††thanks: Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short
Correlations in Singlets and Doublets

T. Araújo Lima  Departamento de Física, Universidade Federal Rural de
Pernambuco, Recife, PE 52171-900, Brazil  R. B. do Carmo  Instituto Federal de
Alagoas, Piranhas, AL 57460-000, Brazil

###### Abstract

Billiards are flat cavities where a particle is free to move between elastic
collisions with the boundary. In chaos theory these systems are simple
prototypes, their conservative dynamics of a billiard may vary from regular to
chaotic, depending only on the border. The results reported here seek to shed
light on the quantization of classically chaotic systems. We present numerical
results on classical and quantum properties in two bi-parametric families of
Billiards, Elliptical Stadium Billiard (ESB) and Elliptical-$C_{3}$ Billiards
(E-$C_{3}$B). Both are elliptical perturbations of chaotic billiards with
originally circular sectors on their borders. Our numerical calculations show
evidence that the elliptical families can present a mixed classical phase
space, identified by a parameter $\rho_{\text{c}}<1$, which we use to guide
our analysis of quantum spectra. We explored the short correlations through
nearest neighbor spacing distribution $p(s)$, which showed that in the mixed
region of the classical phase space, $p(s)$ is well described by the Berry-
Robnik-Brody (BRB) distributions for the ESB. In agreement with the expected
from the so-called ergodic parameter $\alpha=t_{\text{H}}/t_{\text{T}}$, the
ratio between the Heisenberg time and the classical diffusive-like transport
time signals the possibility of quantum dynamical localization when
$\alpha<1$. For the E-$C_{3}$B family, the eigenstates can be split into
singlets and doublets. BRB describes $p(s)$ for singlets as the previous
family in the mixed region. However, the $p(s)$ for doublets are described by
new distributions recently introduced in the literature but only tested in a
few cases for $\rho_{\text{c}}<1$. We observed that as $\rho_{\text{c}}$
decreases, the $p(s)$’s tend to move away simultaneously from the GOE
(singlets) and GUE (doublets) distributions.

Billiards. Chaos. Quantization. GOE. GUE.

††preprint: APS/123-QED

##  I Introduction

The idea that molecules may be behind Thermodynamics (grounded in Statistical
Mechanics) was one of the tremendous scientific advances of the 19th century.
In particular, these particles, constituents of gases, are associated with the
concept of ergodicity, then called _molecular chaos_. The word ergodic came
from the Greek _ergon_ (work) and _odos_ (trajectory) and was used by
Boltzmann to represent the hypothetical visit to all points of the phase space
by a particle of that gas with random microscopic dynamic behavior. The
introduction of the probability in theory that came to be called Statistical
Mechanics of Equilibrium passed by a long probationary regime, with more
convincing results occurring only in the first decades of the 20th century
[1]. The so-called Ergodic Hypothesis only gained the rigor of a theorem with
the work of the Russian mathematician Y. Sinai in the 60s-70s for an ideal gas
of only two particles [2]. A system is chaotic if two neighboring trajectories
in the phase space separate exponentially. Suppose the distance in phase space
between such trajectories is proportional to $e^{\lambda t}$. The $\lambda$
parameter is called the Lyapunov exponent. In reality, $\lambda$ represents
the greatest of Lyapunov’s exponents. Therefore, the existence of at least one
positive Lyapunov exponent characterizes a chaotic system [3]. Billiards
systems are prototypes in the study of chaos and describe the free movement of
a point particle in a closed domain $\Omega$ with elastic reflections on the
boundary $\partial\Omega$ of the domain. The nature of this conservative
dynamical system depends exclusively on the shape of the border
$\partial\Omega$, varying from entirely regular (i.e., ellipses and annular
concentric regions) to completely chaotic (i.e., Sinai billiard). Without loss
of generality, we consider that the particle has mass $m=1$ and velocity of
module $|\bm{v}|$ $=1$. A discrete dynamics well describes this 2-dimensional
motion in time on variables $(\ell,\phi)$, the fraction of perimeter of
$\partial\Omega$, and the incidence angle where a collision happens
parametrizes the discrete-time generally [4]. A primordial example that
deserves to be mentioned here is the Bunimovich stadium. This billiard can
present $\lambda>0$. Its shape consists of two semicircles joined by two
finite-size segments $2t$, forming a stadium. It is chaotic for any $t>0$. In
a pure circular billiard, collisions keep the angular momentum in relation to
its center (focus) constant. The Bunimovich stadium does not present this
property in its dynamics, known as a defocusing [5]. It is hugely relevant to
this work because the Elliptical Stadium Billiard is a perturbation of it,
resulting in classical dynamics with mixed phase space.

Quantum mechanics has been one of the best-tested physical theories since its
emergence. The theory makes excellent predictions not only for the atom of
hydrogen, which is classically integrable, as well as the helium atom, which
is classically not integrable. Nothing is more natural than whether there is
an effect analogous to chaos in quantum mechanics. The term _quantum chaos_ is
generally understood as studying the quantum behavior of classically chaotic
systems [6]. One commonly used means of studying these systems is to
statistically characterize spectral properties in the semiclassical regime and
compare them with results from the random matrices theory [7].

In billiards, obtaining the energy spectrum is an essential step for analysis.
The problem is to solve the time-independent Schrödinger equation with null
potential in the planar region $\Omega$ with Dirichlet boundary conditions at
$\partial\Omega$:

$\left\\{\begin{array}[]{l}\nabla^{2}\varphi_{n}(\bm{r})=-k_{n}^{2}\varphi(\bm{r}),\quad\bm{r}\in\Omega\\\
\varphi_{n}(\bm{r})=0,\quad\bm{r}\in\partial\Omega,\end{array}\right.$ (1)

expression is also known as the Helmholtz Equation [8]. Where
$k_{n}^{2}=2mE_{n}/\hslash^{2}$. In order to characterize universality, one
must first unfold the energy spectrum $\\{E_{n}\\}$ so that a unit means
($\langle s_{n}\rangle=1$) nearest neighbor spacing (nns)
$s_{n}=E_{n+1}-E_{n}$ is obtained. This approach became relevant after two
important conjectures. Namely, the Berry-Tabor (BT) conjecture [9] and the
Bohigas-Giannoni-Schmit (BGS) conjecture [10]. The BT conjecture states that,
in the semiclassical limit, the statistical properties of the energy spectrum
of a classically integrable system must correspond to the prediction of
uncorrelated randomly distributed energy levels. As a result, the
semiclassical nns distribution $p(s)$ must obey Poisson:

$p_{\text{P}}(s)=\exp(-s).$ (2)

On the other hand, according to the BGS conjecture, in the case of a
classically chaotic system, the spectral properties must follow the universal
statistics of the eigenvalues of Gaussian random matrices [7]. Several recent
works have improved the turnover of the BGS conjecture in a theorem [11, 12,
13, 14]. These proofs still have controversies and limitations pointed out by
some authors [15, 16]. The terminology ”BGS conjecture” fits the current
article for quantized billiards. More recently, in [17], the conjecture was
extended to purely ergodic systems. If one disregards spin, in the presence
(absence) of time-reversal symmetry, $p(s)$ must correspond to that of the
GOE, _Gaussian Orthogonal Ensemble_ (GUE, _Gaussian Unitary Ensemble_):

$\left\\{\begin{array}[]{l}p_{\text{GOE}}(s)=(\pi/2)s\exp(-\pi s^{2}/4),\\\
p_{\text{GUE}}(s)=(32/\pi^{2})s^{2}\exp(-4s^{2}/\pi).\end{array}\right.$ (3)

Based on these assumptions, Leyvraz, Schmit, and Seligman (LSS) [18] predicted
and tested numerically that chaotic billiards with only a three-fold ($C_{3}$)
symmetry (without reflection symmetry) have doublets with spectral statistics
of the GUE type, although billiards are by time reversal. LSS considered a
billiard consisting of three straight segments of an equilateral triangle with
rounded corners by two circumferences of different radii, here called
Circular-$C_{3}$ Billiard (C-$C_{3}$B). In particular, LSS showed results for
a double ratio between the radii where there is a satisfactory agreement for
$p(s)$ with the GUE statistics, for a total of approximately 800 doublets.
Later, C. Dembowski _et al._ used microwave billiards with $C_{3}$ symmetry to
check experimentally the result predicted by LSS. Besides, they showed that
singlets follow GOE [19].

The BT [20], BGS [21, 22, 23] and LSS conjectures have been investigated in
the literature, but there have been comparatively fewer studies on the LSS
findings [24, 25, 26, 27]. Until now, little has been said about the
situations of $C_{3}$ symmetric billiards with mixed classical phase space,
where chaotic sea and stable KAM-islands coexist. Here, we propose to shed
light on the quantum properties of billiards with mixed classical phase space.
For this, we perform numerical calculations on the energy spectra of two bi-
parametric families of billiards with elliptical sectors on their boundaries
and analyze the short correlations. The first one is the Elliptical Stadium
Billiard (ESB), a perturbation of the Bunimovich Stadium, whose mixed
classical phase space was studied in [28, 29]. In sequence, we introduce a
perturbation of the C-$C_{3}$B, replacing the circumferences with ellipses.
Recently, billiards with elliptical borders have been studied in other
contexts, i.e., in singular potentials [30], in relativistic limits [31], and
flows that move around chaotic cores [32]. We start the analysis by presenting
the billiards, discussing their classical dynamics, showing some mixed phase
spaces, and calculating the fraction of the chaotic sea on these phase spaces.
Finally, we follow with the quantized billiards’ spectral properties,
investigating the nns distribution $p(s)$ with formulas for intermediate
quantum statistics derived for the doublets recently [27].

##  II The bi-parametric billiards families and Classical Dynamics

The billiards systems studied in this work belong to two bi-parametric
families, the Elliptical Stadium Billiards (ESB) and Elliptical-$C_{3}$
Billiards (E-$C_{3}$B). The first one consists of a perturbation of the
Bunimovich Stadium. It comprises two half-ellipses (major semi-axis $a$ and
minor semi-axis $1$) that bracket a rectangular sector of thickness $2t$ and
height $2$. [28] showed that in the region $a\in(1,\sqrt{2})$ and
$t\in(0,\infty)$ are possible to find chaotic dynamics or a mixed phase space
depending on the parameters. In [29] is presented a critical behavior of the
billiard dynamics near a transition curve, $t(a)=\sqrt{a^{2}-1}$ for the
interval $a\in(1,\sqrt{4/3})$. Based on these previous works, we focus our
analysis on this last interval and $t\in(0,1/\sqrt{3})$. The E-$C_{3}$B is
based on C-$C_{3}$B, but ellipses instead of circumferences curve the corners.
The larger (smaller) ellipse has $A_{\text{e}}$ $(a_{\text{e}})$ and
$B_{\text{e}}$ $(b_{\text{e}})$ semi-axes. In all cases described here, the
relations $A_{\text{e}}=2a_{\text{e}}$ and $B_{\text{e}}=2b_{\text{e}}$ are
maintained, with $a_{\text{e}}$ and $b_{\text{e}}$ in the range
$(0,\sqrt{3}/6)$. The LSS billiard is reproduced with
$a_{\text{e}}=b_{\text{e}}=\sqrt{3}/12$. Here, by our knowledge, we present
for the first time a perturbation on the C-$C_{3}$B resulting in a system that
shows a mixed phase space.

A Fundamental Domain (FD) is a neighborhood in $\Omega$ that contains only one
image for any point in the system. Besides the boundary of the
$\partial\Omega$, there are additional boundaries between adjacent FDs, which
are the symmetry lines. Classically, billiard dynamics can always be reduced
to a FD by assuming specular reflections at the symmetry lines [33, 34]. For
this, we use the FD of each billiard in our calculations on the classical
dynamics. In Fig. 1, we graph billiards in families indicating the parameters
and their respective FDs.

Figure 1: (a) Original Boundaries of Elliptical Stadium Billiard and
Elliptical-$C_{3}$ Billiard. For ESB, the symmetry lines are referent to
reflections on vertical and horizontal axes. While E-$C_{3}$B are referent to
$120^{\circ}$ rotational axes. (b) Fundamental Domains of ESB and E-$C_{3}$B.
The symmetry lines are replaced by additional boundaries forming the planar
region where we analyze these billiards’ classical dynamics.

The global dynamical properties of the ESB with unit mass and velocity may be
characterized through collisions of orbits with the vertical side of its FD
shown in Fig. 1. An additional part of the boundary dictates this edge and
does not change with the variation of parameters. The reduced phase space is
then a rectangle defined by the vertical position $y$, where a collision
occurs at discrete time $n$, and the tangent component of the velocity in a
collision, $v_{y}$, with $0<y<1$ and $-1<v_{y}<1$. The small gray dots in Fig.
2 show the phase plane for some values of parameters $(a,t)$ after $n=10^{5}$
collisions from the initial conditions (ICs), clearly exhibiting a mixed
(regular-irregular) characteristic. We plot one example of a stable trajectory
in red for each one. Quantitative characterization of these mixed-phase spaces
can be made through the chaotic (regular) fraction $\rho_{\text{c}}$
($\rho_{\text{r}}$) of each phase portrait with
$\rho_{\text{c}}+\rho_{\text{r}}=1$ and $0\leqslant\rho_{\text{c}}\leqslant
1$. The phase plane is partitioned into $N_{\text{c}}$ small disjoint cells to
measure these quantities [35, 36, 37, 29, 27]. For a given orbit, let $N(n)$
be the number of different cells in the phase space, which are visited up to
$n$ impacts in the cross-section. The relative measure $r(n)$ is defined as
the fraction of visited cells averaged over a set of ICs, i.e., $r(n)=\langle
N(n)\rangle/N_{\text{c}}$. So the chaotic fraction of the phase space is
obtained via

$\rho_{\text{c}}=\lim_{n\to\infty}\lim_{N_{\text{c}}\to\infty}r(n),$ (4)

for ICs in the chaotic sea. In our numerical approach, we consider
$N_{\text{c}}=10^{6}$, $n=2\cdot 10^{7}$, and averages in 20 random ICs. For
the billiards with mixed phase space in Fig. 2,
$\rho^{(\text{a})}_{\text{c}}=0.991884$ and
$\rho^{(\text{b})}_{\text{c}}=0.857184$. The left panel of Fig. 3 shows a
numerical diagram of $\rho_{\text{c}}$. The ergodic property
$(\rho_{\text{c}}=1)$ is numerically guaranteed in black regions. This diagram
also supports previous works [28, 29], where a critical transition from a
mixed phase space to a fully ergodic was found to cross a critical line
$t(a)=\sqrt{a^{2}-1}$.

Figure 2: Upper Panels: ESB boundaries for some values of parameters $(a,t)$
with stable trajectories in red. Lower Panels: corresponding phase portraits
for $10^{5}$ collisions with the vertical boundary from the IC
$(y_{0},v_{y0})=(0.5,0.0)$ (small gray dots). The red plots correspond to the
trajectories in the upper panels. The mixed phase spaces present
$\rho^{(\text{a})}_{\text{c}}=0.991884$ and
$\rho^{(\text{b})}_{\text{c}}=0.857184$. Figure 3: Left Panel: diagram of the
chaotic fraction of the phase space $\rho_{\text{c}}$ for the ESB. The tiny
green line is the critical line $t(a)=\sqrt{a^{2}-1}$ studied in [28, 29].
Right Panel: same diagram for the E-$C_{3}$B showing a distinguished phase
space behavior depending on the parameters. The ergodic property
$(\rho_{\text{c}}=1)$ is numerically guaranteed in black regions. These maps
will guide us in exploring quantum properties, where these values will be
relevant parameters to our analysis.

The E-$C_{3}$B’s classical dynamical properties will be studied in the same
way but are characterized through the collisions of the orbits with the
horizontal side of its FD shown in Fig. 1, which does not change with the
variation of parameters. The reduced phase space is then a rectangle defined
by the horizontal position $x$, and the tangent component of the velocity in a
collision, $v_{x}$, with $0<x<\sqrt{3}/3$ and $-1<v_{x}<1$. The small gray
dots in Fig. 4 show the phase plane for some values of parameters
$(a_{\text{e}},b_{\text{e}})$ after $n=2\cdot 10^{7}$ collisions from the ICs,
clearly exhibiting mixed (regular-irregular) characteristic. The values of
chaotic fraction are $\rho^{(\text{a})}_{\text{c}}=0.935152$ and
$\rho^{(\text{b})}_{\text{c}}=0.800792$. The right panel of Fig. 3 shows a
numerical diagram of $\rho_{\text{c}}$. The ergodic property
$(\rho_{\text{c}}=1)$ is numerically guaranteed in black regions. This map
will guide us in exploring quantum properties described in the next section,
where these values will be relevant parameters to our analysis.

Figure 4: Upper Panels: E-$C_{3}$B boundaries for some values of parameters
$(a_{\text{e}},b_{\text{e}})$ with stable trajectories in red. Lower Panels:
corresponding phase portraits for $2\cdot 10^{7}$ collisions with the
horizontal boundary from the IC $(x_{0},v_{x0})=(0.5,0.0)$ (small gray dots).
The red plots correspond to the trajectories in the upper panels. The mixed
phase spaces present $\rho^{(\text{a})}_{\text{c}}=0.935152$ and
$\rho^{(\text{b})}_{\text{c}}=0.800792$.

##  III Quantization and Eigenvalues Short Correlations

All Energy spectra $\\{E_{n}\\}$ of eq. (1) were calculated with an algorithm
based on the scaling method introduced by E. Vergini and M. Saraceno (VS) in
[38]. This approach allows us to access high-lying energy eigenvalues that
have been unfolded to obtain a unit mean spacing ($\langle s_{n}\rangle=1$)
for each billiard. Our results are based on sets of approximately 70,000
eigenvalues for a given pair of parameters. According to [6], there is
possibly no more intensely studied spectral statistics more than $p(s)$, the
density of probability of finding two levels nearest neighbor spaced by $s$.

###  III.1 The Singlets Case

Initially, we focused on results for ESB. Some proposes have been made to
describe these distributions for systems whose present mixed-phase space on
its classical counterpart. Here we focus on two of them. They result in
intermediate formulas between Poisson and GOE statistics through parameters
variation. Firstly, we cite the purely phenomenologic approach by Brody [39],
where an exponent $\nu$ is gradually varied to obtain a smooth change between
the integrable ($\nu=0$) and chaotic ($\nu=1$) cases:

$p_{\text{B}}(s)=a_{\nu}(\nu+1)s^{\nu}\exp\left[-a_{\nu}s^{(\nu+1)}\right],$
(5)

where $a_{\nu}=\left[\Gamma\left(\frac{\nu+2}{\nu+1}\right)\right]^{\nu+1}$
and $\Gamma(x)$ is the Gamma function. The second distribution cited here is
the Berry-Robnik-Brody (BRB), a proposal that takes under consideration the
chaotic (regular) fraction of the classical phase space $\rho_{\text{c}}$
($\rho_{\text{r}}$) [40]:

$p_{\text{BRB}}(s)=\\\
\exp(-\rho_{\text{r}}s)\left\\{\frac{\rho_{\text{r}}^{2}}{(\beta+1)\Gamma\left(\frac{\beta+2}{\beta+1}\right)}Q\left[\frac{1}{\beta+1};a_{\beta}(\rho_{\text{c}}s)^{\beta+1}\right]+\right.\\\
\left.[2\rho_{\text{r}}\rho_{\text{c}}+(\beta+1)a_{\beta}\rho_{\text{c}}^{\beta+2}s^{\beta}]\exp[-a_{\beta}(\rho_{\text{c}}s)^{\beta+1}]\right\\}.$
(6)

As in the Brody distribution,
$a_{\beta}=\left[\Gamma\left(\frac{\beta+2}{\beta+1}\right)\right]^{\beta+1}$
and $Q(\kappa;x)$ is the Incomplete Gamma function. This distribution can go
through other distributions varying the free parameters $\rho_{\text{c}}$ and
$\beta$. For $\beta=0$, $p_{\text{BRB}}(s)=p_{\text{P}}(s)$ and for $\beta=1$
it recovers the distribution of Berry-Robnik (BR) [41]. If
$\rho_{\text{c}}=0$, $p_{\text{BRB}}(s)=p_{\text{P}}(s)$ again, while for
$\rho_{\text{c}}=1$, $p_{\text{BRB}}=p_{\text{B}}(s)$.

The nns for ESB were previously studied in [22] with around 3,000 eigenvalues
of eq. (1). We use the VS method to obtain around 65,000 eigenvalues beyond
the first 5,000. The BRB distribution can fit all $p(s)$ obtained for all
parameters tested on ESB. We have two independent parameters for this
distribution, $\rho_{\text{c}}$, and $\beta$. However, we fixed
$\rho_{\text{c}}$ at the value obtained in the diagram of Fig. 3. The upper
panels of Fig. 5 shows representative results. The chaotic case presents
$\beta=1.000\pm 0.020$, the GOE distribution. The mixed
($0<\rho_{\text{c}}<1$) present $\beta=0.978\pm 0.018$ and $\beta=0.191\pm
0.014$, intermediate distributions between Poisson and GOE. These results go
in the direction of the quantum localization, previously studied in other
billiards systems [42, 43] and discussed next.

Figure 5: Representative results for BRB distributions fits for $p(s)$. Upper
Panels: results on ESB with $a=1.04$ and some values of $t$. The chaotic case
$t=0.287$ $(\rho_{\text{c}}=1)$ presents $\beta=1.000\pm 0.020$, the GOE
distribution. The mixed cases $t=0.15$ and $t=0.01$ ($0<\rho_{\text{c}}<1$)
present $\beta=0.978\pm 0.018$ and $\beta=0.191\pm 0.014$ respectively,
intermediate distributions between Poisson and GOE. Lower Panels: results on
E-$C_{3}$B with some values of $(a_{\text{e}},b_{\text{e}})$. The chaotic
case, $(a_{\text{e}},b_{\text{e}})=(0.2,0.25)$ ($\rho_{\text{c}}=1$) presents
$\beta=1.000\pm 0.097$, the GOE distribution. The mixed cases,
$(a_{\text{e}},b_{\text{e}})=(0.2784,0.256)$ and
$(a_{\text{e}},b_{\text{e}})=(0.2886088,0.281522)$ ($0<\rho_{\text{c}}<1$)
present $\beta=0.999\pm 0.057$ and $\beta=0.203\pm 0.073$ respectively, in the
range of intermediate distributions between Poisson and GOE. The fits with the
Brody formula and BRB distribution are indistinguishable in both billiards
families.

Quantum dynamical localization corresponds to a peculiar quantum distribution
of the linear or angular momentum peaked at zero, with walls that decay
exponentially, differently from the classical results, which predicts, for a
chaotic or disordered system, a diffusive transport [44]. The phenomenon can
be reviewed in [45]. An interesting feature of the quantum dynamical
localization is that it allows us to estimate the conditions under which the
comparison with the standard random matrix theory is adequate or, in other
words, whether an energy eigenvalues data set belongs to the deep
semiclassical regime. We follow closely [42] in the short description below.
The key idea is to express the ergodic parameter
$\alpha=t_{\text{H}}/t_{\text{T}}$, where $t_{\text{H}}$ is the (quantum)
Heisenberg time, and $t_{\text{T}}$ is the (classical) transport time, in
terms of accessible magnitudes, such as the (quantum) energy $E$ and the
(classical) number of collisions off the billiard border, $N_{\text{T}}$. From
[42] the ratio is expressed as

$\alpha=\frac{k\mathcal{L}}{\pi N_{\text{T}}},$ (7)

where $\mathcal{L}$ is the perimeter of the boundary and $k^{2}\sim E$. The
condition for quantum dynamical localization in a given energy spectrum,
$\alpha\leqslant 1$, can then be written as $k\leqslant k_{\text{c}}=\pi
N_{\text{T}}/\mathcal{L}$. To estimate $N_{\text{T}}$, we consider an ensemble
of orbits initially directed perpendicularly to $\partial\Omega$ and follow
its random spreading as a function of the discrete time $n$. The symbols in
Fig. 6 illustrate the results for the mean square momentum $\langle
p^{2}\rangle$ as a function of $n$ in a monolog scale (averaged in sets of
$10^{3}$ randomly chosen ICs) for members of two billiards family. Saturation
of $\langle p^{2}\rangle$ occurs at different times $N_{\text{T}}$ depending
on parameters. For the ESB family, all calculated spectra have
$k_{max}\lesssim k_{\text{c}}$ as the largest eigenvalue, equivalent to the
70,000th level at least. These facts are in agreement with the intermediate
statistics well fitted with eq. (6) as in [40, 42, 23, 27]. The same occurs
for the singlets in the E-$C_{3}$B family, where the condition
$k_{max}\lesssim k_{\text{c}}$ is equivalent to the 70,000th level. The
representative results are in the lower panels of Fig. 5. The chaotic case
presents $\beta=1.000\pm 0.097$, the GOE distribution. The mixed
($0<\rho_{\text{c}}<1$) present $\beta=0.999\pm 0.057$ and $\beta=0.203\pm
0.073$, in the range of intermediate distributions between Poisson and GOE. In
the next section, we discuss the doublets subspace.

Figure 6: Calculated mean square of the momentum as a function of the discrete
time $n$ in a monolog scale (number of collisions of the particle off the
billiard boundary). Lines are guides for the eyes. Upper panel: results for
members of the ESB family with $a=1.04$. The red dots are for $t=0.287$ and
present saturation at $N_{\text{T}}\simeq 7.10^{2}$. Blue dots are for
$t=0.15$, and saturation at $N_{\text{T}}\simeq 2.10^{3}$, and black dots are
for $t=0.01$ presenting $N_{\text{T}}\simeq 6.10^{2}$. Lower Panel: same
calculations for the E-$C_{3}$B, the red dots are for
$(a_{\text{e}},b_{\text{e}})=(0.2,0.25)$ and present saturation at
$N_{\text{T}}\simeq 3.10^{2}$. Blue dots are for
$(a_{\text{e}},b_{\text{e}})=(0.2784,0.256)$ and saturation at
$N_{\text{T}}\simeq 7.10^{2}$, and, black dots are for
$(a_{\text{e}},b_{\text{e}})=(0.2886088,0.281522)$ presenting
$N_{\text{T}}\simeq 2.10^{3}$.

###  III.2 The Doublets Case

Consider a classically chaotic system with time-reversal (TR) invariance and a
point-group (PG) symmetry. If the TR and the PG operations do not commute,
non-self-conjugate invariant subspaces of the PG must exhibit GUE spectral
fluctuations instead of GOE ones [18]. For example, consider a billiard in the
$xy$ plane with the $C_{3}$ symmetry. Such a billiard has eigenfunctions
$\varphi_{m}$ $(m=-1,0,+1)$, such that $\varphi_{0}$ is symmetric and repeats
itself after a rotation of $2\pi/3$ about the symmetry axis, whereas
$\varphi_{\pm 1}$ will be repeated only after three consecutive rotations of
$2\pi/3$. In other words, if $R(2\pi/3)$ is the rotation operator for an angle
of $2\pi/3$, one has $R(2\pi/3)\varphi_{m}=\exp(i\frac{2\pi}{3}m)\varphi_{m}$.
Let $\Theta$ be the time reversal operator. $\Theta$ is an antiunitary
operator that commutes with the Hamiltonian $H$, which has eigenvalue $E_{m}$,
_i.e._ , $H\varphi_{m}=E_{m}\varphi_{m}$. It follows that
$H\Theta\varphi_{m}=\Theta H\varphi_{m}=E_{m}\Theta\varphi_{m}$
($\Theta\varphi_{m}$ is also an eigenfunction of $H$ with the same eigenvalue
$E_{m}$). Are $\varphi_{m}$ and $\Theta\varphi_{m}$ the same eigenstate? For
this subspace one may write $\Theta\varphi_{m}=(-1)^{m}\varphi_{-m}$. Thus,
$\Theta\varphi_{0}=\varphi_{0}$, _i.e._ , $\varphi_{0}$ is a singlet. The top
panels in Fig. 7 show cases of the probability density $|varphi_{0}|^{2}$. On
the other hand, $\varphi_{1}$ and $\varphi_{-1}$ must correspond to distinct
states. One refers to this doublet state as a Kramers degeneracy. The middle
panels in Fig. 7 show the real and imaginary parts of the member $\varphi_{1}$
of a doublet, say $(\varphi_{1},\varphi_{-1})$, in the same billiard. The
probability density $|\varphi_{1}|^{2}$ recovers the $C_{3}$ symmetry
(rightmost middle panel in Fig. 7). The bottom panels in Fig. 7 show the same
state under the application under rotation operator $R(2\pi/3)$. A complex
conjugation of the shown state obtains the other member $\varphi_{-1}$ of the
doublet. Since these degenerate states are not TR invariant, they must follow
the GUE of random matrices, providing the billiard is classically chaotic,
according to the LSS results.

Figure 7: Top panels: Density plots of squared eigenfunctions corresponding to
singlet states in the LSS billiard, exhibiting the underlying $C_{3}$
symmetry. In the color scale, $|\varphi_{1,a}|^{2}$ is the maximum probability
in each case. Middle panels: Real and imaginary parts of a member
$\varphi_{1}$ of a doublet. In the color scale, $\pm\varphi_{1,a}$ is the
minimum and maximum of the wave function. The probability density recovers the
$C_{3}$ symmetry (right panels). Bottom panels: Same state in the middle under
the application of the rotation operator $R(2\pi/3)$.

For the E-$C_{3}$B, the degenerate states remain invariant to TR. However, the
spectral distribution will be changed for cases where the classical dynamics
is not completely chaotic ($\rho_{\text{c}}<1$), with a $p(s)$ resultant that
deviates from the GUE case. Thus, it is necessary to use new intermediate
formulas to study the distribution of doublets in billiards with mixed
classical phase space. The following formulas we derived in [27]. Following
the same steps in [39] led to the eq. (5), a Brody-like formula for the
transition between the Poisson and GUE distributions is obtained, namely,

$p_{\text{B,}2}(s)=(\eta+1)b_{\eta}^{2}s^{2\eta}\exp\left(-b_{\eta}s^{\eta+1}\right),$
(8)

where

$b_{\eta}=\left[\Gamma\left(\frac{2\eta+1}{\eta+1}\right)\right]^{-(\eta+1)},$
(9)

and $0\leqslant\eta\leqslant 1$. For $\eta=0$, $p_{\text{B},2}(s)$ reduces to
the Poisson distribution, whereas for $\eta=1$, the Wigner distribution for
the GUE is obtained. In [40], the dynamical localization of chaotic
eigenstates was taken into account and their coupling with the regular ones
through tunneling effects. The so-called BRB distribution previously discussed
in sec. III.1. Following this, the formula that corresponds to the Poisson
$\leftrightarrow$ GUE crossover is

$p_{\text{BRB,2}}(s)e^{\rho_{\text{r}}s}=\\\
\rho_{\text{r}}\rho_{\text{c}}b_{\gamma}^{\frac{1}{\gamma+1}}\left(2-\rho_{\text{r}}s\right)Q\left[\frac{1+2\gamma}{1+\gamma};b_{\gamma}\left(\rho_{\text{c}}s\right)^{\gamma+1}\right]+\\\
\bigg{[}\rho_{\text{r}}^{2}\left(1+b_{\gamma}\rho_{\text{c}}^{\gamma+1}s^{\gamma+1}\right)+\\\
(1+\gamma)\left(\rho_{2}^{\gamma+1}b_{\gamma}s^{\gamma}\right)^{2}\bigg{]}e^{-b_{\gamma}\left(\rho_{\text{c}}s\right)^{\gamma+1}},$
(10)

where $b_{\gamma}$ is defined as in eq. (9) and $Q(\kappa;x)$ is the
incomplete Gamma function. Here, $p_{\text{BRB,2}}(s)=p_{\text{P}}(s)$ if
$\rho_{\text{r}}=1$ or if $\gamma=0$, and
$p_{\text{BRB,2}}(s)=p_{\text{B,}2}(s)$ if $\rho_{\text{r}}=0$. In [27], the
above formula was widely tested only in the regime of full ergodicity
(polygonal cases) and in a single case with $\rho_{\text{c}}<1$. Here, we
detail a non-polygonal billiards family that produces a wide variability of
$\rho_{\text{c}}$ values. In these cases, $p_{\text{BRB,2}}$ well-fitted
distributions of nns for $\rho_{\text{c}}<1$ for all investigated cases. The
representative results are in Fig. 8. As in the previous section, the doublets
subspace is in the region of the spectrum such that $k\lesssim k_{\text{c}}$,
equivalent to 60,000th level.

Figure 8: Representative results for BRB-like distributions, eq. (10), fits
for $p(s)$ in doublets subspace for same members of E-$C_{3}$B family of Fig.
5. In panel (a), the chaotic case $(a_{\text{e}},b_{\text{e}})=(0.2,0.25)$
($\rho_{\text{c}}=1$) presents $\gamma=0.960\pm 0.050$, in the range of a GUE
distribution. In panels (b) and (c), the mixed cases
$(a_{\text{e}},b_{\text{e}})=(0.2784,0.256)$ and
$(a_{\text{e}},b_{\text{e}})=(0.2886088,0.281522)$ ($0<\rho_{\text{c}}<1$)
present $\gamma=1.000\pm 0.032$ and $\gamma=1.00\pm 0.13$ respectively, in the
range of intermediate distributions between Poisson and GUE. Fits with Brody-
like formula (8), and BRB-like distribution (10), are indistinguishable.

##  IV Conclusions ans Perspectives

This paper presents numerical results on classical dynamics and quantization
in two bi-parametric billiard families. The ESB comprises two ellipses of
minor semi-axe unitary, major semi-axe $a$, and a rectangular region of length
$2t$ [28, 29]. The other family, introduced here as E-$C_{3}$B, presents the
$C_{3}$ symmetry [18, 25, 27] and is formed by an equilateral triangle with
rounded corners by two ellipses with semi-axis $A_{\text{e}}=2a_{\text{e}}$
and $B_{\text{e}}=2b_{\text{e}}$. First, we investigate the classical dynamics
of these billiards where we built detailed diagrams for the chaotic fraction
$(\rho_{\text{c}})$ of their phase spaces. After that, we investigated the nns
distributions $p(s)$ for these systems, a measure of short correlations. In
the asymmetric ESB family, the parameters space region $(a,t)$ where the
classical phase space is mixed (regular and chaotic regions coexist), all
found statistics present intermediated results between Poisson and GOE
distributions. The BRB distribution [40], eq. (6), very well fitted all cases.
These results perfectly agree with the expected from the ergodic parameter
$\alpha$ that signals the possibility of quantum dynamical localization when
$\alpha<1$. All sets of eigenvalues used as data are in a range of energy that
satisfies this condition. In the E-$C_{3}$B family, the eigenstates can be
split into singlets and doublets subspaces due the symmetry. The first
subspace presents similar results to the previous family, reinforcing the
agreement with the expected energy range set with $\alpha<1$ [43]. The
doublets subspace, whose for the chaotic cases is expected a GUE distribution
shows the more relevant result in this work. All found statistics present
intermediated results between Poisson and GUE distributions for the parameter
space $(a_{\text{e}},b_{\text{e}})$ where the classical phase space is mixed.
A BRB-like formula [27], eq. (10), well fitted all cases. This formula was
tested for $\rho_{\text{c}}<1$ and $\alpha<1$ in just a few cases.
Particularly in the E-$C_{3}$B family, the minimum value of the chaotic
fraction of the classical phase space is $\rho_{\text{c}}\simeq 0.8$. This
limitation can be avoided if we set free the conditions
$A_{\text{e}}=2a_{\text{e}}$ and $B_{\text{e}}=2b_{\text{e}}$, used here to
follow closer to the C-$C_{3}$B introduced by LSS. In this perspective, a
phase diagram analog to Fig. 3 even more intricate is generated, possible
further explorations of eq. (10).

The parameter $\beta$ in eq. (6) was extensively compared with other
localization metrics, including analyses involving Husimi functions,
calculations of the entropy localization measure [42], and normalized inverse
participation ratio [23]. How the new distribution, eq. (10), uses the same
arguments to include the parameter $\gamma$ is meritorious in a future
comparison between this quantity and other localization metrics. Another theme
meritorious of investigation is the level statistics in an energy range that
$\alpha\gg 1$. The BR formulas are expected to provide a good description of
the deep semiclassical regime [41], an excellent agreement has been found with
numerical experiments in a billiard for which the eigenvalues set is around
1,500,000th level [42], an impressive number. The BR-like formula in [27]
should be tested in a range of high energy in the doublets subspace to close
the comparisons between the short correlations in the singlets sets and
doublets subspace. In addition, our results indicate an intriguing correlation
between singlets and doublets spectra for the E-$C_{3}$B family, producing
$p(s)$’s that move away from the GOE and GUE distributions as
$\rho_{\text{c}}$ decreases, thus requiring a further investigation of the
observed effect. In this perspective, a range opens up to investigate the
correlation of spectra of different subspaces [46, 47, 34, 26, 48] in
billiards that present only rotational symmetries greater than three, which
will give the possibility of performing other tests with the new formulas (8)
and (10).

###### Acknowledgements.

Useful discussions with F. M. de Aguiar and K. Terto are gratefully
acknowledged. This work has been supported by the Brazilian Agencies CNPq,
CAPES and FACEPE.

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