is not a zero divisor and we have
$X\nmid Q$. Let us write $P=X^{n}P^{\prime}$ with $X\nmid P^{\prime}$. We set
$R=X+P^{\prime}Q$. Then $f([R]_{A^{\prime}})=f(x)$. We claim that
$[R]_{A^{\prime}}$ is invertible, which concludes the proof. We have to check
that $\langle R,PQ\rangle=k[X]$. Let $D$ be a generator of $\langle
R,PQ\rangle$. Then $D\mid X^{n}R-PQ=X^{n+1}$. Either $X\mid D$, but then
$X\mid R$ and finally $X\mid P^{\prime}Q$ which is false, or $D\in k^{\times}$
and we are done. ∎

######  Proposition B.7.

Let ${\mathcal{F}},{\mathcal{G}}$ be elements in $D_{lc}(N)$ with coefficients
in a field $k$ and having compact supports. Assume that that
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=0$. Then
${\mathcal{F}}\simeq{\mathcal{G}}$.

######  Question B.8.

Is the Proposition true if we replace the field $k$ by $\mathbb{Z}$ or more
generally a ring ?

###### Proof.

If ${\mathcal{F}}=0$, then $\gamma_{g}(0,{\mathcal{G}})=0$ and we obtain
${\mathcal{G}}=0$ by Lemma 6.4. We assume ${\mathcal{F}}\not=0$, hence
$\tau_{0,1/n}({\mathcal{F}})\not=0$ for $n$ big enough. We use the notation
$B_{n}$ of Lemma B.5. By this lemma the inductive system
$\cdots\xrightarrow[]{}B_{n}\xrightarrow[]{}B_{n-1}\xrightarrow[]{}\cdots$ is
made of surjective maps between non empty sets for $n\gg 0$. Hence its limit
is non empty and we deduce a pair $(u,v)\in{\rm
Hom}({\mathcal{F}},{\mathcal{G}})\times{\rm Hom}({\mathcal{F}},{\mathcal{G}})$
such that $([u]_{n},[v]_{n})\in B_{n}$ for all $n$, that is, $[v\circ
u]_{n}=\tau_{0,1/n}({\mathcal{F}})$ and $[u\circ
v]_{n}=\tau_{0,1/n}({\mathcal{G}})$. Let ${\mathcal{C}}$ be the cone of $u$.
For any $n$ we have the following morphism of triangles given by the morphisms
$\tau_{0,1/n}(-)$:

$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathcal{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[v]_{n}}$$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{1/n}({\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{1/n}({\mathcal{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{1/n}({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{1/n}({\mathcal{F}})[1],}$

where both maps $\tau_{0,1/n}({\mathcal{F}})$ and
$\tau_{0,1/n}({\mathcal{G}})$ factorize through $[v]_{n}$. By the same
argument as in the proof of Lemma 6.8-(1) we deduce that
$\tau_{0,2/n}({\mathcal{C}})=0$. Hence $\gamma_{g}(0,{\mathcal{C}})\leq 2/n$,
for any $n$. Thus the cone of $u$ vanishes and $u$ is an isomorphism. ∎

###  B.2. Decomposition in $D_{lc}(\mathbb{R})$

Let $\operatorname{Mod}(\mathbb{R})$ be the category of sheaves of $k$-vector
spaces on $\mathbb{R}$. We consider it as usual as the full subcategory of
$D(\mathbb{R})$ of complexes concentrated in degree $0$. A sheaf
${\mathcal{F}}\in\operatorname{Mod}(\mathbb{R})$ which is constructible with
respect to a finite stratification of $\mathbb{R}$ is described by a quiver
representation of type $A_{n}$. Hence Gabriel theorem implies that
${\mathcal{F}}$ is decomposed as
${\mathcal{F}}\simeq\bigoplus_{I\in{\mathcal{I}}}k_{I}$ where ${\mathcal{I}}$
is a finite family of intervals. In this section we extend this result to
$\gamma_{g}$-limits of constructible sheaves under the assumption that the
microsupport is contained in $\\{\tau\geq 0\\}$.

Let $Int^{+}$ be the set of intervals of the form $[a,b[$, $a\in\mathbb{R}$,
$b\in\mathbb{R}\cup\\{+\infty\\}$. We endow $Int^{+}$ with the order

(B.1) $[a,b\mathclose{[}\leq[a^{\prime},b^{\prime}[\qquad\text{if and only
if}\qquad a\leq a^{\prime}\text{ and }b\leq b^{\prime}$

By Lemma A.2, for $I,I^{\prime}\in Int^{+}$ we have ${\rm
Hom}(k_{I},k_{I^{\prime}})\simeq k$ canonically if and only if $I\cap
I^{\prime}\not=\emptyset$ and $I\leq I^{\prime}$ and in this case we denote by
$e_{I}^{I^{\prime}}$ its generator. Otherwise ${\rm
Hom}(k_{I},k_{I^{\prime}})\simeq 0$ and we set $e_{I}^{I^{\prime}}=0$. For
$I\leq I^{\prime}\leq I^{\prime\prime}$ we have
$e_{I^{\prime}}^{I^{\prime\prime}}\circ
e_{I}^{I^{\prime}}=e_{I}^{I^{\prime\prime}}$. For two finite families
${\mathcal{I}}$, ${\mathcal{I}}^{\prime}\subset Int^{+}$ and
${\mathcal{G}}=\bigoplus_{I\in{\mathcal{I}}}k_{I}$,
${\mathcal{G}}^{\prime}=\bigoplus_{J\in{\mathcal{I}}^{\prime}}k_{J}$, we
obtain a natural morphism

$E=E_{{\mathcal{I}}}^{{\mathcal{I}}^{\prime}}\colon\operatorname{Mat}_{lt}({\mathcal{I}}^{\prime}\times{\mathcal{I}})\xrightarrow[]{}{\rm
Hom}({\mathcal{G}},{\mathcal{G}}^{\prime}),\quad(a_{I^{\prime},I})_{I^{\prime}\in{\mathcal{I}}^{\prime},I\in{\mathcal{I}}}\mapsto\sum
a_{I^{\prime},I}e_{I}^{I^{\prime}},$

where $\operatorname{Mat}_{lt}$ denotes the space of lower triangular matrices
($a_{I^{\prime},I}\not=0$ implies $I\leq I^{\prime}$). The map $E$ is
surjective and, for three families ${\mathcal{I}}$, ${\mathcal{I}}^{\prime}$,
${\mathcal{I}}^{\prime\prime}$, we have
$E_{{\mathcal{I}}^{\prime}}^{{\mathcal{I}}^{\prime\prime}}(B)\circ
E_{{\mathcal{I}}}^{{\mathcal{I}}^{\prime}}(A)=E_{{\mathcal{I}}}^{{\mathcal{I}}^{\prime\prime}}(BA)$.

######  Lemma B.9.

Let ${\mathcal{I}}$, ${\mathcal{I}}^{\prime}\subset Int^{+}$ be two finite
families and ${\mathcal{G}}=\bigoplus_{I\in{\mathcal{I}}}k_{I}$,
${\mathcal{G}}^{\prime}=\bigoplus_{J\in{\mathcal{I}}^{\prime}}k_{J}$. Let
$\varepsilon>0$ be given and let
$u\colon{\mathcal{G}}\xrightarrow[]{}{\mathcal{G}}^{\prime}$,
$v\colon{\mathcal{G}}^{\prime}\xrightarrow[]{}T_{\varepsilon*}({\mathcal{G}})$
be such that $v\circ u=\tau_{0,\varepsilon}({\mathcal{G}})$. We assume that
each $I\in{\mathcal{I}}$ is of length $>\varepsilon$. Then there exist an
isomorphism
$\phi\colon{\mathcal{G}}^{\prime}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{G}}^{\prime}$
and an injective map
$\sigma\colon{\mathcal{I}}\xrightarrow[]{}{\mathcal{I}}^{\prime}$ such that
$I\leq\sigma(I)$ for all $I\in{\mathcal{I}}$ (for the order $\leq$ defined in
(B.1)) and $\phi\circ
u\colon{\mathcal{G}}\xrightarrow[]{}{\mathcal{G}}^{\prime}$ is the sum of the
natural morphisms $k_{I}\xrightarrow[]{}k_{\sigma(I)}$.

###### Proof.

We set ${\mathcal{I}}^{\prime\prime}=\\{T_{\varepsilon}(I)$;
$I\in{\mathcal{I}}\\}$. In view of the discussion after (B.1) we can find
lower triangular matrices
$A\in\operatorname{Mat}_{lt}({\mathcal{I}}^{\prime}\times{\mathcal{I}})$,
$B\in\operatorname{Mat}_{lt}({\mathcal{I}}^{\prime\prime}\times{\mathcal{I}}^{\prime})$
representing $u$, $v$. Then
$BA\in\operatorname{Mat}_{lt}({\mathcal{I}}^{\prime\prime}\times{\mathcal{I}})$
is lower triangular. We identify ${\mathcal{I}}$ and
${\mathcal{I}}^{\prime\prime}$ through $T_{\varepsilon}$ and we claim that
$BA\in\operatorname{Mat}_{lt}({\mathcal{I}}\times{\mathcal{I}})$ is still
lower triangular. We take $I_{1},I_{2}\in{\mathcal{I}}$ and assume that
$0\not=(BA)_{I_{1},I_{2}}=\sum_{J\in{\mathcal{I}}^{\prime}}B_{T_{\varepsilon}(I_{1}),J}A_{J,I_{2}}$.
There exists $J$ such that $B_{T_{\varepsilon}(I_{1}),J},A_{J,I_{2}}\not=0$,
hence $I_{2}\leq J\leq T_{\varepsilon}(I_{1})$. If we assume moreover that
$I_{2}\not\leq I_{1}$ we must have $T_{\varepsilon}(I_{1})\cap
I_{2}\not=\emptyset$ because $I_{1}$ and $I_{2}$ are of length $>\varepsilon$.
Hence $e_{I_{2}}^{T_{\varepsilon}(I_{1})}\not=0$ and we deduce that the
decomposition of $v\circ u$ on the basis of the non zero
$e_{I}^{I^{\prime\prime}}$’s has a non zero coefficient on
$e_{I_{2}}^{T_{\varepsilon}(I_{1})}$. Since $v\circ
u=\tau_{0,\varepsilon}({\mathcal{G}})$ this means that $I_{1}=I_{2}$,
contradiciting $I_{2}\not\leq I_{1}$. Hence $I_{2}\leq I_{1}$, as claimed.

Finally $BA$ is lower triangular (in
$\operatorname{Mat}_{lt}({\mathcal{I}}\times{\mathcal{I}})$) and has entries
$1$ along the diagonal since it represents
$\tau_{0,\varepsilon}({\mathcal{G}})$. Hence $BA$ is invertible. In particular
$|{\mathcal{I}}|\leq|{\mathcal{I}}^{\prime}|$, $A$ has rank $|{\mathcal{I}}|$,
we can decompose ${\mathcal{I}}^{\prime}$ as
${\mathcal{I}}^{\prime}={\mathcal{I}}\sqcup{\mathcal{I}}^{\prime}_{1}$ and
find an invertible matrix
$A^{\prime}\in\operatorname{Mat}_{lt}({\mathcal{I}}^{\prime}\times{\mathcal{I}}^{\prime})$
such that the block ${\mathcal{I}}\times{\mathcal{I}}$ in $A^{\prime}A$ is the
identity matrix. Now we let $\sigma$ be the map given by the partition
${\mathcal{I}}^{\prime}={\mathcal{I}}\sqcup{\mathcal{I}}^{\prime}_{1}$ and
$\phi$ be the morphism induced by $A^{\prime}$. ∎

######  Proposition B.10.

Let ${\mathcal{F}}\in\operatorname{Mod}(\mathbb{R})$ such that
$SS({\mathcal{F}})\subset\\{(t;\tau)\in T^{*}\mathbb{R};$ $\tau\geq 0\\}$. We
assume that ${\mathcal{F}}$ is a $\gamma_{g}$-limit of constructible sheaves
${\mathcal{F}}_{n}\in\operatorname{Mod}(\mathbb{R})$ such that
$SS({\mathcal{F}}_{n})\subset\\{(t;\tau)\in T^{*}\mathbb{R};$ $\tau\geq 0\\}$.
Then there exists an at most countable set of intervals ${\mathcal{I}}$ such
that ${\mathcal{F}}\simeq\bigoplus_{I\in{\mathcal{I}}}k_{I}$.

We notice that the singular support hypothesis in the proposition implies that
our intervals are of the type $[a,b[$, $[a,\infty[$ or
$\mathopen{]}-\infty,b[$.

###### Proof.

(i) Up to taking a subsequence we can assume
$\gamma_{g}({\mathcal{F}},{\mathcal{F}}_{n})\leq\varepsilon_{n}=2^{-n}$. For
each $n$ there exists a finite family ${\mathcal{I}}_{n}\subset Int^{+}$ such
that ${\mathcal{F}}_{n}:=\bigoplus_{I\in{\mathcal{I}}_{n}}k_{I}$. By
Proposition 6.24 we can assume, up to translating ${\mathcal{F}}_{n}$ by less
than $\varepsilon_{n}$, that there exist compatible morphisms
$f_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{F}}_{n+1}$ and that
$\operatorname{hocolim}{\mathcal{F}}_{n}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}$.
Since ${\mathcal{F}}_{n}\in\operatorname{Mod}(\mathbb{R})$, we have
$\operatorname{hocolim}{\mathcal{F}}_{n}\simeq\varinjlim{\mathcal{F}}_{n}$. We
can also assume (maybe changing $\varepsilon_{n}$ by $4\varepsilon_{n}$) that
there exist morphisms
$g_{n}\colon{\mathcal{F}}_{n+1}\xrightarrow[]{}T_{\varepsilon_{n}*}({\mathcal{F}}_{n})$
such that $g_{n}\circ f_{n}=\tau_{0,\varepsilon_{n}}({\mathcal{F}}_{n})$.

(ii) It remains to check that $\varinjlim{\mathcal{F}}_{n}$ is decomposed. For
this we modify the $f_{n}$’s using Lemma B.9. We let
${\mathcal{J}}_{n}\subset{\mathcal{I}}_{n}$ be the set of intervals of length
$>2\varepsilon_{n}$ and we set
$\widetilde{\mathcal{F}}_{n}=\bigoplus_{I\in{\mathcal{J}}_{n}}k_{I}$. We first
put the $f_{n}$’s in “diagonal form”, $\widehat{f}_{n}$ (see the diagram
(B.2)). Lemma B.9 applied with
$f_{0}|_{\widetilde{\mathcal{F}}_{0}}\colon\widetilde{\mathcal{F}}_{0}\xrightarrow[]{}{\mathcal{F}}_{1}$
gives an injective map
$\sigma_{0}\colon{\mathcal{J}}_{0}\xrightarrow[]{}{\mathcal{I}}_{1}$ and an
isomorphism
$\phi_{1}\colon{\mathcal{F}}_{1}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}_{1}$
such that $\phi_{1}\circ f_{0}|_{\widetilde{\mathcal{F}}_{0}}$ has a “diagonal
form” (that is, it is the sum of the natural morphisms
$k_{I}\xrightarrow[]{}k_{\sigma_{0}(I)}$). We set
$\widehat{f}_{0}=\phi_{1}\circ f_{0}$ and
$f^{\prime}_{1}=f_{1}\circ\phi_{1}^{-1}$. Now we apply the lemma with
$f^{\prime}_{1}|_{\widetilde{\mathcal{F}}_{1}}\colon{\mathcal{F}}_{1}\xrightarrow[]{}{\mathcal{F}}_{2}$
and obtain
$\phi_{2}\colon{\mathcal{F}}_{2}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}_{2}$
and $\sigma_{1}\colon{\mathcal{I}}_{1}\xrightarrow[]{}{\mathcal{I}}_{2}$. We
set $\widehat{f}_{1}=\phi_{2}\circ f^{\prime}_{1}$ and
$f^{\prime}_{2}=f_{2}\circ\phi_{2}^{-1}$. We go on inductively and end up with
a sequence of injective maps
$\sigma_{n}\colon{\mathcal{J}}_{n}\xrightarrow[]{}{\mathcal{I}}_{n+1}$ and
morphisms
$\widehat{f}_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{F}}_{n+1}$
such that $\widehat{f}_{n}$ restricted to $\widetilde{\mathcal{F}}_{n}$ has a
“diagonal form”.

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(iii) With our definition of ${\mathcal{J}}_{n}$ we have moreover
$\sigma_{n}({\mathcal{J}}_{n})\subset{\mathcal{J}}_{n+1}$. Indeed, modifying
$g_{n}$ as $\widehat{g}_{n}=T_{\varepsilon_{n}*}(\phi_{n})\circ
g_{n}\circ\phi_{n+1}^{-1}$, we see that
$\widehat{g}_{n}\circ\widehat{f}_{n}=\tau_{0,\varepsilon_{n}}({\mathcal{F}}_{n})$.
It follows that, for any $[a,b\mathclose{[}\in{\mathcal{J}}_{n}$ and
$[a^{\prime},b^{\prime}\mathclose{[}=\sigma_{n}([a,b[)$, we must have $a\leq
a^{\prime}\leq a+\varepsilon_{n}$ and $b\leq b^{\prime}\leq
b+\varepsilon_{n}$. Hence $b^{\prime}-a^{\prime}\geq b-a-\varepsilon_{n}\geq
2\varepsilon_{n}-\varepsilon_{n}=2\varepsilon_{n+1}$, which shows that
$[a^{\prime},b^{\prime}\mathclose{[}\in{\mathcal{J}}_{n+1}$.

We can then define
$\widetilde{f}_{n}=\widehat{f}_{n}|_{\widetilde{\mathcal{F}}_{n}}\colon\widetilde{\mathcal{F}}_{n}\xrightarrow[]{}\widetilde{\mathcal{F}}_{n+1}$
and $\widetilde{f}_{n}$ has a “diagonal form” with respect to the inclusion
$\sigma_{n}|_{{\mathcal{J}}_{n}}\colon{\mathcal{J}}_{n}\hookrightarrow{\mathcal{J}}_{n+1}$.
For a given $I\in{\mathcal{J}}_{n}$ we write
$\sigma_{k}\circ\sigma_{k-1}\circ\dots\circ\sigma_{n}(I)=[a_{k},b_{k}[$; the
sequences $(a_{k})$, $(b_{k})$ are non decreasing and have limits, say
$a_{\infty}$, $b_{\infty}$. Then $\varinjlim k_{[a_{k},b_{k}[}\simeq
k_{[a_{\infty},b_{\infty}[}$. We set $I_{\infty}=[a_{\infty},b_{\infty}[$. We
let ${\mathcal{J}}$ be the increasing union of the ${\mathcal{J}}_{n}$’s. Then
each element of ${\mathcal{J}}$ corresponds to an interval $I_{\infty}$ as
above and we have proved
$\varinjlim\widetilde{\mathcal{F}}_{n}\simeq\bigoplus_{I_{\infty}\in{\mathcal{J}}}k_{I_{\infty}}$.
By Lemma 6.20 the maps
$\widetilde{\mathcal{F}}_{k}\xrightarrow[]{}\varinjlim\widetilde{\mathcal{F}}_{n}$
and ${\mathcal{F}}_{k}\xrightarrow[]{}\varinjlim{\mathcal{F}}_{n}$ have cones
which go to $0$ when $k\xrightarrow[]{}\infty$. It follows by Lemma 6.4 that
the natural map
$\varinjlim\widetilde{\mathcal{F}}_{n}\xrightarrow[]{}\varinjlim{\mathcal{F}}_{n}$
has a cone which is arbitrarily close to $0$, hence it is an isomorphism. ∎

######  Corollary B.11.

We recall that $k$ is a field. Let ${\mathcal{F}}\in D_{lc}(\mathbb{R})\cap
D_{\tau\geq 0}(\mathbb{R})$ be a limit of constructible objects of
$D_{\tau\geq 0}(\mathbb{R})$. We assume that ${\mathcal{F}}$ is constant
outside $[-A,A]$ for some $A$. Then there exists an at most countable set of
intervals ${\mathcal{I}}$ and integers $d_{I}$, $I\in{\mathcal{I}}$, such that
${\mathcal{F}}\simeq\bigoplus_{I\in{\mathcal{I}}}k_{I}[d_{I}]$.

###### Proof.

If we prove the result for ${\mathcal{F}}\otimes k_{[-A-1,A+1[}$ we deduce the
result for ${\mathcal{F}}$ by gluing the decomposition with the constant
sheaves ${\mathcal{F}}\otimes k_{]-\infty,-A[}$ and ${\mathcal{F}}\otimes
k_{]A,\infty[}$. Hence we can assume ${\mathcal{F}}$ has compact support.

As in the case of constructible sheaves the result follows from Proposition
B.10 and a decomposition of a complex as sum of its cohomology. However we
didn’t prove that limits of constructible sheaves have no
$\operatorname{Ext}^{2}$ so we have to be careful to decompose the cohomology
before we take the limit, as follows.

There exists a sequence of constructible objects ${\mathcal{F}}_{n}\in
D_{\tau\geq 0}(\mathbb{R})$ which $\gamma_{g}$-converges to $F$. Since
$\operatorname{Ext}^{2}({\mathcal{G}},{\mathcal{G}}^{\prime})=0$ for any two
constructible
${\mathcal{G}},{\mathcal{G}}^{\prime}\in\operatorname{Mod}(\mathbb{R})$, we
have
${\mathcal{F}}_{n}\simeq\bigoplus_{i\in\mathbb{Z}}H^{i}{\mathcal{F}}_{n}[-i]$
(see for example [KS06, Cor. 13.1.20]).

In $D_{\tau\geq 0}(\mathbb{R})$ we have $K_{\varphi}^{a}(-)=T_{a*}(-)$, where
$T_{a}$ is as usual the translation by $a$, and this functor commutes with the
cohomology, that is, $H^{i}(T_{a*}({\mathcal{G}}))\simeq
T_{a*}(H^{i}({\mathcal{G}}))$. It follows that
$\gamma_{g}(H^{i}{\mathcal{G}},H^{i}{\mathcal{G}}^{\prime})\leq\gamma_{g}({\mathcal{G}},{\mathcal{G}}^{\prime})$,
for any ${\mathcal{G}},{\mathcal{G}}^{\prime}\in D_{\tau\geq 0}(\mathbb{R})$.
Hence $H^{i}{\mathcal{F}}_{n}$ $\gamma_{g}$-converges to $H^{i}{\mathcal{F}}$.

It follows that
${\mathcal{F}}_{n}\simeq\bigoplus_{i}H^{i}{\mathcal{F}}_{n}[-i]$
$\gamma_{g}$-converges to $\bigoplus_{i}H^{i}{\mathcal{F}}[-i]$. By unicity of
the limit (Proposition B.7) we have
${\mathcal{F}}\simeq\bigoplus_{i}H^{i}{\mathcal{F}}[-i]$. Now the result
follows from Proposition B.10. ∎

##  Appendix C Comparing different interleaving metrics on
$D(N\times{\mathbb{R}})$.

To prove Theorem 1.2 we have used the distance $\gamma_{\tau}$ on sheaves over
$N\times\mathbb{R}$ and the fact that its restriction to sheaves associated
with Hamiltonian maps coincides with the spectral distance (see Proposition
6.13). We can also consider the distance $\gamma_{g}$ on sheaves, where $g$ is
some Riemannian metric on $N$. For the sake of completeness we describe the
distance $\gamma_{g}$ from the point of view of spectral distance and we prove
that the distances $\gamma_{\tau}$ and $\gamma_{g}$ induce equivalent
topologies (when restricted to sheaves associated with Hamiltonian maps).

In the following we let $H\colon T^{*}N\xrightarrow[]{}\mathbb{R}$ be a
function which coincides with $(q,p)\mapsto||p||_{g}$ outside a compact
neighbourhood of $0_{N}$ and $\varphi=\varphi_{H}$ denotes its Hamiltonian
flow. We will use spectral invariants and sheaves for $\varphi^{s}$ but in §4
and §5 we only defined them for compactly supported isotopies. However we can
extend them to a positive Hamiltonian $H$ as follows. Let $H_{r}$ be a
truncation of $H$ to the disc bundle of $T^{*}N$ of radius $r$, so that
$H_{r}$ is an increasing sequence of Hamiltonians converging to $H$. Then the
homogeneous lift of $s\mapsto\varphi_{H_{r}}^{-s}\circ\varphi_{H_{r+1}}^{s}$
to $T^{*}(N\times\mathbb{R})\setminus 0_{N\times\mathbb{R}}$ is a non negative
Hamiltonian isotopy. Hence we have a natural map ${\mathcal{K}}_{{\rm
id}}\xrightarrow[]{}{\mathcal{K}}_{\varphi_{H_{r}}^{-s}}\circ{\mathcal{K}}_{\varphi_{H_{r+1}}^{s}}$,
or equivalently,
${\mathcal{K}}_{\varphi_{H_{r}}^{s}}\xrightarrow[]{}{\mathcal{K}}_{\varphi_{H_{r+1}}^{s}}$.
We define ${\mathcal{K}}_{\varphi_{s}}$ as the homotopy colimit of this
inductive system (say $r$ runs over the integers). In the same way
$c_{+}(\varphi_{r}^{-s}\psi)$ is a decreasing sequence, and we say that
$c_{+}(\varphi^{-s}\psi)=0$ if $\lim_{r}c_{+}(\varphi_{r}^{-s}\psi)=0$. We
shall prove that the sequence is in fact stationary, so this means
$c_{+}(\varphi_{r}^{-s}\psi)=0$ for $r$ large enough.

Now Lemma 5.7 extends as follows: there are morphisms
${\mathcal{K}}_{\psi}\longrightarrow{\mathcal{K}}_{\varphi^{s}}\longrightarrow{\mathcal{K}}_{\varphi^{2s}\circ\psi}$
if and only if $c_{-}(\varphi^{s}\psi)=0$ and $c_{-}(\varphi^{s}\psi^{-1})=0$.
Using Corollary 6.10 we obtain

######  Lemma C.1.

If $c_{-}(\varphi^{s}\psi)=0$ and $c_{-}(\varphi^{s}\psi^{-1})=0$, the
composition of the map of the map
${\mathcal{K}}_{\psi}\longrightarrow{\mathcal{K}}_{\varphi^{s}}$ and
${\mathcal{K}}_{\varphi^{s}}\longrightarrow{\mathcal{K}}_{\varphi^{2s}\circ\psi}$
coincides with the canonical map
${\mathcal{K}}_{\psi}\longrightarrow{\mathcal{K}}_{\varphi^{2s}\psi}$.

Now we may define the distance $\gamma^{g}$ on ${\rm Ham}(T^{*}N)$ as follows

######  Definition C.2.

We set

$c_{-}^{g}(\psi)=-\inf\left\\{s\geq 0\mid c_{-}(\varphi^{s}\psi)=0\right\\}$

and

$c_{+}^{g}(\psi)=-c_{-}^{g}(\psi^{-1})=\inf\left\\{s\geq 0\mid
c_{+}(\varphi^{-s}\psi)=0\right\\}$

Finally we set

$\gamma^{g}(\psi)=c_{+}^{g}(\psi)-c_{-}^{g}(\psi)$

######  Proposition C.3.

We have

  1. (1)

The function $\gamma^{g}$ defines a metric by
$\gamma^{g}(\psi_{1},\psi_{2})=\gamma^{g}(\psi_{1}\psi_{2}^{-1})$.

  2. (2)

The topologies defined by $\gamma^{g}$ and $\gamma$ coincide.

###### Proof.

(1) Apply section 3 of [PS21] to the Kernels ${\mathcal{K}}_{\varphi^{s}}$ (or
the symplectic flow $\varphi^{s}$).

(2) We shall first prove that $c_{+}(\psi)\leq C_{W}c_{+}^{g}(\psi)$ and
$\gamma(\psi)\leq C_{W}\gamma^{g}(\psi)$ for all $\psi$ such that ${\rm
supp}(\psi)\subset K$. We consider $c_{+}(\varphi^{-s}\psi)$ and remember that
$\varphi^{s}$ is generated by a positive homogeneous Hamiltonian $H$. By a
classical argument (see e.g. [Vit92], lemma 4.7 p. 699) the map $s\mapsto
c_{+}(\varphi^{-s}\psi)$ is piecewise $C^{1}$ and where it is $C^{1}$ we have

$\frac{d}{ds}c_{+}(\varphi^{-s}\psi)_{\mid s=s_{0}}=-K_{s}(z)$

where $K_{s}$ is the Hamiltonian associated to the map $\varphi^{-s}\psi$ and
$z$ is some fixed point of $\varphi^{-s}\psi$ having non-negative action.
Indeed, if $S_{s}$ is a Generating function Quadratic at infinity for
$\varphi^{-s}\psi$ we know that $c_{+}(\varphi^{-s}\psi)$ is a critical value
of $S_{s}$ and is non-negative. It thus corresponds to fixed points of
$\varphi^{-s}\psi$ with non-negative action and its derivative is
$\frac{d}{ds}S_{s}(\xi_{z})$ where $\xi_{z}$ corresponds to $z$, one of the
fixed points such that $S_{s}(\xi_{z})=c_{+}(\varphi^{-s}\psi)$.

In our case, if $z$ is outside the support of $\psi$, the fixed point is a
fixed point of $\varphi^{-s}$ and these have action $0$ since $H$ is
$1$-homogeneous (as on an orbit $p\dot{q}-H(q,p)=p\frac{\partial H}{\partial
p}-H(q,p)=0$). So we only need to estimate $H$ in the support of $\psi$ and we
set $W$ to be a neighbourhood of this support. Then if
$C_{W}=\sup\\{H(x,p)\mid(x,p)\in W\\}$ we have

$\frac{d}{ds}c_{+}(\varphi^{-s}\psi)_{\mid s=s_{0}}\geq-C_{W}$

so that

$c_{+}(\varphi^{-s}\psi)-c_{+}(\psi)\geq-C_{W}s$

and if $c_{+}(\varphi^{-s}\psi)=0$ we have

$C_{W}s\geq c_{+}(\psi)$

hence

$c_{+}(\psi)\leq C_{W}c^{g}(\psi)$

Applying the same to $\psi^{-1}$, we get

$\gamma(\psi)\leq C_{W}\gamma^{g}(\psi)$

Note that we also proved that $\gamma(\varphi_{r}^{-s}\psi)$ is stationary for
$r$ such that $B(0,r)$ contains the support of $\psi$.

Conversely, let us write $\varphi^{s}=\xi_{r}^{s}\circ\eta_{r}^{s}$ where
$\xi_{r}^{s}$ (resp. $\eta_{r}^{s}$) is the flow of $f_{r}(H)$ (resp.
$(1-f_{r})(H)$) where $f_{r}$ is a function given by

  1. (1)

$f_{r}(t)=t$ for $t\geq 2r$

  2. (2)

$f_{r}=\frac{3r}{2}$ for $t\leq r$

  3. (3)

$f_{r}$ is non-decreasing

Note that the flows $\xi_{r}^{s},\eta_{r}^{s}$ are autonomous, commute, and
$f_{r}(H)$ is bounded from below by $\frac{3r}{2}$ and $\eta_{r}^{s}$ is
supported in a ball of radius $K\cdot r$.

Then

$c_{+}(\varphi^{-s}\psi)=c_{+}(\xi^{-s}\eta_{r}^{-s}\psi)\leq
c_{+}(\eta_{r}^{-s}\psi)-\frac{3r}{2}s$

as long as $c_{+}(\varphi^{-s}\psi)>0$ so we have

$\frac{3r}{2}c_{+}^{g}(\psi)\leq c_{+}(\eta_{r}^{-s}\psi)\leq\pi K^{2}\cdot
r^{2}+c_{+}(\psi)$

As a result if $c_{+}(\psi_{j})$ converges to zero, let
$r_{j}=\sqrt{c_{+}(\psi_{j)}}$ . Then

$c_{+}^{g}(\psi_{j})\leq\frac{2}{3}\left(\pi K\cdot
r_{j}+\frac{c_{+}(\psi_{j})}{r_{j}}\right)\leq\frac{2}{3}(\pi
K+1)\sqrt{c_{+}(\psi_{j})}$

and we deduce that $\lim_{j}c_{+}^{g}(\psi_{j})=0$.

\begin{overpic}[width=227.62204pt]{f_r2-b.pdf} \put(6.0,85.0){$f_{r}$}
\put(0.0,25.0){$\frac{3r}{2}$} \put(77.0,2.0){$t$} \put(27.0,7.0){$\mid$}
\put(27.0,2.0){$r$} \end{overpic} Figure 6. The function $f_{r}$

∎

######  Remark C.4.

  1. (1)

We do not know if the metrics $\gamma$ and $\gamma^{g}$ are equivalent, only
that they define the same topology.

  2. (2)

Here is an analogy for the difference between $\gamma$ and $\gamma^{g}$.
Consider the following norm on $C_{b}^{0}(X,{\mathbb{R}})$ : we first define

$\max_{f}(g)=\inf\left\\{s\geq 0\mid\max(g-sf)\leq 0\right\\}$

and

$\min_{f}(g)-\max_{f}(-g)=\inf\left\\{s\geq 0\mid\min(g+sf)\geq 0\right\\}$

Then we set $\|g\|_{f}=\max_{f}(g)-\min_{f}(g)$. Provided

$0<\inf_{x\in X}f(x)\leq\sup_{x\in X}f(x)<+\infty$

it is obvious that $\|\bullet\|_{f}$ and $\|\bullet\|$ are equivalent.

##  Appendix D Local Floer cohomology

Our goal is to prove the following statement, which was used in Subsection
5.2.

Let $\varphi$ be a Hamiltonian map supported in a Darboux chart, $V$. Since
$\Gamma(\varphi)$ coincides with the diagonal, $\Gamma({\rm Id})$ outside
$W=V\times V$, we can consider two types of Floer cohomology :

  1. (1)

$FH^{*}(\varphi,W)$ obtained by using an almost complex structure making
$\partial W$ pseudoconvex and building the boundary operator by considering
only the Floer trajectories contained in $W$

  2. (2)

The full Floer cohomology $FH^{*}(\varphi)=FH^{*}(\Gamma(\varphi,\Delta))$

Note that we may also restrict the action to some interval $[a,b]$.

######  Proposition D.1.

The above two Floer cohomologies coincide. The same holds when restricting the
action to $[a,b]$.

This is a special case of the following. Let $L_{t}$ be an exact Lagrangian
isotopy supported in a pseudoconvex domain $W$ (i.e. $L_{t}$ is constant
outside $W$). We can consider $FH_{[a,b]}^{*}(L_{1},L_{0};W)$ and
$FH_{[a,b]}^{*}(L_{1},L_{0})$.

######  Proposition D.2.

We have $FH_{[a,b]}^{*}(L_{1},L_{0};W)=FH_{[a,b]}^{*}(L_{1},L_{0})$.

###### Proof.

We may assume that $W$ is contained in a Weinstein neighbourhood of $L_{0}$.
We shall then use the structure of the cotangent bundle to write in these
coordinates $s\cdot(q,p)=(q,sp)$ and $s\cdot L_{1}$ is the image of $L_{1}$ by
the action of $s$. We have that $FH_{[sa,sb]}^{*}(s\cdot L_{1},L_{0};W)$ does
not depend on $s$, since the intersection points $s\cdot L_{1}\cap L_{0}$ are
constant, their action is multiplied by $s$. We choose an almost complex
structure making $\partial W$ pseudoconvex and such that $L_{0}\cap\partial W$
is a Legendrian, say $\Lambda_{0}$, and for all $x\in\Lambda_{0}$ we have
$JT_{x}\Lambda_{0}\subset T_{x}\partial W$.

Then the holomorphic strips cannot exit from $W$ by pseudoconvexity, since an
interior point of such a curve cannot be tangent to $\partial W$ and if the
boundary of the strip in $L_{j}$ was tangent to $\partial W$ then the
holomorphic curve would be tangent to $\partial W$ from inside, which is again
impossible by pseudoconvexity. As a result the coboundary map defining
$FH_{[sa,sb]}^{*}(s\cdot L_{1},L_{0};W)$ is also constant. Of course the same
holds for $FH_{[a,b]}^{*}(L_{1},L_{0})$, as in that case we do not even have
to worry about the location of the holomorphic strips. Now for $s$ small
enough all holomorphic strips must have area less than any given
$\varepsilon$; but there is a positive $\varepsilon_{0}$ such that no
holomorphic strip of area less than $\varepsilon_{0}$ can exit from $\partial
W$ (by a monotonicity argument). As a result for $s$ small enough we have
$FH_{[sa,sb]}^{*}(s\cdot L_{1},L_{0};W)=FH_{[sa,sb]}^{*}(s\cdot L_{1},L_{0})$
and this implies the Proposition. ∎

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