“Striped” Rectangular Rigid Box with Hermitian and non-Hermitian
$\mathcal{PT}$ Symmetric Potentials

Shailesh Kulkarni***e-mail<EMAIL_ADDRESS>and  Rajeev K.
Pathak†††e-mail<EMAIL_ADDRESS>
Department of Physics, Savitribai Phule Pune University, Ganeshkhind, Pune,
411007, India

ABSTRACT

Eigenspectra of a spinless quantum particle trapped inside a rigid,
rectangular, two-dimensional (2D) box subject to diverse inner potential
distributions are investigated under hermitian, as well as non-hermitian
antiunitary $\mathcal{PT}$ (composite parity and time-reversal) symmetric
regimes. Four sectors or “stripes” inscribed in the rigid box comprising
contiguously conjoined parallel rectangular segments with one side equaling
the entire width of the box are studied. The stripes encompass piecewise
constant potentials whose exact, complete energy eigenspectrum is obtained
employing matrix mechanics. Various striped potential compositions, viz. real
valued ones in the hermitian regime as well as complex, non-hermitian but
$\mathcal{PT}$ symmetric ones are considered separately and in conjunction,
unraveling among typical lowest lying eigenvalues, retention and breakdown
scenarios engendered by the $\mathcal{PT}$ symmetry, bearing upon the strength
of non-hermitian sectors. Some states exhibit a remarkable crossover of
symmetry ‘making’ and ‘breaking’: while a broken $\mathcal{PT}$ gets
reinstated for an energy level, higher levels may couple to continue with
symmetry breaking. Further, for a charged quantum particle a $\mathcal{PT}$
symmetric electric field, furnished with a striped potential backdrop, also
reveals peculiar retention and breakdown $\mathcal{PT}$ scenarios. Depictions
of prominent probability redistributions relating to various potential
distributions both under norm conserving unitary regime for hermitian
Hamiltonians and non-conserving ones post $\mathcal{PT}$ breakdown are
presented.

##  1 Introduction

Among the few problems that pragmatically merit exact quantum mechanical
solutions, the ‘rigid-box’ problems form a special class signifying,
contingent on dimensionality and symmetry, a quantum particle confined within
some standard geometries such as line segments, rectangular and circular
areas, rectangular parallelepipeds (cuboids), cubes, spheroids, spheres and
thus forth. In all these cases, as is well-known, appropriately chosen
coordinate systems permit separation of variables in the configuration space,
yielding analytical solutions. Following its inception by McDonald and Kaufman
[1], ‘Quantum Billiards’ (QB), that has emerged a generic term for analyzing
the dynamics of a particle in confined spaces, has stimulated widespread
interest over the past four decades [2, 3, 4, 5, 6, 7, 8]. Although even a
slightest deviation from standard shapes could render a systematically
tractable solution impossible, cogent numerical methods to tackle such
situations have been developed [9, 10]. Interestingly, particles moving in
polygonal enclosures whose interior angles are rational multiples of $\pi$
were observed to exhibit ‘pseudo-integrability’ embodying tenets of quantum
chaos [11]. Moreover, some customized mesoscopic semiconductor quantum dots
were experimentally observed to exhibit salient QB characteristics in terms of
forward and back-scattered electron wave-packets and their revivals [12].
Later, a case strikingly akin to this effect was impeccably solved exactly by
Robinett [5], for a 2D rigid circular box augmented by an infinite thin
barrier or baffle introduced along one of its radii constituting quantum
billiards. Noticeably, QB were demonstrated to furnish a manifest natural
connection between the complementary phenomena of classical and quantum chaos
[2, 3].

Meanwhile, Bender and Boettcher [14], through their pioneering article in
1998, launched the exotic theme of $\mathcal{PT}$ Symmetry, signifying
invariance of the Hamiltonian under the composite, discrete symmetry
operations of Parity ($\mathcal{P}$) and Time-reversal ($\mathcal{T}$ ),
irrespective of their order of application, for a wide variety of systems.
Introduction of $\mathcal{PT}$ symmetry, a remarkable non-Hermitian extension
of Quantum Mechanics, has evoked inexorable intrigue and has stimulated a
multitude of theoretical [14, 15] as well as experimental [17, 16] ventures.
When in a Hamiltonian system, a perfect balance is struck between gain and
loss mechanisms, $\mathcal{PT}$ symmetry manifests through the corresponding
potential (energy) terms. $\mathcal{PT}$ symmetry also got introduced in
confined quantum systems by several workers: notably, Bittner et al. [18]
examined a two-state Hamiltonian of a dissipative microwave billiard in the
neighborhood of an exceptional point, bringing forth a peculiar $\mathcal{PT}$
symmetry. Further, Dasarathy et al. [19] imposed $\mathcal{PT}$ symmetry not
by parameterizing the Hamiltonian, but rather implicitly, through the boundary
conditions over the standard 1D rigid box segment and established that the
ensuing dynamics invariably conformed to a robust $\mathcal{PT}$ symmetry. In
their recent experiments, Gu and coworkers [20] ingeniously demonstrated
incorporation of $\mathcal{PT}$ symmetry in a “Stripe-LASER” waveguide with
its pumped part functioning as ‘gain’, and the other unpumped one serving as
‘loss’, due to the intrinsic absorption of a deliberately introduced dye.
Interestingly, Kreibich [21] presented a concrete scenario for realizing
$\mathcal{PT}$ symmetry for particle currents in Bose-Einstein condensates,
simulating a two-mode model system embedded in a larger hermitian system
having at least four additional coupled ‘reservoir’ wells equipped with a
balanced gain-loss mechanism. For discrete confined systems, Musslimani et al.
[22] conclusively demonstrated that $\mathcal{PT}$ symmetry can support
soliton solutions in 1D as well as 2D nonlinear optical lattices.
$\mathcal{PT}$ symmetry thus is not limited to a theoretical abstraction, but
also presents tangible perspectives amenable to direct experimentation [15,
16, 17].

Induction of $\mathcal{PT}$ symmetry in the ‘particle-in-a-box’ context was
likewise carried out, in particular, by Yusupov et al. [23], who studied the
quantum dynamics in a 1D box, driven by a $\mathcal{PT}$ symmetric complex
potential, with impulsive, non-Hermitian, delta-function ‘kicks’, and observed
that the otherwise sustained $\mathcal{PT}$ symmetry broke down beyond certain
characteristic threshold strengths of the kicking parameter. Interestingly,
notwithstanding the standard constraints for the elementary 1D rigid box
problem, Adamu [24], by a direct post facto imposition of the $\mathcal{PT}$
symmetric boundary conditions on the eigenfunctions, obtained a distinct class
of eigenfunctions associated with the same eigenvalues as that of the usual 1D
box. Around the same time, Fernàndez and Garcia [25] applied perturbation
theory to a 2D rigid square box incorporating some model parameterized
$\mathcal{PT}$ symmetric potentials, that exhibited retention of
$\mathcal{PT}$ symmetry below some specific threshold parameter-values.

For yet another confined 2D system albeit with circular geometry, Agarwal and
others [26] presented exact solutions to typified $\mathcal{PT}$ symmetric
potentials with sinusoidally oscillating azimuth, revealing some
counterintuitive $\mathcal{PT}$ breakdown scenarios, e.g. raising the
hermitian component sometimes resulted in accelerating, rather than deterring,
breaching of $\mathcal{PT}$. Further, in another exploration on $\mathcal{PT}$
applied to 2D, finite discrete latticework, these workers [27] demonstrated
that for a set of coupled chains in 2D with only two balanced gain-loss
$\mathcal{PT}$ sites interspersed, the transition threshold could be markedly
enhanced as a function of the coupling, providing a handle on tune-ability.
Incidentally, for a 1D rigid box, a family of $\mathcal{PT}$ symmetric complex
potentials isospectral with its eigenspectrum was obtained [28].

The foregoing discussion emphasizes that quantum particles in entrenched, or
confined, quantum billiards type configurations could exhibit uncommon
dynamics, accentuated with application of $\mathcal{PT}$ symmetry. Subscribing
to this premise, the present article is aimed at systematically studying the
behavior of a spinless particle in a ‘striped’ two dimensional rectangular
rigid quantum box, a special type of quantum billiards whose exact quantum
mechanical energy eigenspectrum will be obtained. Piecewise constant
potentials are introduced along the breadth of the box, in chosen sectors or
stripes, where some or all of them are selectively rendered hermitian and/or
$\mathcal{PT}$ symmetric, the latter accomplished by introduction of balanced
gain and loss sectors. We delineate onset of $\mathcal{PT}$ symmetry breaking
at characteristic exceptional points and bring forth the consequent mutations
in the probability density. While the present venture evidently provides exact
solutions to a model quantum mechanical situation, a pertinent physical
problem that could serve as its excellent prototype would be a three
dimensional tubular wave-guide with a uniform rectangular normal cross-
section, with propagation of the particle or light flux in the longitudinal
positive $z$ direction, and a suitably tune-able lateral beam distribution in
the $x-y$ plane: for a 1D analogue, reference may be made to the work of
Moiseyev and others [29], who studied a waveguide with a piecewise constant
complex refractive indices. In the present problem, preemptively factoring out
the plane-wave propagation along positive $z$, we focus on the wave function
projected onto the plane, within the ambit of the rectangle. It will turn out
that the exactly solvable 2D rectangular rigid box, in conjunction with a
stipulated $\mathcal{PT}$ symmetric potential distribution in the interior
also selectively exhibits sustenance and abrogation of the symmetry consequent
to the interplay between the relative strengths of the intervening potentials.

##  2 Two dimensional rectangular rigid box with striped potential
distribution: exact solutions

We consider a rigid rectangular box in the Cartesian $x-y$ plane, bounded by
$\displaystyle 0\leq x\leq a,0\leq y\leq b$, partitioned with piecewise
constant potentials $V_{1},V_{2},V_{3}$ and $V_{4}$ in contiguous rectangular
sectors or stripes throughout their breadth, equaling the breadth of the rigid
box, parallel to the $x$-axis. The inside potentials are finite with possible
finite discontinuities at each interface within the box. The rectangular
sectors, i.e. the stripes are symmetric around the $y$-median as depicted in
the accompanying schematic Figure-1. The region outside the box, of course, is
impenetrable, with positive infinite potential.

$a$$b$$b_{3}$$b_{1}$$b_{2}=b/2$$0$$y$$x$region-IVregion-IIIregion-
IIregion-I$V_{4}$$V_{3}$$V_{2}$$V_{1}$ Figure 1: A schematic “striped”
rectangular box in the $x-y$ plane, with $0<x<a,0<y<b$; partitioned with four
rectangular sectors I, II, III and IV respectively bearing the piecewise
constant, finite potentials $V_{1},V_{2},V_{3}$ and $V_{4}$. The partitions
are symmetric with respect to the median $y=b/2\equiv b_{2}$, whence,
$b_{3}-b_{2}=b_{2}-b_{1}<b/2$. Outside the box, throughout, the potential is
$+\infty$.

The potential $V(x,y)$ within the rigid box is designated as follows:

$V(x,y)=\begin{cases}V_{4},\ b_{3}<y<b\\\ V_{3},\ b_{2}<y<b_{3}\\\ V_{2},\
b_{1}<y<b_{2}\\\ V_{1},\ 0<y<b_{1}\end{cases}$

with $0<x<a$ throughout and $V(x,y)\rightarrow+\infty$, outside the rectangle.
Note that within the box, the only explicit dependence of the potential
$V(x,y)$ is on the $y$-coordinate; therefore inside, $V(x,y)\equiv V(y)$
alone, a fact that will be exploited shortly. The stationary states for a
scalar (spinless) particle of mass $\mu$ trapped inside the planar $2$D
rectangular geometry are the solutions of the time-independent Schrödinger
equation

$\hat{H}\psi(x,y)\equiv-\frac{\hbar^{2}}{2\mu}\Big{[}\nabla^{2}_{2D}+V(x,y)\Big{]}\psi(x,y)=E\psi(x,y)$
(1)

where $\nabla^{2}_{2D}$ is $2$D Laplacian operator and $\psi(x,y)$ designating
the time-independent energy eigenfunctions associated with energy eigenvalues
$E$ of the Hamiltonian operator $\hat{H}$. Let us recall that for the
standard, rudimentary text book case of a particle entrapped in the rigid box,
but otherwise free inside, i.e. $V(x,y)=0$ throughout the box, the
corresponding eigenfunctions $\psi^{(0)}_{n_{x},n_{y}}(x,y)$ factorize
themselves as $\psi^{(0)}_{n_{x},n_{y}}(x,y)=u_{n_{x}}(x)u_{n_{y}}(y)$ within
the box and identically vanish outside the rectangle. The quantum numbers
$n_{x}$ and $n_{y}$ take values $1,2,\cdots$. The normalized forms of the
factor functions are

$u_{n_{x}}(x)=\begin{cases}\sqrt{\frac{2}{a}}\sin\left(\frac{n_{x}\pi
x}{a}\right),0\leq x\leq a\\\ 0,\text{otherwise}\end{cases}$

and

$u_{n_{y}}(y)=\begin{cases}\sqrt{\frac{2}{b}}\sin\left(\frac{n_{y}\pi
y}{b}\right),0\leq y\leq b\\\ 0,\text{otherwise}\end{cases}$

while the energy eigenvalues emerge as

$E^{(0)}_{n_{x},n_{y}}=\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}\Big{)}$

Note that degeneracies, including accidental degeneracies, could occur in this
bound state two dimensional (2D) problem. To solve for the eigenspectrum of
the striped box, we expand the wave function $\Psi(x,y)$ vide the completeness
of the product functions $u_{n_{x}}(x)u_{n_{y}}(y)$ :

$\psi(x,y)=\sum_{n_{x},n_{y}=1}^{\infty}C_{n_{x},n_{y}}u_{n_{x}}(x)u_{n_{y}}(y)$
(2)

Substituting the above in Eq. (1), one is led to

$\displaystyle\sum_{n_{x}=1}^{\infty}u_{n_{x}}(x)\sum_{n_{y}=1}^{\infty}\Big{[}\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}\Big{)}+V(y)\Big{]}C_{n_{x},n_{y}}u_{n_{y}}(y)=E\sum_{n_{x}=1}^{\infty}u_{n_{x}}(x)\sum_{n_{y}=1}^{\infty}C_{n_{x},n_{y}}u_{n_{y}}(y);$
(3)

owing to the explicit dependence of the potential exclusively on the
$y$-coordinate, within the rectangle. This leads to the separation of the
$x$-solution altogether, whence, exploiting the linear independence of the
functions $\\{u_{n_{x}}(x)\\}$ one arrives at

$\sum_{n_{y}=1}^{\infty}\Big{[}\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{n_{x}^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}\Big{)}+V(y)-E\Big{]}C_{n_{x},n_{y}}u_{n_{y}}(y)=0.$
(4)

Since the $x$-dependence separates out (analogous to that coordinate being
“cyclic” in the Hamiltonian defined over the box-region), we can designate the
$n_{x}$ quantum number a definite, fixed value, say $n_{x}\equiv n_{x}^{(0)}$
for the $x$-solution, which is then denoted by $u_{n_{x}^{(0)}}(x)$. Next, we
substitute the appropriate piecewise constant forms for $V(y)$ in the four
sectors, clamp the $n_{x}$ quantum number at the chosen $n_{x}^{(0)}$ value,
pre-multiply the above Eq. (4) throughout by $u^{*}_{n^{\prime}_{y}}(y)$ and
integrate over the $y$ in the domain $[0,b]$. Implementing this in sequential
conjunction and harnessing the orthonormality of $u_{n_{y}}(x)$ functions the
procedure results in the following connection:

$\displaystyle\sum_{n_{y}=1}^{\infty}\Big{\\{}\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{[}\frac{(n_{x}^{(0)})^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}\Big{]}-E\Big{\\}}C_{n_{y}}^{n^{(0)}_{x}}\delta_{n^{\prime}_{y}n_{y}}+\frac{2}{b}\sum_{n_{y}=1}^{\infty}\Big{(}\sum_{i=1}^{4}\
V_{i}\ \int_{b_{i-1}}^{b_{i}}\ dy\ \phi(y)\Big{)}C_{n_{y}}^{n^{(0)}_{x}}=0$
(5)

where $b_{0}\equiv 0$ and $b_{4}\equiv b$ and
$\phi(y)=\sin\Big{(}\frac{n^{\prime}_{y}\pi
y}{b}\Big{)}\sin\Big{(}\frac{n_{y}\pi y}{b}\Big{)}$. The coefficients have
been relabeled as $C_{n^{(0)}_{x},n_{y}}\equiv C_{n_{y}}^{n^{(0)}_{x}}$,
reminding that only the discrete index $n_{y}$ actually varies, the index
$n^{(0)}_{x}$ having been clamped. The wave function then simplifies to

$\psi(x,y)=u_{n^{(0)}_{x}}(x)\sum_{n_{y}=1}^{\infty}C_{n_{y}}^{n^{(0)}_{x}}u_{n_{y}}(y)$
(6)

After evaluation of the integrals the above line of arguments engenders a
matrix eigenvalue equation

$[M][C]=E[C].$ (7)

Here, the matrix $M=[M]_{n^{\prime}_{y},n_{y}}$ is an infinite dimensional
‘square’ matrix, labeled by the respective row and column indices
$n^{\prime}_{y},n_{y}=1,2,3\cdots$; while $C_{n_{y}}^{n^{(0)}_{x}}$ incarnates
as a column vector. The matrix elements of $M$ bear the explicit form

$\displaystyle M_{n_{y},n_{y}}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{(n_{x}^{(0)})^{2}}{a^{2}}+\frac{n_{y}^{2}}{b^{2}}\Big{)}\Big{]}+\frac{1}{b}\sum_{i=1}^{3}b_{i}(V_{i}-V_{i+1})+V_{4}$
(8) $\displaystyle+\frac{1}{2\pi
n_{y}}\Big{[}\sum_{i=1}^{3}(V_{i+1}-V_{i})\sin\Big{(}\frac{2\pi
n_{y}b_{i}}{b}\Big{)}\Big{]};$

for the diagonal entries, and

$\displaystyle M_{n^{\prime}_{y},n_{y}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi(n^{\prime}_{y}-n_{y})}\Big{[}\sum_{i=1}^{3}(V_{i}-V_{i+1})\sin\Big{(}\frac{\pi(n^{\prime}_{y}-n_{y})b_{i}}{b}\Big{)}\Big{]}$
(9)
$\displaystyle+\frac{1}{\pi(n^{\prime}_{y}+n_{y})}\Big{[}\sum_{i=1}^{3}(V_{i+1}-V_{i})\sin\Big{(}\frac{\pi(n^{\prime}_{y}+n_{y})b_{i}}{b}\Big{)}\Big{]};$

for the off-diagonal entries ($n^{\prime}_{y}\neq n_{y})$. We shall put the
foregoing analysis to test with a variegated set of hermitian, and non-
hermitian but $\mathcal{PT}$ symmetric combinations, as carried out in the
next section.

##  3 Results and discussion

To tackle the stationary-state problem for seeking the eigenvalues and
eigenfunctions, we must adopt an appropriate scale. Setting the Bohr radius
$\displaystyle a_{B}\equiv\frac{4\pi\epsilon_{0}\hbar^{2}}{m_{e}e^{2}}$ (in
usual notation) to represent a unit distance, and $\hbar$ and the electron
mass, $m_{e}$ both numerically unity, the energy scale is set by the quantity
$\frac{\hbar^{2}}{2m_{e}a_{B}^{2}}=13.605693\ eV\equiv 1$ Rydberg energy.
Thus, we shall measure distances in units of Bohr radii and energies in units
of Rydbergs. We also have, for convenience (although not direly necessary),
chosen the mass of the quantum particle $\mu$ to be the electron mass,
$m_{e}$. The task now is to solve the matrix equation, Eq. (7), with the
prescriptions in Eqs. (8) and (9) above. For nontrivial solutions, the secular
equation, viz. $det[M-EI]=0$ (with $I$ denoting the identity matrix) must be
satisfied, which determines the energy eigenvalues $E$ and thereafter,
corresponding to each eigenvalue, the column vector $C$ formed of the
expansion coefficients, readily synthesizes the wave function as stipulated by
Eq. (6).

Although in principle, the range spanned by the quantum numbers
$n^{\prime}_{y},n_{y}$ goes from $1$ to $\infty$, in practice, a proper upper
limit $n_{y}^{max}$ must be chosen for numerical implementation of the exact
matrix-eigenvalue-equation, Eq.(7). It turns out that for an accurate
evaluation the lowest $15$-odd energy eigenvalues for maximal magnitude of the
$V_{i}$ (cf. Figure-1) $150\ Ry$ energy units, the upper limit of
$n_{y}^{max}=30$ suffices for convergence to occur. We have chosen a better
stringent and robust upper limit of $n_{y}^{max}=50$, for reaffirming
convergence of the lowest eigenvalues, rendering the computations numerically
completely unequivocal. The convergence is markedly rapid, which may be
attributed to the following: for sufficiently large values of $n^{\prime}_{y}$
and $n_{y}$, the diagonal elements of the Matrix $M$, (cf. Eq.(8)) form the
major contribution, as they scale as $\sim(n^{\prime}_{y})^{2}$, while the
off-diagonal ones far removed from the principal diagonal, scale magnitude
wise as $\sim 1/n^{\prime}_{y}$ or $\sim 1/n_{y}$ (cf. Eq.(9)), endowing a
relative significance of $\sim(n^{\prime}_{y})^{3}$ to the diagonal elements
over the far off-diagonals, thereby imparting a desirable characteristic for
convergence.

If the origin is chosen to be the center of inversion, the parity operator
$\mathcal{P}$, which is a linear and hermitian transformation maps a generic
wave function $\psi(\vec{r},t)$ in accord with
$\mathcal{P}\psi(\vec{r},t)=\psi(-\vec{r},t)$ whereas the antilinenear time-
reversal $\mathcal{T}$ transformation accomplishes the mapping
$\mathcal{T}\psi(\vec{r},t)=\psi^{*}(\vec{r},-t)$. Further, since
$\mathcal{P}=\mathcal{P}^{-1}$, for the present spinless case,
$\mathcal{T}=\mathcal{T}^{-1}$ (while incidentally, for half-odd-integral
spins, due to Kramers’s degeneracy, $\mathcal{T}^{2}=-I$, the negative
identity), in the coordinate representation for the position and canonical
momentum operators respectively follow the operator identities:

$\mathcal{P}\ \vec{r}\ \mathcal{P}^{-1}=-\vec{r},\ \mathcal{T}\ \vec{r}\
\mathcal{T}^{-1}=\vec{r},\ \mathcal{P}\ \hat{\vec{p}}\
\mathcal{P}^{-1}=-\hat{\vec{p}},\ \mathcal{T}\ \hat{\vec{p}}\
\mathcal{T}^{-1}=-\hat{\vec{p}}$

Further $[\mathcal{P},\mathcal{T}]=0$, and $\hat{H}$ is said to be
$\mathcal{PT}$ symmetric if $[\hat{H},\mathcal{PT}]=0$. This requires that for
the single spinless particle [21] :

$\displaystyle(\mathcal{PT})\hat{H}=\Big{[}\frac{(\hat{\vec{p}})^{2}}{2\mu}+V^{*}(-\vec{r})\Big{]}\mathcal{PT}=\hat{H}(\mathcal{PT}),$
(10)

since the antilinear $T$ symmetry deems the transformation for a general
complex-valued, explicitly time independent potential $V$, as $V\equiv
Re(V)+i\ Im(V)\rightarrow Re(V)-i\ Im(V)=V^{*}$. Further, for an eigenstate
$|\psi\rangle$,

$\hat{H}|\psi\rangle=E|\psi\rangle\implies\hat{H}\ \mathcal{PT}\
|\psi\rangle=E^{*}\ \mathcal{PT}\ |\psi\rangle;$ (11)

thus retention of $\mathcal{PT}$ symmetry manifests in reality of the
eigenvalues, while post-$\mathcal{PT}$ breakdown, energy eigenvalues must
occur in complex conjugate-pairs. $\mathcal{PT}$ symmetry is still intriguing,
since, especially for a multivariate problem, it is indiscernible a priori,
exactly in which parameter-regime the symmetry will be broken at the
“exceptional points” and further, the symmetry breaking is spontaneous [33].
Conservation of currents from a general field-theoretic perspective was
established for the $\mathcal{PT}$ operation by Alexandre et al. [34].

If it is required that the regions I through IV have $\mathcal{PT}$ symmetry
incorporated in the potential (energy) part, with the $\mathcal{P}$(parity)
operation carried out around the median $y=b/2\equiv b_{2}$, that is
accomplished through $y-b/2\rightarrow b/2-y$. As is well-known, if the
potential is completely real, $\hat{H}$ is manifestly hermitian with no extra
restrictions for reality of its associated eigenvalues. However, when a real
potential in some sector is accompanied by a complex valued or a pure
imaginary one in another sector, real eigenvalues in the sustained, unbroken
$\mathcal{PT}$ regime could result only when the $Re(V)$ is an even, i.e. a
symmetric function under spatial inversion (parity), while the $Im(V)$ is odd,
i.e. antisymmetric. We shall impose this requirement on the general, complex-
valued potential distributions chosen herein.

For the rectangular $2$D rigid box studied herein we set
$a=\sqrt{3},b=\sqrt{2}$ both in the units of Bohr radii. Such a choice of
irrational distance units should keep the accidental degeneracies to the
minimum, if not completely suppress them. Further, $b_{1}$ and $b_{3}$,
located symmetrically with respect to the median $b_{2}=b/2$ are set to:
$b_{1}=0.4b$ and $b_{3}=0.6b$.

###  3.1 Real valued striped potentials

Since, even the striped real valued $2$D potential distribution seems to have
eluded attention, for an initial orientation, we present seven representative
sets of all completely real-valued potentials with qualitatively different
choices for $V_{1},V_{2},V_{3},V_{4}$ (in Rydberg energy units). The
Hamiltonian operator on the left of Eq. (1) is manifestly hermitian for this
case, with no further symmetry requirements. Table-I presents five lowest
energy eigenvalues, of course all of them real, for each of the seven
adumbrative cases (I-VII); while Figure-2 depicts the pertinent $2$D
probability densities $|\psi(x,y)|^{2}$, where the portrayals are individually
normalized albeit drawn to arbitrary scales. Figure-2 succinctly conveys that
the probability exhibits varying degrees of localization, region-wise.
Hereafter, distances in multiples of Bohr radii and energy values in Rydberg
units will be understood. The “Baseline” refers to the standard problem where
the inside potential identically zero, and as is evident, for which the
$y$-separation also occurs; so that with $n^{(0)}_{x}=1$, $n_{y}$ also has a
fixed value.

Table I: Some typical, all-real-valued, seven sets of “striped” potentials (i.e. potential energies, in Ry) along the four $y$-stripes (Figure-1) and the associated lowest five energy eigenvalues for a rectangular rigid box with $x$ length $a=\sqrt{3}$ and $y$-width $b=\sqrt{2}$ in units of $a_{B}$, the Bohr radius. The parameter $n^{(0)}_{x}=1$ so that the $y$-solution governs the number of peaks (cf. Eq.(6)). For comparison, the “Baseline” refers to the standard rigid box with the inside potential identically zero.  Set | Potentials in $y$-strips (Ry) | Lowest five energy eigenvalues (Ry)
---|---|---
$V_{1}$ | $V_{2}$ | $V_{3}$ | $V_{4}$ | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{5}$
Baseline | 0 | 0 | 0 | 0 | 8.224 | 23.03 | 47.70 | 82.25 | 126.7
I | +100 | -100 | +100 | -100 | -72.61 | -6.131 | 18.88 | 112.7 | 134.9
II | +100 | -100 | -100 | +100 | -43.67 | 83.23 | 130.1 | 145.2 | 203.8
III | -100 | +100 | +100 | -100 | -72.69 | -72.22 | -3.023 | 0.230 | 98.42
IV | +100 | -100 | +100 | +100 | 12.56 | 119.9 | 132.9 | 171.9 | 211.8
V | -100 | +100 | -100 | -100 | -80.60 | -72.45 | -32.74 | -1.211 | 46.75
VI | +100 | -100 | -100 | -100 | -85.05 | -50.30 | 6.566 | 81.70 | 132.2
VII | -100 | +100 | +100 | +100 | -72.45 | -1.436 | 101.7 | 121.6 | 161.1

Figure 2: A collage of normalized ground states $2D$ probability densities
$|\psi(x,y)|^{2}$ (plotted in arbitrary units), respectively corresponding to
the seven sets of striped, all real potentials tabulated in Table-I for the
corresponding ground states (panels I through VII). The familiar “Baseline”
($V=0$ everywhere inside the box) Figure precedes the profiles I-VII, for
comparison.

The baseline energy levels are of course $\displaystyle
E=E^{(0)}_{n_{x},n_{y}}=\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{n^{2}_{x}}{a^{2}}+\frac{n^{2}_{y}}{b^{2}}\Big{)}$;
presented in Table-I: $n_{x}=n^{(0)}_{x}=1$ and $n_{y}$ values sweep through
$1,2,3,4$ and $5$ where the baseline ground state ($n^{(0)}_{x}=1,n_{y}=1$)
probability density displays a single, flat peak (Figure-2). Although we shall
not present in detail here, the expansion coefficients turn out contributing
significantly at best only for the $n_{y}$ values ranging from $n_{y}\sim
1-8$.

###  3.2 Indroducing $\mathcal{PT}$ symmetric potentials

Next, we appraise the effect of introducing $\mathcal{PT}$ symmetry in the
striped regions. We shall consider only the cases with nontrivial scenarios,
where the participant states in $\mathcal{PT}$ sustenance and breakdown often
will, but need not always imperatively involve, the ground-state. Unless
otherwise specified, we employ the eigenvalue-labeling convention from
MATHEMATICA [32] where magnitude-wise the largest eigenvalues are labeled in
diminishing order of their magnitude. Labeling the energy-sequence “p” in this
reverse order, $\mathcal{PT}$ scenarios surrounding $p=50,49,48,\cdots$ will
be considered.

Consider the case where the two outer stripes are held at zero potential
($V_{1}=V_{4}=0$) while the inner two sectors are rendered $\mathcal{PT}$
symmetric, choosing ($V_{2}=i\lambda=V_{3}^{*}$) with the real, positive
parameter $\lambda$ being continuously varied over the range $0$ through
$100$. Figure-3 plots the profiles for $Re(E_{p})$ and $Im(E_{p})$,
respectively the real and imaginary-parts of the energy levels, for $p=50$ as
well as for $p=49$ whose starting points happen to be truly the ground and the
first excited states. Initially, $\mathcal{PT}$ symmetry is retained as the
two levels continue to remain conspicuously real valued and distinct, slowly
start drifting toward each other and merge together at a characteristic
critical threshold value $\lambda_{c}=54.5$, the exceptional ‘critical’ point,
precisely at which the breakdown is triggered. For $\lambda>\lambda_{c}$,
there is a complete $\mathcal{PT}$ breakdown for the two energy levels
$E_{50}$ and $E_{49}$, which thereafter start occurring in complex conjugate
pairs, in consonance with Eq.(11), as demarcated in Figure-3. In similar
fashion, raising the potentials in the two $y$-borderline sectors by setting
$V_{1}=V_{2}=100$, and once again setting $V_{2}=i\lambda=V_{3}^{*}$ sweeping
the range ($0-100$), the starting points here too happen to be exactly the
ground and the first excited states, the exceptional point shifts upward to
$\lambda_{c}=84.0$, exhibiting higher degree of retention of $\mathcal{PT}$
symmetry, as is evident from the transition depicted in Figure-4.

Figure 3: Plots for $Re(E_{p})$ and $Im(E_{p})$ parts of the eigenvalues
$E_{p}$, versus the parameter $\lambda$ in $V_{2}=i\lambda=V^{*}_{3}$;
$V_{1}=0=V_{4}$ for the levels $p=50$ and $p=49$. Blue plot: $Re(E_{50})$, red
plot: $Re(E_{49})$. $\mathcal{PT}$ symmetry is retained, yielding real
eigenvalues, up to the critical value $\lambda_{c}=54.5$ (“exceptional point”)
wherein the orange and green (overlapped) plots for $Im(E_{50})$ and
$Im(E_{49})$ respectively, are zero up to $\lambda_{c}$ beyond which the
symmetry breaks down. Thereafter, the real parts merge together, while the
imaginary parts are rendered equal and opposite, signifying occurrence of the
eigenvalues in complex conjugate pairs. Energies in Ry, distances in Bohr
radii. The quantum number $n^{(0)}_{x}$ has been clamped to unity. Figure 4:
$V_{2}=i\lambda=V^{*}_{3},\lambda\rightarrow(0-100)$ and $n^{(0)}_{x}=1$. The
bordering sectors however are now held at a higher potential
$V_{1}=100=V_{4}$. $Re(E_{p})$ and $Im(E_{p})$ are plotted versus $\lambda$,
blue plot: $Re(E_{50})$, red plot: $Re(E_{49})$. $\mathcal{PT}$ breakdown
occurs at a higher $\lambda_{c}=84.0$. Likewise as in Figure-3, the orange and
green (overlapped) plots represent $Im(E_{50})$ and $Im(E_{49})$. Again,
beyond $\lambda_{c}$, the real parts merge and eigenvalues emerge only as
complex conjugate pairs.

Next, we scrutinize the response to $\mathcal{PT}$ symmetry by interchanging
preceding two blocks: with now the outer sectors held non-hermitian
$\mathcal{PT}$ symmetric: $V_{1}=i\lambda=V^{*}_{4}$ and
$\lambda\rightarrow(0-100)$ while, the interior set to zero: $V_{2}=0=V_{3}$
(the quantum number $n^{(0)}_{x}=1$, as before). A remarkable $\mathcal{PT}$
behavior then becomes manifest: First at the exceptional point
$\lambda_{c_{1}}=1$ (first criticality), a $\mathcal{PT}$ breakdown occurs for
the two lowest two levels $p=50$ (ground-state for $\lambda=0$), $p=49$
(immediately succeeding excited state for $\lambda=0$). Thereafter, instead of
continuing with the symmetry breach throughout the remainder of the
$\lambda$-range, a second criticality emanates for the level $E_{50}$ at
$\lambda_{c_{2}}=51.0$, where, surprisingly, not only there is a complete
restoration of $\mathcal{PT}$ symmetry (i.e. being bestowed with completely
real eigenvalues), for the (erstwhile-) lowest level $E_{50}$, but
concomitantly, at precisely the same juncture, the next two higher states with
$p=49$ and $48$ “collude” together triggering a $\mathcal{PT}$ symmetry-
breaking transition. Further, this ‘crossover’ is particularly a smooth one,
i.e. with no sudden kinks, as is vividly purported by the collages of the
plots in Figures-5 and 6. Note that at the point of second criticality
$\lambda_{c_{2}}$, $E_{50}$ abruptly jumps upward and smoothly continues the
sojourn of the state $E_{48}$, the latter steadily linking itself with the
$\mathcal{PT}$ breakdown curve but now by coupling with the state $E_{49}$,
with $Re(E_{48}),\ Re(E_{49})$ congruently merging together. It is thus clear
that, for real valued eigenvalues, unlike in adiabatic change of the
parameters, the energy hierarchy is not necessarily preserved under
$\mathcal{PT}$ transitions. These counterintuitive tripartite peregrinations
of the transiting participant states cannot be predicted a priori.

Figure 5: $Re(E_{p})$ vs. $\lambda$ behavior for $p=50$ (blue), $p=49$ (red)
and $p=48$ (green), with the outer sectors imparted $\mathcal{PT}$ symmetric:
$V_{1}=i\lambda=V^{*}_{4}$; inside, $V_{2}=0=V_{3}$. The range swept is:
$\lambda\rightarrow(0-100)$. The quantum number $n^{(0)}_{x}=1$, as before. A
remarkable interplay of sustenance and breaking of $\mathcal{PT}$ occurs:
First breakdown for $p=50,49$ at $\lambda_{c_{1}}=11.0$; $\mathcal{PT}$
restoration for $p=50$ and a ‘crossover’ leading to breaking for $p=49$ and
$48$, at $\lambda=51.0$ is evident. See the text for fuller details; cf. also
the next figure (Figure-6). Figure 6: $Im(E_{p})$ vs. $\lambda$ behavior for
$p=50$ (blue), $p=49$ (red) and $p=48$ (green). The other control parameters
are the same as those in Figure-5. On the extreme left, between the first
($\lambda_{c_{1}}=11.0$) and the second ($\lambda_{c_{2}}=51.0$) criticality,
the blue and red plots perfectly overlap, giving a magenta type hue. Post
$\lambda_{c_{2}}$, the eigenvalue $E_{50}$ becomes completely real;
concomitantly preceding two eigenvalues emerge with $E_{49}=E^{*}_{48}$,
suffering a $\mathcal{PT}$ breakdown.

Next, we empower the extreme two sectors with strong $\mathcal{PT}$ symmetry,
vide $V_{1}=50\ i=V^{*}_{4}$ and vary $V_{2}=i\lambda=V^{*}_{3}$ with
$\lambda\rightarrow(0-100)$ while, $n^{(0)}_{x}$ still kept locked at unity,
thus all the four regions are made non-hermitian $\mathcal{PT}$ symmetric. In
response, $\mathcal{PT}$ that is broken right in the beginning dismally fails
to recover and continues to be breached, as the plots in Figure-7 elucidate.

Figure 7: $Re(E_{50})=Re(E_{49})$ (both completely merged, in red) and
$Im(E_{50})=-Im(E_{49}$) (orange-colored plots), for a strong, fixed choice of
$V_{1}=50\ i=V^{*}_{4}$ and varying $V_{2}=i\lambda=V^{*}_{3}$
$\lambda\rightarrow(0-100)$. $\mathcal{PT}$ breakdown, right from the start is
evident throughout.

Incidentally, the present work turns out, in a sense, ‘a spatial counterpart’
to the interesting recent work by Hayrapetyan, Klevansky and Götte [35], who
introduced in a spatially homogeneous optical medium, a time-dependent
$\mathcal{PT}$ symmetric dielectric permittivity and demonstrated within the
framework of classical electrodynamics, the consequent light amplification as
well as attenuation.

###  3.3 Probability density mutations

Probability redistributions surrounding some typical $\mathcal{PT}$
transitions, are presented next. Absolute squared of the ground-level wave
function $|\psi(x,y)|^{2}$, pre (left plot of Figure-8) and post(right plot of
Figure-8) $\mathcal{PT}$ breaking, presented to scale, reflects non-unitarity
(in fact antiunitarity) of the composite $\mathcal{PT}$ operation evidenced
from attenuation in one of the peaks for the state $E_{50}$, after criticality
($\lambda_{c}=55.4$), for the same parameters as in Figure-3.

Figure 8: Pre (left) and Post (right) $\mathcal{PT}$ breaking Probability
density $|\psi(x,y)|^{2}$ for the ground-level $E_{50}$, with the same
parameterization as in Figure-3; with $\lambda=54<54.5=\lambda_{c}$ for
pre-$\mathcal{PT}$ while, $\lambda=56>54.5=\lambda_{c}$ for
post-$\mathcal{PT}$ scenario.

Figure 9: Pre (plot to the left) and Post (plot to the right) $\mathcal{PT}$
breaking Probability density $|\psi(x,y)|^{2}$ for the ground-level $E_{50}$,
$n^{(0)}_{x}=2$ with otherwise the same parameterization as in Figure-3; with
$\lambda=54<54.5=\lambda_{c}$ for pre-$\mathcal{PT}$ and
$\lambda=56>54.5=\lambda_{c}$ for post-$\mathcal{PT}$ scenario.

Not surprisingly, this breakdown also prevails for $n^{(0)}_{x}=2$ where both
the peaks deteriorate (Figure-9); the factored out $x$-part of course having
no direct role to play in the present $\mathcal{PT}$ gamut.

Incidentally, with a small number of partitions as portrayed in Figure-1, it
is straightforward to approach the present problem directly. In the respective
regions, one has the following forms for the wave functions:

$\displaystyle\psi(x,y)$ $\displaystyle\sim$ $\displaystyle
A\sin\Big{(}\frac{n^{(0)}_{x}\pi x}{a}\Big{)}\sin(k_{1}y)\ ;\qquad\qquad\ \ \
\quad\ \ \text{Region-I}$ $\displaystyle\psi(x,y)$ $\displaystyle\sim$
$\displaystyle\sin\Big{(}\frac{n^{(0)}_{x}\pi
x}{a}\Big{)}[B\sin(k_{2}y)+C\cos(k_{2}y)]\ ;\ \text{Region-II}$
$\displaystyle\psi(x,y)$ $\displaystyle\sim$
$\displaystyle\sin\Big{(}\frac{n^{(0)}_{x}\pi
x}{a}\Big{)}[F\sin(k_{3}y)+G\cos(k_{3}y)]\ ;\ \text{Region-III}$
$\displaystyle\psi(x,y)$ $\displaystyle\sim$ $\displaystyle
H\sin\Big{(}\frac{n^{(0)}_{x}\pi x}{a}\Big{)}\sin[k_{4}(y-b)]\ ;\qquad\qquad\
\text{Region-IV}$ (12)

the constant coefficients determinable by matching only the $y$-factors of
wave functions and their derivatives at the respective interfaces, with the
constants $\\{k_{i}\\}_{i=1}^{4}$ prescribed by the connection

$\frac{\hbar^{2}}{2\mu}k_{i}^{2}=E-V_{i}+\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{(}\frac{n^{(0)}_{x}}{a}\Big{)}^{2};\
\forall i=1,\cdots,4$ (13)

Note that the energy eigenvalues $E$ are unknown to begin with; the task is
then tantamount to solving for the eigenvalues and the wave functions, the
latter through determination of the coefficients. However, this approach leads
to a formidable set of coupled transcendental equations tractable only through
numerical techniques; the situation getting increasingly aggravated as the
number of partitions increases. Moreover, if one did venture to solve these
simultaneously, one can in principle obtain but only a specific energy
eigenvalue at a time. For instance, even in the completely Hermitian case (all
real potentials) the energy hierarchy will be indiscernible under this direct
approach, as it is unclear a priori, exactly which eigenvalue - ground or
excited - has resulted. The present matrix approach, on the other hand offers
a holistic method that unequivocally yields the complete eigenspectrum.

###  3.4  $\mathcal{PT}$ symmetric electrostatic potential

$\mathcal{PT}$ symmetry that hinges on an intricate balance between gain and
loss mechanisms is also investigated furnishing a fictitious electric field
with a backdrop of the striped box, for the particle moving in the box
possessing an electric charge $q$. Consider a uniform electric field, say
along the $y$-direction, $\vec{F}=F_{o}\hat{j}$ furnished inside the box, and
zero outside. This engenders the electrostatic potential (energy), once again
with a manifest explicit dependence only on the $y$-coordinate (apart from a
trivial additive constant set to zero here): $V(x,y)\equiv
V(y)=-\alpha(y-b/2)$ with $\alpha=qF_{o}$. A real $\alpha$ imparts reality to
$V$, which is manifestly antisymmetric around $y=b/2$, but still rendering the
Hamiltonian hermitian; whilst a pure imaginary $\alpha=i\lambda,\
\lambda\in\mathbb{R}$ , generates a non-hermitian but a truly $\mathcal{PT}$
symmetric Hamiltonian associated with the corresponding electric field.
Following a similar chain of arguments as was done surrounding Eqs. (1)
through (9) above for the striped potential distribution, the electric field
gives rise to simple prescriptions for the corresponding $M$\- matrix,
$M^{el}$, as described below.

Since $V$ corresponding to the electric field is clearly antisymmetric in the
$y$-coordinate around the median, the pertinent integrals all vanish for
diagonal part of $M^{el}$, resulting strikingly, in exactly the same
contribution as that of the rigid box with identically zero inside potential,

$M^{el}_{n^{\prime}_{y},n_{y}}=\frac{\pi^{2}\hbar^{2}}{2\mu}\Big{[}\frac{(n^{(0)}_{x})^{2}}{a^{2}}+\frac{(n_{y})^{2}}{b^{2}}\Big{]}$
(14)

for the diagonal elements ($n^{\prime}_{y}=n_{y}$), while the off-diagonal
portion can be written as

$M^{el}_{n^{\prime}_{y},n_{y}}=-\frac{\alpha
b}{\pi^{2}}\Big{[}\frac{1}{(n^{\prime}_{y}+n_{y})^{2}}-\frac{1}{(n^{\prime}_{y}-n_{y})^{2}}\Big{]}(1-(-1)^{n^{\prime}_{y}+n_{y}})$
(15)

hence, contributing only for odd combinations of $n^{\prime}_{y}+n_{y}$;
vanishing otherwise.

If the electric field contribution is taken in conjunction with that of the
striped potential, for the combined problem, only the off-diagonal part, Eq.
(9), must simply be augmented by the expression on the right of Eq. (15)
above. Of the number of qualitatively diverse combinations of the striped
potential annexed by the uniform electric field, we shall herein present only
a couple of salient aspects, summarized in Figure-10. For a $\mathcal{PT}$
Symmetric Electric Field borne by $\alpha=20i$ along with a ‘moderate’
$\mathcal{PT}$ symmetric backdrop of $V_{1}=5i=V^{*}_{4}$ and
$V_{2}=-5i=V^{*}_{3}$, $\mathcal{PT}$ is preserved throughout (top panels,
Figure-10) as the resulting energy levels are all real. Enhancing the
background $\mathcal{PT}$ strength with the choice $V_{1}=100i=V^{*}_{4}$ and
$V_{2}=-100i=V^{*}_{3}$ the symmetry, albeit preserved for the ground-level,
does break off for a handful, though not exhaustively all, of the higher ones
(bottom panels, Figure-10), with emergence of equal and opposite imaginary
parts. The $Re(E)$ plots are similar but not identical and practically
indistinguishable on the scale plotted. There thus are interesting trade-offs
between $\mathcal{PT}$ symmetry transitions caused by the fictitious electric
field and of the striped potential.

Figure 10: Top panels: The Real (Left) and Imaginary (Right) parts of the
eigenenergies for a $\mathcal{PT}$ symmetric Electric Field $\alpha=20i$, with
a $\mathcal{PT}$ symmetric background:$V_{1}=5i=V^{*}_{4}$ and
$V_{2}=-5i=V^{*}_{3}$. $\mathcal{PT}$ is preserved throughout: the eigenvalues
are all real. Bottom panels: $\alpha=20i$, with a stronger $\mathcal{PT}$
backdrop, $V_{1}=100i=V^{*}_{4}$ and $V_{2}=-100i=V^{*}_{3}$, the symmetry
breaks for a few states. For convenience and to guide the eye, the eigenvalues
have been labeled from $1$ through $50$ in increasing order of their
magnitude.

##  4 Conclusions

In summary, the present article derives exact quantum mechanical solutions for
a $2$D rectangular rigid box with a piecewise constant set of potentials in
parallel, contiguously connected rectangular sectors or ‘stripes’ that have
their widths the same as that of the box, forming a special type of quantum-
billiards. These potentials are studied to gauge the response of a scalar
particle both under hermitian and non-hermitian yet $\mathcal{PT}$ symmetric
attributes ascribed, and combinations thereof. In the latter, several
interesting$\mathcal{PT}$ symmetry sustenance and breakdown transition
scenarios emerge, including an intriguing reinstating of the broken symmetry
for an energy level, concomitant with a smooth crossover of breakdown
transition transferred to higher states. $\mathcal{PT}$ symmetry that hinges
on an intricate balance between gain and loss mechanisms is also investigated
with a fictitious electric field furnished against a backdrop of the striped
box. It is gratifying all that these spectacular systems are amenable to an
exact quantum mechanical treatment. It would be instructive to investigate the
temporal evolution of an initially localized wave-packet (with compact
support) within the box, as well as under exotic electromagnetic fields,
within the $\mathcal{PT}$ symmetric framework. These studies are currently
under investigation. Acknowledgments
The authors are immensely indebted to Professor Dr. P. Durganandini for
several interesting discussions on the current theme, and her encouragement
throughout. SK is supported by University Grants Commission’s Faculty Recharge
Programme (UGC-FRP), Govt. of India, New Delhi, India.

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