# Exact solutions for analog Hawking effect in dielectric media

S. Trevisan1, F. Belgiorno4,5, and S.L. Cacciatori2,3 1 Department of
Information Engineering, Univerità di Padova, Via Gradenigo 6/b, IT-35131
Padova, Italy.  2 Department of Science and High Technology, Università
dell’Insubria, Via Valleggio 11, IT-22100 Como, Italy  3 INFN sezione di
Milano, via Celoria 16, IT-20133 Milano, Italy  4 Dipartimento di Matematica,
Politecnico di Milano, Piazza Leonardo 32, IT-20133 Milano, Italy  5 INdAM-
GNFM

###### Abstract.

In the framework of the analog Hawking radiation for dielectric media, we
analyze a toy-model and also the 2D reduction of the Hopfield model for a
specific monotone and realistic profile for the refractive index. We are able
to provide exact solutions, which do not require any weak dispersion
approximation. The theory of Fuchsian ordinary differential equations is the
basic tool for recovering exact solutions, which are rigoroulsy identified,
and involve the so-called generalized hypergeometric functions
${}_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$.
A complete set of connection formulas are available, both for the subcritical
case and for the transcritical one, and also the Stokes phenomenon occurring
in the problem is fully discussed. From the physical point of view, we focus
on the problem of thermality. Under suitable conditions, the Hawking
temperature is deduced, and we show that it is in fully agreement with the
expression deduced in other frameworks under various approximations.

##  1\. Introduction

We are interested to focus our attention on analytical calculation of the
analog Hawking effect in dielectric media, and in presence of dispersion.
Analytical calculations for the analog Hawking effect, introduced in the
seminal paper [1] for non-dispersive media, also in the dispersive case have
been largely discussed in literature, see e.g. the following (non-exhaustive)
list of papers [2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23], and for weak dispersion and the transcritical case a
rather general mathematical framework, able to encompass in an unified picture
very relevant models even for the experiments [24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34], has been discussed in [36, 37, 38]. In [39], the authors
introduced a new mathematical perspective in the analog Hawking effect, by
relating the problem to the solution of a fourth-order Fuchsian equation for
the subcritical case. As a remarkable example of the possibilities offered by
Fuchsian equations, we provide here an _exact_ solution of a particular
scattering problem inside a dielectric. We choose a monotonic refractive index
profile traveling inside the dielectric at a fixed velocity. This kind of
background represents a fundamental setting for a good understanding of the
pair-creation process, and monotonic backgrounds are often considered in
analytical computations in transcritical regime and in numerical simulations
of analog systems ( see _e.g._ [40]). We stress that exact solutions for the
dispersive case of the analog Hawking effect are very hard to be obtained, as
even solutions of ordinary differential equations of the fourth order. The
only other example we can find is contained in the paper by Philbin [18],
which provides exact solutions for the Corley model [3, 9]. Therein solutions
are obtained at the price of introducing an interesting but somehow
unrealistic linear velocity profile $v(x)=-\alpha x$.

The plan of the paper is the following. In Section 2 we present the model and
the monotonic background we are going to consider. In Section 3, starting from
the equation of motion for our model, we show that, by means of a suitable
change of the independent spatial variable in the comoving frame of the pulse,
we are able to obtain a fourth order Fuchsian equation. We provide a detailed
characterization of its local monodromy and spectral type. In Section 4 we
provide the exact solution, and recover rigorously that the generalized
hypergeometric functions
${}_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$
are involved. Furthermore, we provide a study of the Stokes phenomenon, and we
study some physical consequences for the scattering problem at hand. In
Section 5, we consider a generalization of the previous analysis to the case
of the original model (the so-called $\phi\psi$-model) to which the Hopfield
model reduces in the 2D case, and again we consider the scattering problem and
the thermality of the spectrum, which is recovered to coincide with the one
deduced in the weak dispersion limit discussed in [36] under suitable
conditions. In Section 6, we summarize our achievements and display future
perpectives for our analysis.

##  2\. The Cauchy model and the choice of background

The model we consider is the modified $\phi$-$\psi$ model (or “Cauchy model”)
introduced by the authors in [39]. In the laboratory frame, it is expressed by
the Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{2}(\partial_{t_{l}}\phi)^{2}+\frac{1}{2}\left((\partial_{t_{l}}\psi)^{2}+\mu^{2}\psi^{2}\right)+g\phi\partial_{x_{l}}\psi-\frac{\lambda}{4!}\psi^{4}\,.$
(2.1)

The linearized EOMs around a background solution $\psi_{B}$, in the lab frame,
are

$\displaystyle\partial_{t_{l}}^{2}\phi-g\partial_{x_{l}}\psi=0,$ (2.2)
$\displaystyle\partial_{t_{l}}^{2}\psi+g\partial_{x_{l}}\psi-\mu^{2}\phi+\frac{\lambda}{2}\psi_{B}^{2}\psi=0.$
(2.3)

The free-field solutions (for $\lambda=0$) are plane waves
$e^{i\omega_{lab}t_{l}-ik_{lab}x_{l}}$ which satisfy

$\displaystyle
n_{0}^{2}(\omega_{lab}):=\frac{k_{lab}^{2}}{\omega_{lab}^{2}}=\frac{\mu^{2}}{g^{2}}+\frac{\omega_{lab}^{2}}{g^{2}}=:A+B\omega_{lab}^{2}\,.$
(2.4)

The dispersion relation, in a reference frame moving with velocity $V$ with
respect to the lab, has four solutions (see Figure 1): expanding $k(\omega)$
for $\omega\rightarrow 0$, the four modes have the following expressions

$\displaystyle k_{H}$ $\displaystyle=\frac{\mu-gV}{g-\mu
V}\,\omega+O(\omega^{3}),$ (2.5) $\displaystyle k_{B}$
$\displaystyle=-\frac{\mu+gV}{g+\mu V}\,\omega+O(\omega^{3}),$ (2.6)
$\displaystyle k_{P}$ $\displaystyle=\frac{\sqrt{g^{2}-\mu^{2}V^{2}}}{\gamma
V^{2}}-\left(\frac{1}{V}+\frac{g^{2}}{\gamma^{2}V(g^{2}-\mu^{2}V^{2})}\right)\,\omega-\frac{g^{2}(2g^{2}+\mu^{2}V^{2})}{2\gamma(g^{2}-\mu^{2}V^{2})^{5/2}}\,\omega^{2}+O(\omega^{3}),$
(2.7) $\displaystyle k_{N}$
$\displaystyle=-\frac{\sqrt{g^{2}-\mu^{2}V^{2}}}{\gamma
V^{2}}-\left(\frac{1}{V}+\frac{g^{2}}{\gamma^{2}V(g^{2}-\mu^{2}V^{2})}\right)\,\omega+\frac{g^{2}(2g^{2}+\mu^{2}V^{2})}{2\gamma(g^{2}-\mu^{2}V^{2})^{5/2}}\,\omega^{2}+O(\omega^{3})\,.$
(2.8)

Figure 1.  The dispersion relation (2.4) represented in the comoving frame
with the background, with $g=1$, $\mu=1.2$; for $0<\omega<\omega_{MAX}$ there
are four real solutions.

The reason of the choice of the model lies in the simple expression of the
dispersion relation: however, in many cases we will refer to $\text{DR}(k)$ as
a generic fourth order polynomial, so the choice of a particular dispersion
relation is not really crucial. We are going to solve the linearized equations
(2.2)-(2.3) with a particular choice of background. In the experiments the
background field is represented by a laser pulse, which is naturally
_localized_ and travels rigidly at a certain velocity $V$. We will instead
consider a monotonic background

$\displaystyle\psi_{b}(x-Vt)^{2}=1-\tanh(\beta(x-Vt))\,.$ (2.9)

We claim that this is a good model for the right-side of a laser pulse;
moreover, a monotonic background represents a better model for an event
horizon and allows to better understand the nature of Hawking Radiation.
Monotonic backgrounds of this type were also used in some previous studies of
analog black holes and white holes (see for example [40]).

The linearized equations (2.2) and (2.3) can be put together to a fourth-order
equation of generalized Orr-Sommerfeld type. It is convenient to write these
equations in the comoving coordinates $t=\gamma(t_{l}-Vx_{l})$,
$x=\gamma(x_{l}-Vt_{l})$. Since the potential term is independent of the
comoving time, we seek for a solution in the form $\psi=e^{-i\omega t}f(x)$,
$\phi=e^{-i\omega t}g(x)$. By applying $(v^{\mu}\partial_{\mu})^{2}$ to the
second equation, we obtain a single fourth order equation for $f(x)$ only:

$\displaystyle 0=V^{4}\gamma^{4}f^{(4)}(x)+4iV^{3}\omega\gamma^{4}f^{(3)}(x)+$
(2.10) $\displaystyle+\frac{1}{2}\gamma^{2}f^{\prime\prime}(x)\left(-\lambda
V^{2}\tanh(\tilde{\beta}x)+2g^{2}+\lambda
V^{2}-2\mu^{2}V^{2}-12V^{2}\omega^{2}\gamma^{2}\right)+$ (2.11)
$\displaystyle+i\gamma^{2}Vf^{\prime}(x)\left(i\tilde{\beta}\lambda
V\text{sech}^{2}(\tilde{\beta}x)+\omega\left(-\lambda\tanh(\tilde{\beta}x)+2g^{2}+\lambda-2\mu^{2}-4\omega^{2}\gamma^{2}\right)\right)+$
(2.12)
$\displaystyle+\frac{1}{2}\gamma^{2}f(x)\left[\omega^{2}\left(\lambda\tanh(\tilde{\beta}x)-2g^{2}V^{2}-\lambda+2\mu^{2}+2\omega^{2}\gamma^{2}\right)\right.$
(2.13) $\displaystyle\left.+2\tilde{\beta}\lambda
V\text{sech}^{2}(\tilde{\beta}x)(\tilde{\beta}V\tanh(\tilde{\beta}x)-i\omega)\right]\,,$
(2.14)

where $\tilde{\beta}=\frac{\beta}{\gamma}$.

##  3\. Reduction to a Fuchsian equation: monodromy and Riemann scheme

As in [39], we perform the following change of variables on Eq. (2.10),

$\displaystyle z$ $\displaystyle=-e^{2\tilde{\beta}x}\,,$ (3.1)

which implies

$\displaystyle\partial_{x}$
$\displaystyle=2\tilde{\beta}\theta_{z}:=2\tilde{\beta}z\frac{d}{dz}\,.$ (3.2)

By defining the rescaled parameters $G=\frac{g}{2\tilde{\beta}}$,
$\Omega=\frac{\omega}{2\tilde{\beta}}$, $M=\frac{\mu}{2\tilde{\beta}}$ and
$\Lambda=\frac{\lambda}{4\tilde{\beta}^{2}}$, we end up with the following
equation:

$\displaystyle
0=V^{4}\gamma^{4}z^{4}f^{(4)}(z)+f^{(3)}(z)\left(6V^{4}\gamma^{4}z^{3}-4iV^{3}\Omega\gamma^{4}z^{3}\right)$
(3.3)
$\displaystyle+f^{\prime\prime}(z)\left(G^{2}\gamma^{2}z^{2}-V^{2}\gamma^{2}z^{2}\left(\frac{\Lambda}{z-1}+M^{2}\right)\right.$
(3.4)
$\displaystyle\left.\qquad-6V^{2}\Omega^{2}\gamma^{4}z^{2}-12iV^{3}\Omega\gamma^{4}z^{2}+7V^{4}\gamma^{4}z^{2}\right)$
(3.5)
$\displaystyle+f^{\prime}(z)\left(-2iG^{2}V\Omega\gamma^{2}z+G^{2}\gamma^{2}z+V^{2}\gamma^{2}z\left(\frac{\Lambda}{(1-z)^{2}}-M^{2}\right)-2iV\Omega\gamma^{2}z\left(\frac{\Lambda}{(1-z)}-M^{2}\right)\right.$
(3.6) $\displaystyle\left.\qquad+\frac{2\Lambda
V^{2}\gamma^{2}z^{2}}{(1-z)^{2}}-6V^{2}\Omega^{2}\gamma^{4}z-4iV^{3}\Omega\gamma^{4}z+V^{4}\gamma^{4}z+4iV\Omega^{3}\gamma^{4}z\right)$
(3.7)
$\displaystyle+f(z)\left(-G^{2}V^{2}\Omega^{2}\gamma^{2}+\Omega^{2}\gamma^{2}\left(\frac{\Lambda}{1-z}+M^{2}\right)\right.$
(3.8) $\displaystyle\left.\qquad+\frac{2\Lambda
V^{2}\gamma^{2}z^{2}}{(1-z)^{3}}+\frac{\Lambda
V^{2}\gamma^{2}z}{(1-z)^{3}}-\frac{2i\Lambda
V\Omega\gamma^{2}z}{(1-z)^{2}}+\Omega^{4}\gamma^{4}\right)\,.$ (3.9)

This equation is of Fuchsian type, with three singular points, $z=0,1,\infty$.

As an alternative form, we can write the EOMs (2.2)-(2.3) as a system of first
order and perform the same change of variables as before. With this procedure
we obtain the system

$\displaystyle\frac{dU}{dz}=A(z)U\,,$ (3.10)

were

$\displaystyle U$ $\displaystyle=\begin{pmatrix}g(z)\\\ g^{\prime}(z)\\\
f(z)\\\ f^{\prime}(z)\end{pmatrix}$ (3.11) $\displaystyle A(z)$
$\displaystyle=\left(\begin{array}[]{cccc}0&1&0&0\\\
\frac{\Omega^{2}}{V^{2}z^{2}}&-\frac{V^{2}\gamma^{2}z-2iV\Omega\gamma^{2}z}{V^{2}\gamma^{2}z^{2}}&\frac{iG\Omega}{V\gamma
z^{2}}&-\frac{G}{V^{2}\gamma z}\\\ 0&0&0&1\\\ -\frac{iG\Omega}{V\gamma
z^{2}}&\frac{G}{V^{2}\gamma
z}&-\frac{-\frac{\Lambda}{1-z}-M^{2}-\Omega^{2}\gamma^{2}}{V^{2}\gamma^{2}z^{2}}&-\frac{V^{2}\gamma^{2}z-2iV\Omega\gamma^{2}z}{V^{2}\gamma^{2}z^{2}}\\\
\end{array}\right)\,.$ (3.16)

We can reduce (3.10) to a _Fuchsian System of Normal Form_ [41] by changing
variable to

$\displaystyle Y(z)$ $\displaystyle=P(z)U(z)\,,$ (3.17) $\displaystyle P(z)$
$\displaystyle=\text{diag}\left(\frac{1}{z},1,\frac{1}{z},1\right)\,.$ (3.18)

The system has now the form

$\displaystyle\frac{dY}{dz}$
$\displaystyle=\left(\frac{A_{1}}{z}+\frac{A_{2}}{z-1}\right)Y\,,$ (3.19)
$\displaystyle A_{1}$ $\displaystyle=\left(\begin{array}[]{cccc}-1&1&0&0\\\
\frac{\Omega^{2}}{V^{2}}&\frac{-V-2i\Omega}{V}&-\frac{iG\Omega}{V\gamma}&-\frac{G}{V^{2}\gamma}\\\
0&0&-1&1\\\
\frac{iG\Omega}{V\gamma}&\frac{G}{V^{2}\gamma}&\frac{\Lambda+M^{2}+\Omega^{2}\gamma^{2}}{V^{2}\gamma^{2}}&\frac{-V-2i\Omega}{V}\\\
\end{array}\right)\,,$ (3.24) $\displaystyle A_{2}$
$\displaystyle=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&-\frac{\Lambda}{V^{2}\gamma^{2}}&0\\\ \end{array}\right)\,.$ (3.29)

The matrices $A_{1}$ and $A_{2}$ are constant and they are, respectively, the
residue at the simple poles $z=0$ and $z=1$. We may also define

$\displaystyle A_{0}:=-A_{1}-A_{2}\,,$ (3.30)

which corresponds to the residue at the simple pole $z=\infty$.

###  3.1. The local solutions and monodromy

We start looking for local solutions of (3.3) around $z=\infty$. After
changing variable to $t=1/z$, we can look for a solution in the form

$\displaystyle f(t)=t^{-i\alpha}\sum_{n=0}^{\infty}c_{n}t^{n}\,.$ (3.31)

The characteristic equation for the exponent $k:=2\tilde{\beta}\alpha$ is

$\displaystyle\text{DR}(k)$
$\displaystyle:=\gamma^{2}(\mu^{2}(kV+\omega)^{2}-g^{2}(k+V\omega)^{2}+(kV+\omega)^{4}\gamma^{2})=0\,,$
(3.32)

which is nothing but the dispersion relation (2.4) as written in the comoving
reference frame with the background. Equation (3.32) as in general four
distinct complex solutions, so we find four independent solutions in the form
(3.31): the spectral type of the equation at $z=\infty$ is thus $(1111)$.
There is the possibility of emergence of a resonant case, where the difference
between eigenvalues is an integer, still in a zero measure set in the space of
available parameters appearing in our model.

A similar behaviour is found at $z=0$: we find four independent local
solutions of the form

$\displaystyle f(z)=z^{i\tilde{\alpha}}\sum_{n=0}^{\infty}c_{n}z^{n}\,,$
(3.33)

where $\tilde{k}:=2\tilde{\beta}\tilde{\alpha}$ satisfies

$\displaystyle\text{DR}_{0}(k)$
$\displaystyle:=\gamma^{2}(-g^{2}(\tilde{k}+V\omega)^{2}+(\tilde{k}V+\omega)^{2}(\lambda+\mu^{2}+(\tilde{k}V+\omega)^{2}\gamma^{2}))=0\,.$
(3.34)

Eq. (3.34) is equivalent to (3.32) if one maps
$\mu^{2}\mapsto\mu^{2}+\lambda$. The spectral type at $z=0$ is again $(1111)$,
again almost everywhere in the space of available parameters appearing in our
model. Some interesting formulas concerning (3.32) and (3.34) are discussed in
Appendix A.

The situation at $z=1$ is different. After defining $y=z-1$, the
characteristic equation for solutions of the form

$\displaystyle f(y)=y^{a}\sum_{n=0}^{\infty}c_{n}y^{n}$ (3.35)

has four integer solutions $a=0,1,2,3$. This situation is known in literature
as the resonant case (cf. e.g. [41]), and requires a particular study. We
refer mostly to [42], where still the discussion is left incomplete, and,
particularly, to the thoroughful analysis appearing in [43], and also to [44].
As suggested in the aforementioned literature, we apply the so-called
Frobenius method for the analysis of solutions at a Fuchsian singularity, also
in the resonant case, and we can also verify if there are logarithmic
contributions (even in the resonant case, they might also not appear).

By means of the Frobenius method, we obtain three independent integer
solutions

$\displaystyle u_{1}(y)$
$\displaystyle=y^{3}+y^{4}\left[-\frac{3}{2}+\frac{\Lambda}{12\gamma^{2}V^{2}}+i\frac{\Omega}{V}\right]+o(y^{4})$
(3.36) $\displaystyle u_{2}(y)$
$\displaystyle=y^{2}+y^{3}\left[-2+\frac{\Lambda}{6\gamma^{2}V^{2}}+i\frac{4\Omega}{3V}\right]+o(y^{3})$
(3.37) $\displaystyle u_{3}(y)$
$\displaystyle=y\;+y^{2}\left[-3+\frac{\Lambda}{2\gamma^{2}V^{2}}+i\frac{2\Omega}{V}\right]+o(y^{2})\,,$
(3.38)

and one logarithmic solution

$\displaystyle
u_{0}(y)=1+y\left[-6-\frac{\Lambda}{V^{2}\gamma^{2}}+\frac{4i\Omega}{V}\right]+o(y)$
(3.39)
$\displaystyle+\log(y)\Big{(}R_{1}u_{1}(y)+R_{2}u_{2}(y)+R_{3}u_{3}(y)\Big{)}\,,$
(3.40)

where

$\displaystyle R_{3}$ $\displaystyle=\frac{\Lambda}{V^{2}\gamma^{2}}$ (3.41)
$\displaystyle R_{2}$
$\displaystyle=\frac{\Lambda\left(5V-2i\Omega\right)}{4V^{3}\gamma^{2}}$
(3.42) $\displaystyle R_{1}$ $\displaystyle=\frac{\Lambda\left(9V^{2}-6i\Omega
V-\Omega^{2}\right)}{18V^{4}\gamma^{2}}\,.$ (3.43)

The study of the monodromy of the solution is important for the
characterization of the equation [41]. Starting from a basis of solutions
$(u_{1}(\bar{z}),...,u_{4}(\bar{z}))$ evaluated at some
$\bar{z}\in\mathbb{C}$, we can prolong these solutions along a path that goes
around a singular point $a\in\mathbb{C}$ and closes back to $\bar{z}$ (without
enclosing other singular points): the new vector
$(u^{\prime}_{1}(\bar{z}),...,u^{\prime}_{4}(\bar{z}))$ that results from this
transformation is related to the initial one by a matrix $M_{a}$. Such matrix
is independent on the point $\bar{z}$ and is called the _monodromy_ matrix of
the solutions $(u_{1},...,u_{4})$ at the point $a$. The monodromy matrix of
the solutions $(u_{0}(y),u_{1}(y),u_{2}(y),u_{3}(y))$ of (3.36)-(3.39) at
$z=1$ is easily computed as

$\displaystyle M_{1}=\left(\begin{array}[]{cccc}1&2\pi iR_{1}&2\pi iR_{2}&2\pi
iR_{3}\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)\,,$ (3.48)

whose Jordan form is

$\displaystyle J_{M_{1}}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&1\\\ 0&0&0&1\\\ \end{array}\right)\,.$ (3.53)

The monodromy at $z=0$ and $z=\infty$ are even more easy to determine and they
are represented respectively by the diagonal matrices

$\displaystyle M_{0}$
$\displaystyle=\text{diag}\left(e^{i2\pi\tilde{\alpha}_{1}},e^{i2\pi\tilde{\alpha}_{2}},e^{i2\pi\tilde{\alpha}_{3}},e^{i2\pi\tilde{\alpha}_{4}}\right)\,,$
(3.54) $\displaystyle M_{\infty}$
$\displaystyle=\text{diag}\left(e^{-i2\pi\alpha_{1}},e^{-i2\pi\alpha_{2}},e^{-i2\pi\alpha_{3}},e^{-i2\pi\alpha_{4}}\right)\,,$
(3.55)

whose Jordan form is $J_{M_{0}}=J_{M_{\infty}}=\mathbb{I}_{4}$. From the
Jordan form we can infer that the spectral type of the equation at $z=1$ is
$(3,1)$ (see [41]). The spectral type of the equation is thus
$((1111),(31),(1111))$: this spectral type is classified as _rigid_. Without
entering into mathematical details, an equation is called rigid if the local
monodromy class of its solutions uniquely determines also the _global_
monodromy class. Another way of expressing the same concept is that the
equation only depends on its _local data_ (_i.e._ the characteristic
exponents) and there are no _accessory parameters_. We can also calculate the
so-called _index of rigidity_ [41]:

$\displaystyle\iota=\sum_{j=0}^{p}\text{dim}Z(M_{j})-(p-1)n^{2}\,,$ (3.56)

where $Z(M_{j})$ is the centralizer of the matrix $M_{j}$ (_i.e._ the
dimension of the vector space of matrices that commute with $M_{j}$) and $p+1$
is the number of distinct singular points of the equation. A known result says
that a Fuchsian system is rigid if and only if $\iota=2$. In our case, a
simple computation leads to

$\displaystyle Z(M_{0})+Z(M_{1})+Z(M_{\infty})-(4)^{2}=4+10+4-16=2\,.$ (3.57)

Rigid equations have thus a simple structure and there are many results
available for their characterization and solution. The rigidity of the
equation allows in principle to find integral representations of the solutions
and write exact expressions of connection coefficients for the local solutions
at the different singular points: for the physical problem of the scattering.
Another consequence of rigidity is that the local monodromy classes uniquely
determine the global monodromy: this is interesting for physics, since the
action of the global monodromy can be interpreted as the result of the
scattering of a wave, so the scattering coefficients may be derived from the
monodromy matrices [45, 46].

###  3.2. Gauge Transformation and Riemann scheme

We start again from Eq. (3.3) and we perform a so-called gauge transformation
in order to put to zero one of the characteristic exponents. We look for a
solution of the form

$\displaystyle f(z):=z^{i\frac{\tilde{k}_{1}}{2\tilde{\beta}}}(z-1)\,u(z)\,,$
(3.58)

where $\tilde{k}_{1}$ satisfies

$\text{DR}_{0}(\tilde{k}_{1})=0\,.$

The exponent of $(z-1)$ was chosen to lower the order of the singularities at
$z=1$. The function $u(z)$ now satisfies

$\displaystyle u(z)$
$\displaystyle\Big{[}\gamma^{2}\left(G^{2}(\tilde{k}_{1}+V\Omega)^{2}-(\tilde{k}_{1}V+\Omega)^{2}\left(\gamma^{2}(\tilde{k}_{1}V+\Omega)^{2}+\Lambda+M^{2}\right)\right)$
(3.59)
$\displaystyle+\gamma^{2}z\left(-G^{2}(\tilde{k}_{1}+V\Omega-i)^{2}+M^{2}(\Omega+(\tilde{k}_{1}-i)V)^{2}+\gamma^{2}(\Omega+(\tilde{k}_{1}-i)V)^{4}\right)\Big{]}+$
(3.60) $\displaystyle u^{\prime}(z)$
$\displaystyle\Big{[}\gamma^{2}z\Big{(}V(2ik_{1}V+V+2i\Omega)\left(-i\gamma^{2}((1+i)k_{1}V-iV+(1+i)\Omega)((1+i)k_{1}V+V+(1+i)\Omega)+L+M^{2}\right)$
(3.61) $\displaystyle\qquad-iG^{2}(2k_{1}+2V\Omega-i)\Big{)}$ (3.62)
$\displaystyle+\gamma^{2}z^{2}\Big{(}-iV(2\Omega+(2k_{1}-3i)V)\left(M^{2}+\gamma^{2}\left((-5+2k_{1}(k_{1}-3i))V^{2}+2(2k_{1}-3i)V\Omega+2\Omega^{2}\right)\right)$
(3.63) $\displaystyle\qquad 2iG^{2}k_{1}+2iG^{2}V\Omega+3G^{2}\Big{)}\Big{]}+$
(3.64) $\displaystyle u^{\prime\prime}(z)$
$\displaystyle\Big{[}\gamma^{2}z^{2}\left(-G^{2}+6\tilde{k}_{1}^{2}V^{4}\gamma^{2}+12\tilde{k}_{1}V^{3}\Omega\gamma^{2}-12i\tilde{k}_{1}V^{4}\gamma^{2}+\Lambda
V^{2}+M^{2}V^{2}+6V^{2}\Omega^{2}\gamma^{2}-12iV^{3}\Omega\gamma^{2}-7V^{4}\gamma^{2}\right)$
(3.65)
$\displaystyle+\gamma^{2}z^{3}\left(G^{2}-6\tilde{k}_{1}^{2}V^{4}\gamma^{2}-12(\tilde{k}_{1}-2i)V^{3}\Omega\gamma^{2}+24i\tilde{k}_{1}V^{4}\gamma^{2}-M^{2}V^{2}-6V^{2}\Omega^{2}\gamma^{2}+25V^{4}\gamma^{2}\right)\Big{]}+$
(3.66) $\displaystyle u^{(3)}(z)$
$\displaystyle\Big{[}\gamma^{2}z^{4}\left(4i\tilde{k}_{1}V^{4}\gamma^{2}+4iV^{3}\Omega\gamma^{2}+10V^{4}\gamma^{2}\right)+\gamma^{2}z^{3}\left(-4i\tilde{k}_{1}V^{4}\gamma^{2}-4iV^{3}\Omega\gamma^{2}-6V^{4}\gamma^{2}\right)\Big{]}+$
(3.67) $\displaystyle u^{(4)}(z)$ $\displaystyle
z^{4}(z-1)V^{4}\gamma^{4}=0\,.$ (3.68)

The last equation can be written in a more convenient form using (3.32) and
(3.34):

$\displaystyle u(z)$
$\displaystyle\big{[}-\text{DR}_{0}(\tilde{k}_{1})+z\text{DR}(\tilde{k}_{1}-i)\big{]}+$
(3.69) $\displaystyle u^{\prime}(z)$
$\displaystyle\Big{[}z(\text{DR}_{0}(\tilde{k}_{1})-\text{DR}_{0}(\tilde{k}_{1}-i))-z^{2}(\text{DR}(\tilde{k}_{1}-i)-\text{DR}(\tilde{k}_{1}-2i))\Big{]}+$
(3.70) $\displaystyle u^{\prime\prime}(z)$
$\displaystyle\Big{[}\frac{1}{2}z^{3}(\text{DR}(\tilde{k}_{1}-i)-2\text{DR}(\tilde{k}_{1}-2i)+\text{DR}(\tilde{k}_{1}-3i))$
(3.71)
$\displaystyle-\frac{1}{2}z^{2}(\text{DR}_{0}(\tilde{k}_{1})-2\text{DR}_{0}(\tilde{k}_{1}-i)+\text{DR}_{0}(\tilde{k}_{1}-2i))\Big{]}+$
(3.72) $\displaystyle u^{(3)}(z)$
$\displaystyle\Big{[}\frac{1}{6}z^{3}(\text{DR}_{0}(\tilde{k}_{1})-3\text{DR}_{0}(\tilde{k}_{1}-i)+3\text{DR}_{0}(\tilde{k}_{1}-2i)-\text{DR}_{0}(\tilde{k}_{1}-3i))$
(3.73)
$\displaystyle-\frac{1}{6}z^{4}(\text{DR}(\tilde{k}_{1}-i)-3\text{DR}(\tilde{k}_{1}-2i)+3\text{DR}(\tilde{k}_{1}-3i)-\text{DR}(\tilde{k}_{1}-4i))\Big{]}+$
(3.74) $\displaystyle u^{(4)}(z)$
$\displaystyle\Big{[}\frac{1}{24}z^{5}(\text{DR}(\tilde{k}_{1}-i)-4\text{DR}(\tilde{k}_{1}-2i)+6\text{DR}(\tilde{k}_{1}-3i)-4\text{DR}(\tilde{k}_{1}-4i)+\text{DR}(\tilde{k}_{1}-5i))$
(3.75) $\displaystyle-\frac{1}{24}$ $\displaystyle
z^{4}(\text{DR}_{0}(\tilde{k}_{1})-4\text{DR}_{0}(\tilde{k}_{1}-i)+6\text{DR}_{0}(\tilde{k}_{1}-2i)-4\text{DR}_{0}(\tilde{k}_{1}-3i)+\text{DR}_{0}(\tilde{k}_{1}-4i))\Big{]}=0\,.$
(3.76)

where $\text{DR}_{0}(\tilde{k}_{j})=0$ and $\text{DR}(k_{j})=0$.

It is easy to verify, by studying the local solutions as in Section 3.1, that
the characteristic exponents of Eq. (3.69) are

$\displaystyle\left[\begin{matrix}z=0&z=1&z=\infty\\\
0&0&1-\frac{i}{2\tilde{\beta}}(k_{1}-\tilde{k}_{1})\\\
\frac{i}{2\tilde{\beta}}(\tilde{k}_{2}-\tilde{k}_{1})&1&1-\frac{i}{2\tilde{\beta}}(k_{2}-\tilde{k}_{1})\\\
\frac{i}{2\tilde{\beta}}(\tilde{k}_{3}-\tilde{k}_{1})&2&1-\frac{i}{2\tilde{\beta}}(k_{3}-\tilde{k}_{1})\\\
\frac{i}{2\tilde{\beta}}(\tilde{k}_{4}-\tilde{k}_{1})&-1&1-\frac{i}{2\tilde{\beta}}(k_{4}-\tilde{k}_{1})\end{matrix}\right]\,.$
(3.78)

Eq. (3.78) is the so-called Riemann Scheme of the equation111More precisely,
the Riemann P-Scheme is usually written as
$\displaystyle\mathcal{P}\left\\{\begin{matrix}w^{\prime}&w^{\prime\prime}&w^{\prime\prime\prime}\\\
a_{1}&b_{1}&c_{1}\\\ a_{2}&b_{2}&c_{2}\\\ a_{3}&b_{3}&c_{3}\\\
a_{4}&b_{4}&c_{4}\end{matrix}\ ;z\right\\}\,,$ (3.79) and indicates
independently the equations and the solutions. It is often also called
Papperitz symbol, but it has been introduced by Riemann first.. By defining

$\displaystyle\alpha_{i}$
$\displaystyle:=1-\frac{i}{2\tilde{\beta}}(k_{i}-\tilde{k}_{1})\,,\quad
i=1,2,3,4$ (3.80) $\displaystyle\beta_{j}$
$\displaystyle:=1-\frac{i}{2\tilde{\beta}}(\tilde{k}_{j+1}-\tilde{k}_{1})\,,\quad
j=1,2,3\,,$ (3.81)

we can write Eq. (3.78) as

$\displaystyle\left[\begin{matrix}z=0&z=1&z=\infty\\\ 0&0&\alpha_{1}\\\
1-\beta_{1}&1&\alpha_{2}\\\ 1-\beta_{2}&2&\alpha_{3}\\\
1-\beta_{3}&-\beta_{4}&\alpha_{4}\end{matrix}\right]\,.$ (3.82)

which corresponds to the Riemann scheme of the hypergeometric function

${}_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$

in the standard form [47]. The exponent $\beta_{4}$ in the hypergeometric
function is defined by $\sum_{i=0}^{4}\alpha_{i}=\sum_{i=0}^{4}\beta_{i}$, and
is indeed equal to 1. Therefore, the spectral type and the Riemann scheme of
our fourth order equation coincide with those of the hypergeometric function
${}_{4}F_{3}$. Since the system is rigid, Eq. (3.59) has to be equivalent to
the hypergeometric equation ${}_{4}E_{3}$ [48], and ${}_{4}F_{3}$ has to be a
solution as we are now going to see.

##  4\. The exact solution: hypergeometric $\,{}_{4}F_{3}$, Stokes phenomenon
and connection formulas

We look for a locally holomorphic solution of Eq. (3.69) around $z=0$

$\displaystyle u(z)=1+c_{1}z+c_{2}z^{2}+...$ (4.1)

We are going to prove the following proposition that gives the explicit
expression of the coefficients $c_{n}$:

######  Proposition 1.

Given any two fourth order polynomials $\text{DR}(k)$ and $\text{DR}_{0}(k)$,
let $\tilde{k}_{1}$ be one of the roots of $\text{DR}_{0}$. Let $u(z)$ be a
meromorphic function which solves Eq. (3.69) and suppose that $u(z)$ is
locally holomorphic around $z=0$. Then, the general term of the series
expansion (4.1) satisfies

$\displaystyle
c_{n}=\frac{\prod_{r=1}^{n}\text{DR}(\tilde{k}_{1}-r\,i)}{\prod_{s=1}^{n}\text{DR}_{0}(\tilde{k}_{1}-s\,i)}\,.$
(4.2)

###### Proof.

See Appendix B.

∎

Using the definitions (3.80)-(3.81) and writing the dispersion relations in
terms of their roots as in (A.1) and (A.2), we easily find

$\displaystyle
c_{n}=\frac{\prod_{i=1}^{4}\alpha_{i}(\alpha_{i}+1)...(\alpha_{i}+n)}{n!\prod_{j=1}^{3}\beta_{j}(\beta_{j}+1)...(\beta_{j}+n)}\,.$
(4.3)

This is precisely the general term of the hypergeometric function
$\,{}_{4}F_{3}$. So we can say that

$\displaystyle
u(z)=\,_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$
(4.4)

is an exact solution of (3.59), and

$\displaystyle
f(z)=z^{i\frac{\tilde{k}_{1}}{2\tilde{\beta}}}(z-1)\,_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$
(4.5)

is a solution of (3.3). Solving the scattering problems now just amounts to
writing the connection coefficients of the hypergeometric function between
$z=0$ and $z=\infty$: for example, the connection coefficient
$\tilde{k}_{1}\rightarrow k_{1}$ is

$\displaystyle C_{\tilde{k}_{1}\rightarrow
k_{1}}=\frac{\Gamma(\beta_{1})\Gamma(\beta_{2})\Gamma(\beta_{3})\Gamma(\alpha_{2}-\alpha_{1})\Gamma(\alpha_{3}-\alpha_{1})\Gamma(\alpha_{4}-\alpha_{1})}{\Gamma(\alpha_{2})\Gamma(\alpha_{3})\Gamma(\alpha_{4})\Gamma(\beta_{1}-\alpha_{1})\Gamma(\beta_{2}-\alpha_{1})\Gamma(\beta_{3}-\alpha_{1})}\,.$
(4.6)

Notice that in the generic case (excluding resonances) a basis of solution is
automatically obtained replacing $\tilde{k}_{1}$ (and $k_{1}$) with any of the
$\tilde{k}_{j}$ (and $k_{j}$). Indeed, the equation is invariant under
permutation of the $j$s. To be more explicit, we have the following basis of
solutions:

$\left(f_{1}(z),f_{2}(z),f_{3}(z),f_{4}(z)\right),$ (4.7)

where $f_{j}(z)$, with $j>1$, are just obtained from (4.5) by replacing
$\tilde{k}_{1}$ (and $k_{1}$) with any of the $\tilde{k}_{j}$ (and $k_{j}$).
As a consequence, we also obtain the general solution of our equation of
motion as follows:

$F(z)=\sum_{i=1}^{4}D_{i}f_{i}(z),$ (4.8)

where the constants $D_{i}$ have to be fixed according to the scattering
process one is considering. It is remarkable that the basis is already
diagonal in the $\tilde{k}_{i}$, in the sense that the physical modes on the
left side (corresponding to $x\to-\infty$, see also the following subsection)
are asymptotically represented by just the element of the basis with index
$j$: $f_{j}(x)\propto e^{i\tilde{k}_{j}x}$ as $x\to-\infty$.
Some physical considerations are mandatory. The aforementioned connection
coefficients are responsible of the phenomenon of mode conversion in the
scattering process, i.e. they show that, from passing from the left, i.e. at
$x=-\infty$, with input mode $\tilde{k}_{1}$, to the right, i.e. $x=\infty$,
with potential output modes $k_{j}$, $j=1,2,3,4$, the S-matrix is not, in
general, diagonal, as output modes with $j\not=1$ are allowed. In making this
possible, a fundamental role is played by the Stokes phenomenon, which is
discussed in the following subsection. It is to be stressed the following
point: the Stokes phenomenon is present when at least an irregular singularity
appears (see e.g. [41]). In the present case, the equation with $z$ as
independent variable displays three Fuchsian singularities, as seen, i.e.
three regular singular points $z=0,z=1,z=\infty$. Still, by coming back to the
original variable $x$, which is the relevant one for the physical problem, one
finds that, actually, $x=\pm\infty$ on the real axis correspond to irregular
singularities, as essential singularities in $\tanh(\beta x)$ and in
$\cosh^{-2}(\beta x)$ appear in the coefficients of the equation itself. This
fact is at the root of the Stokes phenomenon in the physical problem at hand.

###  4.1. Integral representation and Stokes phenomenon

By using the integral representation of the Hypergeometric function and
changing variable back to $x$, we can write the selected solution of the EOM
as

$\displaystyle f(x)=$
$\displaystyle\frac{\Gamma(\beta_{1})\Gamma(\beta_{2})\Gamma(\beta_{3})}{2\pi
i\Gamma(\alpha_{1})\Gamma(\alpha_{2})\Gamma(\alpha_{3})\Gamma(\alpha_{4})}e^{i\tilde{k}_{1}x}(1+e^{2\tilde{\beta}x})\times$
(4.9)
$\displaystyle\int_{\gamma-i\infty}^{\gamma+i\infty}ds\frac{\Gamma(s)\Gamma(\alpha_{1}-s)\Gamma(\alpha_{2}-s)\Gamma(\alpha_{3}-s)\Gamma(\alpha_{4}-s)}{\Gamma(\beta_{1}-s)\Gamma(\beta_{2}-s)\Gamma(\beta_{3}-s)}(-1)^{-s}e^{-2\tilde{\beta}xs}\,,$
(4.10)

with $0<\gamma<1$. The integrand function has simple poles in the $s$-plane
that are disposed on five lines parallel to the real axis. The poles are found
at

$\displaystyle s$ $\displaystyle=\tilde{s}_{n}:=-n\,,$ $\displaystyle s$
$\displaystyle=s_{1,n}:=\alpha_{1}+n\,,$ $\displaystyle s$
$\displaystyle=s_{2,n}:=\alpha_{2}+n\,,$ $\displaystyle s$
$\displaystyle=s_{3,n}:=\alpha_{3}+n\,,$ $\displaystyle s$
$\displaystyle=s_{4,n}:=\alpha_{4}+n\,,$

with $n=0,1,2,3...$. The poles are represented in Figure 2, where the relative
position of the poles is fixed by the following identification of the modes
(see Figure 1):

$\displaystyle``1"=H\,,\quad``2"=B\,,\quad``3"=P\,,\quad``4"=N\,.$ (4.11)

Figure 2. The poles of the integrand function of Eq.(4.9). The grey line
represents the path of integration. This figure holds true just when $k_{j}$
and $\tilde{k}_{j}$ are real for all $j=1,2,3,4$, i.e. in the subcritical
case.

We can analytically continue the function $f$ in the complex $x$-plane, in
order to study the behaviour for $x\rightarrow\infty e^{i\theta}$ for
different angles $\theta$. By writing

$\displaystyle x=|x|e^{i\theta}\,,\quad s=|s|e^{i\phi}\,,$

we see that the integral is convergent only in the half plane defined by

$\displaystyle\cos\theta\cos\phi-\sin\theta\sin\phi\geq 0\,,$

which defines a half-plane delimited by the line $\tan\phi=\cot\theta$.
Namely, as the angle $\theta$ varies, the corresponding half-plane in the
variable $s$ is defined by
$\phi\in\left[\theta-\frac{\pi}{2},\theta+\frac{\pi}{2}\right]$.

Starting from $\theta=\pi$, which represents the solution at $x<0$ (inside the
horizon), the integral is defined for
$\phi\in\left[\frac{\pi}{2},\frac{3}{2}\pi\right]$: in this sector, the
integral reduces to the sum of the residues at $s=\tilde{s}_{n}$. The sum of
the residues gives the series

$\displaystyle
f(x<0)=e^{i\tilde{k}_{1}x}(1+e^{2\tilde{\beta}x})\left(1+\frac{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}{\beta_{1}\beta_{2}\beta_{3}}e^{2\tilde{\beta}x}+\sum_{n\geq
2}O((e^{2\tilde{\beta}x})^{n})\right)\stackrel{{\scriptstyle
x\rightarrow-\infty}}{{\sim}}e^{i\tilde{k}_{1}x}\,.$ (4.12)

As we move $\theta$, the asymptotic expansion (4.12) remains valid until we
encounter new poles in the corresponding half-plane in $s$: this happens, as
one can see from Figure 3, when $\phi=\arg\alpha_{3}$ or
$\phi=\arg\alpha_{4}$. As we pass those lines, a new term appears in the
asymptotic expansion, corresponding to the residue at the pole
$s_{3,0}=\alpha_{3}$ or $s_{4,0}=\alpha_{4}$. The appearance of new terms in
the asymptotic expansion is known as Stokes Phenomenon: by the previous
analysis, thus, we identified a first Stokes sector, given by

$\displaystyle\theta\in\left[\frac{\pi}{2}+\arg\alpha_{4},\frac{3}{2}\pi+\arg\alpha_{3}\right]\,,$

and the boundaries of this sector are two Stokes lines.

Figure 3. Three contours of integration representing three values of $\theta$.
Full line: $\theta=\pi$; dashed line: $\theta=\frac{3}{2}\pi+\arg\alpha_{3}$;
dashed-dotted line: $\theta=\frac{\pi}{2}+\arg\alpha_{4}$. The latest two are
Stokes lines, because the corresponding contour in the $s$-plane (defined by
the $\phi$-angle) includes a new pole, giving rise to an additional term in
the asymptotic expansion. Also this figure holds true just for the subcritical
case.

As we move $\theta$ past the Stokes line
$\theta=\frac{3}{2}\pi+\arg\alpha_{3}$, we include a new pole, $s_{3,0}$, in
the contour: the asymptotic expansion becomes

$\displaystyle
f(x)=e^{i\tilde{k}_{1}x}\left(1+O(e^{-2\tilde{\beta}|x|})\right)+C_{\tilde{k}_{1}\rightarrow
k_{3}}e^{ik_{3}x}\,,$

so the new term introduces mode-mixing. This expansion is true until we reach
the next pole.
Now, we note that the residues at the poles $s_{j,n}$ are

$\text{Res}_{s=s_{j,n}}\sim
e^{(i\tilde{k}_{1}+2\tilde{\beta}-2\tilde{\beta}s_{j,n})x}=e^{ik_{j}x}e^{-nx}\,.$

Therefore, the contributions of the poles with $n\geq 1$ are negligible as
long as we are interested in the asymptotic expansion
($|x|\rightarrow\infty$). From this consideration we understand that the only
poles that are related to the Stokes phenomenon are $s=s_{j,0}$. The next
Stokes lines are thus met at $\theta=\frac{3}{2}\pi+\arg\alpha_{1}$ or
$\theta=\frac{\pi}{2}+\arg\alpha_{2}$.

It is now easy to figure out, by continuing the argument exposed above, all
the Stokes lines of the function $f(x)$, which, ordered by increasing
$\theta$, correspond to

$\displaystyle\theta_{1}$
$\displaystyle=\frac{\pi}{2}+\arg\alpha_{3},\quad\theta_{2}=\frac{\pi}{2}+\arg\alpha_{1},\quad\theta_{3}=\frac{\pi}{2},\quad\theta_{4}=\frac{\pi}{2}+\arg\alpha_{2},\quad\theta_{5}=\frac{\pi}{2}+\arg\alpha_{4},$
$\displaystyle\theta_{6}$
$\displaystyle=\frac{3}{2}\pi+\arg\alpha_{3},\quad\theta_{7}=\frac{3}{2}\pi+\arg\alpha_{1},\quad\theta_{8}=\frac{3}{2}\pi\,\quad\theta_{9}=\frac{3}{2}\pi+\arg\alpha_{2},\quad\theta_{10}=\frac{3}{2}\pi+\arg\alpha_{4}\,.$

The value of $\arg\alpha_{j}$ depends on the values of the momenta $k_{j}$ and
$\tilde{k}_{1}$. The momenta $k_{j}$, being unperturbed by the background, are
always real. On the other hand, as we will discuss also in Section 4.3,
$\tilde{k}_{1}$ is real in subcritical regime and complex in transcritical
regime: in that case we have $\text{Im}\tilde{k}_{1}<0$. We can thus evaluate
$\arg\alpha_{j}$ as

$\displaystyle\arg\alpha_{j}=\begin{cases*}\arctan\left(-\frac{k_{j}-\tilde{k}_{1}}{2\tilde{\beta}}\right)\,,\quad\text{subcritical
regime}\\\
\arctan\left(-\frac{k_{j}-\text{Re}\tilde{k}_{1}}{2\tilde{\beta}-\text{Im}\tilde{k}_{1}}\right)\,,\quad\text{transcritical
regime}\,.\end{cases*}$

###  4.2. Subcritical scattering

The solution (4.5), for $z\rightarrow 0$ (which corresponds to the left
asymptotic region $x\rightarrow-\infty$) is

$\displaystyle f\sim
z^{i\frac{\tilde{k}_{1}}{2\tilde{\beta}}}=e^{i\tilde{k}_{1}x}\,.$ (4.13)

At right infinity $x\rightarrow+\infty$ ($z\rightarrow\infty$) it splits into
a sum of plane waves

$\displaystyle
f\sim\sum_{j=1}^{4}C_{j}z^{i\frac{ik_{j}}{2\tilde{\beta}}}=\sum_{j=1}^{4}C_{j}e^{ik_{j}x}\,,$
(4.14)

where the connection coefficient

$C_{j}:=C_{\tilde{k}_{1}\rightarrow k_{j}}$

can be obtained from (4.6) by switching $\alpha_{1}\leftrightarrow\alpha_{j}$.
If we put

$\tilde{k}_{1}=\tilde{k}_{H}\,.$

the function $f(x)$ represents the scattering of the in-going modes $H$ from
left infinity and $P$, $N$ and $B$ from right infinity, which produces an out-
going $H$ mode at right infinity. This in the process that, since Hawking’s
seminal work, is usually considered in black hole physics to deduce the
spontaneous particle creation. Following backwards the out-going $H$ mode, we
find that it originates from a mixture of modes: in particular, the
coefficient $C_{N}$ represents the mixing with the negative-norm $N$ mode. As
it is shown in [39], the expected number of spontaneously created Hawking
particles is

$\displaystyle|N|=\left|\frac{|C_{N}|^{2}v_{N}\partial_{\omega}\text{DR}(\omega,k)|_{k_{N}}}{|C_{H}|^{2}v_{H}\partial_{\omega}\text{DR}(\omega,k)|_{k_{H}}}\right|\,,$
(4.15)

Notice that the function $\text{DR}(k)$ defined in (3.32) is slightly
different from the function $\text{DR}(\omega,k)$ that appears in (4.15),
which derives from the normalization of the quantum theory: they differ by a
global factor $(\omega+Vk)^{2}$.

(a)

(b)

Figure 4.  (A): The dispersion relation (3.34) in the perturbed region
($x\rightarrow-\infty$) for $\lambda<\lambda_{\text{crit}}$. The modes
$0<\omega<\omega_{MIN}$ do not experience an event horizon (_subcritical
regime_). (B): The dispersion relation (3.34) for
$\lambda>\lambda_{\text{crit}}$. In this case, for any $\omega$, the modes $H$
and $P$ become imaginary as they experience an event horizon: this is referred
to as _transcritical regime_.

The momenta $\tilde{k}_{j}(\omega)$ (which correspond to the normal modes at
$x\rightarrow-\infty$) depend on the background amplitude $\lambda$. For large
enough values of $\lambda$, the momenta $\tilde{k}_{H}(\omega)$ and
$\tilde{k}_{P}(\omega)$ can become imaginary, as can be seen from Figures 4(a)
and 4(b). In the current literature, the distinction between the subcritcal
and transcritical regimes is governed by the presence or absence of a horizon,
corresponding to a turning point in the differential equation of motion: the
subcritical case is when no such turning point appears. On the other hand, the
two regimes can be characterized also in a different way: one identifies the
transcritical case by the fact that, in the asymptotic dispersion relation,
two roots which are real in the unperturbed asymptotic region, become complex
conjugates in the pertubed region. This is exactly the criterion we adopt in
our approach to the problem. See also the discussion in subsection 5.5.

We start considering the case where all $\tilde{k}_{j}$ are real (_i.e._
subcritical regime). In this case, the square module of (4.6) can be written
explicitly:

$\displaystyle|C_{1}|^{2}=$
$\displaystyle\frac{(\tilde{k}_{1}-\tilde{k}_{2})(\tilde{k}_{1}-\tilde{k}_{3})(\tilde{k}_{1}-\tilde{k}_{4})(\tilde{k}_{2}-k_{1})(k_{1}-\tilde{k}_{3})(k_{1}-\tilde{k}_{4})}{(\tilde{k}_{1}-k_{2})(\tilde{k}_{1}-k_{3})(\tilde{k}_{1}-k_{4})(k_{1}-k_{2})(k_{1}-k_{3})(k_{1}-k_{4})}$
(4.16)
$\displaystyle\times\frac{\sinh(\frac{\pi\gamma(\tilde{k}_{1}-k_{2})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{1}-k_{3})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{1}-k_{4})}{2\beta})}{\sinh(\frac{\pi\gamma(\tilde{k}_{1}-\tilde{k}_{2})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{1}-\tilde{k}_{3})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{1}-\tilde{k}_{4})}{2\beta})}$
(4.17)
$\displaystyle\times\frac{\sinh(\frac{\pi\gamma(\tilde{k}_{2}-k_{1})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{3}-k_{1})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{4}-k_{1})}{2\beta})}{\sinh(\frac{\pi\gamma(k_{1}-k_{2})}{2\beta})\sinh(\frac{\pi\gamma(k_{1}-k_{3})}{2\beta})\sinh(\pi(k_{1}-k_{4}))}\,,$
(4.18)

and similarly $|C_{j}|^{2}$ are obtained by rotations of the indices. We want
to compare this results with the perturbative expansion we made in [39]. We
start by writing explicitly

$\displaystyle|N|=$
$\displaystyle\frac{(\tilde{k}_{H}-k_{N})(k_{N}-\tilde{k}_{P})(k_{N}-\tilde{k}_{N})(k_{N}-\tilde{k}_{B})(\omega+k_{H}V)^{2}}{(\tilde{k}_{H}-k_{H})(k_{H}-\tilde{k}_{P})(k_{H}-\tilde{k}_{N})(k_{H}-\tilde{k}_{B})(\omega+k_{N}V)^{2}}$
(4.19)
$\displaystyle\times\frac{\sinh\left(\frac{\pi\gamma(\tilde{k}_{H}-k_{H})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{P}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{N}-k_{N})}{2\beta}\right)}{\sinh\left(\frac{\pi\gamma(\tilde{k}_{H}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{P}-k_{H})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{N}-k_{H})}{2\beta}\right)}$
(4.20)
$\displaystyle\times\frac{\sinh\left(\frac{\pi\gamma(\tilde{k}_{B}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{H}-k_{P})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{H}-k_{B})}{2\beta}\right)}{\sinh\left(\frac{\pi\gamma(\tilde{k}_{B}-k_{H})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{P}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{N}-k_{B})}{2\beta}\right)}\,,$
(4.21)

where we have used the expression for the flux factors

$\displaystyle
v_{i}(\omega)\partial_{\omega}\text{DR}|_{k_{i}}=-\frac{\gamma^{2}V^{4}}{\omega_{lab}^{2}|_{k_{i}}}\prod_{j\neq
i}(k_{i}(\omega)-k_{j}(\omega))\,,$ (4.22)

that was deduced in the Appendix of [39]. We use the low-frequency expressions
of the momenta $k_{j}(\omega)$, written in (2.5)-(2.8). Notice that the
expressions of $\tilde{k}_{j}(\omega)$ are simply obtained by the shift
$\mu^{2}\mapsto\mu^{2}+\lambda$.
We stress that, up to this point, our results are exact, in the sense that no
approximation has been made. Still, in order to provide analytical expressions
to the moments $k_{j},\tilde{k}_{j}$, we are forced to introduce some
approximations, indeed $k_{j},\tilde{k}_{j}$ are roots of a fourth degree
equation: one might provide explicit expressions for the corresponding roots,
but at the price to write very long and by no means perspicuous expressions.
As a consequence, for these roots we use approximate expressions for low
$\omega$, as discussed in the previous sections. With the help of Wolfram
Mathematica, we compute the leading order of $|N|$ both in $\lambda$ and in
$\omega$: we checked that the two limit commute, so the order of the two
expansions makes no difference. We obtain

$\displaystyle|N|=\left(\frac{\pi^{2}\lambda^{2}\omega g(g+\mu
V)\sinh^{2}\left(\frac{\pi\sqrt{g^{2}-\mu^{2}V^{2}}}{2\beta
V^{2}}\right)}{16\beta^{2}\gamma\mu(g-\mu
V)\left(g^{2}-\mu^{2}V^{2}\right)^{3/2}}+O(\omega^{2})\right)+O(\lambda^{3})\,.$
(4.23)

We notice that the qualitative behaviour is the same as in [39], and in
particular we have $N\sim\omega$: this behaviour confirms what was found for
the subcritical case also in [20]. This is a strong confirmation that such a
behaviour should be expected in subcritical systems, and it seem not to depend
neither on particular approximations, nor on the characteristics of the
background function.

An even more interesting comparison is the estimation of the “effective
temperature” that the authors found in [39] for the subcritical case. The
ratio $\frac{|P|}{|N|}$, in the case $\tilde{k}_{j}\in\mathbb{R}$ as before,
becomes

$\displaystyle\frac{|P|}{|N|}=$
$\displaystyle\frac{(\tilde{k}_{H}-k_{P})(k_{P}-\tilde{k}_{P})(k_{P}-\tilde{k}_{N})(k_{P}-\tilde{k}_{B})(k_{N}V+\Omega)^{2}}{(\tilde{k}_{H}-k_{N})(\tilde{k}_{P}-k_{N})(\tilde{k}_{N}-k_{N})(\tilde{k}_{B}-k_{N})(k_{P}V+\Omega)^{2}}$
(4.24)
$\displaystyle\times\frac{\sinh(\frac{\pi\gamma(\tilde{k}_{H}-k_{N})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{N}-k_{P})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{P}-k_{P})}{2\beta})}{\sinh(\frac{\pi\gamma(\tilde{k}_{H}-k_{P})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{P}-k_{N})}{2\beta})\sinh(\frac{\pi\gamma(\tilde{k}_{N}-k_{N})}{2\beta})}$
(4.25)
$\displaystyle\times\frac{\sinh(\frac{\pi\gamma(\tilde{k}_{B}-k_{P})}{2\beta})\sinh(\frac{\pi\gamma(k_{H}-k_{N})}{2\beta})\sinh(\frac{\pi\gamma(k_{N}-k_{B})}{2\beta})}{\sinh(\frac{\pi\gamma(\tilde{k}_{B}-k_{N})}{2\beta})\sinh(\frac{\pi\gamma(k_{H}-k_{P})}{2\beta})\sinh(\frac{\pi\gamma(k_{P}-k_{B})}{2\beta})}\,.$
(4.26)

To estimate the temperature we compute the leading order in $\omega$ of
$\log\left(\frac{|P|}{|N|}\right)$. We use the low-$\omega$ expressions of the
modes $k_{j}$ that we have written in Eq. (2.5)-(2.8). The momenta
$\tilde{k}_{j}$ can be obtained by the switch
$\mu^{2}\rightarrow\mu^{2}+\lambda$: they can be also written as follows:

$\displaystyle\tilde{k}_{N}$
$\displaystyle=\frac{\sqrt{\mu^{2}+\lambda}-gV}{g-\sqrt{\mu^{2}+\lambda}V}\,\omega+O(\omega^{3}),$
(4.27) $\displaystyle\tilde{k}_{B}$
$\displaystyle=-\frac{\sqrt{\mu^{2}+\lambda}+gV}{g+\sqrt{\mu^{2}+\lambda}V}\,\omega+O(\omega^{3}),$
(4.28) $\displaystyle\tilde{k}_{P}$
$\displaystyle=\frac{\sqrt{\lambda_{\text{crit}}-\lambda}}{\gamma
V}-\left(\frac{1}{V}+\frac{g^{2}}{\gamma^{2}V^{3}(\lambda_{\text{crit}}-\lambda)}\right)\,\omega-\frac{g^{2}(2g^{2}+(\mu^{2}+\lambda)V^{2})}{2\gamma
V^{5}(\lambda_{\text{crit}}-\lambda)^{5/2}}\,\omega^{2}+O(\omega^{3}),$ (4.29)
$\displaystyle\tilde{k}_{H}$
$\displaystyle=-\frac{\sqrt{\lambda_{\text{crit}}-\lambda}}{\gamma
V}-\left(\frac{1}{V}+\frac{g^{2}}{\gamma^{2}V^{3}(\lambda_{\text{crit}}-\lambda)}\right)\,\omega+\frac{g^{2}(2g^{2}+(\mu^{2}+\lambda)V^{2})}{2\gamma
V^{5}(\lambda_{\text{crit}}-\lambda)^{5/2}}\,\omega^{2}+O(\omega^{3})\,,$
(4.30)

where

$\displaystyle\lambda_{\text{crit}}=\frac{g^{2}-\mu^{2}V^{2}}{V^{2}}\,.$
(4.31)

These expressions make clear that the subcritical regime (i.e.
$\tilde{k}_{j}\in\mathbb{R}$) corresponds to $\lambda<\lambda_{\text{crit}}$.

The leading order of $\log\left(\frac{|P|}{|N|}\right)$ is

$\displaystyle\log\left(\frac{|P|}{|N|}\right)=\frac{\pi g\omega}{\beta\gamma
V\left(g^{2}-\mu^{2}V^{2}\right)^{3/2}\left(g^{2}-V^{2}\left(\lambda+\mu^{2}\right)\right)}\Bigg{[}g\lambda
V^{2}\sqrt{g^{2}-\mu^{2}V^{2}}\times$ (4.32)
$\displaystyle\left(\coth\left(\frac{\pi\left(\sqrt{g^{2}-\mu^{2}V^{2}}-\sqrt{g^{2}-V^{2}\left(\lambda+\mu^{2}\right)}\right)}{2\beta
V^{2}}\right)+\coth\left(\frac{\pi\left(\sqrt{g^{2}-V^{2}\left(\lambda+\mu^{2}\right)}+\sqrt{g^{2}-\mu^{2}V^{2}}\right)}{2\beta
V^{2}}\right)\right)$ (4.33)
$\displaystyle+2\left(g^{2}V\sqrt{\left(\lambda+\mu^{2}\right)\left(g^{2}-\mu^{2}V^{2}\right)}-2gV^{2}\left(\lambda+\mu^{2}\right)\sqrt{g^{2}-\mu^{2}V^{2}}\right)$
(4.34)
$\displaystyle-2\left(\mu^{2}V^{3}\sqrt{\left(\lambda+\mu^{2}\right)\left(g^{2}-\mu^{2}V^{2}\right)}+2g^{3}\sqrt{g^{2}-\mu^{2}V^{2}}\right)\coth\left(\frac{\pi\sqrt{g^{2}-\mu^{2}V^{2}}}{2\beta
V^{2}}\right)\Bigg{]}+O(\omega^{2})\,.$ (4.35)

In order to compare it to the perturbative result, we take the limit
$\lambda\rightarrow 0$:

$\displaystyle\frac{2g\omega\left(\pi(2g+\mu
V)\sqrt{g^{2}-\mu^{2}V^{2}}\coth\left(\frac{\pi\sqrt{g^{2}-\mu^{2}V^{2}}}{2\beta
V^{2}}\right)+2\beta gV^{2}\right)}{\beta\gamma
V\left(g^{2}-\mu^{2}V^{2}\right)^{3/2}}.$ (4.37)

For $\beta\sim 0$, which amounts physically to considering small values for
the derivative of the dielectric pulse, we find:

$\displaystyle\log\left(\frac{|P|}{|N|}\right)\approx\beta_{pert}\omega\,,$
(4.38) $\displaystyle\beta_{pert}=\frac{2\pi g(2g+\mu V)}{\beta\gamma
V(g^{2}-\mu^{2}V^{2})}=\frac{\pi\gamma}{\beta}\lim\limits_{\omega\rightarrow
0}\frac{2k_{H}-k_{P}-k_{N}}{\omega}\,.$ (4.39)

This results confirms the validity of the prediction made in [39] and that the
value ot $T_{pert}$ is not strongly dependent on the peculiarities of the
background. We notice, however, that the expression (4.32) allows to study how
the temperature depends on $\lambda$: in particular, for
$\lambda=\lambda_{\text{crit}}$, we have

$\displaystyle
T(\lambda=\lambda_{\text{crit}})=\left.\frac{\omega}{\log(P/N)}\right|_{\lambda=\lambda_{\text{crit}}}=0.$
(4.40)

The vanishing of the temperature at $\lambda=\lambda_{\text{crit}}$ is very
puzzling, in the sense that thermal particle creation may be found both in the
subcritical and in the transcritical case, whereas a discontinuous behaviour
between the two regimes is just suggested by such a result when
$\lambda=\lambda_{\text{crit}}$. It has also been stressed that, in the above
scheme of approximation for low $\omega$, for $\lambda=\lambda_{\text{crit}}$
one finds a degeneracy, at least at the leading order, of $\tilde{k}_{P}$ with
$\tilde{k}_{N}$, and a singular behaviour of the subleading ones. This kind of
phenomenon will be investigated in future analysis.

###  4.3. Transcritical scattering

We now consider the solution in the transcritical case, that is
$\lambda>\lambda_{\text{crit}}$. In this case, the $\tilde{k}_{H}$ and
$\tilde{k}_{P}$ become complex, as it is shown in Figure 4(b). This fact is
the direct consequence of the presence of an event horizon: these modes cannot
propagate to the left-infinity. The low-$\omega$ expressions are found from
Eq. (4.27)-(4.30) for $\lambda>\lambda_{\text{crit}}$: notice, however, that
the modes have now the wrong label, since the mode that is labeled $N$ becomes
complex, while the $H$-mode is real, in contradiction with the visual
interpretazion of Figure 4(b). Thus, for the transcritical regime, we need to
rename the modes in the following way:

$\displaystyle\tilde{k}_{N}$
$\displaystyle=\frac{\sqrt{\mu^{2}+\lambda}-gV}{g-\sqrt{\mu^{2}+\lambda}V}\,\omega+O(\omega^{3}),$
(4.41) $\displaystyle\tilde{k}_{B}$
$\displaystyle=-\frac{\sqrt{\mu^{2}+\lambda}+gV}{g+\sqrt{\mu^{2}+\lambda}V}\,\omega+O(\omega^{3}),$
(4.42) $\displaystyle\tilde{k}_{P}$
$\displaystyle=i\frac{\sqrt{\lambda-\lambda_{crit}}}{\gamma
V}-\left(\frac{1}{V}-\frac{g^{2}}{\gamma^{2}V^{3}(\lambda-\lambda_{crit})}\right)\,\omega+i\frac{g^{2}(2g^{2}+(\mu^{2}+\lambda)V^{2})}{2\gamma
V^{5}(\lambda-\lambda_{crit})^{5/2}}\,\omega^{2}+O(\omega^{3}),$ (4.43)
$\displaystyle\tilde{k}_{H}$
$\displaystyle=-i\frac{\sqrt{\lambda-\lambda_{crit}}}{\gamma
V}-\left(\frac{1}{V}-\frac{g^{2}}{\gamma^{2}V^{3}(\lambda-\lambda_{crit})}\right)\,\omega-i\frac{g^{2}(2g^{2}+(\mu^{2}+\lambda)V^{2})}{2\gamma
V^{5}(\lambda-\lambda_{crit})^{5/2}}\,\omega^{2}+O(\omega^{3})\,.$ (4.44)

Notice that it holds

$\displaystyle(\tilde{k}_{H})^{*}=\tilde{k}_{P}\,.$ (4.45)

This is not true just in the low-$\omega$ limit, but for all $\omega$. Indeed,
the modes $\tilde{k}$ are the roots of a fourth order polynomial with real
coefficients: since the roots $\tilde{k}_{N}$ and $\tilde{k}_{B}$ are always
real, the other two roots must be either real or complex conjugates. Another
very relevant observation is that, being the basis (4.7) asymptotically
diagonal in the $\tilde{k}_{i}$, as discussed in the previous sections, we
have also the possibility to get rid of the unwanted complex and exponentially
growing mode, say $\tilde{k}_{4}$ (about the growing mode cf. e.g. the
discussion in [9]), simply by imposing that the corresponding coefficient
$D_{4}$ is zero. Actually, in our following discussion, we put only
$D_{1}\not=0$. Note also that the connection coefficients, being connecting
$\tilde{k}_{1}$ to $k_{i}$, $i=1,2,3,4$, cannot resume the aforementioned
growing mode.

Thanks to (4.45), we can simplify the expression of $\frac{|P|}{|N|}$: indeed,
when computing $\frac{C_{P}C_{P}^{*}}{C_{N}C_{N}^{*}}$, one finds a factor

$\displaystyle\frac{\Gamma\left(1+\frac{i(\tilde{k}_{H}-k_{P})\gamma}{2\beta}\right)\Gamma\left(-\frac{i(\tilde{k}_{P}-k_{N})\gamma}{2\beta}\right)\Gamma\left(1+\frac{i(k_{P}-\tilde{k}_{H}^{*})\gamma}{2\beta}\right)\Gamma\left(\frac{i\left(\tilde{k}_{P}^{*}-k_{N}\right)\gamma}{2\beta}\right)}{\Gamma\left(1+\frac{i(\tilde{k}_{H}-k_{N})\gamma}{2\beta}\right)\Gamma\left(-\frac{i(\tilde{k}_{P}-k_{P})\gamma}{2\beta}\right)\Gamma\left(1+\frac{i(k_{N}-i\tilde{k}_{H}^{*})\gamma}{2\beta}\right)\Gamma\left(\frac{i\gamma\left(\tilde{k}_{P}^{*}-k_{P}\right)}{2\beta}\right)}\,,$

which, using (4.45) and recalling $\Gamma(1+z)/\Gamma(z)=z$, reduces to

$\displaystyle\frac{(k_{P}-\tilde{k}_{P})(\tilde{k}_{P}^{*}-k_{P})}{(\tilde{k}_{P}-k_{N})(\tilde{k}_{P}^{*}-k_{N})}=\frac{|k_{P}-\tilde{k}_{P}|^{2}}{|k_{N}-\tilde{k}_{P}|^{2}}\,.$

The final exact expression is

$\displaystyle\frac{|P|}{|N|}=\frac{(k_{P}-\tilde{k}_{N})(k_{P}-\tilde{k}_{B})\,|k_{P}-\tilde{k}_{P}|^{2}\,(k_{N}V+\omega)^{2}}{(k_{N}-\tilde{k}_{N})(k_{N}-\tilde{k}_{B})\,|k_{N}-\tilde{k}_{P}|^{2}\,(k_{P}V+\omega)^{2}}\times\quad\qquad\quad\qquad$
(4.46)
$\displaystyle\times\frac{\sinh\left(\frac{\pi\gamma(\tilde{k}_{N}-k_{P})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{B}-k_{P})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{H}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{N}-k_{B})}{2\beta}\right)}{\sinh\left(\frac{\pi\gamma(\tilde{k}_{N}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(\tilde{k}_{B}-k_{N})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{H}-k_{P})}{2\beta}\right)\sinh\left(\frac{\pi\gamma(k_{P}-k_{B})}{2\beta}\right)}$
(4.47)

As we did in the previous Section, we compute, to the leading order in
$\omega$,

$\displaystyle\log\left(\frac{|P|}{|N|}\right)=\frac{2\pi\,\omega\,g^{2}\coth\left(\frac{\pi\sqrt{g^{2}-\mu^{2}V^{2}}}{2\beta
V^{2}}\right)}{\beta\gamma
V\left(g^{2}-\mu^{2}V^{2}\right)(1-\lambda_{\text{crit}}/\lambda)}+O\left(\omega^{2}\right)\,.$
(4.48)

We can now write the Hawking temperature

$\displaystyle
T(\lambda)=\frac{\omega}{\log\left(\frac{|P|}{|N|}\right)}=\frac{\beta\gamma
V\left(g^{2}-\mu^{2}V^{2}\right)}{2\pi\,g^{2}\coth\left(\frac{\pi\sqrt{g^{2}-\mu^{2}V^{2}}}{2\beta\gamma
V^{2}}\right)}\left(1-\frac{\lambda_{\text{crit}}}{\lambda}\right)\,.$ (4.49)

This result confirms what was found for the subcritical case, that is

$\displaystyle T(\lambda_{\text{crit}})=0\,.$

More interestingly, in the far critical case
$\lambda\gg\lambda_{\text{crit}}$, if we consider $\beta\sim 0$ as previously
done, we find

$\displaystyle T_{H}=\frac{\beta\gamma
V\left(g^{2}-\mu^{2}V^{2}\right)}{2\pi\,g^{2}}\,,$ (4.50)

which coincides with the far critical limit which was obtained in [39] using
the Orr-Sommerfeld approach.

In Figure 5(b) we plot the temperature in units of $T_{H}$, namely

$\displaystyle\frac{T(\lambda)}{T_{H}}=\frac{\omega}{T_{H}\log\left(\frac{|P|}{|N|}\right)}\,,$
(4.51)

for various values of $\lambda$, both in subcritical and transcritical case.
As in the perturbative approach of the previous section, for the plot we
choose $g=1$, $\mu=1.2$; we than choose a near critical pulse velocity $V=0.8$
and a low value $\beta=0.02$. We clearly observe what was predicted in [39]
using a perturbative approach: the effective temperature computed for
$\lambda\ll\lambda_{\text{crit}}$ (subcritical regime) is

$\displaystyle T(\lambda\sim 0)\approx\frac{T_{H}}{3}\,.$ (4.52)

Starting from this value, the temperature decreases for increasing $\lambda$
until it reaches zero at $\lambda=\lambda_{\text{crit}}$. For
$\lambda>\lambda_{\text{crit}}$ the temperature starts growing again, and
$\lambda\gg\lambda_{\text{crit}}$ it stabilizes at the value $T_{H}$.

(a)

(b)

Figure 5. The temperature $T(\lambda)$ in units of $T_{H}$, as defined in
(4.51), for different values of $\lambda$, in subcritical regime (A) and
transcritical regime (B). The plots are made for a near critical pulse
velocity and a low value of the parameter $\beta$. Starting from
$T(\lambda\sim 0)\approx T_{H}/3$, the temperature decreases till reaching
zero for $\lambda=\lambda_{\text{crit}}$; for $\lambda>0$ (transcritical) the
temperature increases again, reaching $T_{H}$ for
$\lambda\gg\lambda_{\text{crit}}$.

So far, the exact solution we provided has confirmed the predictions that were
made using different approximations in different regimes. In the future, an
even deeper study of this solution may allow to describe precisely the
transition between the subcritical and the transcritical regime, the onset of
termality and formation of the event horizon.

##  5\. The original $\phi\psi$-model, exact solutions and thermality

In this section, we provide results concerning a reduction of the so-called
Hopfield model, which represents an effective description of the interaction
between the electromagnetic field and a dielectric medium. In particular,
atoms and molecules of the dielectric are replaced by a mesoscopic
polarization field, still providing an efficient physical description. The
electromagnetic Lagrangian for the full Hopfield model is quite involved, and
has been discussed, by means of different theoretical tools, in [49, 19]. A
simplified model, introduced in [16], can be related to the two dimensional
reduction of the Hopfield model adopted in [50], and is such that the
electromagnetic field and the polarization field are simulated by a pair of
scalar fields, $\varphi$, and $\psi$ respectively, in the so-called
$\varphi\psi$–model. Despite its simplification, it is still set-up in such a
way that we get exactly the same dispersion relation and, moreover, we can
simulate the same coupling as in the full case. Its Lagrangian is

$\displaystyle{\mathcal{L}}_{\varphi\psi}=\frac{1}{2}(\partial_{\mu}\varphi)(\partial^{\mu}\varphi)+\frac{1}{2\chi\omega_{0}^{2}}\left[(v^{\alpha}\partial_{\alpha}\psi)^{2}-\omega_{0}^{2}\psi^{2}\right]-\frac{g}{c}(v^{\alpha}\partial_{\alpha}\psi)\varphi+\frac{\lambda}{4!}\psi^{4},$
(5.1)

where $\chi$ plays the role of the dielectric susceptibility, $v^{\mu}$ is the
usual four-velocity vector of the dielectric, $\omega_{0}$ is the proper
frequency of the medium, and $g$ is the coupling constant between the fields.
The latter constant is henceforth put equal to one, as its original motivation
(see [16]) can be relaxed without problems in a more advanced discussion (cf.
also [36]). As shown in [38], we may introduce the above fourth-order non-
linear term in the polarization field $\psi$ in the Lagrangian. Herein, we
assume $\lambda>0$.
By extending our analysis, and on the grounds of the previous sections, we
adopt a phenomenological model where we can leave room for a spacetime
dependence of the microscopic parameters $\chi,\omega_{0}$, in such a way that
$\chi\omega_{0}^{2}$ is a constant. The equations of motion are

$\displaystyle\square\varphi+\frac{1}{c}(v^{\alpha}\partial_{\alpha}\psi)=$
$\displaystyle 0,$ (5.2)
$\displaystyle\frac{1}{\chi\omega_{0}^{2}}(v^{\alpha}\partial_{\alpha})^{2}\psi+\frac{1}{\chi}\psi-\frac{1}{c}v^{\alpha}\partial_{\alpha}\varphi=$
$\displaystyle 0.$ (5.3)

In particular, we define

$\epsilon^{2}:=\frac{1}{\chi\omega_{0}^{2}},$ (5.4)

which corresponds to the parameter appearing in the Orr-Sommerfeld like
equation (master equation, cf. [36]). We can separate the above system,
obtaining equations involving only one of the fields $\varphi,\psi$. We can
also separate the equations for $\varphi,\psi$, and, in order to maintain the
same line of reasoning as in the previous sections, we focus on $\psi$,
obtaining

$\epsilon^{2}\square(v^{\alpha}\partial_{\alpha})^{2}\psi+\square\frac{1}{\chi}\psi+\frac{1}{c^{2}}(v^{\alpha}\partial_{\alpha})^{2}\psi=0.$
(5.5)

Let us also choose, as usual, the comoving frame; If we put
$\psi(t,x)=e^{-i\omega t}h(x)$, we obtain

$\displaystyle\epsilon^{2}h^{(4)}(x)+2i\epsilon^{2}\frac{\omega}{v}h^{(3)}(x)+\left(\frac{1-\gamma^{2}\frac{v^{2}}{c^{2}}\chi(x)}{\gamma^{2}v^{2}\chi(x)}-\epsilon^{2}\frac{w^{2}}{\gamma^{2}v^{2}}\right)h^{(2)}(x)+\left(-\frac{2i\omega}{c^{2}v}-\frac{2\chi^{\prime}(x)}{\gamma^{2}v^{2}\chi^{2}(x)}+\frac{2\epsilon^{2}i\omega^{3}}{c^{2}v}\right)h^{(1)}(x)+$
(5.6)
$\displaystyle\left(\frac{\omega^{2}}{c^{2}\gamma^{2}v^{2}\chi(x)}(1+\gamma^{2}\chi(x))+\frac{2\chi^{\prime
2}(x)-\chi(x)\chi^{\prime\prime}(x)}{\gamma^{2}v^{2}\chi^{3}(x)}-\epsilon^{2}\frac{\omega^{4}}{v^{2}c^{2}}\right)h(x)=0,$
(5.7)

where $\chi^{\prime}(x),\chi^{\prime\prime}(x)$ indicate the first and the
second derivative wrt $x$. We stress that, with respect to [36], we do not
eliminate the third order term, as we don’t need to grant a Orr-Sommerfeld
form for our equation of motion, as we are going to compute exact solutions,
i.e., solutions which do not depend on the smallness of the parameter
$\epsilon$. In the following, we assume the monotone profile in the comoving
frame

$\frac{1}{\chi(x)}=\frac{1}{\chi_{0}}-\frac{1}{2}\lambda(1-\tanh(\tilde{\beta}x)),$
(5.8)

where $\chi_{0}$ is a constant value of the dielectric susceptibility and we
define the parameter $\tilde{\beta}=\frac{\beta}{\gamma}$ as in the previous
model. We discuss some physical consequence of our choice in Appendix C.

###  5.1. A Fuchsian framework for the equation of motion

We consider the following change of variable: $z=-e^{2\tilde{\beta}x}$. As a
consequence, we obtain

$\frac{1}{\chi(z)}=\frac{1}{\chi_{0}}-\frac{\lambda}{1-z},$ (5.9)

and

$\displaystyle
16\epsilon^{2}\tilde{\beta}^{4}z^{4}h^{(4)}(z)+16\epsilon^{2}\frac{(6\tilde{\beta}v+i\omega)}{v}\tilde{\beta}^{3}z^{3}h^{(3)}(z)+$
(5.10) $\displaystyle
4\left(\frac{-1+\epsilon^{2}\omega^{2}}{c^{2}}+\frac{1-z+\chi_{0}\left(-\lambda+\epsilon^{2}\gamma^{2}(28\tilde{\beta}^{2}v^{2}+12i\tilde{\beta}v\omega-\omega^{2})\right)(1+z)}{\chi_{0}\gamma^{2}v^{2}(1-z)}\right)\tilde{\beta}^{2}z^{2}h^{(2)}(z)+$
(5.11) $\displaystyle
4\left(4\tilde{\beta}^{3}\epsilon^{2}+\frac{4i\tilde{\beta}^{2}\epsilon^{2}\omega}{v}+\frac{\tilde{\beta}(-1+\epsilon^{2}\omega^{2})}{c^{2}}+\frac{i\omega(-1+\epsilon^{2}\omega^{2})}{c^{2}v}+\frac{\tilde{\beta}}{\gamma^{2}v^{2}}\left(\frac{1}{\chi_{0}}-\frac{\lambda(1+z)}{(1-z)^{2}}-\epsilon^{2}\gamma^{2}\omega^{2}\right)\right)\tilde{\beta}zh^{(1)}(z)+$
(5.12) $\displaystyle\left(-\frac{4\tilde{\beta}^{2}\lambda
z(z+1)}{\gamma^{2}v^{2}(1-z)^{3}}-\frac{\epsilon^{2}\omega^{4}}{v^{2}c^{2}}+\frac{\omega^{2}(1-z-\chi_{0}\lambda)}{c^{2}\chi_{0}\gamma^{2}v^{2}(1-z)}+\frac{\omega^{2}}{c^{2}v^{2}}\right)h(z)=0.$
(5.13)

We start looking for local solutions, along the path sketched for (3.3),
around $z=\infty$. By introducing $t=1/z$, a series expansion for the
solutions can be provided in the following form:

$\displaystyle f(t)=t^{-i\alpha}\sum_{n=0}^{\infty}c_{n}t^{n}\,.$ (5.14)

The characteristic equation for the exponent $k:=2\tilde{\beta}\alpha$ is

$\displaystyle\text{DR}(k)$
$\displaystyle:=(\omega^{2}-\chi_{0}\gamma^{2}(kv+\omega)^{2}(-1+\epsilon^{2}\omega^{2})+c^{2}k^{2}(-1+\chi_{0}\epsilon^{2}\gamma^{2}(kv+\omega)^{2}))=0.$
(5.15)

As in the previous sections, except possibly for a zero measure set in the
space of available parameters in our model, we have four distinct roots which,
moreover, not differ each other by an integer value. As a consequence, the
spectral type is $(1111)$.

At $z=0$, we get four independent local solutions of the form

$\displaystyle f(z)=z^{i\tilde{\alpha}}\sum_{n=0}^{\infty}c_{n}z^{n}\,,$
(5.16)

where $\tilde{k}:=2\tilde{\beta}\tilde{\alpha}$ satisfies

$\displaystyle\text{DR}_{0}(\tilde{k})$
$\displaystyle:=(\omega^{2}-\chi_{0}\lambda\omega^{2}-\chi_{0}\gamma^{2}(\tilde{k}v+\omega)^{2}(-1+\epsilon^{2}\omega^{2})+c^{2}{\tilde{k}}^{2}(-1+\chi_{0}(\lambda+\epsilon^{2}\gamma^{2}(\tilde{k}v+\omega)^{2})))=0,$
(5.17)

The spectral type at $z=0$ is again $(1111)$, again almost everywhere in the
space of available parameters appearing in our model.

Also in this case, at $z=1$ the so-called resonant case[41] is verified. Let
us define $y:=z-1$. Then, the characteristic equation for solutions of the
form

$\displaystyle f(y)=y^{a}\sum_{n=0}^{\infty}c_{n}y^{n}$ (5.18)

has four integer solutions $a=0,1,2,3$. This situation, again, requires a
particular study, that we perform by means of the Frobenius method.

We can show that there exist three independent integer solutions

$\displaystyle u_{1}(y)$
$\displaystyle=y^{3}+y^{4}\left[-\frac{3}{2}-\frac{\lambda}{48\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}-i\frac{\omega}{4\tilde{\beta}v}\right]+o(y^{4})$
(5.19) $\displaystyle u_{2}(y)$
$\displaystyle=y^{2}+y^{3}\left[-2-\frac{\lambda}{24\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}-i\frac{\omega}{3\tilde{\beta}v}\right]+o(y^{3})$
(5.20) $\displaystyle u_{3}(y)$
$\displaystyle=y\;+y^{2}\left[-3-\frac{\lambda}{8\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}-i\frac{\omega}{2\tilde{\beta}v}\right]+o(y^{2})\,,$
(5.21)

and one logarithmic solution

$\displaystyle
u_{0}(y)=1+y\left[-6+\frac{\lambda}{4\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}-i\frac{\omega}{\tilde{\beta}v}\right]+o(y)$
(5.22)
$\displaystyle+\log(y)\Big{(}R_{1}u_{1}(y)+R_{2}u_{2}(y)+R_{3}u_{3}(y)\Big{)}\,,$
(5.23)

where

$\displaystyle R_{3}$
$\displaystyle=-\frac{\lambda}{4\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}$
(5.24) $\displaystyle R_{2}$
$\displaystyle=-\frac{5\lambda}{16\tilde{\beta}^{2}\epsilon^{2}\gamma^{2}v^{2}}$
(5.25) $\displaystyle R_{1}$
$\displaystyle=-\frac{\lambda\left(36\tilde{\beta}^{2}c^{2}+\omega^{2}\right)}{288\tilde{\beta}^{4}c^{2}\epsilon^{2}\gamma^{2}v^{2}}\,.$
(5.26)

###  5.2. Monodromy and rigidity

The monodromy matrix of the solutions $(u_{0}(y),u_{1}(y),u_{2}(y),u_{3}(y))$
at $z=1$ is easily computed as

$\displaystyle M_{1}=\left(\begin{array}[]{cccc}1&2\pi iR_{1}&2\pi iR_{2}&2\pi
iR_{3}\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)\,,$ (5.31)

whose Jordan form is

$\displaystyle J_{M_{1}}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&1\\\ 0&0&0&1\\\ \end{array}\right)\,.$ (5.36)

The monodromy at $z=0$ and $z=\infty$ are, also in this case, represented
respectively by the diagonal matrices

$\displaystyle M_{0}$
$\displaystyle=\text{diag}\left(e^{i2\pi\tilde{\alpha}_{1}},e^{i2\pi\tilde{\alpha}_{2}},e^{i2\pi\tilde{\alpha}_{3}},e^{i2\pi\tilde{\alpha}_{4}}\right)\,,$
(5.37) $\displaystyle M_{\infty}$
$\displaystyle=\text{diag}\left(e^{-i2\pi\alpha_{1}},e^{-i2\pi\alpha_{2}},e^{-i2\pi\alpha_{3}},e^{-i2\pi\alpha_{4}}\right)\,,$
(5.38)

whose Jordan form is $J_{M_{0}}=J_{M_{\infty}}=\mathbb{I}_{4}$. As a
consequence, the spectral type of the equation at $z=1$ is $(3,1)$, and the
spectral type of the equation is $((1111),(31),(1111))$, i.e. it is rigid as
in the case discussed in the previous section.

###  5.3. Gauge Transformation and Riemann scheme

As in the previous sections for the simplified model, we can obtain a solution
of the form

$\displaystyle f(z):=z^{i\frac{\tilde{k}_{1}}{2\tilde{\beta}}}(z-1)\,u(z)\,,$
(5.39)

where $\tilde{k}_{1}$ satisfies

$\text{DR}_{0}(\tilde{k}_{1})=0\,.$

The function $u(z)$ now satisfies an equation which is analog to (3.59), which
we avoid to write explicitly, as it is quite long. What happens is that one
may verify that for the solution $u(z)$, also in the case of the standard
$\phi\psi$-model, the same equation (3.69) holds true, where now the
dispersion relations are the ones in (5.15) and in (5.17) respectively. Also
the so-called Riemann Scheme of the equation is the same as in (3.78). Also in
this case, by letting

$\displaystyle\alpha_{i}$
$\displaystyle:=1-\frac{i}{2\tilde{\beta}}(k_{i}-\tilde{k}_{1})\,,\quad
i=1,2,3,4$ (5.40) $\displaystyle\beta_{j}$
$\displaystyle:=1-\frac{i}{2\tilde{\beta}}(\tilde{k}_{j+1}-\tilde{k}_{1})\,,\quad
j=1,2,3\,,$ (5.41)

we can write Eq. (3.78) as

$\displaystyle\left[\begin{matrix}z=0&z=1&z=\infty\\\ 0&0&\alpha_{1}\\\
1-\beta_{1}&1&\alpha_{2}\\\ 1-\beta_{2}&2&\alpha_{3}\\\
1-\beta_{3}&-\beta_{4}&\alpha_{4}\end{matrix}\right]\,.$ (5.42)

which, as discussed in the previous sections, corresponds to the Riemann
scheme of the hypergeometric function

${}_{4}F_{3}(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4};\beta_{1},\beta_{2},\beta_{3};z)$

in the standard form.

###  5.4. The Hawking temperature

The scattering coefficients $C_{j}$ for the Hopfield model have the same form
as the ones of the previous Sections, a part for the different expressions of
the momenta $k_{j}(\omega)$ and $\tilde{k}_{j}(\omega)$. As in the previous
section, one is able to deal with analytical expressions for the momenta only
in suitable limits, and this is the strategy that is adopted also in the
present case.

In the low-$\omega$ limit, the right-infinity modes are

$\displaystyle k_{H}$
$\displaystyle=\frac{\chi_{0}\gamma^{2}v+\sqrt{c^{2}(1+\chi_{0})}}{c^{2}-\chi_{0}\gamma^{2}v^{2}}\,\omega+o(\omega)$
(5.43) $\displaystyle k_{B}$
$\displaystyle=\frac{\chi_{0}\gamma^{2}v-\sqrt{c^{2}(1+\chi_{0})}}{c^{2}-\chi_{0}\gamma^{2}v^{2}}\,\omega+o(\omega)$
(5.44) $\displaystyle k_{P}$
$\displaystyle=\frac{\sqrt{c^{2}-\chi_{0}\gamma^{2}v^{2}}}{c\sqrt{\chi_{0}}{\epsilon}\gamma
v}+\frac{c^{2}\omega}{v\left(\chi_{0}\gamma^{2}v^{2}-c^{2}\right)}+o(\omega)$
(5.45) $\displaystyle k_{N}$
$\displaystyle=-\frac{\sqrt{c^{2}-\chi_{0}\gamma^{2}v^{2}}}{c\sqrt{\chi_{0}}{\epsilon}\gamma
v}+\frac{c^{2}\omega}{v\left(\chi_{0}\gamma^{2}v^{2}-c^{2}\right)}+o(\omega)\,,$
(5.46)

while the left-infinity modes are given by

$\displaystyle\tilde{k}_{H}$
$\displaystyle=\frac{\chi_{0}\gamma^{2}v+\sqrt{c^{2}\left(1+\chi_{0}+\chi_{0}^{2}(\lambda-1)\lambda-2\chi_{0}\lambda\right)}}{c^{2}-\chi_{0}\gamma^{2}v^{2}-c^{2}\chi_{0}\lambda}\,\omega+o(\omega)$
(5.47) $\displaystyle\tilde{k}_{B}$
$\displaystyle=\frac{\chi_{0}\gamma^{2}v-\sqrt{c^{2}\left(1+\chi_{0}+\chi_{0}^{2}(\lambda-1)\lambda-2\chi_{0}\lambda\right)}}{c^{2}-\chi_{0}\gamma^{2}v^{2}-c^{2}\chi_{0}\lambda}\,\omega+o(\omega)$
(5.48) $\displaystyle\tilde{k}_{P}$
$\displaystyle=+\frac{\sqrt{\chi_{0}(\lambda_{\text{crit}}-\lambda)}}{{\epsilon}\gamma
v}-\frac{\left(1-\chi_{0}\lambda\right)\,\omega}{v\chi_{0}\left(\lambda_{\text{crit}}-\lambda\right)}$
(5.49) $\displaystyle\tilde{k}_{N}$
$\displaystyle=-\frac{\sqrt{\chi_{0}(\lambda_{\text{crit}}-\lambda)}}{{\epsilon}\gamma
v}-\frac{\left(1-\chi_{0}\lambda\right)\,\omega}{v\chi_{0}\left(\lambda_{\text{crit}}-\lambda\right)}$
(5.50)

The transition between subcritical and transcritical regime happens at
$\lambda=\lambda_{\text{crit}}$, which in this case is

$\displaystyle\lambda_{\text{crit}}=\frac{1-\chi_{0}\gamma^{2}\frac{v^{2}}{c^{2}}}{\chi_{0}}\,.$
(5.52)

We assume the condition $\lambda_{\text{crit}}>0$.

As we did in Sections 4.2 and 4.3, we expand the factor $\frac{|P|}{|N|}$ in
order to find an expression of the Hawking temperature. The flux factors that
appear in the expressions of $|P|$ and $|N|$ (see Eq. (4.15)) for the Hopfield
model are given by

$\displaystyle
v_{i}(\omega)\partial_{\omega}\text{DR}|_{k_{i}}=-\frac{\gamma^{2}V^{4}}{\omega^{2}-k_{i}(\omega)^{2}}\prod_{j\neq
i}(k_{i}(\omega)-k_{j}(\omega))\,.$ (5.53)

#### Subcritical case

In the subcritical case, expanding for low-$\omega$ and low-$\lambda$ we find

$\displaystyle\log\left(\frac{|P|}{|N|}\right)=\left(\beta_{sub}+o(\lambda)\right)\omega+o(\omega)\,,$
(5.54)

where

$\displaystyle\beta_{sub}=\frac{2\left(\pi\left(1-v^{2}(\text{$\chi_{0}$}+1)\right)\left(\gamma^{2}v^{2}\text{$\chi_{0}$}+v\sqrt{\text{$\chi_{0}$}+1}+1\right)\coth\left(\frac{\pi\gamma\sqrt{1-v^{2}(\text{$\chi_{0}$}+1)}}{2\beta
v\sqrt{\text{$\chi_{0}$}}\epsilon}\right)+2\beta\gamma
v^{3}\text{$\chi_{0}$}^{3/2}\epsilon\sqrt{1-v^{2}(\text{$\chi_{0}$}+1)}\right)}{\beta\gamma
v\left((\chi_{0}+1)v^{2}-1\right)^{2}}$ (5.55)

Once again, for $\beta\sim 0$ we find

$\displaystyle\beta_{sub}\approx\frac{2\pi\left(\gamma^{2}v^{2}\text{$\chi_{0}$}+v\sqrt{\text{$\chi_{0}$}+1}+1\right)}{\beta\gamma
v\left(1-v^{2}(\text{$\chi_{0}$}+1)\right)}\,,$ (5.56)

which is the same result that was found in [39] with a perturbative approach.
Notice that, for this model, taking the limit $\beta\sim 0$ in (5.55) is the
same as taking the limit $\epsilon\sim 0$: this corresponds to the weak
dispersion limit, a situation that is often studied in literature.

We can expand $\log{\frac{|P|}{|N|}}$ for
$\lambda\lesssim\lambda_{\text{crit}}$, we find

$\displaystyle\log{\frac{|P|}{|N|}}=\left(\frac{A}{\frac{\lambda_{crit}}{\lambda}-1}+B+O\left(\frac{\lambda_{crit}}{\lambda}-1\right)\right)\omega+o(\omega)\,,$
(5.57)

with determined factors $A$ e $B$. From this expansion we deduce the
temperature in the near critical regime is

$\displaystyle
T_{nc}(\lambda)\approx\frac{\beta\left(1-v^{2}(\text{$\chi_{0}$}+1)\right)\tanh\left(\frac{\pi\gamma\sqrt{1-v^{2}(\text{$\chi_{0}$}+1)}}{2\beta
v\sqrt{\text{$\chi_{0}$}}\epsilon}\right)}{4\pi\gamma
v\text{$\chi_{0}$}}\left(\frac{\lambda_{crit}}{\lambda}-1\right)\,.$ (5.58)

In particular, notice as before that

$T(\lambda=\lambda_{\text{crit}})=0\,.$

The same considerations as in the previous section hold true about this point.

#### Transcritical case

The transcritical regime is reached for $\lambda>\lambda_{\text{crit}}$. In
this case, the low-$\omega$ expansion gives a much simpler result,

$\displaystyle\log{\frac{|P|}{|N|}}=\beta(\lambda)\omega+o(\omega)\,,$ (5.59)

where

$\displaystyle\beta(\lambda)=\frac{2\pi\gamma\,v\chi_{0}\coth\left(\frac{\pi\gamma\sqrt{1-v^{2}(\chi_{0}+1)}}{2\beta
v\sqrt{\chi_{0}}\epsilon}\right)}{\beta\left(1-v^{2}(\chi_{0}+1)\right)(1-\frac{\lambda_{\text{crit}}}{\lambda})}\,.$
(5.60)

Thus, the Haking temperature in transcritical regime is

$\displaystyle
T_{c}(\lambda)=\frac{\beta\left(1-v^{2}(\chi_{0}+1)\right)\tanh\left(\frac{\pi\gamma\sqrt{1-v^{2}(\chi_{0}+1)}}{2\beta
v\sqrt{\chi_{0}}\epsilon}\right)}{2\pi\gamma
v\chi_{0}}\left(1-\frac{\lambda_{\text{crit}}}{\lambda}\right)$ (5.61)

In the limit $\lambda\gg\lambda_{\text{crit}}$ (see also Appendix C), we reach
the limit temperature

$\displaystyle
T_{H}=\frac{\beta\left(1-v^{2}(\chi_{0}+1)\right)\tanh\left(\frac{\pi\gamma\sqrt{1-v^{2}(\chi_{0}+1)}}{2\beta
v\sqrt{\chi_{0}}\epsilon}\right)}{2\pi\gamma v\chi_{0}}\,.$ (5.62)

This result, for $\beta\sim 0$, again coincides with what was found in [39]
using the Orr-Sommerfeld approach. It is also to be stressed, as for the
subcritical case, that if, in place of $\beta\sim 0$, one considers in the
last equation $\epsilon\sim 0$, i.e. the usual weak dispersion limit which is
commonly adopted in the literature on the dispersive analog Hawking effect, we
get the same result. In order to provide a more extensive comparison with the
Orr-Sommerfeld approach and some more insights, in the following subsection we
sketch the basic calculations involved.

###  5.5. The transcritical case in the Orr-Sommerfeld picture

The separated equation of motion for the spatial part of the polarization
field $h(x)$, has been displayed in (5.6), where, in the present case, the
specific profile (5.8) is understood. We eliminate, as usual [36], the third
order term by putting $h(x)=\exp(-2i\frac{\omega}{v}x)f(x)$. Then we obtain
the following equation:

$\epsilon^{2}f^{(4)}(x)+\sum_{i=0}^{2}p_{3-i}(x,\epsilon)f^{(2-i)}(x)=0,$
(5.63)

where the coefficients $p_{i}(x,\epsilon)$ are, in the Orr-Sommerfeld
approach, analytic functions in $\epsilon$:

$p_{i}(x,\epsilon)=\sum_{n=0}^{\infty}\epsilon^{n}p_{in}(x).$

A real turning point $x=x_{tp}$, i.e. an horizon, is found when
$p_{30}(x_{tp})=0$. See [36] and references therein, with particular focus on
the papers by Nishimoto. We get

$p_{3}(x,\epsilon)=\left[\left(1-\chi_{0}\gamma^{2}\frac{v^{2}}{c^{2}}\right)-\frac{\lambda}{2\gamma^{2}v^{2}}(1-\tanh(\tilde{\beta}x))\right]+\epsilon^{2}\omega^{2}\frac{1}{2v^{2}}(1+2\frac{v^{2}}{c^{2}}).$
(5.64)

We can easily identify the turning point by solving $p_{30}(x=x_{tp})=0$.We
find

$\tilde{\beta}x_{tp}=\operatorname{arctanh}\left(1-2\frac{\lambda_{\text{crit}}}{\lambda}\right)$
(5.65)

It interesting to notice that, by assuming $\lambda$ and
$\lambda_{\text{crit}}$ both positive, as in our previous analysis, the
condition $\lambda>\lambda_{\text{crit}}$ amounts to the reality of the
critical point at hand, i.e. to
$(1-2\frac{\lambda_{\text{crit}}}{\lambda})\in(-1,1)$. This is in agreement
with the assumption of transcritical case and with the interpretation of
$\lambda_{\text{crit}}$. We also have [36]

$T_{H}=\frac{\gamma^{2}v^{2}|n^{\prime}(x_{tp})|}{2\pi},$ (5.66)

where the refractive index is given by $n(x)=\sqrt{1+\chi(x)}$. Then, after
restoring the parameter $\beta=\gamma\tilde{\beta}$, we obtain

$T_{H}=\frac{\beta
c^{2}\left(1-(1+\chi_{0})\frac{v^{2}}{c^{2}}\right)\left(1-\frac{\lambda_{\text{crit}}}{\lambda}\right)}{2\pi
v\gamma\chi_{0}},$ (5.67)

which is in perfect agreement with (5.61) if one considers the limit as
$\epsilon\to 0$ in (5.61), because, trivially, the contribution of the factor
involving the hyperbolic tangent goes to 1 in that limit.

##  6\. Conclusions

In the framework of the analog Hawking effect in dielectric media, we have
taken into account both the Cauchy model, which has the characteristic to be
as simple as possible, and the original $\phi\psi$-model, with the explicit
aim to find out exact solutions for the scattering problem for a suitable but
physically meaningful monotone profile for the dielectric refractive index
perturbation. On the one hand, this has required to embed the physical
problem, from a mathematical point of view, in the framework of Fuchsian
equations on the Riemann sphere. We have first introduced the complex variable
$z$, and obtained a fourth order equation displaying three regular singular
points $z=0,1,\infty$. We have determined the monodromy properties of the
solutions near the aforementioned singular points, and also found that our
equations satisfies the so-called rigidity properties, which have eventually
allowed us to conclude that exact global solutions are available and involve
the generalized hypergeometric function $\,{}_{4}F_{3}$. For this
hypergeometric function, a study of the Mellin-Barnes integral representation
has allowed us to reach two fundamental goals: a complete analysis of the
Stokes phenomenon and also a complete set of connection formulas, which are at
the root of the description of the $S$-matrix for the scattering process
associated with the analog Hawking effect.
On the other hand, we have taken into account some fundamental physical
problems, which, of course, involve, as a focal point, the determination of
the analog Hawking temperature. This part of the analysis has required some
approximations, as fully analytical calculations are hard to be managed
successfully. In particular, for the asymptotic expressions of the momenta of
the modes involved in the scattering, we have adopted an expansion for low
frequencies $\omega$, which is still standard in analytical calculations in
literature. We have also considered both the subcritical regime and the
transcritical one, and found explicit expressions for the Hawking temperature
which are compatible both with the ones obtained in a perturbative framework
in [39] and, in the limit of weak dispersive effects, in [36]. The
aforementioned analysis, form a physical point of view, is just a very
interesting but still incomplete one, as other regimes (beyond the low
frequency one) can be investigated, and further amplitudes can be calculated,
for a complete description of the full scattering matrix involved in the
problem. We deserve a depeening and an extension of our study to future
investigations.

## Acknowledgements

The authors are grateful to professor Yoshishige Haraoka for his precious
advice and insight in the mathematical aspects that are related to this work,
especially concernng rigid Fuchsian equations.

##  Appendix A Useful relations

We write (3.32) and (3.34) as

$\displaystyle\text{DR}(k)$
$\displaystyle=\gamma^{4}V^{4}(k-k_{1})(k-k_{2})(k-k_{3})(k-k_{4})\,,$ (A.1)
$\displaystyle\text{DR}_{0}(k)$
$\displaystyle=\gamma^{4}V^{4}(k-\tilde{k}_{1})(k-\tilde{k}_{2})(k-\tilde{k}_{3})(k-\tilde{k}_{4})\,,$
(A.2)

where $k_{j}$ is a solution of $\text{DR}(k)=0$ and $\tilde{k}_{j}$ is a
solutions of $\text{DR}_{0}(\tilde{k})=0$. By confronting (3.32) with (A.1) we
deduce the following useful relations,

$\displaystyle\frac{1}{(2\beta)^{4}}k_{1}k_{2}k_{3}k_{4}$
$\displaystyle=\frac{\Omega^{2}\left(-G^{2}V^{2}+M^{2}+\Omega^{2}\gamma^{2}\right)}{V^{4}\gamma^{2}}\,,$
(A.3) $\displaystyle\frac{1}{(2\beta)^{3}}\sum_{i\neq j\neq l}k_{i}k_{j}k_{l}$
$\displaystyle=-\frac{2\Omega\left(-G^{2}+M^{2}+2\Omega^{2}\gamma^{2}\right)}{V^{3}\gamma^{2}}\,,$
(A.4) $\displaystyle\frac{1}{(2\beta)^{2}}\sum_{i\neq j}k_{i}k_{j}$
$\displaystyle=\frac{V^{2}\left(M^{2}+6\Omega^{2}\gamma^{2}\right)-G^{2}}{V^{4}\gamma^{2}}\,,$
(A.5) $\displaystyle\frac{1}{(2\beta)}\sum_{i}k_{i}$
$\displaystyle=-\frac{4\Omega}{V}\,,$ (A.6)

and similarly from (3.34) and (A.2)

$\displaystyle\frac{1}{(2\beta)^{4}}\tilde{k}_{1}\tilde{k}_{2}\tilde{k}_{3}\tilde{k}_{4}$
$\displaystyle=\frac{\Omega^{2}\left(-G^{2}V^{2}+\Lambda+M^{2}+\Omega^{2}\gamma^{2}\right)}{V^{4}\gamma^{2}}\,,$
(A.7) $\displaystyle\frac{1}{(2\beta)^{3}}\sum_{i\neq j\neq
l}\tilde{k}_{i}\tilde{k}_{j}\tilde{k}_{l}$
$\displaystyle=-\frac{2\Omega\left(-G^{2}+\Lambda+M^{2}+2\Omega^{2}\gamma^{2}\right)}{V^{3}\gamma^{2}}\,,$
(A.8) $\displaystyle\frac{1}{(2\beta)^{2}}\sum_{i\neq
j}\tilde{k}_{i}\tilde{k}_{j}$
$\displaystyle=\frac{V^{2}\left(\Lambda+M^{2}+6\Omega^{2}\gamma^{2}\right)-G^{2}}{V^{4}\gamma^{2}}\,,$
(A.9) $\displaystyle\frac{1}{(2\beta)}\sum_{i}\tilde{k}_{i}$
$\displaystyle=-\frac{4\Omega}{V}\,.$ (A.10)

##  Appendix B Proof of Proposition 1

We prove the Theorem by induction. It is easy to verify that Eq. (4.2) holds
for $n=1,2,3,4$: indeed, substituting (4.1) into (3.69) and truncating at
order $4$ (using $\text{DR}_{0}(\tilde{k}_{1})=0$) gives

$\displaystyle 0=$ $\displaystyle
z(-c_{1}\text{DR}_{0}(\tilde{k}_{1}-i)+\text{DR}(\tilde{k}_{1}-i))$ (B.1)
$\displaystyle+$ $\displaystyle
z^{2}\big{(}c_{1}\text{DR}(\tilde{k}_{1}-2i)+3c_{1}\text{DR}_{0}(\tilde{k}_{1}-i)-c_{2}\text{DR}_{0}(\tilde{k}_{1}-2i)-3\text{DR}(\tilde{k}_{1}-i)\big{)}$
(B.2) $\displaystyle+$ $\displaystyle
z^{3}\Big{(}-3c_{1}(\text{DR}(\tilde{k}_{1}-2i)+\text{DR}_{0}(\tilde{k}_{1}-i))+c_{2}\text{DR}(\tilde{k}_{1}-3i)$
(B.3)
$\displaystyle+3c_{2}\text{DR}_{0}(\tilde{k}_{1}-2i)-c_{3}\text{DR}_{0}(\tilde{k}_{1}-3i)+3\text{DR}(\tilde{k}_{1}-i)\Big{)}$
(B.4) $\displaystyle+$ $\displaystyle
z^{4}\Big{(}3c_{1}\text{DR}(\tilde{k}_{1}-2i)+c_{1}\text{DR}_{0}(\tilde{k}_{1}-i)-3c_{2}(\text{DR}(\tilde{k}_{1}-3i)+\text{DR}_{0}(\tilde{k}_{1}-2i))$
(B.5)
$\displaystyle+c_{3}\text{DR}(\tilde{k}_{1}-4i)+3c_{3}\text{DR}_{0}(\tilde{k}_{1}-3i)-c_{4}\text{DR}_{0}(\tilde{k}_{1}-4i)-\text{DR}(\tilde{k}_{1}-i)\Big{)}+O\left(z^{5}\right)\,,$
(B.6)

from which one can compute $c_{1}$, …, $c_{4}$ by annihilating the coefficient
of each order. Even though it is not necessary for the sake of the proof, we
can verify (4.2) also for some further $n$, by using the following identity,
that is true for any fourth order polynomial222The identity holds more
generally for any polynomial of order $k$,  $\sum_{m=0}^{n}\begin{pmatrix}n\\\
m\end{pmatrix}(-1)^{m}P_{k}(m)=0\,,\quad n>k\,,$ and it derives immediately
from the following property of binomial coefficients:
$\sum_{m=0}^{n}\begin{pmatrix}n\\\ m\end{pmatrix}(-1)^{m}m^{k}=0\,,\quad
n>k\,.$ $P_{4}(k)$,

$\displaystyle 0=\sum_{m=0}^{n}\begin{pmatrix}n\\\
m\end{pmatrix}(-1)^{m}P_{4}(-im)\,,\quad n>4\,.$ (B.8)

The identity (B.8) allows to write $\text{DR}(\tilde{k}_{1}-in)$ ($n\geq 5$)
as a linear combination of $\text{DR}(\tilde{k}_{1}-i)$,
$\text{DR}(\tilde{k}_{1}-2i)$, $\text{DR}(\tilde{k}_{1}-3i)$ and
$\text{DR}(\tilde{k}_{1}-4i)$, and similarly for $\text{DR}_{0}$.
Now, for $n$ generic, assume that $c_{n}$, $c_{n+1}$, $c_{n+2}$, $c_{n+3}$
satisfy (4.2). Take any two fourth degree polynomials

$\displaystyle\text{DR}(k)$
$\displaystyle=a_{0}+a_{1}k+a_{2}k^{2}+a_{3}k^{3}+a_{4}k^{4}\,,$
$\displaystyle\text{DR}_{0}(k)$
$\displaystyle=b_{0}+b_{1}k+b_{2}k^{2}+b_{3}k^{3}+b_{4}k^{4}\,.$

Substituting (4.1) into (3.69), we find that the coefficient $c_{n+4}$
satisfies the recurrence relation

$\displaystyle
A_{0}\,c_{n}+A_{1}\,c_{n+1}+A_{2}\,c_{n+2}+A_{3}\,c_{n+3}+A_{4}\,c_{n+4}=0\,,$
(B.9)

where

$\displaystyle
A_{0}=\left(b_{0}-ib_{1}(n+1)-b_{2}(n+1)^{2}+ib_{3}(n+1)^{3}+b_{4}(n+1)^{4}\right)$
(B.10)
$\displaystyle+k_{1}\left(b_{1}-2ib_{2}(n+1)-3b_{3}(n+1)^{2}+4ib_{4}(n+1)^{3}\right)$
(B.11)
$\displaystyle+k_{1}^{2}\left(b_{2}-3ib_{3}(n+1)-6b_{4}(n+1)^{2}\right)$
(B.12) $\displaystyle+k_{1}^{3}(b_{3}-4ib_{4}(n+1))+b_{4}k_{1}^{4}\,,$ (B.13)
$\displaystyle
A_{1}=-\Big{(}a_{0}-ia_{1}(n+1)-a_{2}n^{2}-2a_{2}n-a_{2}+ia_{3}n^{3}+3ia_{3}n^{2}+3ia_{3}n+ia_{3}+a_{4}n^{4}$
(B.14)
$\displaystyle+4a_{4}n^{3}+6a_{4}n^{2}+4a_{4}n+a_{4}+3b_{0}-3ib_{1}n-6ib_{1}-3b_{2}n^{2}-12b_{2}n-12b_{2}+3ib_{3}n^{3}$
(B.15)
$\displaystyle+18ib_{3}n^{2}+36ib_{3}n+24ib_{3}+3b_{4}n^{4}+24b_{4}n^{3}+72b_{4}n^{2}+96b_{4}n+48b_{4}\Big{)}$
(B.16) $\displaystyle-
k_{1}\Big{(}a_{1}-2ia_{2}n-2ia_{2}-3a_{3}n^{2}-6a_{3}n-3a_{3}+4ia_{4}n^{3}+12ia_{4}n^{2}+12ia_{4}n+4ia_{4}$
(B.17)
$\displaystyle+3b_{1}-6ib_{2}n-12ib_{2}-9b_{3}n^{2}-36b_{3}n-36b_{3}+12ib_{4}n^{3}+72ib_{4}n^{2}+144ib_{4}n+96ib_{4}\Big{)}$
(B.18) $\displaystyle-
k_{1}^{2}\left(a_{2}-3ia_{3}n-3ia_{3}-6a_{4}n^{2}-12a_{4}n-6a_{4}+3b_{2}-9ib_{3}n-18ib_{3}-18b_{4}n^{2}-72b_{4}n-72b_{4}\right)$
(B.19) $\displaystyle-
k_{1}^{3}(a_{3}-4ia_{4}n-4ia_{4}+3b_{3}-12ib_{4}n-24ib_{4})-k_{1}^{4}(a_{4}+3b_{4})\,,$
(B.20) $\displaystyle
A_{2}=3\Big{(}a_{0}-ia_{1}(n+2)-a_{2}n^{2}-4a_{2}n-4a_{2}+ia_{3}n^{3}+6ia_{3}n^{2}+12ia_{3}n+8ia_{3}+a_{4}n^{4}$
(B.21)
$\displaystyle+8a_{4}n^{3}+24a_{4}n^{2}+32a_{4}n+16a_{4}+b_{0}-ib_{1}n-3ib_{1}-b_{2}n^{2}-6b_{2}n-9b_{2}+ib_{3}n^{3}+9ib_{3}n^{2}$
(B.22)
$\displaystyle+27ib_{3}n+27ib_{3}+b_{4}n^{4}+12b_{4}n^{3}+54b_{4}n^{2}+108b_{4}n+81b_{4}\Big{)}$
(B.23)
$\displaystyle+3k_{1}\Big{(}a_{1}-2ia_{2}n-4ia_{2}-3a_{3}n^{2}-12a_{3}n-12a_{3}+4ia_{4}n^{3}+24ia_{4}n^{2}+48ia_{4}n$
(B.24)
$\displaystyle+32ia_{4}+b_{1}-2ib_{2}n-6ib_{2}-3b_{3}n^{2}-18b_{3}n-27b_{3}+4ib_{4}n^{3}+36ib_{4}n^{2}+108ib_{4}n+108ib_{4}\Big{)}$
(B.25)
$\displaystyle+3k_{1}^{2}\Big{(}a_{2}-3ia_{3}n-6ia_{3}-6a_{4}n^{2}-24a_{4}n-24a_{4}+b_{2}-3ib_{3}n-9ib_{3}-6b_{4}n^{2}-36b_{4}n-54b_{4}\Big{)}$
(B.26)
$\displaystyle+3k_{1}^{3}(a_{3}-4ia_{4}n-8ia_{4}+b_{3}-4ib_{4}n-12ib_{4})+3k_{1}^{4}(a_{4}+b_{4})\,,$
(B.27) $\displaystyle
A_{3}=-3\Big{(}3a_{0}-3ia_{1}(n+3)-3a_{2}n^{2}-18a_{2}n-27a_{2}+3ia_{3}n^{3}+27ia_{3}n^{2}+81ia_{3}n+81ia_{3}$
(B.28)
$\displaystyle+3a_{4}n^{4}+36a_{4}n^{3}+162a_{4}n^{2}+324a_{4}n+243a_{4}+b_{0}-ib_{1}n-4ib_{1}-b_{2}n^{2}-8b_{2}n-16b_{2}$
(B.29)
$\displaystyle+ib_{3}n^{3}+12ib_{3}n^{2}+48ib_{3}n+64ib_{3}+b_{4}n^{4}+16b_{4}n^{3}+96b_{4}n^{2}+256b_{4}n+256b_{4}\Big{)}$
(B.30) $\displaystyle-
k_{1}\Big{(}3a_{1}-6ia_{2}n-18ia_{2}-9a_{3}n^{2}-54a_{3}n-81a_{3}+12ia_{4}n^{3}+108ia_{4}n^{2}+324ia_{4}n$
(B.31)
$\displaystyle+324ia_{4}+b_{1}-2ib_{2}n-8ib_{2}-3b_{3}n^{2}-24b_{3}n-48b_{3}+4ib_{4}n^{3}+48ib_{4}n^{2}+192ib_{4}n+256ib_{4}\Big{)}$
(B.32) $\displaystyle-
k_{1}^{2}\Big{(}3a_{2}-9ia_{3}n-27ia_{3}-18a_{4}n^{2}-108a_{4}n-162a_{4}+b_{2}-3ib_{3}n-12ib_{3}$
(B.33) $\displaystyle-6b_{4}n^{2}-48b_{4}n-96b_{4}\Big{)}$ (B.34)
$\displaystyle-
k_{1}^{3}(3a_{3}-12ia_{4}n-36ia_{4}+b_{3}-4ib_{4}n-16ib_{4})-k_{1}^{4}(3a_{4}+b_{4})\,,$
(B.35) $\displaystyle
A_{4}=\left(a_{0}-ia_{1}(n+4)-a_{2}(n+4)^{2}+ia_{3}(n+4)^{3}+a_{4}(n+4)^{4}\right)$
(B.36)
$\displaystyle+k_{1}\left(a_{1}-2ia_{2}(n+4)-3a_{3}(n+4)^{2}+4ia_{4}(n+4)^{3}\right)$
(B.37)
$\displaystyle+k_{1}^{2}\left(a_{2}-3ia_{3}(n+4)-6a_{4}(n+4)^{2}\right)+k_{1}^{3}(a_{3}-4ia_{4}(n+4))+a_{4}k_{1}^{4}\,.$
(B.38)

Substituting the expressions for $c_{n}$, …, $c_{n+3}$ into Eq. (B.9), we find
that $c_{n+4}$ satisfies (4.2) if and only if

$\displaystyle
A_{0}\,\text{DR}_{0}(\tilde{k}_{1}-i(n+1))\text{DR}_{0}(\tilde{k}_{1}-i(n+2))\text{DR}_{0}(\tilde{k}_{1}-i(n+3))\text{DR}_{0}(\tilde{k}_{1}-i(n+4))$
(B.39)
$\displaystyle+A_{1}\,\text{DR}(\tilde{k}_{1}-i(n+1))\text{DR}_{0}(\tilde{k}_{1}-i(n+2))\text{DR}_{0}(\tilde{k}_{1}-i(n+3))\text{DR}_{0}(\tilde{k}_{1}-i(n+4))$
(B.40)
$\displaystyle+A_{2}\,\text{DR}(\tilde{k}_{1}-i(n+1))\text{DR}(\tilde{k}_{1}-i(n+2))\text{DR}_{0}(\tilde{k}_{1}-i(n+3))\text{DR}_{0}(\tilde{k}_{1}-i(n+4))$
(B.41)
$\displaystyle+A_{3}\,\text{DR}(\tilde{k}_{1}-i(n+1))\text{DR}(\tilde{k}_{1}-i(n+2))\text{DR}(\tilde{k}_{1}-i(n+3))\text{DR}_{0}(\tilde{k}_{1}-i(n+4))$
(B.42)
$\displaystyle+A_{4}\,\text{DR}(\tilde{k}_{1}-i(n+1))\text{DR}(\tilde{k}_{1}-i(n+2))\text{DR}(\tilde{k}_{1}-i(n+3))\text{DR}(k_{1}-i(n+4))=0\,,$
(B.43)

which indeed is true for any $n$, as can be checked by direct algebra or using
Wolfram Mathematica. $\square$

##  Appendix C Physical consequences of assumption (5.8)

We define, as in the non-dispersive case and in the weakly dispersive one, the
refractive index to be

$n(x)=\sqrt{\chi(x)+1}.$ (C.1)

Also, it is useful to define $n_{0}^{2}:=\chi_{0}+1$. We can rewrite (5.8) as
follows:

$n^{2}(x)-1=\frac{n_{0}^{2}-1}{1-\frac{\lambda}{2}(n_{0}^{2}-1)(1-\tanh(\beta
x))}.$ (C.2)

We obtain

$\displaystyle\lim_{x\to+\infty}(n^{2}(x)-1)$ $\displaystyle=n_{0}^{2}-1,$
(C.3) $\displaystyle\lim_{x\to-\infty}(n^{2}(x)-1)$
$\displaystyle=\frac{n_{0}^{2}-1}{1-\lambda(n_{0}^{2}-1)}.$ (C.4)

In standard materials we expect $n^{2}>1$. As a consequence, (C.3) implies
$n^{2}>1$ as $x\to+\infty$ for $n_{0}^{2}>1$. The same request leads to
$1-\lambda(n_{0}^{2}-1)>0$, which means $\lambda<\lambda_{\text{sup}}$, where

$\lambda_{\text{sup}}:=\frac{1}{n_{0}^{2}-1}=\frac{1}{\chi_{0}}.$ (C.5)

We can also wonder if we are assuming a black hole condition (decreasing
$n(x)$) or a white hole one (increasing $n(x)$). Cf. e.g. [36]. Given our
monotone profile, we find that, by assuming, as we did, $\lambda>0$ we obtain
a black hole geometry. A white hole one would be allowed by a negative
$\lambda$. It is also to be noted that it is possible to satisfy both
$\lambda<\lambda_{\text{sup}}$ and $\lambda\gg\lambda_{\text{crit}}$, as in
the discussion following (5.61), indeed we have

$\lambda_{\text{crit}}=\lambda_{\text{sup}}\gamma^{2}(1-n_{0}^{2}\frac{v^{2}}{c^{2}}),$
(C.6)

so that, for $n_{0}^{2}$ very near $\frac{c^{2}}{v^{2}}-\delta$, for
$0<\delta\ll 1$, we get $\lambda_{\text{sup}}\gg\lambda_{\text{crit}}$, and
then $\lambda_{\text{sup}}>\lambda\gg\lambda_{\text{crit}}$ is allowed.

## References

  * [1] W.G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).
  * [2] R.Brout, S.Massar, R.Parentani and Ph. Spindel, Phys. Rev. D, 52, 4559 (1995).
  * [3] S. Corley, Phys. Rev. D, 57, 6280 (1998).
  * [4] Y. Himemoto, and T. Tanaka, Phys. Rev. D, 61, 064004 (2000).
  * [5] H. Saida, and M. Sakagami, Phys. Rev. D, 61, 084023 (2000).
  * [6] R. Schutzhold and W. G. Unruh, Phys. Rev. D 78, 041504 (2008).
  * [7] W.G. Unruh, and R. Schützhold, Phys. Rev. D 71, 024028 (2005).
  * [8] R. Balbinot, A. Fabbri, S. Fagnocchi and R. Parentani, Riv. Nuovo Cim. 28, 1 (2005).
  * [9] A. Coutant, R. Parentani and S. Finazzi, Phys. Rev. D 85, 024021 (2012).
  * [10] U. Leonhardt and S. Robertson, New J. Phys. 14, 053003 (2012).
  * [11] A. Coutant, A. Fabbri, R. Parentani, R. Balbinot and P. Anderson, Phys. Rev. D 86, 064022 (2012).
  * [12] A. Coutant and R. Parentani, Phys. Fluids 26, 044106 (2014).
  * [13] R. Schutzhold and W. G. Unruh, Phys. Rev. D 88, 124009 (2013).
  * [14] M. Petev, N. Westerberg, D. Moss, E. Rubino, C. Rimoldi, S. L. Cacciatori, F. Belgiorno and D. Faccio, Phys. Rev. Lett. 111, 043902 (2013).
  * [15] A. Coutant and R. Parentani, Phys. Rev. D 90, no. 12, 121501 (2014).
  * [16] F. Belgiorno, S. L. Cacciatori and F. Dalla Piazza, Phys. Rev. D 91, no. 12, 124063 (2015).
  * [17] M. F. Linder, R. Schutzhold and W. G. Unruh, Phys. Rev. D 93, no. 10, 104010 (2016).
  * [18] T. G. Philbin, Phys. Rev. D 94, no. 6, 064053 (2016).
  * [19] F. Belgiorno, S. L. Cacciatori, F. Dalla Piazza and M. Doronzo, Phys. Rev. D 96, no. 9, 096024 (2017).
  * [20] A. Coutant and S. Weinfurtner, Phys. Rev. D 94, no. 6, 064026 (2016).
  * [21] A. Coutant and S. Weinfurtner, Phys. Rev. D 97, no. 2, 025005 (2018).
  * [22] A. Coutant and S. Weinfurtner, Phys. Rev. D 97, no. 2, 025006 (2018).
  * [23] M. J. Jacquet, F. König, Physical Review A, 102(1), 013725 (2020).
  * [24] G. Rousseaux, C. Mathis, P. Maissa, T. G. Philbin and U. Leonhardt, New J. Phys. 10, 053015 (2008).
  * [25] F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino and V. G. Sala et al., Phys. Rev. Lett. 105, 203901 (2010).
  * [26] E. Rubino, F. Belgiorno, S. L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi and V. G. Sala et al., New J. Phys. 13, 085005 (2011).
  * [27] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh and G. A. Lawrence, Phys. Rev. Lett. 106, 021302 (2011).
  * [28] J. Chaline, G. Jannes, P. Maissa and G. Rousseaux, Lect. Notes Phys. 870, 145 (2013).
  * [29] S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh and G. A. Lawrence, Lect. Notes Phys. 870, 167 (2013).
  * [30] J. Steinhauer, Nature Phys. 10, 864 (2014).
  * [31] L.-P. Euvé, F. Michel, R. Parentani, T. G. Philbin and G. Rousseaux, Phys. Rev. Lett. 117, no. 12, 121301 (2016).
  * [32] J. R. Muñoz de Nova, K. Golubkov, V. I. Kolobov and J. Steinhauer, Nature 569, no. 7758, 688 (2019).
  * [33] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. Leonhardt, Phys. Rev. Lett. 122, 010404 (2019).
  * [34] L. P. Euvè, S. Robertson, N. James, A. Fabbri and G. Rousseaux, Phys. Rev. Lett. 124, no. 14, 141101 (2020).
  * [35] J. Fourdrinoy, S. Robertson, N. James, A. Fabbri and G. Rousseaux, Phys. Rev. D 105, no.8, 085022 (2022).
  * [36] F. Belgiorno, S. L. Cacciatori and A. Viganò, Phys. Rev. D 102, no. 10, 105003 (2020).
  * [37] F. Belgiorno, S. L. Cacciatori, A. Farahat and A. Viganò, Phys. Rev. D 102, no. 10, 105004 (2020).
  * [38] F. Belgiorno and S. L. Cacciatori, UNIVERSE 6, 127 (2020).
  * [39] S. Trevisan, F. Belgiorno and S.L. Cacciatori, Phys. Rev. D 108, 025001 (2023).
  * [40] Michel, Parentani, Phys. Rev. D 90, 044033 (2014).
  * [41] Y. Haraoka, Linear Differential Equations in the Complex Domain, Springer International Publishing, 2020.
  * [42] E.A.Coddington, N.Levinson, Theory of Linear Differential Equations. McGraw Hill, New York (1955).
  * [43] A.R.Forsyth,The Theory of Differential Equations. Part III. Ordinary Differential Equations. Vol. IV. Cambridge: at the University Press (1902).
  * [44] T.Craig, A Treatise on Linear Differential Equations. Vol. I. John Wiley and Sons, New York (1889).
  * [45] A .Castro, J.M. Lapan, A. Maloney, M.J. Rodriguez, Class. Quantum Grav. 30 (2013) 165005.
  * [46] B.C. da Cunha, F. Novaes, Journal of High Energy Physics volume 2015, 144 (2015).
  * [47] T. Oshima, Fractional Calculus of Weyl Algebra and Fuchsian Differential Equations, MSJ Meomoirs, vol. 28 (Mathematical Society of Japan, Tokyo, 2012).
  * [48] K. Aomoto, M. Kita, Theory of hypergeometric functions, Springer Tokyo (2011).
  * [49] F. Belgiorno, S. L. Cacciatori and F. Dalla Piazza, Phys. Scripta 91, no.1, 015001 (2016).
  * [50] S. Finazzi and I. Carusotto, Phys. Rev. A 87, 023803 (2013).