# On complex dynamics in a Suris’s integrable map

Yasutaka Hanada<EMAIL_ADDRESS>Department of Information Science,
Faculty of Arts and Sciences, Showa University, Yamanashi 403-0005, Japan
Department of Physics, Faculty of Science, Tokyo Metropolitan University,
Tokyo 192-0397  Akira Shudo  Department of Physics, Faculty of Science, Tokyo
Metropolitan University, Tokyo 192-0397

###### Abstract

Quantum tunneling in a two-dimensional integrable map is studied. The orbits
of the map are all confined to the curves specified by the one-dimensional
Hamiltonian. It is found that the behavior of tunneling splitting for the
integrable map and the associated Hamiltonian system is qualitatively the
same, with only a slight difference in magnitude. However, the tunneling tails
of the wave functions, obtained by superposing the eigenfunctions that form
the doublet, exhibit significant difference. To explore the origin of the
difference, we observe the classical dynamics in the complex plane and find
that the existence of branch points appearing in the potential function of the
integrable map could play the role for yielding non-trivial behavior in the
tunneling tail. The result highlights the subtlety of quantum tunneling, which
cannot be captured in nature only by the dynamics in the real plane.

dynamical tunneling, integrable map, complex classical dynamics

##  I Introduction

Figure 1: (a) Plot of the functions $V_{\lambda}^{\prime}(q)$ (upper) and
$V_{\lambda}(q)$ (lower) for $\lambda=-1/3$, $-2/3$, and $-1$. Phase space
portrait for (b) $\lambda=-1/3$ and (c) $\lambda=-2/3$. In (b) and (c), dots
and solid curves represent the orbits for $f_{\lambda}$ and the associated
contour curve $H_{\lambda}(q,p)$, respectively. Black dashed and black dash-
dotted curves represent the separatrix starting from $(q,p)=(0,0)$ and $(0,\pm
1/2)$, respectively.

Quantum tunneling is a subtle phenomenon because it cannot be captured by any
power series, such as the WKB (Wentzel-Kramers-Brillouin) expansion, in terms
of the Planck constant. This unique aspect can also be understood by
recognizing that the power series expansion diverges, meaning that the
tunneling effect is fundamentally non-perturbative [1].

Recent advances in resurgence theory have revealed that divergent series
contain a wealth of information, even though they are divergent, and have
provided a mathematical framework for dealing with exponentially small
quantities [2, 3, 1, 4, 5]. Historically, the instanton method, proposed in
the field theory and the theory of chemical reactions, is established when
describing tunneling effects of the system with a single degree of freedom [6,
7, 8, 9, 10]. The instanton is a path running along the imaginary time, so it
is not the solution in the real plane. Along the same lines, it has long been
recognized that it is necessary to consider complex paths in order to treat
exponentially small effects. In resurgence theory, complex paths, including
the instanton path, can be formulated as singularities in the Borel plane,
which is obtained by applying the Borel transform to given divergent series
[2, 3, 1, 4, 5].

This paper discusses the subtlety of tunneling effects in a class of
integrable maps for which one would expect nothing special since they are
completely integrable. However, as shown below, this is not necessarily the
case. The work presented here can be regarded as one of a series of studies on
tunneling effects in nonintegrable systems based on the complex semiclassical
analysis [11, 12, 13]. Since only energetic barrier tunneling but also
dynamical tunneling [14, 15, 16] are exponentially small effects, the
importance of complex paths is obvious and the use of complex orbits is
unavoidable if one intends to extract components originating from tunneling
effects. The integrable map studied here shares a certain characteristic with
nonintegrable maps, although it is integrable.

To study tunneling effects in nonintegrable systems, discrete-time dynamical
systems, commonly referred to as maps, are often used as model systems. This
is supported by several reasons: among them, is that the behavior of complex
paths has been well studied for discrete-time dynamical systems compared to
continuous-time (Hamiltonian) dynamical systems [17, 18, 19]. However, we
should keep in mind that it is not immediately clear whether the results
derived for discrete-time dynamical systems, especially those based on complex
dynamics, are applicable to continuous-time flow systems. The difficulty
arises from the lack of a formal connection between the complex dynamics for
the maps and continuous-time flow systems in which time is complexified.

On the other hand, one can find nonlinear integrable Hamiltonian flow systems,
while nonlinear integrable maps are rather rare. The instanton is a path
describing the tunneling transition in continuous Hamiltonian flow systems,
and it is sometimes used as a reference when comparing the behavior of
tunneling in integrable and nonintegrable systems [20]. In contrast, to the
authors’ knowledge, there has not yet been an analysis of the tunneling effect
for the map that is obtained by perturbing an integrable map. In most
analyses, a continuous-time Hamiltonian system is taken as the integrable
limit [21, 22]. Such a treatment is somewhat self-inconsistent, and it is
advisable to seek alternative solutions, if possible.

To fill this gap, here we study tunneling for a class of integrable maps found
by Suris [23]. The integrability of the map is defined similarly to the case
of continuous-time Hamiltonian systems. In particular, to verify the
integrability for the two-dimensional symplectic map, it is sufficient to find
a constant of motion under the discrete-time evolution. In Ref. [23], it was
also shown that the maps with different types of potential function provide
‘Hamiltonians’, and it is easy to check that they are conserved under the time
evolution.

Here we compare quantum tunneling for the integrable map and the associated
‘Hamiltonian’. We quantize the map as usual by introducing the one-step
unitary operator, while Weyl quantization is applied to the continuous
Hamiltonian. For a given classical Hamiltonian, recall that there are
infinitely many quantizations resulting from the choice of the operator
ordering. The differences are known to be of the order of $\hbar^{2}$, which
is larger than exponentially small effects. Thus, even within quantizations of
a continuous Hamiltonian, each of which is determined by the chosen operator
ordering, tunneling effects are in principle uncontrollable. It is therefore
not surprising that the difference in quantization between integrable maps and
the associated integrable Hamiltonian systems can lead to different tunneling
properties.

Our strategy in this paper is to perform high-precision numerical calculations
of quantum systems and to observe the behavior of the complex dynamics
generated by both the integrable map and the corresponding Hamiltonian flow.
We do not perform any semiclassical analysis here, but the comparison of the
complex classical dynamics with the phase space profile of the time-evolved
wave packet reveals non-trivial signatures in the tunneling components, even
though the map is integrable.

Recently, the ultra-near integrable system has been studied in order to
explore the difference of tunneling between integrable and nonintegrable
systems [24, 13]. Ultra-near integrable systems are defined as those systems
in which the classical phase spaces do not exhibit any invariant structures
inherent in nonintegrability in the size of the Planck cell. Even without
islands of stability or chaotic seas, the tail of the wave function generates
non-monotonic step structures [24], and the ergodicity of complex dynamics has
been shown to play a key role in reproducing non-trivial tunneling behavior in
nonintegrable maps [13]. Since there are no symptoms in the real phase space,
the origin of the non-trivial tunneling tails comes down to the question of
how chaos is born in the complex plane. The study of the integrable map is
expected to be helpful in understanding this question.

The organization of this report is as follows. In section II, we introduce the
symplectic map which was found to be integrable by Suris [23], together with
the Hamiltonian to which the orbits generated by the map are confined. We then
define the unitary operator that provides the quantum dynamics for the map.
section III gives numerical results showing the difference in tunneling
signatures between the integrable and the corresponding Hamiltonian system.
Although the tunneling splitting exhibits only a tiny difference, the
tunneling tails for the localized states constructed from the eigenfunctions
forming the doublet differ significantly. In section IV we observe the
behavior of the classical dynamics in the complex plane. This is only a
preliminary study towards the semiclassical understanding of the non-trivial
tunneling revealed in the integrable map, but the result provides important
implications for understanding the observed phenomenon. What distinguishes the
two systems is the fact that the solution of the Hamiltonian flow has no
singularities in the complex plane, while the derivative of the potential
function has branch points, which lead to multivalued dynamics in the
integrable map. We also discuss issues that are expected to arise in the
development of semiclassical analysis in the complex domain. section V
summarizes the results obtained in the present report and gives some
arguments, especially in relation to quantum tunneling in nonintegrable
systems.

##  II Model

Figure 2:  (a) Eigenstates $\psi_{n}(q)$ (cyan) and $\Psi_{n}(q)$ (black) with
$\hbar=1/200\pi$ for (i) $n=0$, (ii) $n=50$, (iii) $n=150$, and (iv) $n=199$.
(b) Husimi representation of $\psi_{n}(q)$ in normal scale for (i) $n=0$, (ii)
$n=50$, (iii) $n=150$, and (iv) $n=199$. The black dotted curve shows the
contour curve with $H_{\lambda}(q,p)=E_{n}$. The black broken curve shows the
separatrix starting from $(q,p)=(0,0)$.

To begin with, we introduce a two-dimensional area-preserving map
$f:\mathbb{T}^{2}\to\mathbb{T}^{2}$,

$f:\begin{pmatrix}q_{k+1}\\\
p_{k+1}\end{pmatrix}=\begin{pmatrix}q_{k}+p_{k}-V^{\prime}(q_{k})\\\
p_{k}-V^{\prime}(q_{k})\end{pmatrix}.$ (1)

Suris [23] has shown that there exists a conserved quantity for the map:

$H(q_{k+1},p_{k+1})=H(q_{k},p_{k})$ (2)

for some specific choices of the potential function $V(q)$. Among several
possibilities, we here take a potential function defined by [25]

$V_{\lambda}(q):=\frac{1}{\pi}\int_{0}^{q}\arctan\left(\frac{\lambda\sin 2\pi
q^{\prime}}{1+\lambda\cos 2\pi q^{\prime}}\right)\mathrm{d}{q^{\prime}}.$ (3)

Note that the potential function $V_{\lambda}(q)$ is periodic with period 1.
For this potential, the Hamiltonian

$H_{\lambda}(q,p):=-\cos 2\pi p-\lambda\left[\cos 2\pi q+\cos
2\pi(q-p)\right],$ (4)

satisfies the condition (2) for any $\lambda$. Below, we denote the map with
the potential (3) by $f_{\lambda}$ to specify the map depends on the parameter
$\lambda$.

Figure 3:  (a) Plot of the tunneling splittings $\Delta\mathcal{E}_{0}$
(solid) and $\Delta E_{0}$ (dashed) vs of $1/\hbar$. (b) Plot of the tunneling
splittings $\Delta\mathcal{E}_{0}$ (solid) and $\Delta E_{0}$ (dashed) vs
$-\lambda$. (c) Plot of $1-|\langle\psi_{0}|\Psi_{0}\rangle|$ vs $-\lambda$.

Since the condition (2) holds for $k\in\mathbb{Z}$, $H_{\lambda}$ can be
regarded as a conserved quantity (Hamiltonian) for the map $f_{\lambda}$. In
addition, the Hamiltonian flow $F_{\lambda}(q(t),p(t))$ generated by the
Hamiltonian equations of motion,

$\dot{q}(t)=\frac{H_{\lambda}(q,p)}{\partial
p},~{}~{}\dot{p}(t)=-\frac{H_{\lambda}(q,p)}{\partial q},$ (5)

is obviously integrable. Therefore, the map $f_{\lambda}$ does not exhibit
chaotic motion, and can be said to be integrable. In this report, we refer
$f_{\lambda}$ as the Suris’s integrable map.

Figure 1(a) shows the functions $V_{\lambda}^{\prime}(q)$ and $V_{\lambda}(q)$
for different values of $\lambda$ 111The integration of $V_{\lambda}(q)$ is
numerically evaluated by “double exponential quadrature” [takahasi1973]
implemented in Mathematica until numerical convergence is achieved to at least
with 200 digits of mantissa.. The profile of $V_{\lambda}^{\prime}(q)$ for
$|\lambda|<1$ looks a “skewed” sine function. As $\lambda\to\pm 1$,
$V_{\lambda}^{\prime}(q)$ tends to a piecewise linear function, which is
discontinuous at $q=1/2$ for $\lambda=1$ and $q=0$ for $\lambda=-1$. A fixed
point $(q,p)=(0,0)$ is elliptic-type if $\lambda>0$ and hyperbolic if
$\lambda<0$. For $|\lambda|>1$ there are several discontinuous points in
$q\in[0,1)$. In this report we take $\lambda\in(-1,0)$.

Figures 1(b) and 1(c) give phase space portraits for $\lambda=-1/3$ and
$-2/3$. Each dot and solid curve of the same color represents a single orbit
$\left\\{f_{\lambda}^{k}(q_{0},p_{0})\right\\}_{k\in\mathbb{N}}$ starting from
different initial conditions $(q_{0},p_{0})$ and the corresponding contour
curves of $H_{\lambda}(q,p)=H_{\lambda}(q_{0},p_{0})$, respectively. There are
stable fixed points at $(q,p)=(\pm 1/2,0)$ and an unstable fixed point at
$(q,p)=(0,0)$. In addition, the map has stable periodic points with period 2
at
$(q,p)=(\mp\arccos(-\frac{\lambda}{2})/2\pi,\pm\arccos(\frac{\lambda^{2}}{2}-1)/2\pi)$
and unstable periodic points with period 2 at $(q,p)=(0,\pm 1/2)$ (when we
take $\textrm{mod}\ 1$ for $q$). Separatrices starting from
$(q,p;E)=(0,0;-1-2\lambda)$ and $(q,p;E)=(0,1/2;1)$ are shown in black dashed
and black dash-dotted curve, respectively. As expected, the orbits
$\left\\{f_{\lambda}^{k}(q_{0},p_{0})\right\\}_{k\in\mathbb{N}}$ for any
$(q_{0},p_{0})\in\mathbb{R}^{2}$ lie on the contour curves of
$H_{\lambda}(q,p)=H_{\lambda}(q_{0},p_{0})$.

We next introduce quantum dynamics associated with the classical map
$f_{\lambda}$. We here assume the periodic boundary condition for both $q$\-
and $p$-directions, which leads to the finite-dimensional Hilbert space whose
dimension is given by, say, $N$, and apply the canonical quantization to the
one-step time evolution, which gives the quantum map
$\hat{U}:|\psi\rangle_{k}\mapsto|\psi\rangle_{k+1}$ [27, 28],

$\hat{U}_{\lambda}:=\mathrm{e}^{-\frac{\mathrm{i}\mkern
1.0mu}{\hbar}\hat{p}^{2}}\mathrm{e}^{-\frac{\mathrm{i}\mkern
1.0mu}{\hbar}V_{\lambda}(\hat{q})}.$ (6)

Floquet theorem leads to the eigenvalue equation,

$\hat{U}_{\lambda}|\psi_{n}\rangle=\mathrm{e}^{-\frac{\mathrm{i}\mkern
1.0mu}{\hbar}\mathcal{E}_{n}}|\psi_{n}\rangle.$ (7)

Correspondingly, we introduce the quantized Hamiltonian
$\hat{H}_{\lambda}(\hat{q},\hat{p})$ by adopting the canonical quantization
for the classical Hamiltonian (5). Note here that we take the Weyl’s ordering
for the product of $q$ and $p$. The eigenvalue equation for
$\hat{H}_{\lambda}$ then reads

$\hat{H}_{\lambda}|\Psi_{n}\rangle=E_{n}|\Psi_{n}\rangle.$ (8)

In the following, the quantum number $m$ of the quasi-eigenstate
$|\psi_{m}\rangle$ is determined by the quantum number of the eigenstates
$|\Psi_{n}\rangle$ attaining the maximal overlap
$|\langle\psi_{m}|\Psi_{n}\rangle|^{2}$.

In this paper, numerical computations for the quantum map are performed by
using arbitrary precision arithmetics implemented in Mathematica, MATLAB with
ADVANPIX toolbox [29].

##  III Difference between $|\Psi_{n}\rangle$ and $|\psi_{n}\rangle$

The discrete map $f_{\lambda}$ and the continuous Hamiltonian flow
$F_{\lambda}$ share the same constant of motion $H_{\lambda}(q,p)$, but the
quantization procedures are different: the former is quantized by introducing
the single-step unitary operator $\hat{U}_{\lambda}$ while the Weyl’s
quantization is applied to the Hamiltonian $H_{\lambda}$. Thus, it is by no
means obvious whether the eigenvalues and eigenfunctions are identical and
have the same properties.

In particular, we are interested here in tunneling effects, so the difference
in quantization can be crucial. This is because, as mentioned above,
exponentially small effects slip through ordinary semiclassical arguments,
which are made by expressing related quantities in terms of the expansion of
$\hbar$. Even within quantizations for the Hamiltonian $H_{\lambda}$, there is
ambiguity about the operator ordering, leading to an uncertainty of the order
of $O(\hbar^{2})$, so it is all the more unclear what will happen if the
quantization method differs from each other.

Here we examine tunneling splittings generated in the nearly degenerate energy
levels of the quantum map (6). To obtain a double-well-like setting, we impose
the periodic boundary condition on the phase space region
$(q,p)=[-1,1)\times[-1/2,1/2)$. The periodicity of $V(q)$ leads to the point-
symmetric phase space around $(0,0)$ (see fig. 1), and the ‘libration’ motion
appears for $-1+2\lambda<H_{\lambda}(q,p)<-1-2\lambda$. Due to the symmetry
with respect to the origin, the energy levels of the quantum map produce
tunneling splittings, similar to the one-dimensional double-well potential
system.

Figure 4:  (a) Wave functions $\psi_{L}(q)$ (solid curves) and $\Psi_{L}(q)$
(dashed curves) for different values of $1/\hbar$. The “$\star$” mark is put
to indicate the deviation point $q^{\star}_{-}$. (b) Husimi representation of
(i) $\Psi_{L}(q)$ and (ii) $\psi_{L}(q)$ for $1/\hbar=200\pi$ in the
$\log_{10}$ scale. The dashed gray curve represents the separatrix. Closed
cyan curves show the contour curve associated with the ground state energy
level $E_{0}=-1.639\,057\,02$. Black and white “$\times$” symbols indicate the
position of the turning points for $q(t)$; $(q,p)=(\pm 0.450\,847,\pm
0.012\,215)$, respectively. (b i) The cyan curve connected to the two turning
points shows the instanton curve projected onto the $(x,u)$-plane.

Figure 2(a) gives some eigenfunctions $\psi_{n}(q)$ and $\Psi_{n}(q)$ for
$\hbar=1/200\pi$ in semi-logarithmic scale, and fig. 2(b) shows the
corresponding Husimi representation for $|\Psi_{n}\rangle$ in normal scale.
There is no noticeable difference between $\psi_{n}(q)$ and $\Psi_{n}(q)$, and
the Husimi representation has support on the contour curves
$H_{\lambda}(q,p)=E_{n}$. Figures 3(a) and 3(b) plot the tunneling splitting
defined by

$\Delta\mathcal{E}_{0}=|\mathcal{E}_{1}-\mathcal{E}_{0}|,\qquad\Delta
E_{0}=E_{1}-E_{0},$ (9)

as a function of $1/\hbar$ and $-\lambda$, respectively. In contrast to the
almost perfect agreement of the eigenfunctions shown in fig. 2(a), we can find
a few orders of magnitude difference between the tunneling splittings
$\Delta\mathcal{E}_{0}$ and $\Delta E_{0}$, while the parameter dependence of
$\Delta\mathcal{E}_{0}$ and $\Delta E_{0}$ on $1/\hbar$ and $-\lambda$ is
qualitatively the same; both are expected from the semiclassical analysis for
integrable systems.

A tiny difference in the magnitude implies that some discrepancy should be
hidden in the eigenfunctions. Figure 3(c) plots the magnitude
$1-|\langle\psi_{n}|\Psi_{n}\rangle|^{2}$ as a function of $-\lambda$, which
shows that the difference between $|\psi_{n}\rangle$ and $|\Psi_{n}\rangle$
does indeed exist, although it is exponentially small, less than $10^{-8}$ for
$\lambda\in(-1,0)$. Note also that the difference
$1-|\langle\psi_{n}|\Psi_{n}\rangle|^{2}$ increases as $-\lambda$ tends to 1.

The tunneling splitting appears as a result of the overlap of the wave
function localized at the left $(q<0)$ and the wave function localized at the
right $(q>0)$:

$\displaystyle|\psi_{L(R)}\rangle$
$\displaystyle:=\frac{1}{\sqrt{2}}(|\psi_{1}\rangle\pm|\psi_{0}\rangle),$ (10)
$\displaystyle|\Psi_{L(R)}\rangle$
$\displaystyle:=\frac{1}{\sqrt{2}}(|\Psi_{1}\rangle\pm|\Psi_{0}\rangle).$ (11)

Figures 4(a) and 4(b) present the wave functions $\Psi_{L}(q)$ and
$\psi_{L}(q)$ and the corresponding Husimi representation, respectively. Let
$q^{\times}_{-}$ and $q^{\times}_{+}$ be the turning points of the manifold
$H_{\lambda}(q,p)=E_{0}$ projected onto the $q$-axis. As seen in fig. 4(a),
$|\Psi_{L}(q)|^{2}$ has an exponentially decaying tunneling tail in the region
$q\in(q^{\times}_{-},q^{\times}_{+})$, as predicted by the WKB argument.

The Husimi representation provides more detailed information. As shown in fig.
4(b i), the amplitude of the Husimi representation for $|\Psi_{L}\rangle$
decays exponentially along the instanton curve The instanton curve is obtained
here by the purely imaginary time evolution $t\in\mathrm{i}\mkern
1.0mu\mathbb{R}$ of the Hamiltonian flow $F_{\lambda}$ starting from the
turning point $q^{\times}_{-}$.

$|\psi_{L}(q)|^{2}$ also shows an exponentially decaying tunneling tail from
$q=q^{\times}_{-}$ down to $q=0$. However, $|\psi_{L}(q)|^{2}$ stops decaying
just beyond $q=0$ and then forms a plateau with an oscillatory pattern in the
region $q>0$. As shown in fig. 4(b ii), the Husimi representation of
$|\psi_{L}\rangle$ tells us that the localization along the separatrix
starting at $(q,p)=(0,0)$ is responsible for the oscillatory plateau observed
in the plot of $|\psi_{L}(q)|^{2}$.

Figure 5:  The dots represent the deviation point $q^{\star}_{-}$ as a
function of $1/\hbar$ in case of $\lambda=-1/3$. The dashed line is obtained
by applying linear regression to the plot of $1/q^{\star}_{-}$ vs $1/\hbar$.

Figure 6:  Plot of $\hat{U}_{\lambda}^{k}|\Psi_{L}\rangle$ (a) in the
$q$-representation and (b) in the Husimi representation.

The plateau found in $|\psi_{L}(q)|^{2}$ and the localized component along the
separatrix are reminiscent of “tunneling across the separatrix”. Note that its
importance for our understanding of the anomalous enhancement of the tunneling
probability has been pointed out in Ref. [22]. However, they are not
necessarily the same phenomenon. The main difference is that similar
oscillation patterns in nonintegrable systems penetrate into the region $q<0$,
while the plateau found in the Suris’s integrable map does not. To confirm
this, we examine the interval $[q^{\star}_{-},q^{\star}_{+}]$ in which
$|\psi_{L}(q)|^{2}$ is 10 times larger than $|\Psi_{L}(q)|^{2}$. Here, the
edges $q^{\star}_{-}$ and $q^{\star}_{+}$ are defined by

$\displaystyle q^{\star}_{-}$ $\displaystyle:={\rm
min}\left\\{q\in[-1,1):\log_{10}\left|\frac{\psi_{L}(q)}{\Psi_{L}(q)}\right|^{2}>1\right\\},$
(12) $\displaystyle q^{\star}_{+}$
$\displaystyle:={\rm\max}\left\\{q\in[-1,1):\log_{10}\left|\frac{\psi_{L}(q)}{\Psi_{L}(q)}\right|^{2}>1\right\\}.$
(13)

Figure 5 plots $q^{\star}_{-}$ as a function $1/\hbar$. Dots and a black
dashed curve in fig. 5 indicate numerical data from $q^{\star}_{-}$ and a
linear fit to the obtained data, respectively. The plot shows that the value
of $1/q^{\star}_{-}$ increases linearly with $1/\hbar$, meaning that
$q^{\star}_{-}\to 0$ for $1/\hbar\to\infty$. Note in particular that the decay
rate of $1/q^{\star}_{-}$ with $1/\hbar$ is extremely slow compared to
nonintegrable cases [22].

##  IV Toward semiclassical understanding for plateau with oscillatory pattern

While the WKB (smiclassical) analysis for eigenstates of integrable
Hamiltonian systems is well established [30], there is no semiclassical
prescription to analyze eigenstates of nonintegrable systems. On the other
hand, the semiclassical analysis in the time domain is available even for
nonintegrable systems, and the methodology incorporating the complex domain is
now in a usable state [31, 32, 33, 12, 34]. Here, we discuss the origin of the
plateau accompanied by oscillations observed in the quantum Suris’s integrable
map by taking the time-domain semiclassical approach.

Among possible options for initial conditions, we choose the state
$|\Psi_{L}\rangle$, an eigenstate of the Hamiltonian $\hat{H}_{\lambda}$,
since we are particularly interested in the discrepancy between
$|\psi_{L}\rangle$ and $|\Psi_{L}\rangle$. Figures 6(a) and 6(b) illustrate
the propagation of the wave function $\hat{U}_{\lambda}^{k}|\Psi_{L}\rangle$
in the $q$-representation and the Husimi representation, respectively. We can
see that the amplitude of the wave function
$\hat{U}_{\lambda}^{k}|\Psi_{L}\rangle$ for
$q\in[q^{\star}_{-},q^{\star}_{+}]$ increases rapidly with time, and after
four steps it already reaches a profile similar to $\psi_{L}(q)$. Note that
the saturated amplitude of $\hat{U}_{\lambda}^{k}|\Psi_{L}\rangle$ in the
interval $q\in[q^{\star}_{-},q^{\star}_{+}]$ is slightly higher than the
amplitude of $\psi_{L}(q)$.

The Husimi representation shows us an explicit wave packet propagation in
phase space. As time evolves, the dominant part of the wave function
$\hat{U}_{\lambda}^{k}|\Psi_{L}\rangle$ passes through the fixed point
$(q,p)=(0,0)$ and propagates along the separatrix. After five steps, the
amplitude spreads almost uniformly along the separatrix. This tells us that
the plateau with oscillation observed in $\psi_{L}(q)$ appears as the result
of the tunneling transport along the separatrix associated with the fixed
point $(q,p)=(0,0)$.

###  IV.1 Complex dynamics of $F_{\lambda}$ and $f_{\lambda}$

In this section we explore the classical dynamics $F_{\lambda}$ and
$f_{\lambda}$ in the complex domain. To do this, we introduce notations for
the real and imaginary parts as $q=x+\mathrm{i}\mkern 1.0muy$, and
$p=u+\mathrm{i}\mkern 1.0muv$, respectively.

First we observe the solution curves in the complex plane under the flow
$F_{\lambda}$. They lie on the equi-energy surface specified by the condition
$H_{\lambda}=E_{n}\in{\mathbb{R}}$. To obtain these curves, we first find the
turning points $q^{\times}_{\pm}$, marked by the black “$\times$” in fig. 7,
and then solve the equations of motion for the Hamiltonian $H_{\lambda}$ along
the imaginary time axis ${\rm i}{\mathbb{R}}$ to connect the turning points
$q^{\times}_{\pm}\in{\mathbb{R}}$. Recall that this solution curve, shown by
the thick black curve, is nothing more than the instanton. Along the instanton
curve we place equally spaced initial conditions, from each of which we solve
the equations of motion along the real time axis ${\mathbb{R}}$, yielding a
set of closed curves shown in green.

Figure 7:  The solution curves obtained under the continuous Hamiltonian flow
$F_{\lambda}$. They are confined in the equi-energy surface given by the
condition $H_{\lambda}=E_{0}=-1.639\,057\,02$. The curves are projected onto
(a) the $(x,u,v)$-space and (b) the $(x,u)$-plane. In (a) and (b) the black
closed curve and the black semicircular curve represent the equi-energy curves
in the real plane and the instanton curve, respectively. The symbol “$\times$”
stands for the turning point of $x(t)$. (a) The cyan line tending to infinity
shows a solution curve of $H_{\lambda}$ starting from the point $(x,u)=(0,0)$,
marked by the cyan point on the instanton. (b) The cyan dots show the orbits
of $f_{\lambda}$ starting from the point $(x,u)=(0,0)$ on the instanton.

As for the dynamics of the map $f_{\lambda}$ in the complex plane, the
multivaluedness of the potential function $V_{\lambda}^{\prime}(q)$ generates
non-trivial behavior. To see this, we focus on the dynamics at the points
$(x,u)=(0,0)$, $(0,\pm 1/2)$, and $(\pm 1/2,\pm 1/2)$, respectively. The
equations of motion of the Hamiltonian flow $F_{\lambda}$ are written
explicitly as

$\begin{split}&\dot{y}(t)=2\pi\left[\alpha_{1}\sinh 2\pi
v+\alpha_{2}\lambda\sinh 2\pi(y-v)\right],\\\
&\dot{v}(t)=-2\pi\lambda\left[\alpha_{3}\sinh 2\pi y+\alpha_{4}\sinh
2\pi(y-v)\right],\\\ &\dot{x}(t)=\dot{u}(t)=0,\end{split}$ (14)

where $(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(1,-1,1,1)$ for
$(x,u)=(0,0)$, $(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(-1,-1,-1,1)$
for $(x,u)=(\pm 1/2,\pm 1/2)$, and
$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(-1,1,1,-1)$ for $(x,u)=(0,\pm
1/2)$. Since $x(t),u(t)=const.$, the solution curves shown in eq. 14 are
restricted to the purely imaginary $(y,v)$-plane. Note that the fixed points
$(y,v)=(0,0)$ of eq. 14 are all hyperbolic, and the right-hand sides of eq. 14
consist of monotonic and entire functions, so that the solution curves around
$(y,v)=(0,0)$ diverge monotonically to $y(t),v(t)\to\pm\infty$ under forward
time evolution (see fig. 8).

The greens dots in fig. 7 show the classical orbits
$\left\\{f_{\lambda}^{k}(q_{0},p_{0})\right\\}_{k\in\mathbb{Z}}$ for which the
initial points $(q_{0},p_{0})$ are set along the instanton. We observe that
all the orbits
$\left\\{f_{\lambda}^{k}(q_{0},p_{0})\right\\}_{k\in\mathbb{Z}}$ preserve the
energy $H_{\lambda}=E$ as expected, but there are exceptions: the orbits
starting at $(x_{0},u_{0})=(0,0)$ are not bounded on the Hamiltonian flow
$F_{\lambda}$. As explained below, the multivaluedness of the function
$V_{\lambda}^{\prime}(q)$ for $q\in\mathbb{C}$ leads to such anomalous
behavior. In fig. 7(b), the cyan dots show the projection of the itinerary of
the orbit whose initial condition is placed on the instanton orbit onto the
$(x,u)$-plane. We can see that the real part $(x_{k},u_{k})$ of the orbit
jumps among $(x_{k},v_{k})=(0,0)$, $(\pm 1/2,\pm 1/2)$, and $(0,\pm 1/2)$,
while the real part ($x(t),u(t)$) does not change under the Hamiltonian flow
$F_{\lambda}$ [see eq. 14].

Figure 8:  Flow of eq. 14 on the $(y,v)$-plane for $\lambda=-1/3$. Vertical
dashed lines represent $y=y^{\diamond}$ for $\lambda=-1/3$.

This itinerary of the orbit can be understood by considering the derivative
$V_{\lambda}^{\prime}(q)$ of the potential function in the complex $q$-plane.
Introducing the new variable,

$\displaystyle z=\frac{\lambda\sin 2\pi q}{1+\lambda\cos 2\pi q}~{},$ (15)

we find that the derivative $V_{\lambda}^{\prime}(z)$ can be expressed as

$\displaystyle V_{\lambda}^{\prime}(z)=\frac{1}{\pi}\arctan z~{},$ (16)

where the arctangent function has an alternative expression,

$\displaystyle\arctan z=\frac{1}{2\mathrm{i}\mkern
1.0mu}\log\left(\frac{1+\mathrm{i}\mkern 1.0muz}{1-\mathrm{i}\mkern
1.0muz}\right)=\int\frac{\mathrm{d}z}{1+z^{2}}~{},$ (17)

which obviously has the branch points on the $z$-plane at
$z=\pm\mathrm{i}\mkern 1.0mu$, originating from the logarithmic function.
Correspondingly, on the $q$-plane $V_{\lambda}^{\prime}(q)$ has the branch
points at $q=\pm\mathrm{i}\mkern
1.0mu\frac{1}{2\pi}\log(-\lambda)=:\pm\mathrm{i}\mkern 1.0muy^{\diamond}$.

Now, we consider the itinerary of the orbit starting from
$(x,y,u,v)=(0,y_{0},0,v_{0})$. In other words, we set the real part of the
initial points to $(x_{0},u_{0})=(0,0)$, but take various values for the
imaginary parts $(y,v)=(y_{0},v_{0})$, assuming that $|y_{0}|<y^{\diamond}$.
Since the $V_{\lambda}^{\prime}(q)$ is a multivalued function in the
logarithmic type, the one-step iteration leads to an infinite number of
images. Several studies have explored the complex dynamics associated with
multivalued functions [35], but here we consider only the one-step iteration
to get rid of the complication caused by the multi-step iteration.

Since the stability at the origin $(y,v)=(0,0)$ is hyperbolic [see fig. 8(a)],
for any given $v_{0}$, there exists an initial condition
$|y_{0}|<y^{\diamond}$ such that the next step satisfies the condition
$|y_{1}|>y^{\diamond}$. Recall that the Hamiltonian flow follows eq. 14, so in
the time evolution the $x(t)$ remains constant ($x(t)=0$) and only the value
of $y(t)$ changes. Therefore, in the $(x,y)$-plane, the point $(0,y_{0})$
passes over the branch points $(0,\pm y^{\diamond})$ before and after being
mapped to the point $(0,y_{1})$ . After passing one of the branch points, the
mapped point gains either $1/2+m$ and $-1/2-m^{\prime}$
$(m,m^{\prime}\in{\mathbb{Z}})$, reflecting the multivalued nature of the
potential function (16). As a result, the initial point $(x_{0},u_{0})=(0,0)$
is mapped to the lattice points $(1/2+m,1/2+m^{\prime})$
$(m,m^{\prime}\in\mathbb{Z})$. However, these lattice points can be identified
as $(1/2,\pm 1/2)$ and $(-1/2,\pm 1/2)$ by applying the periodic boundary
condition. So the real parts $(x_{0},u_{0})=(0,0)$ are mapped to
$(x_{1},u_{1})=(1/2,\pm 1/2)$ and $(-1/2,\pm 1/2)$.

For the initial points $(0,y_{0})$ in the $(x,y)$-plane that do not pass the
branch points within the single step iteration, they simply move along the
solution curve governed by eq. 14. The points generated by the multivalued
nature of the map are again identified by applying the periodic boundary
condition. Depending on the initial condition $(x_{0},u_{0})$, the single step
iteration in the $(y,v)$-plane gives the map from $(y_{0},v_{0})$ to
$(y_{1},v_{1})$ on the same flow curve. Note that the transition between
different $(x,u)$-planes does not occur because we here assume that the
initial points $(0,y_{0})$ in the $(x,y)$-plane do not pass the branch points.

Figure 9:  (a) The function $V_{\lambda}^{\prime}(q)$ in the $(x,y)$-plane
with $\lambda=-1/3$. The argument of $V_{\lambda}^{\prime}(q)$ and the
absolute value $|V_{\lambda}^{\prime}(q)|$ are distinguished by color and
brightness, respectively. The blue diamond “$\diamond$” symbol represents the
branch point of $V^{\prime}(q)$. The white line starting from the branch point
represents the branch cut. The black curve shows a contour curve
$\mathrm{Im}(V_{\lambda}^{\prime}(q))=0$.

###  IV.2 One-step propagation

Figure 10:  (a) Plot of the wave functions $\hat{U}_{\lambda}|\Psi_{L}\rangle$
(red curve) and $|\Psi_{L}\rangle$ (black dashed curve) for (i)
$\lambda=-0.3$, (ii) $\lambda=-0.5$, (iii) $\lambda=-0.7$, and (iv)
$\lambda=-0.9$. (b) Husimi representation of
$\hat{U}_{\lambda}|\Psi_{L}\rangle$. The cyan dotted line represents a line
$(q,p)=(0,0)$ to $(q,p)=(1/2,1/2)$ for a guide. The gray solid curve and the
gray dashed curve represent a contour curve of $H_{\lambda}(q,p)=E_{0}$ and
the separatrix starting from $(q,p)=(0,0)$, respectively.

As illustrated in fig. 6, the plateau with oscillation observed in the
eigenfunction $|\psi_{n}\rangle$ is reproduced by a wave function with
relatively short time steps. This should make it possible to develop the
semiclassical analysis in the time domain.

As explained in the previous section, the multivaluedness of the map leads to
the transition that never occurs in the integrable Hamiltonian flow. The
orbits of the Hamiltonian flow $F_{\lambda}$ starting from the unstable fixed
point $(x(0),u(0))=(0,0)$ do not change the real part, i.e.,
$(x(t),u(t))=(0,0)~{}(t>0)$ independent of the initial conditions
$(y_{0},v_{0})$, while the orbits generated by the map $f_{\lambda}$ jump from
$(x_{0},u_{0})=(0,0)$ to either $(x_{1},u_{1})=(1/2,\pm 1/2)$ or $(-1/2,\pm
1/2)$ when they pass over the branch points, otherwise they stay at
$(x_{1},u_{1})=(0,0)$. Such a transition should explain the propagation of the
wave function along the separatrix, but some issues need to be clarified
before proceeding.

The first one is that the orbit in the complex plane is bifurcated even in
one-step propagation, reflecting the multivaluedness of the map. The
transition from $(0,0)$ to $(1/2,1/2)$ is observed in the one-step propagation
of the quantum wave packet (see fig. 10), while the orbit can jump from
$(0,0)$ to $(\pm 1/2,\mp 1/2)$ as explained in the previous section. We expect
that this can be solved by properly treating the Stokes phenomenon when
performing the saddle point approximation. If the path from $(0,0)$ to $(\pm
1/2,\mp 1/2)$ gives rise to a divergent solution, then it should be removed
from the final solution due to the Stokes phenomenon. For multi-step analysis,
however, the multivaluedness of the map must be treated in each step, which
makes semiclassical analysis difficult. As can be seen from fig. 6, the
plateau along the separatrix becomes more visible as time evolves. It may be a
relatively short time, but we have to perform a semiclassical calculation with
multiple time steps. This requires a more sophisticated approach to the Stokes
phenomenon.

On the other hand, fig. 10 shows that the plateau can also be made visible by
increasing the parameter $-\lambda$. This may give us the impression that the
one-step semiclassical analysis is sufficient to explain the observed
phenomenon, and one can get rid of multi-step calculations. However, the
problem is not simple, because as the parameter $-\lambda$ is increased, the
potential $V_{\lambda}(q)$ and its derivative $V_{\lambda}^{\prime}(q)$ become
sharper, as shown in fig. 1(a). This reminds us of the diffraction effect
[36], which is expected to occur when the potential function has
discontinuities. We should include higher-order terms in the saddle
approximations, or more legitimately develop uniform approximation type
arguments to cope with such situations.

##  V Conclusion and discussion

In this report, we have studied quantum tunneling for a nonlinear integrable
map found by Suris, with a particular focus on the comparison with the
corresponding continuous Hamiltonian system. Although the Suris’s map is
completely integrable in the sense of Liouville-Aronld in Hamiltonian systems,
and the orbits in the real plane generated by the Suris’s map lie on the equi-
energy curve for the continuous Hamiltonian, our numerical computations
performed with arbitrary precision arithmetic have revealed that the
characteristics of the tunneling tails in the eigenfunctions differ from each
other. This means that the behavior of the tunneling tail cannot be deduced
from the classical dynamics in real phase space.

The discrepancy found here between the map and the corresponding Hamiltonian
system is not surprising. As noted in the introduction, even for the same
Hamiltonian, there is an ambiguity in the quantization arising from the choice
of the operator ordering, while the magnitude of the tunneling effects is in
general exponentially small with respect to the Planck constant $\hbar$, which
cannot be captured by any power series expansion.

A lesson we learn from this example is that it is necessary to investigate the
dynamics in the complex plane to understand the tunneling effect properly.
Here, we have not performed a fully semiclassical calculation with complex
orbits, but it would be reasonable to expect that the difference in the
behavior of the complex dynamics gives rise to the difference in the tunneling
tail. The branch points, appearing in the potential function of the map,
produce multivaluedness of the complex map whose treatment is one of the
topics in the theory of complex dynamical systems [35]. We have shown that the
real part of the orbits at the fixed point of the Hamiltonian flow remains the
same under the dynamics, while the orbit changes its real part discontinuously
before and after crossing the branch point. This must be the origin of the
localization of the wave function along the separatrix for the fixed point,
and also reminiscent of the tunneling transition across the separatrix found
in Refs. [21, 22].

The map $f_{\lambda}$ is slightly complicated compared to the flow
$F_{\lambda}$ due to the presence of singularities producing multivaluedness
of the dynamics. This fact reminds us of the so-called Poinlev$\grave{\rm e}$
conjecture: in integrable systems, singularities appearing in the time plane
are at most limited to the poles [37]. This in turn suggests that the system
can be nonintegrable if branch points appear in the complex time plane of the
solutions. One can say that the Suris’s integrable map is completely
integrable as far as the dynamics in the real plane is concerned, whereas it
is one step outside of integrable systems. Our present result shows that this
difference is reflected in the tunneling effect.

It is important to note that such subtlety is not limited to the system
studied in this report. If the system is close enough to the integrable limit,
the phase space is almost covered by Kolmogorov-Arnold-Moser curves. For the
classical-to-quantum correspondence, the size of the Planck cell is an
important spatial scale; even if invariant structures such as nonlinear
resonances or stochastic layers appear in phase space, it does not matter if
their sizes are smaller than the size of the Planck cell under consideration.
This is a fundamental working hypothesis when considering the manifestation of
classical chaos in the corresponding systems [38]. According to this
principle, the nearly integrable system, in which the invariant structures
inherent in the nonintegrability are all on the sub-Planck cell scale, should
be identified with the completely integrable system.

This must hold true in the real plane, but not necessarily in the complex
plane. This is because the phase space profile for the nonintegrable system,
no matter how close it is to an integrable system, is dramatically different
from that of the completely integrable system. The KAM curves are analytic in
the real plane, but they have natural boundaries in the complex plane [39, 40,
41, 42]. Furthermore, it is highly probable that homoclinic intersections
between stable and unstable manifolds occur beyond the natural boundaries
[12]. Chaos in the complex plane developed there is difficult to $`$see’ from
the phase space profile in the real plane, but tunneling transport is driven
by chaotic orbits in such a deep complex plane [13].

Quantum tunneling is a phenomenon that does not exist in classical mechanics.
There are approaches that try to understand quantum tunneling in terms of
invariant objects in the real phase space [43, 44]. These implicitly assume
that tunneling tails of wave functions in nonintegrable systems can be
expressed in terms of tunneling tails of wave functions supported by invariant
manifolds in the real phase space. In other words, the broadening of the wave
functions [45] is assumed to be sufficient to capture the signature of
nonintegrable tunneling. However, there is a situation in which such an
approach does not work [13], and there one needs to make full use of chaos in
the complex plane. Chaos in the complex plane appears beyond natural
boundaries and thus is not reachable from the real plane. We believe that this
fact is not necessarily limited to the situation where no visible invariant
structures appear compared to the Planck cell as reported in Ref. [13], but
the mechanism found there works in tunneling processes in nonintegrable
systems in general. The phase spaces used so far to study the tunneling in
nonintegrable systems are too complex to specify purely quantum phenomena,
which could slip through any kind of analysis based on classical dynamics in
the real plane. The question of what are the unique characteristics of
tunneling effects in nonintegrable systems should be revisited.

###### Acknowledgements.

The authors are grateful to Yutaka Ishii for helpful comments on the complex
dynamical systems. This research was funded by JSPS KAKENHI Grants No.17K05583
and 22H01146.

## References

  * Voros [1983] A. Voros, The return of the quartic oscillator. the complex wkb method, Annales de l’I.H.P. Physique théorique 39, 211 (1983).
  * Dingle [1973] R. Dingle,  _Asymptotic expansions: their derivation and interpretation_ , Vol. 521 (Academic Press London, 1973).
  * Écalle [1981] J. Écalle,  _Les fonctions résurgentes:(en trois parties)_ , Vol. 1 (Université de Paris-Sud, Département de Mathématique, Bât. 425, 1981).
  * Delabaere  _et al._ [1997] E. Delabaere, H. Dillinger, and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, Journal of Mathematical Physics 38, 6126 (1997).
  * Kawai and Takei [2005] T. Kawai and Y. Takei,  _Algebraic analysis of singular perturbation theory_ , Vol. 227 (American Mathematical Society, 2005).
  * Callan Jr and Coleman [1977] C. G. Callan Jr and S. Coleman, Fate of the false vacuum. ii. first quantum corrections, Physical Review D 16, 1762 (1977).
  * Coleman [1977] S. Coleman, Fate of the false vacuum: Semiclassical theory, Physical Review D 15, 2929 (1977).
  * Coleman [1988] S. Coleman,  _Aspects of Symmetry: Selected Erice Lectures_ (Cambridge University Press, 1988).
  * George and Miller [1972] T. F. George and W. H. Miller, Complex-valued classical trajectories for linear reactive collisions of h + $\mathrm{H}_{2}$ below the classical threshold, The Journal of Chemical Physics 56, 5722 (1972).
  * Miller [1974] W. H. Miller, Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants, The Journal of Chemical Physics 61, 1823 (1974).
  * Shudo and Ikeda [1995] A. Shudo and K. S. Ikeda, Complex classical trajectories and chaotic tunneling, Physical Review Letters 74, 682 (1995).
  * Shudo and Ikeda [2011] A. Shudo and K. S. Ikeda, Complex semiclassical approach to chaotic tunneling, Dynamical Tunneling: Theory and Experiment , 139 (2011).
  * Koda  _et al._ [2023] R. Koda, Y. Hanada, and A. Shudo, Ergodicity of complex dynamics and quantum tunneling in nonintegrable systems, Physical Review E 108, 054219 (2023).
  * Davis and Heller [1981] M. J. Davis and E. J. Heller, Quantum dynamical tunneling in bound states, The Journal of Chemical Physics 75, 246 (1981).
  * Creagh [1998] S. C. Creagh, Tunneling in two dimensions, in  _Tunneling in Complex Systems_ , edited by S. Tomsovic (World Scientific, 1998) p. 35.
  * Keshavamurthy and Schlagheck [2011] S. Keshavamurthy and P. Schlagheck,  _Dynamical tunneling: theory and experiment_ (CRC Press, 2011).
  * Milnor [2011] J. Milnor,  _Dynamics in One Complex Variable_ , Vol. 160 (Princeton University Press, 2011).
  * Beardon [2000] A. F. Beardon,  _Iteration of rational functions: Complex analytic dynamical systems_ , Vol. 132 (Springer Science & Business Media, 2000).
  * Morosawa  _et al._ [2000] S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda,  _Holomorphic dynamics_ , Vol. 66 (Cambridge University Press, 2000).
  * Takahashi [2011] K. Takahashi, Semiclassical analysis of multidimensional barrier tunneling, Dynamical Tunneling: Theory and Experiment , 95 (2011).
  * Hanada  _et al._ [2015] Y. Hanada, A. Shudo, and K. S. Ikeda, Origin of the enhancement of tunneling probability in the nearly integrable system, Physical Review E 91, 042913 (2015).
  * Hanada  _et al._ [2023] Y. Hanada, K. S. Ikeda, and A. Shudo, Dynamical tunneling across the separatrix, Physical Review E 108, 064210 (2023).
  * Suris [1989] Y. B. Suris, Integrable mappings of the standard type, Functional Analysis and Its Applications 23, 74 (1989).
  * Iijima  _et al._ [2022] R. Iijima, R. Koda, Y. Hanada, and A. Shudo, Quantum tunneling in ultra-near-integrable systems, Physical Review E 106, 064205 (2022).
  * Meiss [1992] J. D. Meiss, Symplectic maps, variational principles, and transport, Reviews of Modern Physics 64, 795 (1992).
  * Note [1] The integration of $V_{\lambda}(q)$ is numerically evaluated by “double exponential quadrature.
  * Berry  _et al._ [1979] M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros, Quantum maps, Annals of Physics 122, 26 (1979).
  * Casati  _et al._ [1979] G. Casati, B. V. Chirikov, F. M. Izraelev, and J. Ford, Stochastic behavior of a quantum pendulum under a periodic perturbation, in  _Stochastic Behavior in Classical and Quantum Hamiltonian Systems_, Lecture Notes in Physics, edited by G. Casati and J. Ford (Springer, Berlin, Heidelberg, 1979) pp. 334–352.
  * [29] Advanpix LLC., Multiprecision computing toolbox for matlab, http://www.advanpix.com/.
  * Berry and Mount [1972] M. V. Berry and K. E. Mount, Semiclassical approximations in wave mechanics, Reports on Progress in Physics 35, 315 (1972).
  * Shudo  _et al._ [2009a] A. Shudo, Y. Ishii, and K. S. Ikeda, Julia sets and chaotic tunneling: I, Journal of Physics A 42, 265101 (2009a).
  * Shudo  _et al._ [2009b] A. Shudo, Y. Ishii, and K. S. Ikeda, Julia sets and chaotic tunneling: Ii, Journal of Physics A 42, 265102 (2009b).
  * Shudo and Ikeda [2008] A. Shudo and K. S. Ikeda, Stokes geometry for the quantum hénon map, Nonlinearity 21, 1831 (2008).
  * Shudo and Ikeda [2016] A. Shudo and K. S. Ikeda, Toward pruning theory of the stokes geometry for the quantum hénon map, Nonlinearity 29, 375 (2016).
  * Bullett  _et al._ [2019] S. Bullett, L. Lomonaco, and C. Siqueira, Correspondences in complex dynamics, New Trends in One-Dimensional Dynamics: In Honour of Welington de Melo on the Occasion of His 70th Birthday IMPA 2016, Rio de Janeiro, Brazil, November 14–17 , 51 (2019).
  * Ishikawa  _et al._ [2012] A. Ishikawa, A. Tanaka, K. S. Ikeda, and A. Shudo, Diffraction and tunneling in systems with mixed phase space, Physical Review E 86, 036208 (2012).
  * Ablowitz  _et al._ [1980] M. J. Ablowitz, A. Ramani, and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of p-type. ii, Journal of Mathematical Physics 21, 1006 (1980).
  * Haake [1991] F. Haake,  _Quantum signatures of chaos_ (Springer, 1991).
  * Greene and Percival [1981] J. M. Greene and I. C. Percival, Hamiltonian maps in the complex plane, Physica D 3, 530 (1981).
  * Percival [1982] I. C. Percival, Chaotic boundary of a hamiltonial map, Physica D 6, 67 (1982).
  * Berretti and Chierchia [1990] A. Berretti and L. Chierchia, On the complex analytic structure of the golden invariant curve for the standard map, Nonlinearity 3, 39 (1990).
  * Berretti and Marmi [1992] A. Berretti and S. Marmi, Standard map at complex rotation numbers: creation of natural boundaries, Physical Review Letters 68, 1443 (1992).
  * Tomsovic and Ullmo [1994] S. Tomsovic and D. Ullmo, Chaos-assisted tunneling, Physical Review E 50, 145 (1994).
  * Brodier  _et al._ [2002] O. Brodier, P. Schlagheck, and D. Ullmo, Resonance-assisted tunneling, Annals of Physics 300, 88 (2002).
  * Balazs and Voros [1990] N. Balazs and A. Voros, Wigner’s function and tunneling, Annals of Physics 199, 123 (1990).