#  $\mathcal{PT}$-symmetric quantum mechanics

Carl M. Bender  Department of Physics, Washington University, St. Louis,
Missouri 63130, USA<EMAIL_ADDRESS>Daniel W. Hook  Centre for Complexity
Science, Imperial College London, London SW7 2AZ, UK<EMAIL_ADDRESS>
###### Abstract

It is generally assumed that a Hamiltonian for a physically acceptable quantum
system (one that has a positive-definite spectrum and obeys the requirement of
unitarity) must be Hermitian. However, a $\mathcal{PT}$-symmetric Hamiltonian
can also define a physically acceptable quantum-mechanical system even if the
Hamiltonian is not Hermitian. The study of $\mathcal{PT}$-symmetric quantum
systems is a young and extremely active research area in both theoretical and
experimental physics. The purpose of this Review is to provide established
scientists as well as graduate students with a compact, easy-to-read
introduction to this field that will enable them to understand more advanced
publications and to begin their own theoretical or experimental research
activity. The ideas and techniques of $\mathcal{PT}$ symmetry have been
applied in the context of many different branches of physics. This Review
introduces the concepts of $\mathcal{PT}$ symmetry by focusing on elementary
one-dimensional $\mathcal{PT}$-symmetric quantum and classical mechanics and
relies in particular on oscillator models to illustrate and explain the basic
properties of $\mathcal{PT}$-symmetric quantum theory.

September 1, 2024

###### Contents

  1. I Introduction to $\mathcal{PT}$ symmetry
    1. I.1 Mathematical background
    2. I.2 $\mathcal{CPT}$ theorem
  2. II Examples of $\mathcal{PT}$-symmetric Hamiltonians
    1. II.1 An elementary $\mathcal{PT}$-symmetric Hamiltonian
    2. II.2 Some $\mathcal{PT}$-symmetric Hamiltonians have complex spectra
    3. II.3 Reality of eigenvalues
    4. II.4 Multiplicative deformation
    5. II.5 $2\times 2$ Matrix Hamiltonian with a $\mathcal{PT}$ transition
    6. II.6 Other multiplicative deformations
  3. III Deformation of eigenvalue problems
    1. III.1 Complex asymptotic approximations
    2. III.2 Boundary conditions for the Sextic Hamiltonian
    3. III.3 Complex deformation of the harmonic oscillator
    4. III.4 Boundary conditions for $H=p^{2}+x^{2}(ix)^{\varepsilon}$
    5. III.5 Sextic potential revisited
    6. III.6 Measuring the position of a particle
    7. III.7 $x^{10}$ potential
  4. IV Upside-down potentials
    1. IV.1 Classical particle in a quartic upside-down potential
    2. IV.2 Quantum particle in a quartic upside-down potential
    3. IV.3 Elementary proof of spectral reality
    4. IV.4 Discussion of the parity anomaly
    5. IV.5 Comparison of $x^{4}$ and $-x^{4}$ Potentials
  5. V Anharmonic oscillators
    1. V.1 Particle-physics analysis of $\mathcal{PT}$-symmetric bound states
    2. V.2  Classical symmetric double-well potentials
    3. V.3  $\mathcal{PT}$-symmetric quasi-exactly solvable Hamiltonians
    4. V.4 Hermitian and $\mathcal{PT}$-symmetric anharmonic oscillators
      1. V.4.1 Hermitian anharmonic oscillator
      2. V.4.2 Elementary model of topological quantization
      3. V.4.3 Large-order behavior of perturbation theory
      4. V.4.4 Summation of divergent perturbation series
      5. V.4.5 Perturbation series for $\mathcal{PT}$-symmetric oscillators
  6. VI This is quantum mechanics (not just complex analysis)
    1. VI.1 Finding the inner product
    2. VI.2 The $\mathcal{C}$ operator and the $\mathcal{CPT}$ inner product
    3. VI.3 Discussion of the $\mathcal{CPT}$-inner product
    4. VI.4 Example: $\mathcal{PT}$-symmetric $2\times 2$ matrix Hamiltonian
    5. VI.5  Calculating the $\mathcal{C}$ Operator
    6. VI.6  Algebraic calculation of the $\mathcal{C}$ operator
    7. VI.7  Isospectral Hermitian and $\mathcal{PT}$-symmetric Hamiltonians
    8. VI.8 Observables
  7. VII Opportunities for Research
  8. A Increasing role of complex variables in physics

In realms unseen, where quantum whispers speak,

A novel symmetry doth enter stage,

With parity and time, a dance unique,

To bind the quantum actors, new age.

The $\mathcal{PT}$-symmetry, a twofold guise,

Where $\mathcal{P}$ reflects in space, yet $\mathcal{T}$, in time,

Together sing a tale of compromise,

In quantum world, a balance, most sublime.

Unbroken, this domain of spectral dance,

The loss and gain of energy entwined,

Their quantum states, a harmonizing trance,

In eerie beauty, to each other bind.

And thus, in deep embrace of mystic schemes,

$\mathcal{PT}$-symmetric quantum magic gleams.

—Sonnet from ChatG$\mathcal{PT}$

##  I Introduction to $\mathcal{PT}$ symmetry

The study of $\mathcal{PT}$-symmetric quantum mechanics began in 1998 with the
observation that a quantum-mechanical Hamiltonian that is invariant under
combined space reflection (parity) $\mathcal{P}$ and time reversal
$\mathcal{T}$ can have a spectrum that is entirely real and bounded below even
if the Hamiltonian is not Hermitian [35]. This letter was followed shortly
thereafter by a second which demonstrated that when the spectrum of a
$\mathcal{PT}$-symmetric Hamiltonian is real and bounded below, the
Hamiltonian defines a unitary (probability-conserving) quantum-mechanical
theory [45]. Thus, the mathematical constraint that a Hamiltonian operator be
Hermitian in the Dirac sense $H^{\dagger}=H$, where ${\dagger}$ represents
combined complex conjugation and matrix transposition, can be weakened to the
more physically intuitive constraint of spacetime reflection invariance
($\mathcal{PT}$ symmetry):

$[H,\mathcal{PT}]=0.$ (1.1)

Because the condition of $\mathcal{PT}$ invariance is a weaker constraint than
Hermiticity, it allows one to study vastly many new non-Hermitian quantum
theories, some of which exhibit surprising physical properties. Thus,
$\mathcal{PT}$ symmetry offers a rich area for theoretical and experimental
investigation. There are already over 10,000 papers published and numerous
reviews, books, and theses written on the subject of $\mathcal{PT}$ symmetry.
Many research grants have been awarded and numerous conferences and symposia
have been held. Research activity on $\mathcal{PT}$ symmetry is accelerating
(see Fig. 1).

Figure 1: Annual publications referencing $\mathcal{PT}$ symmetry since its
inception in 1998 (not including papers on the arXiv). The total number of
research publications is growing rapidly and currently exceeds 10,000. Source:
Dimensions from Digital Science [161].

###  I.1 Mathematical background

The study of $\mathcal{PT}$-symmetric quantum mechanics draws heavily on
complex analysis because the condition of $\mathcal{PT}$ symmetry arises when
we deform a Hermitian theory, such as the quantum harmonic oscillator, into
the non-Hermitian complex domain. We will see that $\mathcal{PT}$ symmetry
marries the fundamental concepts of quantum physics with the mathematical
theory of complex variables.

While complex numbers have been studied by mathematicians for many centuries,
complex numbers did not play an essential role in the formulation of physical
theories until the twentieth century when the study of quantum mechanics
began. The complex number $i$ does not even appear in early studies of special
and general relativity. It was not until many years later, when the
representations of the homogeneous Lorentz group were analyzed in depth, did
it become clear that fermionic spinor representations, which require complex
numbers, were physically relevant. In particular, the fundamental
spin-$\tfrac{1}{2}$ representation of the Lorentz group is complex [147, 148].

The homogeneous Lorentz group is defined as the set of all real $4\times 4$
matrices (matrices whose 16 elements are all real numbers) that map the
spacetime vector $x^{\mu}$ into another spacetime vector $x^{\prime\mu}$ in
such a way that $x^{\mu}x_{\mu}$ remains invariant:

$x^{2}+y^{2}+z^{2}-t^{2}=x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-t^{\prime 2}.$
(1.2)

This condition guarantees that the speed of light is the same in all inertial
frames of reference.

In the context of the Lorentz group, $\mathcal{PT}$ symmetry appears naturally
and at a fundamental level. The parity matrix $\mathcal{P}$ is an element of
the Lorentz group because it flips the sign of the spatial coordinates,

$\mathcal{P}:\,(x,y,z,t)\to(-x,-y,-z,t),$

and thus leaves $x^{2}+y^{2}+z^{2}-t^{2}$ invariant. The time-reversal matrix
$\mathcal{T}$ is also an element of the Lorentz group because it flips the
sign of the time coordinate,

$\mathcal{T}:\,(x,y,z,t)\to(x,y,z,-t),$

and again leaves $x^{2}+y^{2}+z^{2}-t^{2}$ invariant. The $\mathcal{PT}$
matrix changes the sign of all four spacetime coordinates,

$\mathcal{PT}:\,(x,y,z,t)\to(-x,-y,-z,-t),$

and thus it too is an element of the Lorentz group.

Mathematically, the real Lorentz group is a six-parameter continuous group
consisting of four disconnected components, as indicated schematically in Fig.
2. The first part, as shown at the top of Fig. 2, is called the proper
orthochronous Lorentz group (POLG). The POLG contains all the elements of the
homogeneous Lorentz group that are continuously connected in group space to
the identity element, which is the transformation that leaves the vector
$(x,y,z,t)$ unchanged. The POLG is a subgroup of the homogeneous Lorentz
group. The second component of the Lorentz group (the left part of Fig. 2)
consists of all of the Lorentz transformations in the POLG multiplied by
$\mathcal{P}$. This part of the Lorentz group is not a subgroup because it
does not contain an identity element. The third component of the Lorentz group
(right part of Fig. 2) consists of all of the Lorentz transformations in the
POLG multiplied by $\mathcal{T}$. Again, this part of the Lorentz group is not
a subgroup. The fourth part of the Lorentz group consists of all of the
elements of the POLG multiplied by the spacetime reflection operator
$\mathcal{PT}$ and it too is not a subgroup.

Figure 2: Schematic picture of the group space of the homogeneous real Lorentz
group. This continuous group consists of four disconnected parts: At the top
of the diagram is the proper orthochronous Lorentz group (POLG); the POLG
contains the identity element and is a subgroup of the Lorentz group. The
other three components of the homogeneous real Lorentz group are the POLG
multiplied by the parity (space-reflection) operator $\mathcal{P}$, the time-
reversal operator $\mathcal{T}$, and the spacetime reflection operator
$\mathcal{PT}$. If we extend the real Lorentz group to the complex Lorentz
group [147], there are now only two disconnected parts. Continuous paths
through complex group space (indicated as vertical and horizontal lines)
connect the top and bottom components and the left and right components of the
real Lorentz group.

For many years after the discovery of special relativity, the homogeneous
Lorentz group pictured in Fig. 2 was believed to be the fundamental symmetry
group of the natural world. However, Lee and Yang theorized that space
reflection (parity) $\mathcal{P}$ is not a fundamental symmetry of the
universe [175]; the 1957 Nobel Prize was awarded for this work. The work of
Lee and Yang implies that an experiment can be designed that achieves
different experimental results in left-handed and right-handed laboratories;
that is, two laboratories that are spatial reflections (mirror images) of one
another. Such an experiment was performed by Wu [264], for which she was
awarded the Wolf Prize in 1978.

Subsequently, Cronin and Fitch were awarded the 1980 Nobel Prize for
demonstrating that time reversal $\mathcal{T}$ is also not a symmetry of
nature [113]. Their experiment demonstrates, in principle, that a laboratory
traveling forward in time is experimentally distinguishable from a physically
identical laboratory that is traveling backward in time. The implication of
all this work is that the Lorentz transformations in the left and the right
portions of the Lorentz group illustrated in Fig. 2 are not fundamental
symmetries of the universe.

These discoveries regarding parity and time reversal raise an obvious question
regarding the Lorentz transformations in the lower portion of Fig. 2. If we
have two laboratories, one traveling forward in time and the other space-
reflected and traveling backward in time, is there an experiment that will
yield different results in these two laboratories? Are these two laboratories
experimentally distinguishable or is combined $\mathcal{PT}$ reflection
(spacetime reflection) a fundamental discrete symmetry of nature?

To examine this question, it helps to extend the real homogeneous Lorentz
group to the complex homogeneous Lorentz group, which is the set of all
complex $4\times 4$ matrices that leave $x^{2}+y^{2}+z^{2}-t^{2}$ invariant.
The complex homogeneous Lorentz group consists of two, and not four,
disconnected parts. Figure 2 shows that there is a continuous path (horizontal
line) through complex-Lorentz-group space connecting the left and right
portions of the real Lorentz group. There is also a continuous path (vertical
line) connecting the upper and lower parts of the real Lorentz group [250].
This complex group structure suggests that since it is natural to extend the
real number system to the complex number system and since complex numbers
appear commonly in modern physics, one should consider $\mathcal{PT}$
reflection as a possible symmetry of some physical theories.

###  I.2 $\mathcal{CPT}$ theorem

Extending the real homogeneous Lorentz group to the complex Lorentz group is
the key ingredient in the proof of a profound theorem in quantum field theory
and particle physics known as the $\mathcal{CPT}$ theorem [250]. This theorem
states that two identical physical measurements, where the first measurement
is done in a laboratory traveling forward in time and consisting of ordinary
matter and the second is done in a space-reflected laboratory traveling
backward in time and consisting of antimatter, will return identical results.

The proof of the $\mathcal{CPT}$ theorem requires several crucial assumptions
made in axiomatic quantum field theory about the nature of particle
interactions, such as unitarity, locality, causality, positivity of the
particle spectrum, and Hermiticity of the Hamiltonian. These assumptions allow
one to extend the real Lorentz group to the complex Lorentz group and to
perform an analytic continuation in group space from the upper to the lower
portion of Fig. 2. Along this path the state of an elementary particle
undergoes a transition from the original particle to its corresponding
antiparticle. [The proof of the $\mathcal{CPT}$ theorem fills an entire book
[250].]

We do not discuss quantum field theory, antiparticles, and the charge-
conjugation operator in particle physics here; this Review is limited to a
discussion of quantum mechanics. However, we will see that there is a
fascinating connection between the $\mathcal{CPT}$ theorem and the results on
$\mathcal{PT}$ symmetry presented in this Review. The development in this
Review can be viewed as establishing the quantum-mechanical converse of the
$\mathcal{CPT}$ theorem.

We begin by explaining how to construct various non-Hermitian quantum-
mechanical theories (not field theories) that are $\mathcal{PT}$ invariant and
we examine their properties. The spectra of such theories are often entirely
real and bounded below. We then show that when the spectrum of a
$\mathcal{PT}$-symmetric Hamiltonian is real, the Hamiltonian has a hidden
invariance; there exists a new quantum-mechanical operator $\mathcal{C}$ that
commutes with the $\mathcal{PT}$-symmetric Hamiltonian. (We call this operator
$\mathcal{C}$ because its properties closely resemble those of the charge-
conjugation operator $\mathcal{C}$ used in the $\mathcal{CPT}$ theorem of
particle physics.) We use this $\mathcal{C}$ operator to construct an inner
product endowed with a positive-definite norm and establish that the
$\mathcal{PT}$-symmetric Hamiltonian defines a unitary quantum-mechanical
theory.

The organization of this Review is straightforward: Section II discusses the
construction and properties of $\mathcal{PT}$-symmetric Hamiltonians with
special emphasis on the calculation of eigenvalues. It is shown how to obtain
$\mathcal{PT}$-symmetric quantum-mechanical theories as complex deformations
of Hermitian theories. These examples emphasize the intimate connection
between the physical constraint of $\mathcal{PT}$ symmetry and the
mathematical theory of complex variables. Section III examines the process of
complex analytical deformation in detail and Sec. IV gives a quantum and
classical analysis of $\mathcal{PT}$-symmetric upside-down potentials. Section
V describes the remarkable properties of the $\mathcal{PT}$-symmetric quartic
anharmonic oscillator at both the quantum-mechanical and the classical-
mechanical levels.

Having discussed the properties of $\mathcal{PT}$-symmetric eigenvalue
problems for a variety of quantum-mechanical models, we then address some
crucial theoretical issues regarding $\mathcal{PT}$-symmetric quantum
mechanics. In Sec. VI we construct the $\mathcal{C}$ operator, which in turn
is used to define the $\mathcal{PT}$-symmetric inner product and the Hilbert
space. (In conventional Hermitian quantum mechanics we already know what the
inner product is even before we know the Hamiltonian, but in
$\mathcal{PT}$-symmetric quantum mechanics the inner product is determined by
the Hamiltonian itself.) Section VII briefly summarizes some of the current
research on $\mathcal{PT}$-symmetric quantum-mechanical systems and suggests
possibilities for future research.

Because the study of $\mathcal{PT}$-symmetric quantum theory involves heavy
use of complex-variable theory, we include an Appendix that briefly summarizes
the historical interest in extending real mathematics to complex mathematics,
the study of which began in earnest in the sixteenth century. Complex calculus
is powerful and can often solve problems that the calculus of real variables
is incapable of solving. Physicists have long understood the power of complex
analysis as a calculational tool, but complex numbers played no role in
formulating physical theories until the 20th century. Since its appearance in
quantum theory, the number $i$ has played an increasingly important role in
physics.

##  II Examples of $\mathcal{PT}$-symmetric Hamiltonians

How does one construct a $\mathcal{PT}$-symmetric quantum-mechanical theory?
An elementary recipe for creating a $\mathcal{PT}$-symmetric Hamiltonian is to
begin with a quantum-mechanical Hamiltonian that is both Hermitian and
$\mathcal{PT}$ symmetric, and then to perform a $\mathcal{PT}$-symmetric
deformation of this Hamiltonian [35, 33, 51]. The resulting Hamiltonian, which
is a complex analytic continuation of the original Hermitian Hamiltonian, is
still $\mathcal{PT}$ symmetric but it is no longer Hermitian. Nevertheless,
the deformed Hamiltonian operator may continue to have an entirely real
spectrum whose energy levels remain well separated and do not cross. [For
further examples see [143, 271, 176, 177]].

###  II.1 An elementary $\mathcal{PT}$-symmetric Hamiltonian

To illustrate the construction of a $\mathcal{PT}$-symmetric Hamiltonian we
perform a simple $\mathcal{PT}$ deformation of the quantum harmonic-oscillator
Hamiltonian

$H=\mbox{$\textstyle{\frac{1}{2}}$}p^{2}+\mbox{$\textstyle{\frac{1}{2}}$}x^{2}.$
(2.1)

This Hamiltonian is Hermitian $H^{\dagger}=H$ and the eigenvalues $E_{n}$ of
this Hamiltonian are all real and positive:

$E_{n}=n+\mbox{$\textstyle{\frac{1}{2}}$}\quad(n=0,\,1,\,2,\,...).$ (2.2)

The $\mathcal{P}$ and $\mathcal{T}$ operators are conventionally defined as
follows: The parity operator $\mathcal{P}$ is a linear Hermitian operator that
performs space reflection, so $\mathcal{P}^{2}=I$ ($I$ is the unity operator)
and $\mathcal{P}^{-1}=\mathcal{P}$. Under parity reflection both the position
operator $x$ and the momentum operator $p$ change sign:

$\mathcal{P}x\mathcal{P}^{-1}=-x,\qquad\mathcal{P}p\mathcal{P}^{-1}=-p.$ (2.3)

The time-reversal operator $\mathcal{T}$ is also a reflection operator and for
bosonic systems $\mathcal{T}^{2}=I$, so $\mathcal{T}^{-1}=\mathcal{T}$. Time
reversal leaves $x$ invariant but changes the sign of $p$:

$\mathcal{T}x\mathcal{T}^{-1}=x,\qquad\mathcal{T}p\mathcal{T}^{-1}=-p.$ (2.4)

However, unlike the $\mathcal{P}$ operator, the $\mathcal{T}$ operator is not
linear; $\mathcal{T}$ is said to be antilinear because it changes the sign of
the complex number $i$:

$\mathcal{T}i\mathcal{T}^{-1}=-i.$ (2.5)

The operations of reversing space and reversing time are independent and thus
$\mathcal{P}$ and $\mathcal{T}$ commute $[\mathcal{P},\mathcal{T}]=0$.

According to the definitions above we see immediately that the Hamiltonian
(2.1) satisfies (1.1) and thus this Hamiltonian is $\mathcal{PT}$ symmetric as
well as Hermitian. (In this case, $H$ is independently invariant under
$\mathcal{P}$ and $\mathcal{T}$ reflections.) The combination $ix$ is not
Hermitian, but it is $\mathcal{PT}$ symmetric. Therefore, we can use the $ix$
operator to perform a $\mathcal{PT}$-symmetric deformation of the Hamiltonian
$H$: Let $\varepsilon$ be a real parameter called a deformation parameter. The
simplest kind of deformation is an additive deformation, so we use an additive
deformation to construct the deformed Hamiltonian

$H=\mbox{$\textstyle{\frac{1}{2}}$}p^{2}+\mbox{$\textstyle{\frac{1}{2}}$}x^{2}+\varepsilon
ix.$ (2.6)

The Hamiltonian (2.6) is not Hermitian when $\varepsilon\neq 0$ because $ix$
is not Hermitian, but $H$ is $\mathcal{PT}$ symmetric because $H$ commutes
with the $\mathcal{PT}$ operator.

A brief calculation in which we replace $x$ by $x-i\varepsilon$ in the time-
independent Schrödinger equation associated with this Hamiltonian verifies the
surprising result that the eigenvalues of $H$ in (2.6) are all real:

$E_{n}=n+\mbox{$\textstyle{\frac{1}{2}}$}\big{(}1+\varepsilon^{2})\quad(n=0,\,1,\,2,\,...).$
(2.7)

Although the eigenvalues of $H$ in (2.6) are real, the corresponding
eigenfunctions $\psi_{n}(x)$ are complex. However, they still satisfy the
original undeformed boundary condition on $\psi_{n}(x)$; namely, that
$\psi_{n}(x)$ vanishes exponentially as $x\to\pm\infty$. An important property
of these eigenfunctions is that they are all $\mathcal{PT}$ symmetric; that
is, $\psi_{n}(x)$ remains invariant (apart from an overall multiplicative
phase) if we replace $i$ by $-i$ (that is, take the complex conjugate) and
replace $x$ by $-x$. Furthermore, the deformed eigenfunctions continue to be
orthogonal,

$\int_{-\infty}^{\infty}dx\,\psi_{m}(x)\psi_{n}(x)=0\quad(m\neq n),$

for all real $\varepsilon$.

This example suggests that there are infinitely many more
$\mathcal{PT}$-symmetric Hamiltonians than Hermitian Hamiltonians. In place of
the single Hermitian harmonic-oscillator Hamiltonian in (2.1), we now have a
continuously infinite class of non-Hermitian Hamiltonians in (2.6)
parametrized by $\varepsilon$. All of these Hamiltonians have entirely real
positive spectra and they define perfectly acceptable quantum-mechanical
theories. The Hermitian Hamiltonian corresponding to $\varepsilon=0$ is just
one special case that is not fundamentally different from the other
Hamiltonians. This shows that while Hermiticity is a sufficient condition for
having a real spectrum, it is not a necessary condition. This also suggests
that we might adopt the view that $\mathcal{PT}$ symmetry is a fundamental
property of a quantum Hamiltonian with Hermiticity being a mere special case.

###  II.2 Some $\mathcal{PT}$-symmetric Hamiltonians have complex spectra

The example above might lead one to conjecture that the property of
$\mathcal{PT}$ symmetry in (1.1) is strong enough to establish that the
eigenvalues of any $\mathcal{PT}$-symmetric Hamiltonian are all real. This
conjecture is false, but it is nevertheless instructive to examine the
following incorrect attempt to prove this conjecture [33, 51].

Let $\psi$ be a (time-independent) eigenstate of the Hamiltonian operator $H$
with eigenvalue $E$:

$H\psi=E\psi.$ (2.8)

Assume that $H$ is $\mathcal{PT}$ symmetric; that is, $H$ commutes with
$\mathcal{PT}$: $[H,\mathcal{PT}]=0$. Next, recall a theorem from linear
algebra that is often quoted in elementary quantum-mechanics courses: If two
operators $A$ and $B$ commute, then they are simultaneously diagonalizable;
that is, an eigenstate of $A$ is also an eigenstate of $B$. For example, if
$H$ commutes with $\mathcal{P}$, then the energy eigenstates of $H$ have
definite parity. On the basis of this theorem, one might think that if $H$
commutes with $\mathcal{PT}$, then $\psi$ must also be an eigenstate of
$\mathcal{PT}$:

$\mathcal{PT}\psi=\lambda\psi,$ (2.9)

where $\lambda$ is an eigenvalue of the $\mathcal{PT}$ operator. (Warning:
This assumption is wrong!)

Let us examine the eigenvalue equation (2.9). Because
$\mathcal{P}^{2}=\mathcal{T}^{2}=I$ and $[\mathcal{P},\mathcal{T}]=0$ it
follows that $(\mathcal{PT})^{2}=I$. Also, $\mathcal{T}$ performs complex
conjugation. Thus, if we multiply (2.9) on the left by $\mathcal{PT}$, we
obtain

$\psi=\lambda^{*}\lambda\psi.$ (2.10)

This proves that $|\lambda|=1$. Thus, the eigenvalues of the $\mathcal{PT}$
operator lie on the unit circle in the complex plane. This is an interesting
complex generalization of the result that the eigenvalues of the $\mathcal{P}$
operator are $\pm 1$; that is, that the eigenvalues of $\mathcal{P}$ lie on a
one-dimensional unit circle on the real axis (instead of on a two-dimensional
circle in the complex plane).

Finally, we multiply the eigenvalue equation (2.8) on the left by the
$\mathcal{PT}$ operator and divide the resulting equation by $\lambda$, which
is allowed because we have shown that $\lambda$ is nonzero. We are therefore
tempted to conclude that

$E=E^{*}.$ (2.11)

While it is correct that $|\lambda|=1$, this conclusion is false!

The mistake in this argument is that we misquoted the elementary theorem
above; we neglected to say that $A$ and $B$ must be linear operators. It is
not necessarily true that an eigenstate of $H$ is also an eigenstate of
$\mathcal{PT}$ because while $\mathcal{PT}$ commutes with $H$, $\mathcal{PT}$
is not a linear operator; $\mathcal{PT}$ is antilinear because $\mathcal{T}$
performs complex conjugation. However, we have established an important
result: If an eigenstate of a $\mathcal{PT}$-symmetric Hamiltonian is also an
eigenstate of $\mathcal{PT}$, then the corresponding energy eigenvalue is
real.

Unfortunately, proving that an eigenstate of a $\mathcal{PT}$-symmetric
Hamiltonian is simultaneously an eigenstate of the $\mathcal{PT}$ operator can
be nontrivial; this must be done on a case-by-case basis for each
$\mathcal{PT}$ symmetric Hamiltonian. In some cases the proof may be difficult
and one might have to resort to a convincing numerical analysis instead of a
mathematical proof. Proving that the eigenvalues of $H$ in (2.6) are real is
easy, but a general technique for proving spectral reality for a given
$\mathcal{PT}$ symmetric Hamiltonian is unlikely to be found.

Nevertheless, there are infinitely many more non-Hermitian
$\mathcal{PT}$-symmetric Hamiltonians that have entirely real spectra than
Hermitian Hamiltonians. These $\mathcal{PT}$-symmetric Hamiltonians often
describe physical systems having novel and experimentally verifiable
properties. This is why the theoretical and experimental study of
$\mathcal{PT}$ symmetry has become so active.

###  II.3 Reality of eigenvalues

An energy measurement performed on an isolated quantum system will always
return a real number. This number is the expectation value of the Hamiltonian
$H$ in the quantum state of the system and it is equal to one of the
eigenvalues of $H$. Thus, the eigenvalues of a Hamiltonian that describes an
isolated quantum system are always real.

Let us recall the standard proof that the energy eigenvalues of a Hermitian
Hamiltonian are real. For simplicity, we assume that the quantum system is
one-dimensional and defined on the whole real line and that the time-
independent Schrödinger eigenvalue equation (2.8) has the simple form

$-\psi^{\prime\prime}(x)+V(x)\psi(x)=E\psi(x)\quad(-\infty<x<\infty).$ (2.12)

We then multiply (2.12) on the left by $\psi^{*}(x)$, integrate from $-\infty$
to $\infty$, and integrate the first term on the left side by parts. The
resulting equation reads

$\displaystyle\int_{-\infty}^{\infty}dx\,|\psi^{\prime}(x)|^{2}+\int_{-\infty}^{\infty}dx\,V(x)|\psi(x)|^{2}$
$\displaystyle\quad=E\int_{-\infty}^{\infty}dx\,|\psi(x)|^{2}.$ (2.13)

Finally, we note that the potential $V(x)$ is real if $H$ is Hermitian, so all
three integrals in (2.13) are real. This implies that $E$ is real.

The key step in this argument is the integration by parts. Integration by
parts is possible only if the eigenfunction $\psi(x)$ vanishes sufficiently
fast as $|x|\to\infty$. Thus, the proof that the eigenspectrum is real
requires more than just the Hermiticity of $H$. Hermiticity, like the
condition of $\mathcal{PT}$ symmetry, is a local condition that holds (or does
not hold) at each value of $x$; thus, one can immediately see if a Hamiltonian
is Hermitian or $\mathcal{PT}$ symmetric by mere inspection. However, the
condition that the eigenfunctions vanish fast enough as $|x|\to\infty$ to
allow integration by parts is a nontrivial global (nonlocal) condition. These
local and global conditions taken together are called self-adjointness. (Note
that there is no difference between the condition of Hermiticity and self-
adjointness for a matrix Hamiltonian because in that case the proof that the
eigenvalues are real does not require an integration by parts.)

This condition of self-adjointness is crucial in quantum mechanics because it
implies that the quantum theory is unitary (probability-conserving). In Sec.
VI we show that if the eigenspectrum of a $\mathcal{PT}$-symmetric Hamiltonian
$H$ is entirely real, then $H$ defines a unitary theory of quantum mechanics.
For many nontrivial $\mathcal{PT}$-symmetric Hamiltonians, the spectrum has
been proved to be entirely real, as we see in the next example.

###  II.4 Multiplicative deformation

Let us perform a multiplicative deformation of the harmonic-oscillator
Hamiltonian (2.1) to obtain the non-Hermitian $\mathcal{PT}$-symmetric class
of Hamiltonians

$H=p^{2}+x^{2}(ix)^{\varepsilon}.$ (2.14)

This nontrivial Hamiltonian is not Hermitian, but the spectrum of $H$ is
entirely real and positive if $\varepsilon\geq 0$ [35]. In addition, the
eigenfunctions are all $\mathcal{PT}$ symmetric if $\varepsilon\geq 0$.
However, if $\varepsilon<0$, only a finite number of the lowest-lying
eigenvalues are real and the corresponding eigenfunctions are $\mathcal{PT}$
symmetric. The rest of the eigenvalues are complex and the corresponding
eigenfunctions are not $\mathcal{PT}$ symmetric.

A plot of the lowest seven eigenvalues for $-1<\varepsilon<0.5$ is shown in
Fig. 3. The range $\varepsilon>0$, in which the eigenvalues are all real, is
called the region of unbroken $\mathcal{PT}$ symmetry. In this region the
eigenvalues are positive, discrete, and increase monotonically with increasing
$\varepsilon$. At $\varepsilon=0$, the low edge of the region of unbroken
$\mathcal{PT}$ symmetry, the eigenvalues are just the harmonic-oscillator
eigenvalues given in (2.2). The range $\varepsilon<0$ is called the region of
broken $\mathcal{PT}$ symmetry. The boundary between these two regions, which
we call the $\mathcal{PT}$ transition, occurs at the unique value of
$\varepsilon$ at which the Hamiltonian is Hermitian. For larger positive
values of $\varepsilon$ the energy levels all continue to rise and remain well
separated. In Fig. 4 the upper range of $\varepsilon$ is extended from $0.5$
to $4$ and the first twenty-one real eigenvalues are shown.

Figure 3: Lowest seven eigenvalues of $H$ in (2.14) plotted for
$-1<\varepsilon<0.5$. These eigenvalues are real, positive, discrete, and
monotonically increasing for $\varepsilon\geq 0$ and they are displayed as
solid lines. As $\varepsilon$ decreases below 0, the ground-state energy
remains real and becomes singular at $\varepsilon=-1$. The other eigenvalues
merge pairwise in a regular and orderly fashion, starting with the highest
eigenvalues, and immediately bifurcate sequentially into complex-conjugate
pairs (whose real and imaginary parts are indicated by dashed/broken lines).
The values of $\varepsilon$ at which pairs of real eigenvalues merge are
called exceptional points. These exceptional points are square-root
singularities, sometimes referred to as pitchfork bifurcations, in the
complex-$\varepsilon$ plane. Figure 4: First twenty-one eigenvalues of $H$ in
(2.14) plotted for $-1<\varepsilon\leq 4$. The eigenvalues are all real,
positive, discrete, and monotonically increasing for $\varepsilon\geq 0$. This
plot only shows the eigenvalues when they are real. Note that as $\varepsilon$
decreases below 0 the eigenvalues become degenerate pairwise and enter the
complex plane as complex-conjugate pairs, but the ground-state energy
continues to remain real and becomes singular as $\varepsilon$ approaches
$-1$.

In general, whenever a $\mathcal{PT}$-symmetric deformation of a
$\mathcal{PT}$-symmetric Hermitian Hamiltonian $H$ is performed, the
eigenvalues of the deformed Hamiltonian are either real or else they occur in
complex-conjugate pairs. It is easy to see why this is true: The eigenvalues
of a Hamiltonian are the roots of the secular-determinant equation

${\rm det}(H-IE)=0,$ (2.15)

which is a polynomial in $E$ if $H$ is a finite-dimensional matrix. We insert
the unit operator $I=\mathcal{PT}\mathcal{PT}$ in the determinant, commute one
factor of $\mathcal{PT}$ through $H-IE$, and then permute the other factor of
$\mathcal{PT}$ cyclically around so that the operator
$\mathcal{PT}\mathcal{PT}$ reappears and can be replaced by the unit operator.
Remember that if $H$ is $\mathcal{PT}$ symmetric it commutes with
$\mathcal{PT}$, but when $\mathcal{PT}$ commutes past $E$ it replaces $E$ with
$E^{*}$. Thus, we get

${\rm det}(H-IE^{*})=0.$ (2.16)

Thus, if $E$ is a solution to (2.15), then so is $E^{*}$. Hence the secular
equation is real, and as stated above, the eigenvalues $E$ are either real or
come in complex-conjugate pairs, which is consistent with the plot in Fig. 3.

The entire spectrum of a $\mathcal{PT}$-deformed Hamiltonian may remain real
for all $\varepsilon$ or for some region of $\varepsilon$; there may also be
more than one region of broken and of unbroken $\mathcal{PT}$ symmetry. It was
several years before it was established rigorously that the eigenvalues of the
Hamiltonian $H$ in (2.14) are real and positive if $\varepsilon>0$ [122, 123,
120, 124, 125, 121]. This proof requires the use of powerful theorems of
complex analysis.

###  II.5 $2\times 2$ Matrix Hamiltonian with a $\mathcal{PT}$ transition

The $\mathcal{PT}$ transition described in the previous subsection is a
feature of many $\mathcal{PT}$-symmetric Hamiltonians. As was demonstrated in
a beautiful series of optics experiments done by Christodoulides et al. [128,
215, 195, 153, 196, 236], one can use optical fibers to observe and measure
the $\mathcal{PT}$ transition in $\mathcal{PT}$-symmetric matrix Hamiltonians.
Much research activity on $\mathcal{PT}$ symmetry was inspired by these
pioneering and prize-winning experiments.111 Christodoulides was awarded the
2023 Schawlaw Laser Prize by the APS in part for his experimental studies of
$\mathcal{PT}$-symmetric optical fibers. His citation reads: “For pioneering
several areas in laser sciences, among them, the fields of parity-time non-
Hermitian optics, accelerating Airy waves, and discrete solitons in periodic
media.” Following these early experiments, many more theoretical and
experimental studies have been done on $\mathcal{PT}$-symmetric systems
involving lasers and photonics [191, 111, 136, 159, 160, 112, 263, 135, 129,
224, 8, 185, 17], atomic systems [178], quantum Zeno effect [179],
optomechanics [232, 233], phononics [226, 267, 223], acoustics [269, 25, 116,
137], metamaterials [9, 245, 257], optical lattices [165, 182, 192, 193, 230],
exceptional points and enhanced sensing [172, 108, 150], graphene [268, 27,
105], critical phenomena [21], metasurfaces [249, 140, 237, 204], nonlinear
waves and solitons [251, 162, 266], and mechanical systems [28, 180, 181, 34].

The original optical-fiber experiments make use of two identical and parallel
fibers [153, 236]. One fiber has loss; light traveling through this fiber is
passively absorbed by the material in the fiber and it can also be actively
absorbed by placing metallic substances in contact with the fiber to absorb
some of the light energy traveling through the fiber. The other fiber has
gain; light energy can be pumped into this fiber by shining a laser on it. The
two fibers are placed in contact with one another so that they can exchange
energy. Furthermore, the energy flow into the fiber with gain is adjusted so
that it balances the energy flowing out of the fiber with loss (see Fig. 5).

Figure 5: Schematic picture of the light fiber experiment.

There are two possibilities: (i) If the fiber system is strongly coupled, the
light energy that flows into the fiber with gain readily flows across the
boundary between the two fibers, where it then flows out of the fiber with
loss. In this case the optical-fiber system is in dynamic equilibrium; that
is, the total energy at each point along the fiber remains constant in time.
(ii) If the fiber system is weakly coupled, the energy that is flowing into
the fiber with gain is unable to flow easily across the boundary between the
fibers and into the fiber with loss. In this case the optical-fiber system is
not in dynamic equilibrium; the energy in the fiber with gain grows in time
and the energy in the fiber with loss decays in time.

This optical-fiber system can be interpreted as being $\mathcal{PT}$ symmetric
if we define the $\mathcal{P}$ operation as an exchange of the two fibers. The
system is not $\mathcal{P}$ symmetric because the fiber with loss becomes the
fiber with gain, and the fiber with gain becomes the fiber with loss. However,
if we also reverse time, the inflow of energy becomes outflow, outflow becomes
inflow, and the original configuration is recovered.

The behavior of this classical optical-fiber system can be described
mathematically by a $2\times 2$ matrix that is $\mathcal{PT}$-symmetric. This
matrix contains a parameter $g$ that represents the strength of the optical
coupling between the two fibers. We will reinterpret this matrix as a
Hamiltonian that describes a two-state quantum-mechanical system in which the
coupling parameter $g$ plays the role of a deformation parameter. An
elementary study of this matrix predicts that the physical system will exhibit
a $\mathcal{PT}$ transition.

Let us consider first a simple quantum-mechanical system described by the
$1\times 1$ matrix Hamiltonian

$H_{1}=[a+ib],$ (2.17)

where $a$ and $b$ are real numbers. If the parameter $b$ is nonzero, this
Hamiltonian is non-Hermitian. The system described by $H_{1}$ has no spatial
extent; we can think of it as existing at just one point in space. The time-
dependent Schrödinger equation associated with $H_{1}$ is the first-order
differential equation

$i\psi^{\prime}(t)=(a+ib)\psi,$ (2.18)

whose solution is

$\psi(t)=Ce^{(-ai+b)t}\quad(C\leavevmode\nobreak\ {\rm constant}).$

Note that the wave function $\psi(t)$ oscillates with frequency $a$, and if
$b\neq 0$, $\psi(t)$ grows or decays in time so the system described by
$H_{1}$ in (2.17) is not in equilibrium; $\psi(t)$ grows or decays
exponentially when $b$ is positive or negative. (If $b$ is negative, we have
an elementary model of radioactive decay, where $b$ is a measure of the half-
life of the state.) The system described by $H_{1}$ is a simple quantum-
mechanical analog of a single classical optical fiber with loss or gain.

Next, we construct a $2\times 2$ quantum-mechanical matrix Hamiltonian that
describes two noninteracting systems like that in (2.17), one whose wave
function grows in time and the other whose wave function decays in time:

$H_{2}=\left[\begin{array}[]{cc}a+ib&0\\\ 0&a-ib\end{array}\right].$ (2.19)

This matrix Hamiltonian describes two subsystems, one with loss and the other
with gain. Thus, it serves as a quantum-mechanical analog of two independent
optical fibers, one with gain and the other with loss, which are not in
contact and thus cannot exchange energy.

While the physical system described by the Hamiltonian $H_{2}$ is not in
equilibrium if $b\neq 0$, we can achieve dynamic equilibrium by coupling the
two subsystems. We model this coupling mathematically by replacing the zeros
in the off-diagonal elements of $H_{2}$ with the real coupling parameter $g$:

$H_{\rm coupled}=\left[\begin{array}[]{cc}a+ib&g\\\ g&a-ib\end{array}\right].$
(2.20)

Now the subsystems with loss and with gain are symmetrically coupled and so
this Hamiltonian models the situation in which the two optical fibers are in
contact and light energy can leak from one fiber into the other.

The Hamiltonian matrix $H_{\rm coupled}$ is not Hermitian, but it is
$\mathcal{PT}$ symmetric. To see why this is so, we define the parity operator
$\mathcal{P}$, which interchanges the two subsystems, as the linear matrix
operator

$\mathcal{P}=\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right].$ (2.21)

Observe that $\mathcal{P}$ behaves as a reflection operator because
$\mathcal{P}^{2}=I$ ($I$ is the identity matrix). Also, $\mathcal{P}$ is
conventionally Hermitian $\mathcal{P}^{\dagger}=\mathcal{P}$. Next, we define
the time-reversal operation as complex conjugation because this changes the
sign of $t$ in the coupled Schrödinger equations for these matrix
Hamiltonians. (As mentioned earlier in this section, $\mathcal{T}$ is an
antilinear operator.) With these definitions, both $\mathcal{P}$ and
$\mathcal{T}$ are reflection operators:

$\mathcal{P}^{2}=I\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
\mathcal{T}^{2}=I.$

Also, as discussed earlier, parity reflection and time reversal are
independent operations, so $\mathcal{P}$ and $\mathcal{T}$ commute:

$[\mathcal{P},\mathcal{T}]=0.$

With these definitions of $\mathcal{P}$ and $\mathcal{T}$ it is clear that
while the Hamiltonian $H_{\rm coupled}$ is not Hermitian,

$H_{\rm coupled}^{\dagger}\neq H_{\rm coupled},$

it is $\mathcal{PT}$ symmetric:

$\big{[}H_{\rm coupled},\mathcal{PT}\big{]}=0.$ (2.22)

Even though $H_{\rm coupled}$ is not Hermitian, if $g$ is sufficiently large,
the eigenvalues of $H_{\rm coupled}$ become real. The eigenvalues $E$ are the
roots of the secular polynomial

$\det\left(H_{\rm coupled}-IE\right)=E^{2}-2aE+a^{2}+b^{2}-g^{2},$ (2.23)

where $I$ is the identity matrix. As we expect from the discussion in Subsec.
II.4, this secular polynomial is real.

The roots of the polynomial (2.23) are

$E_{\pm}=a\pm\sqrt{g^{2}-b^{2}}.$ (2.24)

Hence, if the two subsystems are weakly coupled ($g^{2}<b^{2}$), then the
eigenvalues are complex. In this case the total system is not in equilibrium
because one eigenstate grows in time and the other decays in time. This is the
region of broken $\mathcal{PT}$ symmetry. However, if the subsystems are
strongly coupled ($g>b$ or $g<-b$), then the eigenvalues in (2.24) become real
and the entire composite system is in dynamic equilibrium; the eigenstates
oscillate in time and do not grow or decay. These are the regions of unbroken
$\mathcal{PT}$ symmetry. So, as a function of coupling parameter $g$, there
are two transition points and there are two regions of unbroken $\mathcal{PT}$
symmetry with a region of broken $\mathcal{PT}$ symmetry sandwiched between
them.

Interestingly, the situation here is unlike that for the Hamiltonian (2.14)
discussed in Subsec. II.4. For that Hamiltonian there is just one transition
point and at that transition point the Hamiltonian is Hermitian. In the
current case, at each of the two transition points there is an abrupt
transition between nonequilibrium and equilibrium states, but at these
transition points the Hamiltonian does not become Hermitian.

This matrix model illustrates an important distinction between Hermitian and
$\mathcal{PT}$-symmetric systems. An unbroken $\mathcal{PT}$-symmetric system
resembles a closed and isolated Hermitian system because its energy levels are
real and it is in dynamic equilibrium. However, it is not closed because it is
still in contact with the external environment. In this optical model light
energy is flowing (in a balanced way) into and out of the system. In contrast,
a broken $\mathcal{PT}$-symmetric system resembles an open system because it
is not in dynamic equilibrium; however, unlike a typical open system the net
probability flux vanishes and the system has $\mathcal{PT}$ symmetry. Thus,
$\mathcal{PT}$-symmetric systems are a new class of systems whose physical
properties are intermediate between open and closed systems.

###  II.6 Other multiplicative deformations

The quantum harmonic-oscillator Hamiltonian $H=p^{2}+x^{2}$ is special in many
ways. For example, the time-independent Schrödinger eigenvalue equation
associated with this Hamiltonian can be solved exactly in terms of Hermite
polynomials, the leading-order WKB approximation gives the exact values of its
eigenvalues, the field-theoretic analog of the harmonic-oscillator Hamiltonian
describes free particles, and so on. However, while the eigenvalue problems
arising from a multiplicative deformation of the harmonic-oscillator
Hamiltonian are not exactly solvable, the spectral behavior in the unbroken
$\mathcal{PT}$-symmetric region is generic. Indeed a plot of the eigenvalues
of $H$ in (2.14) for $\varepsilon\geq 0$, as shown in Fig. 4, resembles the
spectral plots of infinite numbers of other $\mathcal{PT}$-deformed Hermitian
Hamiltonians. For example, if we perform multiplicative deformations of the
huge class of Hermitian Hamiltonians $H=p^{2m}+x^{2n}$, where
$m,\,n=1,\,2,\,3,\,\cdots$, we obtain the infinite class of
$\mathcal{PT}$-symmetric Hamiltonians [51]

$H=p^{2m}+x^{2n}(ix)^{\varepsilon}\quad(m,\,n=1,\,2,\,3,\,\cdots).$ (2.25)

Numerical studies indicate that there is universal behavior when $\varepsilon$
is positive [64]: For all $m$ and $n$ this is a region of unbroken
$\mathcal{PT}$ symmetry. The eigenvalues of these deformed Hamiltonians are
all real, positive, discrete, and monotonically increasing in this region and
there is a $\mathcal{PT}$ transition at $\varepsilon=0$. The low-lying
eigenvalues for the cases $(m=1,\,n=2)$ and $(m=1,\,n=3)$ are plotted in Fig.
6 and the low-lying eigenvalues for $(m=2,\,n=1)$ and $(m=3,\,n=1)$ are
plotted in Fig. 7 for $\varepsilon\geq 0$. However, in the broken
$\mathcal{PT}$-symmetric region $\varepsilon<0$ these models exhibit a variety
of interesting behaviors: While there are an infinite number of exceptional
points just below $\varepsilon=0$ at which high-lying energy levels become
complex, as $\varepsilon$ becomes more negative new finite-width regions of
unbroken $\mathcal{PT}$ symmetry may emerge. For example, for the case
$m=2,\,n=6$ there is an additional region of unbroken $\mathcal{PT}$ symmetry
when $-3<\varepsilon<-2$.

##  III Deformation of eigenvalue problems

We have seen in Sec. II that $\mathcal{PT}$-symmetric deformations of
Hermitian quantum-mechanical theories can lead to new non-Hermitian theories
that exhibit remarkable and unexpected behaviors. Deformation is a powerful
mathematical tool that can provide mathematical insights and establish
relationships between seemingly unrelated mathematical systems. However,
deforming an eigenvalue problem is a nontrivial global procedure because it
requires a simultaneous deformation of the differential equation and the
accompanying boundary conditions. This section discusses in depth the process
of complex deformation of eigenvalue problems.

Figure 6: Left panel: First eleven eigenvalues of $H$ in (2.25) with
$m=1,\,n=2$ plotted for $0\leq\varepsilon\leq 2$. Right panel: First eight
eigenvalues of $H$ in (2.25) with $m=1,\,n=3$ plotted for
$0\leq\varepsilon\leq 2$. This figure emphasizes the generic features of the
spectra of multiplicatively deformed Hermitian Hamiltonians in the unbroken
$\mathcal{PT}$-symmetric regions. In both cases the entire spectrum is real,
positive, discrete, and rising monotonically for $\varepsilon\geq 0$. When
$\varepsilon$ decreases below 0 the eigenvalues begin to become complex, but
the behavior of the eigenvalues in the broken-$\mathcal{PT}$ symmetric region
differs from potential to potential. For example, there is one new unbroken
$\mathcal{PT}$-symmetric region in which the entire spectrum of
$H=p^{2}+x^{6}(ix)^{\varepsilon}$ becomes real again when
$-3\leq\varepsilon\leq-2$ (not shown). Other Hamiltonians may have several
different unbroken regions for negative $\varepsilon$ or no unbroken regions
at all. Figure 7: Left panel: First eleven eigenvalues of $H$ in (2.25) with
$m=2,\,n=1$ plotted for $0\leq\varepsilon\leq 2$. Right panel: First eight
eigenvalues of $H$ in (2.25) for $m=3,\,n=1$ plotted for $0\leq\varepsilon\leq
2$. As in Fig. 6, the eigenvalues are all real, positive, discrete, and
monotonically increasing in the unbroken $\mathcal{PT}$-symmetric region
$\varepsilon\geq 0$. The eigenvalues become complex at exceptional points when
$\varepsilon<0$.

###  III.1 Complex asymptotic approximations

To explain complex deformation and to understand the differences between
$\mathcal{PT}$-symmetric quantum mechanics and Hermitian quantum mechanics it
is essential to understand the nature of asymptotic approximations in the
complex plane. An asymptotic approximation is a relation between two functions
that is expressed as a limit [87]. This relation approximates the behavior of
a function $f(x)$ by a simpler function $g(x)$ as $x$ approaches a point $a$
(which may be $\infty$). The symbol $\sim$ is used to express an asymptotic
approximation:

$f(x)\sim g(x)\quad(x\to a).$

This statement means that $\lim_{x\to a}[f(x)/g(x)]=1$.

In this Review we are specifically interested in the behavior of
eigenfunctions as they approach the boundary of a region where the
eigenfunctions are required to obey boundary conditions. The boundary is often
the point at $|x|=\infty$. Usually, these eigenfunctions are so complicated
that only very rarely are we able to solve for them in closed form and thus we
need to use asymptotic methods to approximate the eigenfunctions.

Complex asymptotic relations are generally not valid as $x$ approaches $a$ in
all complex directions; asymptotic approximations are typically valid in
wedge-shaped regions of the complex plane called Stokes sectors. To describe a
Stokes sector we specify two pieces of information: First, we specify the
complex limit; for example, $|x|\to\infty$ or $|x|\to a$. Second, we give the
angular range of $x$ in which this limit is valid: $\theta_{1}<{\rm
arg}\,x<\theta_{2}$.

As an elementary example of a complex asymptotic approximation, let us
approximate the function
$\sinh(x)=\mbox{$\textstyle{\frac{1}{2}}$}\big{(}e^{x}-e^{-x}\big{)}$ for
large $|x|$. For large positive $x$, $\sinh(x)$ is well approximated by the
simpler function $\mbox{$\textstyle{\frac{1}{2}}$}e^{x}$ because $e^{-x}$ is
negligible compared with $e^{x}$. This approximation continues to be valid for
complex $x$ if ${\rm Re}\,x$ is large and positive; that is, if $|x|\to\infty$
in a complex Stokes sector that is centered about the positive-real-$x$ axis
and that has an opening angle of $\pi$:

$\sinh(x)\sim\mbox{$\textstyle{\frac{1}{2}}$}e^{x}\quad\big{(}|x|\to\infty,\leavevmode\nobreak\
-\tfrac{\pi}{2}<{\rm arg}\,x<\tfrac{\pi}{2}\big{)}.$

Outside this Stokes wedge $\sinh(x)$ has a different asymptotic behavior
because for large $|x|$ with ${\rm Re}\,x<0$, $e^{x}$ is negligible compared
with $e^{-x}$:

$\sinh(x)\sim-\mbox{$\textstyle{\frac{1}{2}}$}e^{-x}\quad\big{(}|x|\to\infty,\leavevmode\nobreak\
\tfrac{\pi}{2}<{\rm arg}\,x<\tfrac{3\pi}{2}\big{)}.$

This abrupt change in the complex asymptotic behavior of $\sinh(x)$ for large
$|x|$, which occurs at the edges ${\rm arg}\,x=\pm\tfrac{\pi}{2}$ of the
Stokes sectors, is a general phenomenon that is typical of asymptotic
approximations. The edges of Stokes sectors are called Stokes lines.

As we see in the next subsection, because the boundary conditions on solutions
to the time-independent Schrödinger equation are valid inside Stokes sectors,
these boundary conditions distinguish $\mathcal{PT}$-symmetric quantum
theories from conventional Hermitian quantum theories. In conventional quantum
mechanics we are ordinarily interested in the behavior of solutions to the
time-independent Schrödinger equation on the real axis subject to boundary
conditions that are also imposed on the real axis, and we are usually not
concerned with the behavior of solutions for complex $x$. However, for the
case of $\mathcal{PT}$-symmetric quantum mechanics the boundary conditions are
often specified in the complex domain.

###  III.2 Boundary conditions for the Sextic Hamiltonian

Let us consider the Hamiltonian $H$ in (2.14) for the sextic case
$\varepsilon=4$:

$H=p^{2}+x^{6}.$ (3.1)

The time-independent Schrödinger equation associated with $H$ in (3.1) is

$-\psi^{\prime\prime}(x)+x^{6}\psi(x)=E\psi(x).$ (3.2)

The exact solutions to this deceptively simple-looking differential equation
are much too complicated to be expressed in closed form. However, we will show
that elementary asymptotic analysis using WKB techniques easily gives the
possible asymptotic behaviors of the solutions $\psi(x)$ to this Schrödinger
equation for large $|x|$ [87].

For the Schrödinger differential equation $y^{\prime\prime}(x)=Q(x)y(x)$ where
$\lim_{|x|\to\infty}|Q(x)|=\infty$, WKB theory provides a simple formula for
the two possible exponential components of the asymptotic behavior: The
solutions can either grow or decay exponentially like

$\exp\left[\pm\int^{x}dt\,\sqrt{Q(t)}\right].$ (3.3)

The expression in (3.3) is called a geometrical-optics approximation [87]. We
emphasize that (3.3) is not a leading-order WKB approximation (called a
physical-optics approximation); it is just the exponential component of the
WKB approximation. From (3.3) we see that the possible exponential behaviors
of solutions to (3.2) for large $|x|$ are
$\exp\big{(}\pm\tfrac{1}{4}x^{4}\big{)}$.

If we were to treat $H$ in (3.1) as a conventional Hermitian Hamiltonian, we
would require that the eigenfunctions vanish as $|x|\to\infty$ on the real-$x$
axis. A numerical calculation would then show that the eigenvalues are all
discrete, real, and positive. The approximate values of the lowest five of
these eigenvalues are

$\displaystyle E_{0}=1.145...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=4.339...,\leavevmode\nobreak\ E_{2}=9.073...,$ $\displaystyle\quad
E_{3}=14.935...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{4}=21.714...\,.$
(3.4)

However, if we treat $H$ as a $\mathcal{PT}$-symmetric Hamiltonian, we find
that while the eigenvalues are again all discrete, real, and positive, they
are different from and larger than those in (3.4). The approximate values of
the lowest five eigenvalues are

$\displaystyle E_{0}=2.439...,\leavevmode\nobreak\
E_{1}=11.882...,\leavevmode\nobreak\ E_{2}=25.412...,$ $\displaystyle\quad
E_{3}=42.024...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{4}=61.222...\,.$
(3.5)

These are the $\mathcal{PT}$ eigenvalues that are shown in Fig. 4.

It is remarkable that one Hamiltonian can have two completely different
spectra. The reason for this is that a Hamiltonian specifies only the local
behavior but not the global behavior of the eigenfunctions. That is, the
Hamiltonian $H$ specifies the Schrödinger differential equation, which in turn
determines the relationships between the eigenfunctions and their derivatives
at the point $x$. However, $H$ does not specify the global boundary conditions
on the solutions to the differential equation.

Treating (3.1) as a conventional Hermitian Hamiltonian, we require that the
solutions to (3.2) be normalizable (square-integrable) on the real-$x$ axis.
This means that the solutions must decay like
$\exp\big{(}-\mbox{$\textstyle{\frac{1}{4}}$}x^{4}\big{)}$ as $x\to\infty$ and
as $x\to-\infty$; we exclude the solutions to the Schrödinger equation that
blow up like $\exp\big{(}\mbox{$\textstyle{\frac{1}{4}}$}x^{4}\big{)}$ as
$x\to\infty$ or as $x\to-\infty$. Like the function ${\rm sinh}\,x$ discussed
in Subsec. III.1, the validity of these exponentially decaying behaviors is
not limited to the real-$x$ axis; these behaviors continue to be valid in the
complex-$x$ plane in two Stokes sectors, each having an angular opening of
$\tfrac{\pi}{4}$. One sector is centered about the positive-real axis and the
other is centered about the negative-real axis.

To summarize, the conventional quantum-mechanical eigenfunctions $\psi(x)$,
which satisfy the differential equation (3.2) and vanish asymptotically as $x$
approaches $\pm\infty$, continue to vanish exponentially as $|x|\to\infty$ in
two Stokes sectors whose opening angles are given by

$-\tfrac{\pi}{8}<\rm{arg}\,x<\tfrac{\pi}{8}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\
-\tfrac{9\pi}{8}<{\rm arg}\,x<-\tfrac{7\pi}{8}.$ (3.6)

These Stokes sectors are displayed in Fig. 8 and are colored yellow.

Figure 8: Stokes sectors for the eigenfunctions of the sextic Hamiltonian
(3.1). If the eigenfunctions are required to vanish exponentially in the
yellow pair of Stokes sectors containing the positive-real axis and the
negative-real axis (3.6), the associated eigenvalues are those of the
conventional Hermitian $x^{6}$ potential that are listed in (3.4). However, if
the eigenfunctions are required to vanish exponentially in the blue pair of
Stokes sectors adjacent to and below the yellow Stokes sectors (3.7), the
associated eigenvalues, which are listed in (3.5), are those of the
$\mathcal{PT}$-symmetric sextic theory in (2.14) with $\varepsilon=4$. Since
the sextic Hamiltonian is real, the Schrödinger eigenvalue problem is
symmetric under complex conjugation, so the array of Stokes sectors is up-down
symmetric. Hence the eigenvalues associated with the gray pair of Stokes
sectors are identical to those associated with the blue pair. The dots
indicate the locations of the classical turning points, which are the
solutions to the equation $x^{6}=E$. Each horizontal (left-right symmetric)
pair of turning points corresponds to a real spectrum. A fourth pair of Stokes
sectors (red) is centered about the positive-imaginary and negative-imaginary
axes. This pair of Stokes sectors, which does not contain turning points, is
obtained by rotating the yellow pair by $90^{\circ}$. The eigenvalues
associated with the eigenfunctions that vanish exponentially in the red Stokes
sectors are the negatives of the Hermitian eigenvalues listed in (3.4).

On the Stokes lines at the edges of the Stokes sectors (3.6) the character of
the conventional Hermitian eigenfunctions, whose eigenvalues are given in
(3.4), changes abruptly. If we cross the Stokes lines at

${\rm arg}\,x=\pm\tfrac{\pi}{8}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
arg}\,x=-\pi\pm\tfrac{\pi}{8},$

the conventional eigenfunctions no longer vanish exponentially for large $|x|$
and instead grow exponentially. However, if we impose new boundary conditions
that require the solution to the Schrödinger equation (3.2) to vanish
exponentially in two Stokes sectors of angular opening $\tfrac{\pi}{4}$ and
centered about $-\tfrac{\pi}{4}$ and $-\tfrac{3\pi}{4}$,

$-\tfrac{3\pi}{8}<{\rm arg}\,x<-\tfrac{\pi}{8}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\
-\tfrac{7\pi}{8}<{\rm arg}\,x<-\tfrac{5\pi}{8},$ (3.7)

we find an entirely new spectrum of eigenvalues; the lowest five are listed in
(3.5). The Stokes sectors in (3.7) are colored blue in Fig. 8. As will be
explained in Subsec. III.4, these Stokes sectors are obtained when we
continuously deform the boundary conditions for the $\mathcal{PT}$-symmetric
Hamiltonian in (2.14) as $\varepsilon\to 4$.

There are two more pairs of Stokes sectors shown in Fig. 8, the gray pair and
the red pair. The gray Stokes sectors are complex conjugates of the blue
sectors. Note that the sextic Hamiltonian in (3.1) is special because it is
real. Thus, in addition to $\mathcal{PT}$ symmetry (left-right symmetry in the
complex plane), $H$ exhibits another discrete symmetry, namely, complex-
conjugation symmetry, which, as explained in Sec. II, is time-reversal
symmetry. The operator $\mathcal{T}$ performs a reflection about the real
axis, so Fig. 8 is up-down symmetric. As a consequence of this symmetry, the
eigenvalues associated with the gray pair of Stokes sectors are identical to
those in the blue Stokes sectors. These eigenvalues are listed in (3.5).

The eigenvalues associated with the red pair of Stokes sectors are the
negatives of the eigenvalues listed in (3.4). To see why this is so, we rotate
the eigenvalue problem associated with the yellow Stokes wedges by
$90^{\circ}$; that is, we replace $x$ in the eigenvalue equation (3.2) by
$ix$. The new eigenvalue equation reads

$-\psi^{\prime\prime}(x)+x^{6}\psi(x)=-E\psi(x).$

We have recovered the original Hermitian eigenvalue problem except that $E$ is
multiplied by $-1$ and $\psi(x)$ vanishes as $x\to\pm i\infty$. The
eigenspectrum associated with this boundary-value problem is unbounded below.
This sign change occurs for all Hamiltonians of the form

$H=p^{2}+x^{2+4n}\quad(n=0,\,1,\,2,\,...).$ (3.8)

The simplest case ($n=0$) of this spectral sign-flip phenomenon is discussed
in the next subsection.

###  III.3 Complex deformation of the harmonic oscillator

As explained earlier in this section, when we perform a deformation of an
eigenvalue problem associated with a Hamiltonian, we must be careful to deform
the boundary conditions on the eigenfunctions in addition to deforming the
Schrödinger equation. If we neglect to do so, we may encounter contradictions
and paradoxes.

For example, let us consider the quantum harmonic-oscillator Hamiltonian

$H=p^{2}+\omega^{2}x^{2},$ (3.9)

where $\omega$ is a positive parameter associated with the classical spring
constant. The Schrödinger eigenvalue problem for this elementary Hamiltonian
reads

$-\psi^{\prime\prime}(x)+\omega^{2}x^{2}\psi(x)=E\psi(x)\,.$ (3.10)

The conventional Hermitian boundary conditions on the eigenfunctions are

$\lim_{x\to\pm\infty}\psi(x)=0\,.$ (3.11)

This is an exactly solvable eigenvalue problem. As we saw in Subsec. II.1, the
eigenfunctions are Hermite functions (Hermite polynomials multiplied by
Gaussians) and the eigenvalues are all positive:

$E_{n}=(2n+1)\omega\quad(n=0,\,1,\,2,\,\dots)\,.$ (3.12)

Let us now perform a smooth complex deformation of the Hamiltonian (3.9) by
rotating the parameter $\omega$, which is initially real and positive, into
the complex plane:

$\omega\to\omega e^{i\theta}.$ (3.13)

If we rotate $\omega$ by an angle $\pi$, that is, $\theta:\,0\to\pi$, then
$\omega$ changes sign. Under this rotation the Hamiltonian (3.9) remains
unchanged. However, its eigenvalues all change sign! Thus, this Hamiltonian
possesses two different spectra; it has an entirely negative spectrum in
addition to the usual positive spectrum (3.12). Given our previous discussion
of the $x^{6}$ potential (Subsec. III.2), this conclusion is not surprising.

The advantage of this harmonic-oscillator example is that the simplicity of
the Hamiltonian allows us to understand clearly what happens as we
continuously deform the boundary conditions (3.11) together with the
differential equation (3.10). Initially, when $\theta=0$, the eigenfunctions
$\psi_{n}(x)$ all vanish exponentially on the positive-$x$ and the
negative-$x$ axes like

$\exp\big{(}-\mbox{$\textstyle{\frac{1}{2}}$}\omega
x^{2}\big{)}\quad(x\to\pm\infty).$ (3.14)

This exponentially-decaying Gaussian behavior persists into a pair of Stokes
sectors of angular opening $\tfrac{\pi}{2}$ that are centered about the
positive-$x$ and negative-$x$ axes. However, when we rotate $\omega$ into the
complex plane according to (3.13), we must also replace the formula for the
exponential Gaussian decay in (3.14) by

$\exp\big{(}-\mbox{$\textstyle{\frac{1}{2}}$}e^{i\theta}\omega x^{2}\big{)}.$
(3.15)

Thus, as $\theta$ increases from 0 to $\pi$ and $\omega$ rotates
anticlockwise, the pair of Stokes sectors (in which the eigenfunctions decay
exponentially) correspondingly rotates clockwise like a propeller by an angle
of $\mbox{$\textstyle{\frac{1}{2}}$}\pi$. The opening angles of the two
sectors remain constant during this rotation, but the orientation of the
Stokes sectors changes and the wedges end up centered about the positive-
imaginary-$x$ and negative-imaginary-$x$ axes when $\theta=\pi$.

This argument makes it clear that the Hamiltonian (3.9) has two independent
spectra, one entirely positive and the other entirely negative, just like the
energy levels associated with the red pair of Stokes sectors in Fig. 8. The
positive-spectrum case is physically relevant. However, while a negative
spectrum may be interesting from a mathematical point of view, a negative
spectrum is physically unacceptable because the energy levels are not bounded
below. This violates the basic physical requirement of stability, namely, that
a physical system must have a lowest-energy (ground) state.

###  III.4 Boundary conditions for $H=p^{2}+x^{2}(ix)^{\varepsilon}$

Let us now return to the sextic Hamiltonian discussed in Subsec. III.2. The
question that we must answer is, Why do we require that the eigenfunctions of
the $\mathcal{PT}$-symmetric Hamiltonian $H=p^{2}+x^{2}(ix)^{\varepsilon}$ in
(2.14) with $\varepsilon=4$ satisfy the vanishing boundary conditions shown in
Fig. 8? (And more generally for any Hamiltonian, What determines the
appropriate pair of Stokes sectors in which to impose the boundary conditions
on the eigenfunctions?) To address this question, we go back to the WKB
formula in (3.3). This formula implies that the eigenfunctions of $H$ either
grow or decay exponentially for large $|x|$ like

$\exp\left(\pm\tfrac{2}{4+\varepsilon}i^{\varepsilon/2}x^{2+\varepsilon/2}\right).$
(3.16)

To determine the appropriate choice of signs for particular values of
$\varepsilon$, we begin by setting $\varepsilon=0$, which reduces $H$ to the
harmonic-oscillator Hamiltonian. As explained in Subsec. III.2, we get a real
positive spectrum if we require that the eigenfunctions of the harmonic-
oscillator Hamiltonian vanish in Stokes sectors centered about the positive-
real-$x$ and negative-real-$x$ axes. The angular opening of these sectors is
$\tfrac{\pi}{2}$. From (3.16) we see that if we continuously deform $H$ by
smoothly increasing the real parameter $\varepsilon$, both Stokes sectors
rotate downward towards the negative-imaginary-$x$ axis.

Specifically, for $\varepsilon\geq 0$, the right Stokes sector rotates
clockwise and is centered about the angle

$\theta_{\rm center,\,right}=-\tfrac{\pi\varepsilon}{8+2\varepsilon}$ (3.17)

and the left Stokes sector rotates anticlockwise and is centered about the
angle

$\theta_{\rm center,\,left}=-\pi+\tfrac{\pi\varepsilon}{8+2\varepsilon}.$
(3.18)

In addition, the angular opening $\theta_{\rm opening\,angle}$ of each Stokes
sector is given by

$\theta_{\rm opening\,angle}=\tfrac{2\pi}{4+\varepsilon}.$ (3.19)

Thus, the sectors become thinner ($\theta_{\rm opening\,angle}$ decreases) as
$\varepsilon$ increases.

This pair of Stokes sectors is shown in Fig. 9 for four values of
$\varepsilon$. Note that at $\varepsilon=4$, we recover the
$\mathcal{PT}$-symmetric orientation of the blue Stokes sectors shown in Fig.
8, which was used in the discussion of the sextic $\mathcal{PT}$-symmetric
Hamiltonian in Subsec. III.1. As $\varepsilon\to\infty$, the sectors become
infinitely thin and lie along the negative-imaginary axis. In this limit we
obtain the $\mathcal{PT}$-symmetric square well [36]. For all $\varepsilon>0$
the pair of Stokes sectors in Fig. 9 is $\mathcal{PT}$ symmetric (left-right
symmetric), which enforces the global $\mathcal{PT}$ symmetry of the
eigenfunction problem.

Figure 9: Pairs of Stokes sectors in the complex-$x$ plane in which the
eigenfunctions of the $\mathcal{PT}$-symmetric Hamiltonian
$H=p^{2}+x^{2}(ix)^{\varepsilon}$ vanish as $|x|\to\infty$. The sectors are
displayed for $\varepsilon=0$ (harmonic oscillator), $\varepsilon=1$
(imaginary cubic oscillator), $\varepsilon=2$ (upside-down quartic
oscillator), and $\varepsilon=4$ (right-side-up sextic oscillator). As
$\varepsilon$ increases smoothly, the sectors rotate downward and become
narrower. The dots indicate the locations of classical turning points. Note
that the turning points also rotate downward as $\varepsilon$ increases (they
rotate slightly faster than the sectors) and reach the positions shown in Fig.
8.

Figure 9 also shows that as $\varepsilon$ increases, there are more and more
classical turning points. These turning points are solutions to the equation

$x^{2}(ix)^{\varepsilon}=E.$

If $\varepsilon$ is not an integer, the function $x^{2}(ix)^{\varepsilon}$ is
defined on a Riemann surface with a branch cut running from $x=0$ to $\infty$,
and because we want the eigenvalue problem associated with the Hamiltonian
(2.14) to respect $\mathcal{PT}$ symmetry (left-right symmetry), we take this
branch cut to lie along the positive-imaginary-$x$ axis. (The four
configurations in Fig. 9 correspond to integer values of $\varepsilon$, so
there are no branch cuts.) As $\varepsilon$ increases smoothly through real
values, the density of turning points increases and new pairs of turning
points continue to emerge through the branch cut on the positive-imaginary-$x$
axis. One of the turning points in each new pair rotates clockwise and the
other rotates anticlockwise as $\varepsilon$ increases.

When $\varepsilon$ ranges from 0 to 1, there are only two turning points on
the principal sheet of the Riemann surface (top diagram in Fig. 9). Just as
$\varepsilon$ approaches 1, the first new pair of turning points appears on
the positive-imaginary-$x$ axis. When $\varepsilon$ is exactly 1, these
turning points coincide (second diagram in Fig. 9). As $\varepsilon$ increases
above 1, the coincident turning points split with one moving east and the
other moving west. When $\varepsilon$ reaches 2, the four turning points on
the principal sheet exhibit left-right and up-down symmetry (third diagram in
Fig. 9). At $\varepsilon=3$ another pair of turning points enters the
principal sheet, and when $\varepsilon=4$ the six turning points are now
arranged in a regular hexagon (fourth diagram in Fig. 9).

We emphasize that the continuous deformation described by (3.17) and (3.18),
in which we smoothly increase $\varepsilon$, preserves the global
$\mathcal{PT}$ symmetry of the eigenvalue problem. The Hamiltonian $H$ is
locally $\mathcal{PT}$ symmetric for any real value of $\varepsilon$. But the
boundary conditions also must remain $\mathcal{PT}$ symmetric (left-right
symmetric) in order that the eigenfunctions vanish in the left and right
sectors displayed in Fig. 9.

Remember that $\mathcal{PT}$ symmetry corresponds to left-right symmetry in
the complex-$x$ plane because $\mathcal{P}$ reflects $x$ through the origin,

$\mathcal{P}:\,x\to-x,$

and $\mathcal{T}$ performs complex conjugation, which reflects $x$ about the
real axis:

$\mathcal{T}:\,x\to x^{*}.$

Thus, the combined effect of a $\mathcal{PT}$ reflection corresponds to a
reflection about the imaginary axis. It is noteworthy that $\mathcal{PT}$
(left-right) reflection and complex conjugation (up-down reflection) are an
orthogonal pair of independent discrete symmetries.

###  III.5 Sextic potential revisited

Let us return to the Hamiltonian (3.1) discussed in Subsec. III.1. We have
stated that this sextic Hamiltonian has two positive real spectra depending on
the choice of boundary conditions; that is, depending on the pair of Stokes
sectors inside of which the eigenfunctions are required to vanish as
$|x|\to\infty$. The conventional Hermitian eigenvalues are associated with the
left-right-symmetric pair of Stokes sectors that contain the real axis and the
first five of these eigenvalues are listed in (3.4). The
$\mathcal{PT}$-symmetric eigenvalues are associated with the left-right-
symmetric pair of Stokes sectors below and adjacent to the Hermitian pair and
the first five eigenvalues associated with this pair of Stokes sectors are
listed in (3.5). These two pairs of sectors are shown in Fig. 8 and the second
pair of Stokes sectors is also shown in the fourth diagram of Fig. 9.

There is a third pair of left-right-symmetric Stokes sectors not shown in Fig.
9. This pair (shown as gray in Fig. 8) lies above and adjacent to the
Hermitian pair of Stokes sectors. The eigenvalues associated with this third
pair of sectors are identical to those associated with the second pair of
Stokes sectors. This is because the sextic Hamiltonian is real, and therefore
in addition to $\mathcal{PT}$ symmetry (left-right symmetry), it also has
complex-conjugate (up-down) symmetry in the complex plane.

An easy way to verify that the sextic Hamiltonian has two different spectra is
simply to calculate the eigenvalues. Of course, the Schrödinger equation (3.2)
cannot be solved analytically. However, WKB theory gives a good semiclassical
approximation to both spectra. The WKB procedure is straightforward [87]:
First, we find the turning points $x_{\pm}$ that separate the classically
allowed and classically forbidden regions; these turning points satisfy the
equation $V(x)=E$. Then, we evaluate the integral in the WKB quantization
condition

$\int_{x_{-}}^{x_{+}}dx\,\sqrt{V(x)-E}\sim\big{(}n+\tfrac{1}{2}\big{)}\pi\quad(n\to\infty).$
(3.20)

Finally, we solve the resulting equation algebraically for $E_{n}$. The
accuracy of this WKB calculation increases rapidly with increasing $n$.

For the conventional Hermitian theory, the turning points are the real roots
of the equation $x^{6}=E$:

$x_{\pm}=\pm E^{1/6}.$

Thus, the quantization condition (3.20) for the Hermitian theory reads

$\int_{-E^{1/6}}^{E^{1/6}}dx\,\sqrt{E-x^{6}}=\big{(}n+\tfrac{1}{2}\big{)}\pi.$

We evaluate this integral in terms of Gamma functions and solve for $E_{n}$:

$E_{n}\sim\left[\frac{(4n+2)\sqrt{\pi}\,\Gamma(2/3)}{\Gamma(1/6)}\right]^{3/2}\quad(n\to\infty).$
(3.21)

From this formula we numerically read off the first five WKB approximations to
the Hermitian eigenvalues:

$\displaystyle E_{0}=0.801...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=4.161...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{2}=8.954...,$
$\displaystyle\quad E_{3}=14.832...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{4}=21.622...\,.$ (3.22)

These approximations are in good agreement with the numerical results in (3.4)
and the accuracy of the WKB calculation improves rapidly with increasing $n$.

For the $\mathcal{PT}$-symmetric sextic theory in Fig. 9 the turning points
lie inside the lower pair of Stokes sectors at $x_{-}=E^{1/6}e^{-2i\pi/3}$ and
$x_{+}=E^{1/6}e^{-i\pi/3}$. To evaluate the integral in the WKB quantization
formula (3.20), it is simplest to integrate along a radial line from $x_{-}$
to the origin and then along a second radial line from the origin out to
$x_{+}$. Evaluating these two integrals in terms of Gamma functions and then
combining the results gives a formula that differs from that in (3.21) for the
Hermitian case by a multiple of $2^{3/2}$:

$E_{n}\sim\left[\frac{(8n+4)\sqrt{\pi}\,\Gamma(2/3)}{\Gamma(1/6)}\right]^{3/2}\quad(n\to\infty).$
(3.23)

From this formula we read off the first five WKB approximations to the
$\mathcal{PT}$-symmetric eigenvalues:

$\displaystyle E_{0}=2.265...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=11.770...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{2}=25.954...,$
$\displaystyle\quad E_{3}=41.950...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{4}=61.157...\,.$ (3.24)

Once again, these approximations are in good agreement with the numerical
calculations reported in (3.5). As noted in Subsec. III.2, the $\mathcal{PT}$
eigenvalues are larger than the corresponding Hermitian eigenvalues.

As explained in Subsec. III.2, any pair of Stokes sectors (regardless of
whether they have a $\mathcal{PT}$-symmetric orientation) provides a setting
for a specific eigenvalue problem with its own tower of eigenvalues. However,
we cannot impose exponentially vanishing boundary conditions on solutions in
adjacent pairs of Stokes sectors. This is because an exponentially decaying
solution becomes an exponentially growing solution as we cross the Stokes line
separating the two sectors. Thus, we must cross at least two Stokes lines to
formulate an eigenvalue problem in which the solutions decay exponentially in
two directions.

There are twenty nonadjacent pairs of Stokes sectors in Fig. 8 inside of which
we can formulate eigenvalue problems. Three left-right pairs give rise to
positive spectra (blue-blue, yellow-yellow, gray-gray), three up-down pairs
give rise to negative spectra (gray-blue, red-red, gray-blue), and the
fourteen other nonadjacent pairs give rise to complex spectra.

Having examined the multiple-spectrum phenomenon at the quantum and at the
semiclassical (WKB) level, it intriguing to study it at the classical level.
For the sextic Hamiltonian $H=p^{2}+x^{6}$, Hamilton’s equations of motion
read

$\displaystyle x^{\prime}(t)=\frac{\partial H}{\partial p}=2p,$ $\displaystyle
p^{\prime}(t)=-\frac{\partial H}{\partial x}=-6x^{5}\,.$ (3.25)

These equations imply that the classical energy $E=H$ is a constant of the
motion, so from the first equation in (3.25) we obtain a first-order
differential equation for the motion of a classical particle in the complex
plane:

$E=\sqrt{\tfrac{1}{4}[x^{\prime}(t)]^{2}+[x(t)]^{6}}\,.$ (3.26)

Without loss of generality we can fix the value for $E$, say $E=1$, and choose
an initial position $x(0)$ for the particle. Then, (3.26) determines the
classical path $x(t)$ of the particle for all times $t>0$.

Figure 10 displays three topological classes of paths for $E=1$. These
positive-energy classical paths are all closed and periodic. Each class of
paths is associated with one of the three pairs of Stokes sectors in Fig. 8
for which the quantum theory has positive spectra and thus each class is
associated with a pair of turning points.

Figure 10: Classical paths in the complex-$x$ plane for the sextic Hamiltonian
$H=p^{2}+x^{6}$ for which the energy $E=1$. These paths are all closed and
periodic. The classical turning points are indicated by black dots. The
central group of paths is associated with the classical turning points at $\pm
1$ in the yellow Stokes sectors in Fig. 8. All of these paths enclose the
turning points at $\pm 1$ except for the special path that oscillates between
them; these paths all have the same period $P$, which is given in (3.27). The
upper and lower groups of paths are associated with the classical turning
points in the gray and blue Stokes sectors in Fig. 8. These paths all have the
same period, which is given in (3.28), but this period is half as long as the
period of the paths in the central region because the particles are moving
faster. This is consistent with the quantized version of this Hamiltonian
because at the quantum level the corresponding energies are higher. This
figure emphasizes an important feature of $\mathcal{PT}$-symmetric quantum
theory: In the classical analog of the quantum theory the particles move
through the complex plane. Thus, unlike the Hamiltonian operator, the operator
$x(t)$, which represents the instantaneous position of a quantum particle, is
not a quantum observable (see Subsec. III.6).

The classical orbits associated with the yellow sectors all have the same
period

$P_{1}=\frac{\Gamma(1/6)\sqrt{\pi}}{3\Gamma(2/3)}.$ (3.27)

However, the periods of the classical orbits associated with the gray-gray and
blue-blue sectors are half as long:

$P_{2}=\frac{\Gamma(1/6)\sqrt{\pi}}{6\Gamma(2/3)}.$ (3.28)

Thus, these classical particles move faster than the particles whose orbits
have period $P_{1}$. This is not surprising because in the associated quantum
theories the energy eigenvalues (3.5) of the gray-gray and blue-blue sectors
are larger than the corresponding energy eigenvalues (3.4) of the yellow-
yellow sectors.

If the classical energy is negative, there are still three families of closed
paths, but the orientation of these paths is rotated by $90^{\circ}$ compared
to those in Fig. 10. The paths for $E=-1$ are shown in Fig. 11. These paths
correspond to the Stokes-sector pairs in Fig. 8 that are colored gray-blue,
red-red, and blue-gray and for which the quantum energy levels are negative
[66].

The classical-particle trajectories shown in Figs. 10 and 11 fall into three
different families in which each orbit associated with a distinct initial
condition. The three families of orbits together cover the entire complex
plane with limiting separatrix curves at the edges of the regions. A classical
path that originates in one of the three regions never crosses into another
region.

Figure 11: Classical paths in the complex-$x$ plane for the sextic Hamiltonian
$H=p^{2}+x^{6}$ for the energy $E=-1$. The black dots show the locations of
the classical turning points. Like the paths in Fig. 10, there are three
families of closed periodic orbits; the periods of these orbits are the same
as those of the corresponding orbits in Fig. 10. The paths shown in this
figure begin at $i$ (connecting the turning points) and $2i$ in the central
region and at $\pm 1.167$ (connecting the turning points) and $\pm 2$ and $\pm
4$ in the left and right regions.

The six regions of particle trajectories displayed in Figs. 10 and 11
correspond to the six real quantum spectra that one obtains by imposing
vanishing boundary conditions in the six pairs of Stokes sectors (three
horizontal and three vertical) in Fig. 8. However, there are 14 more quantum-
mechanical eigenvalue problems corresponding to other nonadjacent pairs of
Stokes sectors. All of these eigenvalue problems have complex energies, which
suggests that the quantum states corresponding to these eigenvalue problems
grow or decay in time and tunnel to different regions of the complex domain.

The classical analog of this quantum behavior is shown in Fig. 12, which is a
complex-energy version of Fig. 10. For this figure we choose a complex energy
$E=1+0.2\,i$ that has a relatively small imaginary part and we plot the
trajectory of a classical particle that begins on the positive imaginary axis
at $x(0)=1.167\,i$. This trajectory remains in the upper region for several
cycles but eventually spirals outward and enters the middle region. It then
spirals inward as it encircles the central pair of turning points. The
particle then goes between the two turning points and spirals outward. At this
point the particle enters another region, spirals inward and outward again,
and continues in this manner to enter and leave one region after another. The
path of this classical motion is uniquely determined by the initial condition
$x(0)$ and this trajectory does not cross itself. The plot in Fig. 12 suggests
that the path is ergodic and space-filling.

The path plotted in Fig. 12 exhibits deterministic random behavior as it
travels from region to region. To verify this, we plot in Fig. 13 the path of
the particle from $t=0$ to $t=250$. The horizontal gray lines indicate the
region the particle visits as a function of time $t$. As is typical of chaotic
behavior, this pattern is exquisitely sensitive to the choice of initial
condition: For an arbitrarily small change in $x(0)$ the pattern eventually
becomes completely different.

Figure 12: Complex-energy version of Fig. 10. If the energy of the classical
particle is complex, the particle is no longer confined to one of the three
regions in Fig. 10. In this figure the energy of the particle is taken to be
$E=1+0.2\,i$. As a result, the pairs of turning points are no longer
horizontal (left-right symmetric). The upper pair of turning points is located
at $-0.514743+0.858312\,i$ and $0.485949+0.874936\,i$, the middle pair is
located at $\pm(1.00069+0.0166245\,i)$, and the lower pair is located at
$-0.485949-0.874936\,i$ and $0.514743-0.858312\,i$. The initial position of
the particle is $x(0)=1.167\,i$. The particle exhibits deterministic random
behavior as it tunnels from region to region. The trajectory is uniquely
determined by its initial position and the path of the particle never crosses
itself. Each time the particle enters a region, it spirals inward and then
outward, and then enters an adjacent region in a deterministic random fashion.
The trajectory of the particle is plotted from $t=0$ until $t=37.70$. The
color scheme below the plot enables one to trace the path of the particle from
region to region. The particle in this plot goes from the upper to the middle
to the lower and back to the middle region. The detailed motion of this
particle up to the longer time $t=250$ is shown in Fig. 13. Figure 13: A plot
of the classical particle in Fig. 12 for $t=0$ until $t=250$. The path cycles
through the color scheme that is used in Fig. 12. To provide additional
assistance with tracking the complex motion of the particle, only the
imaginary part of the position of the particle is plotted. Also, the
horizontal gray lines are provided to identify the region that the particle
occupies. The height of the gray lines is the average of the imaginary parts
of the turning points in each region. Note that the particle exhibits
deterministic random behavior as it goes from region to region; when it leaves
the middle region, it makes a deterministic random choice of whether to enter
the upper or the lower region. If we label the upper, middle, and lower
regions by U, M, and L, we see that as $t$ ranges from 0 to 250, the particle
is in 17 regions: U, M, L, M, L, M, U, M, U, M, U, M, U, M, U, M, L, $...\,.$
The behavior pictured this figure can be interpreted as being a classical
analog of the time-energy quantum uncertainty principle. If we increase (or
decrease) the imaginary part of the energy ${\rm Im}\,E$ of the particle, the
average time $T$ spent in each of the three regions decreases (or increases)
in such a way that the product $({\rm Im}\,E)T$ remains constant.

###  III.6 Measuring the position of a particle

The central region in Fig. 10 contains a continuously infinite number of
classical paths that are all up-down and left-right symmetric. At the quantum
level, the expectation value of the $x$ operator is the functional average
over all classical paths, so by symmetry we know immediately that this
expectation value is 0. This result is correct: The particle spends as much
time to the right of the origin as to the left of the origin and as much time
above the real axis as below the real axis, so if we could perform many
experiments in which we measure the instantaneous complex position of the
particle and then average these measurements, we would find that $\langle
x\rangle=0$. Indeed, for the Hermitian $x^{6}$ theory the expectation value of
the $x$ operator is 0.

However, if we were to average the $\mathcal{PT}$-symmetric paths in the
upper-half or lower-half plane in Fig. 10, we would find that the expectation
value of the $x$ operator for these $\mathcal{PT}$-symmetric theories is a
positive-imaginary or a negative-imaginary number. How can the expectation
value of an operator be imaginary? The short answer to this question is simply
that in $\mathcal{PT}$-symmetric quantum theory the position operator is not
an observable. Indeed, when we extend a Hermitian quantum theory to a non-
Hermitian $\mathcal{PT}$-symmetric quantum theory, particle trajectories are
no longer confined to the real-$x$ axis and particles travel through the
complex-$x$ plane. Since the outcome of a position measurement in
nonrelativistic quantum mechanics should be a real number, we can no longer
say that the $x$ operator is an observable.

This statement should not be surprising because in general the position of a
quantum particle is not an observable. In quantum electrodynamics, for
example, the expectation value of the electron field is not measurable because
the result would be complex. We can, in principle, measure the instantaneous
position of the energy density or the charge density of the electron even
though we cannot measure the actual position of the electron.

In general, in relativistic particle physics the location of a particle cannot
be an observable because if we attempt to trap the particle in an arbitrarily
small box, the uncertainty principle implies that the momentum of the
particle, and thus the energy of the particle, would be arbitrarily large. The
particle would therefore pair-produce and not be in the box at all! The center
of energy of a system, such as an atom, is measurable, but not the positions
of the components of the system.

Even in elementary nonrelativistic Hermitian quantum mechanics, when a
particle tunnels it is a virtual particle and its instantaneous position or
its tunneling trajectory cannot be observed or measured. One can view the
process of tunneling as one in which the particle is following a complex
trajectory [44, 63, 255]. Only in elementary quantum mechanics, which is a
nonrelativistic approximation to relativistic quantum mechanics, can the $x$
operator be treated as an observable. (We examine these issues further in
Sec.VI when we discuss the $\mathcal{PT}$-symmetric inner product and
expectation values.)

Nevertheless, in $\mathcal{PT}$-symmetric quantum mechanics there are other
measurable observables, the most important being the Hamiltonian operator. In
any physical theory the ability to make an energy measurement is a fundamental
requirement. In Hermitian quantum mechanics when we try to observe the
position of a particle, we are actually performing an energy measurement.
Energy absorbed and emitted from the particle if the particle is on the real
axis enters a detector and is transferred to a film (or a nerve cell in a
retina). We then infer from this energy transfer where the particle is
located. In any viable physical theory, whether or not it is possible to
measure the position of a particle, it is crucial that we can still perform an
energy measurement, and the outcome of such a measurement must be a real
number. (Observables are discussed further in Subsec. VI.8.)

Because energy is a measurable observable, one may wonder how an upside-down
potential such as $-x^{4}$ can have a positive-energy spectrum (see Fig. 4 for
$\varepsilon=2$, for example). One might think that an upside-down potential
would be unstable and could not bind positive-energy bound states. We will see
in Sec. IV that an upside-down potential can indeed have stable positive-
energy states. This is because extending the real numbers to the complex
numbers gives up an important property of the real numbers, namely, the
ordering principle. That is, if $a$ and $b$ are complex numbers, inequalities
such as $a>b$ or $a<b$ no longer make sense. When we say that a potential is
unstable because it is unbounded below, we are implicitly using a real
inequality that, as we emphasize in Sec. IV, no longer applies in the context
of $\mathcal{PT}$-symmetric quantum theory!

###  III.7 $x^{10}$ potential

This section has focused on Hamiltonians of the form (3.8) whose right-side-up
potentials $x^{2+4n}$ ($n\geq 0$ integer) are real and positive. In the
complex-$x$ domain these Hamiltonians all possess two independent symmetries,
left-right symmetry ($\mathcal{PT}$ symmetry) and up-down symmetry (complex-
conjugation symmetry). For $n=0$ (the quantum harmonic oscillator) there is
one positive real (Hermitian) spectrum; for $n=1$ (the sextic oscillator)
there are two different positive real spectra, the Hermitian spectrum
associated with the yellow-yellow pair of Stokes sectors and the
$\mathcal{PT}$-symmetric spectrum associated with the gray-gray and blue-blue
pairs of Stokes sectors in Fig. 8. There is no positive spectrum associated
with the red-red pair of sectors; this is because there are no turning points
inside these sectors (see Fig. 8).

The pattern is clear: For a $x^{2+4n}$ potential there are $2n+1$ left-right
symmetric Stokes sectors, but because of up-down (complex-conjugation)
symmetry there are $n+1$ different real positive discrete spectra. The central
pair of sectors gives the Hermitian energy levels, which are the smallest
levels of the multiple-eigenvalue spectrum. Moving downward or upward, the
energy levels associated with each successive pair of left-right sectors
continue to be real and become correspondingly larger.

To illustrate this pattern, we consider briefly the case $n=2$, which
corresponds to the $x^{10}$ potential. There are now five horizontal pairs of
Stokes sectors in the complex-$x$ plane for the Schrödinger equation

$-y^{\prime\prime}(x)+x^{10}y(x)=Ey(x)$

and each sector has an opening angle of $\tfrac{\pi}{6}$ (see Fig. 14).
Because of up-down symmetry there are now three distinct positive-energy
spectra. There are always two fewer turning points than Stokes sectors, and as
with Fig. 8, the red up-down pair of Stokes sectors is not associated with a
positive spectrum because there are no turning points in this sector.

Figure 14: Stokes sectors for the $x^{10}$ theory. There are six pairs of
Stokes sectors, each having an opening angle of $\tfrac{\pi}{6}$. There are
also ten classical turning points (dots) separated by angles of
$\tfrac{\pi}{5}$. Each left-right-symmetric ($\mathcal{PT}$-symmetric) pair of
turning points is associated with a positive discrete quantum-mechanical
spectrum. There are ten turning points (black dots) and twelve Stokes sectors,
and as a consequence, the red up-down-symmetric pair of Stokes sectors does
not contain any turning points. These sectors are not associated with a
positive spectrum. This figure is the analog of Fig. 8 for the $x^{6}$ theory.

As we saw in the sextic case, the WKB quantization condition (3.20) gives good
approximations for each of the three eigenspectra. There are five pairs of
turning points, which are the roots of the equation $x^{10}=E$. The central
(Hermitian) turning points lie on the real axis at

$x_{\pm}=\pm E^{1/10}$

and the next two $\mathcal{PT}$-symmetric pairs of turning points are situated
at

$x_{\pm}=e^{-4i\pi/5}E^{1/10}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ \leavevmode\nobreak\ e^{-i\pi/5}E^{1/10}$

and at

$x_{\pm}=e^{-3i\pi/5}E^{1/10}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ \leavevmode\nobreak\ e^{-2i\pi/5}E^{1/10}.$

The semiclassical WKB approximation to the Hermitian energy levels is

$E_{n}\sim\left[\frac{(6n+3)\sqrt{\pi}\,\Gamma(3/5)}{\Gamma(1/10)}\right]^{5/3}\quad(n\to\infty).$

From this formula we can read off approximations to the first five Hermitian
eigenvalues:

$\displaystyle E_{0}=0.737...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=4.596...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{2}=10.768...,$
$\displaystyle\quad E_{3}=18.866...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{4}=28.680...\,.$ (3.29)

The middle pair of turning points leads to the WKB approximation to the
$\mathcal{PT}$-symmetric energy spectrum

$E_{n}\sim\left[\frac{(6n+3)\sqrt{\pi}\,\Gamma(3/5)}{\cos(\pi/5)\Gamma(1/10)}\right]^{5/3}\quad(n\to\infty)$

for which the first five eigenvalues are approximately

$\displaystyle E_{0}=1.049...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=6.543...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{2}=15.330...,$
$\displaystyle\quad E_{3}=26.858...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{4}=40.831...\,.$ (3.30)

These eigenvalues are larger than the corresponding Hermitian eigenvalues in
(3.29) by a factor of $1.424$.

The lower pair of turning points leads to the WKB approximation

$E_{n}\sim\left[\frac{(6n+3)\sqrt{\pi}\,\Gamma(3/5)}{\cos(2\pi/5)\Gamma(1/10)}\right]^{5/3}\quad(n\to\infty).$

These $\mathcal{PT}$-symmetric eigenvalues are larger than the corresponding
eigenvalues in (3.30) by a factor of $7.780$:

$\displaystyle E_{0}=5.214...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{1}=32.539...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{2}=76.235...,$
$\displaystyle\quad E_{3}=133.568...,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{4}=203.053...\,.$ (3.31)

We obtain the spectrum in (3.30) by analytically continuing the Hamiltonian
$H=p^{2}+x^{6}(ix)^{\varepsilon}$, whose eigenvalues are shown in Fig. 6
(right panel), from the Hermitian case ($\varepsilon=0$) to the non-Hermitian
$\mathcal{PT}$-symmetric case ($\varepsilon=4$). The $\mathcal{PT}$-symmetric
spectrum (3.31) is obtained by analytically continuing the Hamiltonian (2.14)
in $\varepsilon$, as shown in Fig. 9, from the Hermitian case
($\varepsilon=0$) to the non-Hermitian $\mathcal{PT}$-symmetric case
($\varepsilon=8$).

The appearance of three real spectra suggests a plausible particle-physics
interpretation. As of now, there are three known families of leptons (the
electron, the muon, and the tau), which all have very similar physical
properties except for one obvious feature, namely, that the masses of each
successive family are greater. It is interesting that by merely extending a
Hermitian theory into the complex domain, we obtain similarly structured
families [78]. Furthermore, as we saw in Subsec. III.6, if the energies of the
states are slightly complex, the particles become unstable and can decay into
one another.

The five real quantum spectra of the $x^{10}$ potential, one Hermitian
spectrum and two identical up-down symmetric pairs of spectra, correspond to
the five left-right-symmetric pairs of turning points in Fig. 14. These
spectra have clear classical analogs. Figure 15 shows all classical
trajectories corresponding to a real positive energy $E=1$. These trajectories
are all closed and periodic and they fill five distinct regions in the complex
plane. There are two left-right pairs of regions that are symmetric under
complex conjugation (up-down reflection) and which correspond to
$\mathcal{PT}$-symmetric theories, and a central region, which corresponds to
the Hermitian quantum theory. The trajectories in each region all have the
same periods. The trajectories in the central region have the longest periods
and the trajectories in the upper and lower regions have shorter periods.
These five regions along with the separatrices that divide the regions cover
the entire complex plane.

Figure 15: Complex classical trajectories for the Hamiltonian $H=p^{2}+x^{10}$
corresponding to the energy $E=1$. There are five regions of periodic
trajectories and all trajectories in each region have the same period. Three
trajectories are shown in each region, one trajectory joining the turning
points, which represents a particle that oscillates between the turning
points, and two others, which represent a particle that encircles the turning
points. The periods in the central region are the longest and those in the
outer regions are shorter. The regions are divided by separatrices that run
off to $\infty$. These regions along with the separatrices cover the entire
complex plane. The black dots indicate the locations of the classical turning
points. The classical paths shown in this figure begin at $\pm 6\,i$, $\pm
3\,i$, $\pm 1.21\,i$, $\pm 0.368\,i$, $\pm 0.5\,i$, $\pm 0.665\,i$, 3/2, and
1.

Replacing $x$ by $ix$ in the Schrödinger equation
$-y^{\prime\prime}(x)+x^{10}y(x)=Ey(x)$ corresponds to changing the sign of
$E$. Correspondingly, at the classical level, if we plot the classical
trajectories for $E=-1$, the picture in Fig. 15 becomes rotated by
$90^{\circ}$ (see Fig. 16).

Figure 16: Complex classical trajectories for the $x^{10}$ theory with $E=-1$.
This picture is identical to that in Fig. 15 except that it is rotated by
$90^{\circ}$. The paths shown in this figure begin at $i$, $\pm 4\,i$, $\pm
6\,i$, $\pm 6$, $\pm 4$, $\pm 0.66$, $\pm 0.3682$, $\pm 0.3688$, and $\pm
1.21$.

In summary, there are 54 nonadjacent pairs of Stokes sectors in Fig. 14 in
which we can formulate eigenvalue problems. Five pairs of sectors lead to
positive real spectra, of which three are distinct, and these are associated
with the five regions of classical orbits plotted in Fig. 15. Five more pairs
of sectors lead to negative real spectra, of which three are distinct, and
these are associated with the five regions of classical orbits shown in Fig.
16. The 44 remaining pairs of Stokes sectors are associated with complex
eigenvalues [66].

##  IV Upside-down potentials

This section examines Hamiltonians whose potentials have the general form

$V(x)=x^{4n}\quad(n=1,\,2,\,3,\,...).$

Like the potentials discussed in Sec. III, these potentials are real if $x$ is
real and they have an infinite number of positive-energy
$\mathcal{PT}$-symmetric bound states (see Figs. 3 and 4). We obtain these
upside-down potentials by setting $\varepsilon=2,\,6,\,10,\,...$ in (2.14).
These potentials are profoundly different from the right-side-up $x^{2}$,
$x^{6}$, and $x^{10}$ potentials discussed in Sec. III because if the theories
associated with these upside-down potentials are treated as Hermitian quantum
theories, they do not have real energy levels. However, as
$\mathcal{PT}$-symmetric theories, their energy levels are all real, positive,
and discrete.

The conventional (real-axis-only) treatment of upside-down potentials
precludes the possibility that such potentials could have bound states.
Rather, upside-down potentials are normally used as one-dimensional models of
tunneling. One might think that an upside-down potential cannot confine a
bound state simply because at the classical level the potential exerts a real
force on a classical particle that tends to push the particle away from the
peak of the potential. However, this argument fails in the context of
$\mathcal{PT}$-symmetric quantum mechanics because in the complex plane the
intuitive concept of a potential being unbounded below does not apply. As
explained in Subsec. III.6, the complex number system is not ordered; one
cannot say that a complex number is greater than or less than another complex
number. The notion of “unbounded below” relies on the ordering principle,
which only holds in the real number system.

This section examines the subtle question of how an upside-down potential can
actually confine particles and possess bound states and we explain this both
physically and mathematically. We begin with a detailed examination of the
$-x^{4}$ potential.

###  IV.1 Classical particle in a quartic upside-down potential

Setting $\varepsilon=2$ in (2.14) leads to the quartic Hamiltonian

$H=p^{2}-x^{4}.$ (4.1)

This real $\mathcal{PT}$-symmetric Hamiltonian is special because it is
independently symmetric under space-reflection $\mathcal{P}$ and time reversal
$\mathcal{T}$. As discussed above, treating $H$ conventionally, we would say
that a particle with positive energy and subject to forces provided by the
upside-down $-x^{4}$ potential cannot be in a stable bound state. If a
classical particle with positive energy and subject to this potential is
initially located at a point $x$, the particle feels a positive force if $x>0$
and a negative force if $x<0$. Regardless of its initial velocity, the
particle is pushed away from the origin, and as it moves further from the
origin, the potential steepens so the force on the particle gets stronger.
Thus, the particle accelerates to infinity. But is it gone forever?

Let us use complex analysis to examine this deceptively simple problem. At the
classical level it is easy to solve the equations of motion and to calculate
the time for a classical particle under the influence of an upside-down
quartic potential to escape to infinity.222Classical studies of
$\mathcal{PT}$-symmetric systems include [39, 48, 50, 217, 218, 61, 60, 18,
117, 20, 62, 66, 68, 44, 67, 69, 65, 55, 94, 247, 98, 214, 262] Hamilton’s
equations of motion for $H$ in (4.1) read

$\frac{dx}{dt}=\frac{\partial H}{\partial p}=2p,\leavevmode\nobreak\
\leavevmode\nobreak\ \quad\frac{dp}{dt}=-\frac{\partial H}{\partial
x}=4x^{3}.$ (4.2)

The Hamiltonian $H$ is a constant of the motion and the initial energy of the
particle is $E$. Following the classical analysis in Subsec. III.5, we use the
first equation of motion to eliminate $p$ and then write $H=E$ as a nonlinear
first-order differential equation:

$\frac{dx}{dt}=2\sqrt{E+x^{4}},$

where we have taken a square root and used the initial conditions $x(0)=0$ and
$p(0)=\sqrt{E}$. This differential equation is separable, and thus the time
$T=\int\\!dt$ that it takes for the particle to reach $x=\infty$ is

$T=\int_{0}^{\infty}\frac{dx}{2\sqrt{E+x^{4}}}.$ (4.3)

We can evaluate (4.3) in terms of Gamma functions,

$T=\frac{[\Gamma(1/4)]^{2}}{8\sqrt{\pi}}\,E^{-1/4},$ (4.4)

but the point of this calculation is merely that this integral exists, so the
particle reaches infinity in finite time.

If we repeat this calculation for a general upside-down $-|x|^{N}$ potential,
we merely replace $x^{4}$ in (4.3) with $x^{N}$. Evidently, there is a
transition at $N=2$, which corresponds to an upside-down harmonic (parabolic)
potential. If $N>2$, an evaluation of the integral shows that the particle
reaches infinity in finite time, but if $N\leq 2$, the particle takes
infinitely long to reach infinity. Thus, we have a peculiar version of the
Zeno paradox: If $N\leq 2$, the particle cannot reach infinity even though it
is approaching infinity at an ever increasing speed, while if $N>2$, the
particle gets to infinity in finite time $T$.

This result raises a fascinating question. If $N\leq 2$, we know the location
of the particle for all time; it is continuing forever on its journey to
infinity. But if $N>2$, where is the particle when $t>T$?

This is a question that is extremely difficult to answer using real analysis.
There is a singularity on the real-$N$ axis, so we cannot keep $N$ real and
examine what happens if we smoothly increase $N$ past 2. However, Fig. 9
provides insight and suggests that we should use complex-variable theory to
analyze this problem. This figure shows that at the quantum level the boundary
conditions on the eigenvalue problem for $\varepsilon=2$ are imposed in Stokes
sectors whose upper edges lie on the real axes.

In general, the angular opening of a Stokes sector does not include the edges
of the sector. In this case the opening angular ranges

$-\tfrac{\pi}{3}<\theta_{\rm right}<0\,\quad{\rm and}\quad-\pi<\theta_{\rm
left}<-\tfrac{2\pi}{3}$

do not include the angles 0 and $-\pi$, so the Stokes sectors lie just below
the real axis. Thus, at the quantum level one should not think of a particle
as traveling to infinity exactly along the real axis; rather, the quantum
particle travels infinitesimally below the real axis in the complex plane. We
first study this behavior by understanding the classical dynamical motion in
the complex plane and then approach the real axis as a limiting process. In
effect, this analysis allows us to bypass and evade the singularity on the
real axis at $N=2$. We then find that even at the classical level a complex
upside-down quartic potential confines particles.

Let us examine the complex classical particle trajectories for the general
$x^{2}(ix)^{\varepsilon}$ potential in (2.14). At $\varepsilon=0$ the complex
trajectories for a classical particle of energy $E=1$ in the harmonic
potential are pictured in Fig. 17. All but one of these trajectories are
ellipses whose foci are the two turning points on the real axis at $x=\pm 1$.
The trajectory on the real axis between the turning points is a degenerate
ellipse. The notable feature of this complex dynamical system is that the
periods of all the ellipses are the same because of the Cauchy integral
theorem. (Keplerian orbits are also ellipses, but particles traveling in outer
orbits go faster than particles in inner orbits.)

Figure 17: Complex orbits for a classical particle in the harmonic-oscillator
potential $x^{2}$. The particle has energy $E=1$. The possible paths of the
particle are all ellipses whose foci are located at the classical turning
points $x=\pm 1$. Particles move slower in areas of heavier shading. (The
classical turning points are indicated by dots, which are located at the
centers of the shaded regions.) The main feature of these orbits is that
because of Cauchy’s integral theorem their periods are all the same.
Therefore, particles in the outer orbits move faster than particles in the
inner orbits. This is unlike the case of Keplerian gravitational orbits, which
are also ellipses, because particles in the distant Keplerian orbits move more
slowly than particles in the inner orbits. (A year on Pluto is much longer
than an Earth year.)

The situation becomes more complicated when $\varepsilon=2$ because, as we
already know from Fig. 9, there are four turning points rather than two (see
Fig. 18). There are two turning points in the upper-half $x$-plane and two in
the lower-half $x$-plane. The arrangement of these turning points is up-down
symmetric because of complex-conjugation symmetry (reality of the Hamiltonian)
and also left-right symmetric because of $\mathcal{PT}$ symmetry. As in Fig.
17, all classical orbits are closed and periodic, and the set of orbits is
both left-right and up-down symmetric. The classical orbits in the lower-half
$x$-plane correspond to the Stokes sectors in the lower-half plane for the
case $\varepsilon=2$ in Fig. 9. Like the classical orbits in Fig. 17, the
orbits in Fig. 18 all have the same period $P$, which is twice the value of
$T$ in (4.4):

$P=\frac{[\Gamma(1/4)]^{2}}{4\sqrt{\pi}}\,E^{-1/4}.$ (4.5)

Thus, as in Fig. 17, particles in the outer orbits go faster than particles in
the inner orbits.

Figure 18: Complex classical orbits for the upside-down quartic $-x^{4}$
potential. All orbits are closed and periodic and all orbits have the same
period. As in Fig. 17, particles move slower in areas of heavier shading and
the classical turning points are indicated by dots. The set of all orbits
exhibits up-down and left-right symmetry and no orbit crosses the real axis.
As the orbits get closer to the real axis, they get bigger and approach the
shape of a $D$ lying on its side. In the limit as its orbit approaches the
real axis, the classical particle zooms out to infinity along a path that is
infinitesimally close to the real axis, travels at infinite speed around the
rounded part of the $D$, and comes back up the real axis to its starting
point, all in finite time. Since the particle moves slowly when it is near the
origin and rapidly when it is far from the origin, at any given time the
particle is most likely to be found very close to the origin. Thus, the
classical particle is in a localized dynamical bound state centered about the
origin.

Figure 18 shows that as the orbits increase in size, the lower parts of the
upper orbits and the upper parts of the lower orbits become flat and run
parallel and very close to the real axis. As an orbit approaches the real
axis, a particle that starts at the origin runs off to $\infty$ along a
straight line infinitesimally close to the real axis and reaches $\infty$ in
finite time $\tfrac{P}{2}$. It then runs at infinite speed in a D-shaped
roughly semicircular path from $+\infty$ to $-\infty$ in no time, and finally
returns to the origin in time $\tfrac{P}{2}$. The particle does not stay at
infinity; rather, it periodically returns to the origin!

How can a particle that travels to $+\infty$ immediately come back from
$-\infty$? Isn’t $x=+\infty$ infinitely far from $x=-\infty$? The answer is
that on the real axis $+\infty$ is indeed infinitely far from $-\infty$, but
we are studying classical particle motion in the complex plane. In the complex
plane the point at infinity is unique; $+\infty$ and $-\infty$ are the same
point. Thus, the particle does not jump from $+\infty$ to $-\infty$; if it is
at $+\infty$, it is already at $-\infty$.

Most importantly, the particle moves slowly when it is near the origin and
rapidly when it is far from the origin. Since the probability of finding the
particle in a given segment of its orbit is inversely proportional to the
speed of the particle, a particle that begins at the origin will never be
found in the complex plane where it is moving infinitely fast, and it is most
likely to be found on the real axis near the origin. For this $-x^{4}$
potential the probability density $\rho(x)$ of finding the particle of energy
$E$ on the real axis at any point $x$ is simply [see Chap. 1 in Bender  _et
al._ [51]]

$\rho(x)\propto 1/\sqrt{E+x^{4}}.$ (4.6)

Since the integral of $\rho(x)$ is finite, the classical particle behaves as
if it is in a normalized dynamical bound state that is localized at the
origin. We emphasize that even though the particle is subject to a repulsive
potential, it is most likely to be found at the origin.333A classical particle
in a right-side-up potential, such as $V(x)=+x^{4}$, oscillates between the
walls of the potential. Such a particle is most likely to be found near the
turning points at the edges of the potential, where it moving slowly.

###  IV.2 Quantum particle in a quartic upside-down potential

In the previous subsection we used complex-variable methods to show that a
classical particle subject to a $\mathcal{PT}$-symmetric $-x^{4}$ potential
can be in a dynamically stable bound state that is localized at the origin.
However, it is not obvious how a quantum particle would behave in such a
potential. One might try to analyze this problem on the real axis by
approximating the $-x^{4}$ potential with an infinite descending-staircase
potential or with a washboard potential [109]. However, once again, the
simplest way to solve the quantum eigenvalue problem associated with this
upside-down quartic potential is to examine it in the complex plane.

The Schrödinger equation associated with the Hamiltonian (4.1) is

$-\psi^{\prime\prime}(x)-x^{4}\psi(x)=E\psi(x).$ (4.7)

Following the classical analysis in Subsec. IV.1, we require that the solution
$\psi(x)$ to this equation satisfy complex boundary conditions; namely, that
$\psi(x)$ vanishes in the Stokes sectors shown in Fig. 9 for $\varepsilon=2$
as $|x|\to\infty$. A numerical calculation shows that the spectrum is real,
positive, and discrete (see Fig. 4). The first five eigenvalues are

$\displaystyle E_{0}=1.477...,\leavevmode\nobreak\
E_{1}=6.003...,\leavevmode\nobreak\ E_{2}=11.804...,$ $\displaystyle\quad
E_{3}=18.459...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{4}=25.792...\,.$
(4.8)

This eigenvalue problem cannot be solved in closed form. However, we can
perform a leading-order semiclassical WKB calculation using the turning points
in the lower-half $x$-plane that are shown in Fig. 18. The standard WKB
quantization condition is

$\int_{x_{-}}^{x_{+}}dx\,\sqrt{E+x^{4}}\sim\big{(}n+\mbox{$\textstyle{\frac{1}{2}}$}\big{)}\quad(n\to\infty),$

where in this case the turning points are given by

$x_{-}=E^{1/4}e^{-3i\pi/4}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ \leavevmode\nobreak\ x_{+}=E^{1/4}e^{-i\pi/4}.$

To evaluate the WKB integral we integrate from $x_{-}$ to the origin and then
from the origin to $x_{+}$ along radial lines in the complex-$x$ plane. The
resulting integrals are easy to evaluate in terms of Gamma functions:

$E_{n}\sim\left[\frac{(6n+3)\sqrt{2\pi}\,\Gamma(3/4)}{2\Gamma(1/4)}\right]^{4/3}\quad(n\to\infty).$
(4.9)

This WKB formula (4.9) gives the following approximations to the first five
eigenvalues:

$\displaystyle E_{0}=1.377...,\leavevmode\nobreak\
E_{1}=5.956...,\leavevmode\nobreak\ E_{2}=11.769...,$ $\displaystyle\quad
E_{3}=18.432...,\leavevmode\nobreak\ \leavevmode\nobreak\ E_{4}=25.769...\,.$
(4.10)

As we can see, the WKB approximation in (4.9) provides increasingly accurate
approximations to the numerical values of the eigenvalues in (4.8) as $n$
increases.

The remarkable feature of the eigenvalues in (4.8), of course, is that these
quantum-mechanical energy levels are all real. Thus, as in the classical case,
a quantum particle in this upside-down potential is in a stable bound state.
Even though the potential is upside down, the particle wave function remains
localized at the origin; the quantum particle is permanently confined and does
not escape to infinity.

###  IV.3 Elementary proof of spectral reality

We cannot calculate the eigenvalues and eigenfunctions of the Hamiltonians
$H=p^{2}+x^{2}(ix)^{\varepsilon}$ $(\varepsilon>0)$ analytically but, as
mentioned in Sec. II, it was proved using sophisticated mathematical
techniques that the spectra of these Hamiltonians are all real, positive, and
discrete [122]. However, the upside-down quartic Hamiltonian (4.1) is special;
the unique value $\varepsilon=2$ is the only known case for which one can
prove reality, positivity, and discreteness by using nothing more advanced
than elementary calculus. An early study of the $\varepsilon=2$ case in
Buslaev and Grecchi [99] was followed by two more papers [169, 43]. We present
a short proof of reality below.

We begin with the time-independent Schödinger eigenvalue differential equation
associated with $H$ in (4.1),

$\displaystyle-\frac{\hbar^{2}}{2m}\psi^{\prime\prime}(x)-gx^{4}\psi(x)=E\psi(x),$
(4.11)

where we have inserted the dimensional parameters $\hbar$, $g$, and $m$ for
clarity in the subsequent discussion. The boundary conditions on $\psi(x)$, as
shown in Fig. 9 for the case $\varepsilon=2$, are that
$\lim_{|x|\to\infty}\psi(x)=0$ in two Stokes sectors of angular opening
$\tfrac{\pi}{3}$ adjacent to and below the positive and negative real-$x$
axis.

For the differential equation (4.11) we impose vanishing boundary conditions
on a specially chosen curve in the complex plane that we express in parametric
form as

$x=-2iL\sqrt{1+iy/L}\quad(-\infty<y<+\infty),$

where $y$ is a real variable. Note that this curve terminates inside the
Stokes sectors at the angles $-\tfrac{\pi}{4}$ and $-\tfrac{3\pi}{4}$.

We use the parametric representation above to replace the independent variable
$x$ in (4.11) by $y$ and get the differential equation

$\displaystyle\\!\\!\\!\\!\\!-\frac{\hbar^{2}}{2m}\left(1+\frac{iy}{L}\right)\phi^{\prime\prime}(y)-\frac{i\hbar^{2}}{4Lm}\phi^{\prime}(y)$
$\displaystyle\qquad-16gL^{4}\left(1+\frac{iy}{L}\right)^{2}\phi(y)=E\phi(y),$
(4.12)

where $\psi(x)=\phi(y)$. The parameter $L$ has units of length,

$L=\lambda\left(\frac{\hbar^{2}}{mg}\right)^{1/6},$

where $\lambda$ is an arbitrary dimensionless number. The constant $L$ serves
to make the subsequent equations dimensionally consistent. (This parameter
will drop out at the end of the calculation.)

Next, we take a Fourier transform of (4.12). This transform has the general
form

$\tilde{f}(p)\equiv\int_{-\infty}^{\infty}dy\,e^{-iyp/\hbar}f(y).$

Since the Fourier transforms of $f^{\prime}(y)$ and $yf(y)$ are
$ip\tilde{f}(p)/\hbar$ and $i\hbar\tilde{f}^{\prime}(p)$, the Fourier
transform of (4.12) is

$\displaystyle\\!\\!\\!\\!\\!\frac{1}{2m}\left(1-\frac{\hbar}{L}\frac{d}{dp}\right)p^{2}\tilde{\phi}(p)+\frac{\hbar}{4Lm}p\tilde{\phi}(p)$
$\displaystyle\qquad-16gL^{4}\left(1-\frac{\hbar}{L}\frac{d}{dp}\right)^{2}\tilde{\phi}(p)=E\tilde{\phi}(p).$
(4.13)

We simplify (4.13) and get

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-16gL^{2}\hbar^{2}\tilde{\phi}^{\prime\prime}(p)+\left(-\frac{\hbar
p^{2}}{2mL}+32gL^{3}\hbar\right)\tilde{\phi}^{\prime}(p)$
$\displaystyle+\left(\frac{p^{2}}{2m}-\frac{3p\hbar}{4mL}-16gL^{4}\right)\tilde{\phi}(p)=E\tilde{\phi}(p).$
(4.14)

Note that the variable $p$ in this equation is not a conventional momentum
variable. Here, considered as an operator, $p$ represents
$-i\hbar\tfrac{d}{dy}$ and not $-i\hbar\frac{d}{dx}$.

The next step is to convert (4.14) to a conventional Schrödinger equation by
eliminating the one-derivative term. This is done by making the elementary
change of dependent variable

$\tilde{\phi}(p)=e^{Q(p)/2}\Phi(p).$

Then, to eliminate $\Phi^{\prime}(p)$ from the resulting equation we require
that $Q(p)$ satisfy a first-order differential equation whose solution is

$Q(p)=\frac{2L}{\hbar}p-\frac{1}{96gmL^{3}\hbar}p^{3}.$

The Schrödinger equation satisfied by $\Phi(p)$ is then

$\displaystyle\\!\\!\\!\\!\\!\\!-16gL^{2}\hbar^{2}\Phi^{\prime\prime}(p)+\left(-\frac{\hbar
p}{4mL}+\frac{p^{4}}{256gm^{2}L^{4}}\right)\Phi(p)$
$\displaystyle\qquad=E\Phi(p).$

Finally, we perform a scaling transformation of the independent variable,

$p=zL\sqrt{32mg}.$

This replaces the $p$ variable, which has units of momentum, by $z$, which is
a coordinate variable having units of length. The resulting eigenvalue
equation, posed on the real-$z$ axis, is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!-\frac{\hbar^{2}}{2m}\Phi^{\prime\prime}(z)+\left(-\hbar\sqrt{\frac{2g}{m}}\,z+4gz^{4}\right)\Phi(z)=E\Phi(z).$
(4.15)

This new eigenvalue equation, which no longer contains the length parameter
$L$, is the final result of the calculation. The new independent variable $z$
has dimensions of length although it is not a conventional coordinate variable
because it is odd under time reversal.

The eigenvalue equation (4.15) is similar in structure to the original
eigenvalue equation (4.11), but there are two significant differences: (i) The
quartic term in the potential has changed sign, so the upside-down quartic
potential has become a conventional right-side-up quartic potential. (ii)
There is a new linear potential term proportional to Planck’s constant
$\hbar$. This linear term is called a quantum anomaly, which means that it has
no classical analog. (The effect of this parity-anomaly term is discussed in
Subsec. IV.4.)

The Hamiltonian $\tilde{H}$ for which (4.15) is the eigenvalue equation is

$\tilde{H}=\frac{\tilde{p}^{2}}{2m}-\hbar\sqrt{\frac{2g}{m}}\,z+4gz^{4}.$
(4.16)

This Hamiltonian is quite different from

$H=\frac{p^{2}}{2m}-gx^{4},$ (4.17)

which is the Hamiltonian $H$ associated with (4.11). At the quantum level the
Hamiltonian $\tilde{H}$ in (4.16) is conventionally Hermitian and the
potential is bounded below on the real-$z$ axis; $\tilde{H}$ is also
$\mathcal{PT}$-symmetric because the sequence of transformations used to
obtain $\tilde{H}$ all respect $\mathcal{PT}$ symmetry. Note that the
potential in (4.16) is $\mathcal{PT}$-symmetric because while $z$ and
$\tilde{p}$ are canonically conjugate operators satisfying the usual
commutation relation $[z,\tilde{p}]=i$, the new variable $z$ behaves like a
momentum rather than a coordinate variable.

The point of this calculation is simply that while we cannot calculate exactly
the eigenvalues of $H$ (with $\mathcal{PT}$-symmetric boundary conditions
imposed in the Stokes sectors shown in Fig. 9 for $\varepsilon=2$) or the
eigenvalues of $\tilde{H}$ (with conventional boundary conditions imposed on
the real axis), we have shown that these two Hamiltonians are isospectral;
that is, they have exactly the same eigenvalues. Because the potential of
$\tilde{H}$ rises to $+\infty$ as $x\to\pm\infty$, we know that the entire
spectrum of $\tilde{H}$ is real and discrete. This proves that the eigenvalues
of the $\mathcal{PT}$-symmetric Hamiltonian $H$ are also real and discrete.

This elementary calculation easily generalizes if we include a quadratic term
in the $\mathcal{PT}$-symmetric quartic Hamiltonian:

$H=p^{2}+\mu^{2}x^{2}-gx^{4}.$ (4.18)

The corresponding isospectral Hermitian Hamiltonian is just slightly more
complicated than that in (4.16):

${\tilde{H}}=p^{2}-2\hbar\sqrt{g}z+4g\left(z^{2}-\frac{\mu^{2}}{4g}\right)^{2}.$
(4.19)

[This isospectral Hamiltonian is also derived in Buslaev and Grecchi [99],
Jones and Mateo [169], Bender  _et al._ [43].] However, the calculation in
this subsection does not generalize to any other Hamiltonians of the form
(2.14); it only works for $\varepsilon=2$. The techniques used in this
calculation do extend to the higher-derivative $\mathcal{PT}$-symmetric class
of Hamiltonians of the form

$H_{n}=p^{n}-g(ix)^{n^{2}}\quad(n=2,\,3,\,4,\,...)$

but the resulting isospectral Hamiltonians are quite complicated [62].

###  IV.4 Discussion of the parity anomaly

The equations of motion for the quartic Hamiltonian (4.17) are parity
symmetric. The explicit violation of parity symmetry that emerges in the
transformed Hamiltonian in (4.16) occurs at large values of $x$ in the
$\mathcal{PT}$-symmetric Hamiltonian $\tilde{H}$, where the boundary
conditions on the eigenfunction $\psi(x)$ in (4.11) are imposed. (These
boundary conditions are $\mathcal{PT}$ symmetric, but they are not parity
symmetric.) Because we have taken a Fourier transform to obtain the
Schrödinger eigenvalue equation (4.15), the parity-breaking anomaly that we
see in the Hamiltonian (4.16) now manifests itself at small values of $z$ in
the Hamiltonian (4.16) and not in the boundary conditions at $z=\pm\infty$.

The parity-breaking term in $\tilde{H}$ at $z=0$ is proportional to $\hbar$,
so there is no classical analog of this parity-breaking anomaly. Thus, while
the two Hamiltonians $H$ and $\tilde{H}$ have identical quantum-mechanical
spectra, these two Hamiltonians determine different classical behaviors, which
we demonstrate by comparing the classical trajectories for these two
Hamiltonians. First, we plot in Fig. 19 a pair of classical trajectories for
the $\mathcal{PT}$-symmetric Hamiltonian $H$ in (4.17) for the case $m=1$,
$g=1$, and $E=1$. These trajectories begin at $x=\pm 2.0\,i$ and they resemble
the trajectories in Fig. 18 although they have a classical period of
$P=2.622060$.

Figure 19: Typical complex trajectories for $H$ in 4.17 with $m=1$, $g=1$, and
$E=1$. These trajectories begin at $x=\pm 2.0\,i$. The trajectories are
similar to the closed trajectories in Fig. 18, but they have a classical
period of $2.622060$. The dots indicate the locations of the classical turning
points.

Next, using the same values $m=1$, $g=1$, and $E=1$ along with $\hbar=1$, we
plot in Fig. 20 a typical closed classical trajectory for the isospectral
Hermitian Hamiltonian $\tilde{H}$ in (4.16); this trajectory begins at $x=-1$.
The period of this trajectory is $2.495467$, which is shorter than the periods
of the trajectories in Fig. 19.

Figure 20: Complex classical trajectory for the isospectral Hamiltonian
$\tilde{H}$ in (4.16) with $m=1$, $g=1$, $\hbar=1$, and $E=1$. The trajectory
begins at $x=-1$. The quantum energy levels of this Hamiltonian are the same
as those of $H$, but the classical period of the trajectory is $2.495467$,
which is shorter than the periods of the trajectories in Fig. 19. The dots
indicate the locations of the classical turning points.

Finally, we plot in Fig. 21 a classical trajectory of $\tilde{H}$ in (4.16)
for the values $m=1$, $g=1$, and $E=1$ except that now we set $\hbar=0$. The
period of this trajectory is exactly the same as the period of the classical
trajectory of $H$ in Fig. 19. To demonstrate the effect of the quantum
anomaly, we plot a trajectory beginning at $x=-1$ for the shorter period