---
title: PT-Symmetric Klein-Gordon Equations
url: https://www.emergentmind.com/topics/pt-symmetric-klein-gordon-equations
type: topic
---

# PT-Symmetric Klein-Gordon Equations

PT-symmetric Klein-Gordon equations are Klein-Gordon models in which the combined action of parity and time reversal constrains either local gain-loss perturbations, nonlocal reductions, operator domains, or effective potentials. In the formulations considered here, PT symmetry appears in several mathematically distinct ways: as a localized viscous term such as $\epsilon\,\gamma(x)\,u_t$ or $\varepsilon\,\gamma(x,y)\,u_t$ in classical field theory, as an imaginary velocity-dependent term $iW(x)u_t$ for standing waves, as a nonlocal coupling to $u(-x,-t)$ in integrable systems, and as a structural constraint on non-Hermitian Klein-Gordon Hamiltonians and relativistic oscillators [1402.2942, 1512.01103, 1409.7095, 2202.08990, 1801.09602, 1005.3633, 1911.11182]. Across these settings, the central issue is spectral behavior: whether PT symmetry preserves purely imaginary or real spectra, induces eigenvalue splitting, or selects physically distinguished metastable or exactly solvable states.

## 1. PT symmetry as a structural constraint in Klein-Gordon theory

In the classical field-theoretic setting, a standard one-dimensional Klein-Gordon equation,
\[
u_{tt}-u_{xx}+V'(u)=0,
\]
is replaced by a PT-symmetric variant
\[
u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,
\]
where $\epsilon\ge 0$ is small and $\gamma(x)$ is smooth, exponentially localized, and odd, $\gamma(-x)=-\gamma(x)$. Under $\mathcal{PT}:(x,t)\to(-x,-t)$, one has $u_t\mapsto -u_t$ and $\gamma(x)\mapsto-\gamma(-x)$, so the equation is invariant [1402.2942]. A higher-dimensional analogue replaces $\gamma(x)$ by an exponentially localized $\gamma(x,y)$ that is odd in one or two coordinates, again producing a PT-symmetric gain-loss defect without modifying the static kink profile [1512.01103].

A different local realization uses a complex field $u\in\mathbb C$ and the perturbed equation
\[
u_{tt}+i\,W(x)\,u_t-\Delta u+u-f(|u|^2)u=0,
\]
with $W(x)$ real, even, and of moderate decay. Here the term $iW(x)u_t$ models balanced gain and loss; the system is non-Hermitian yet PT-symmetric, and the spectral problem takes the form of a quadratic operator pencil [1409.7095].

PT symmetry also appears in genuinely nonlocal Klein-Gordon systems. Imposing the reduction
\[
q(x,t)=\hat P\hat T\,p(x,t)=p(-x,-t)
\]
on an integrable $3\times 3$ Klein-Gordon system yields an equation in which $u(x,t)$ is coupled directly to $u(-x,-t)$ through the mixed derivative $u_{xt}$ [2202.08990]. In operator-theoretic and quantum-mechanical formulations, PT symmetry is encoded by relations such as
\[
V(x)=V^*(-x), \qquad [H,PT]=0,
\]
or by the construction of closed PT-symmetric extensions of formal Klein-Gordon operators [1801.09602, 1005.3633].

These variants share a common theme: PT symmetry does not restore ordinary self-adjointness, but it imposes a spectral organization that can often be analyzed explicitly.

## 2. One-dimensional kink stability under localized gain and loss

For the one-dimensional PT-symmetric Klein-Gordon field theory studied by Demirkaya et al., the static kink solves
\[
-\phi_{xx}+f(\phi)=0, \qquad \phi(\pm\infty)=u_\pm,
\]
because the dashpot term vanishes for time-independent states. Two explicit examples are the sine-Gordon kink
\[
\phi(x)=4\arctan\!\bigl(e^{x-x_0}\bigr),
\]
and the $\phi^4$ kink
\[
\phi(x)=\tanh(x-x_0),
\]
with analogous anti-kinks [1402.2942].

Linearization about a kink, $u(x,t)=\phi(x)+v(x,t)$, gives
\[
v_{tt}-v_{xx}+\epsilon\,\gamma(x)\,v_t+f'(\phi)\,v=0.
\]
With the modal ansatz $v(x,t)=e^{\lambda t}v(x)$ and
\[
H=-\partial_x^2+f'(\phi(x))=-\partial_x^2-W(x)+\sigma^2,
\]
one obtains the quadratic pencil
\[
\lambda^2 v+\epsilon\,\lambda\,\gamma(x)\,v+Hv=0.
\]
Since $W(x)\to0$ exponentially, the essential spectrum of $H$ is $[\sigma^2,\infty)$, and hence
\[
\sigma_{ess}(L)=(-i\infty,-i\sigma]\cup[i\sigma,i\infty).
\]
If $H$ has simple eigenvalues
\[
0=\sigma_0<\sigma_1^2<\cdots<\sigma_n^2<\sigma^2,
\]
then for $\epsilon=0$ the point spectrum consists of pairs $\lambda=\pm i\sigma_j$; the translational mode at $j=0$ has algebraic multiplicity two for the quadratic pencil [1402.2942].

For small $\epsilon>0$, the discrete spectral points shift according to explicit first-order perturbation formulas. For the translational mode,
\[
\lambda_0
=
-\,\epsilon\,
\frac{\langle\gamma\,\psi_0,\psi_0\rangle}
{\langle\psi_0,\psi_0\rangle}
+O(\epsilon^2),
\]
while for internal modes,
\[
\lambda_j
=
i\,\sigma_j
-
\epsilon\,
\frac{\langle\gamma\,\psi_j,\psi_j\rangle}
{2\,\langle\psi_j,\psi_j\rangle}
+O(\epsilon^2),
\qquad j=1,\dots,n.
\]
The sign of $\langle\gamma\psi_j,\psi_j\rangle$ therefore controls whether the perturbed eigenvalue acquires positive or negative real part [1402.2942].

The location of the kink center relative to the gain-loss interface is decisive. If $x_0=0$, then $W(x)$ is even, the linearization inherits the full Hamiltonian symmetry $\{\pm\lambda,\pm\overline{\lambda}\}$, and $\langle\gamma\psi_j,\psi_j\rangle=0$ for all $j$; all discrete eigenvalues remain purely imaginary to all orders $O(\epsilon)$, so the kink is spectrally neutral. If $x_0<0$, the kink lies on the lossy side and the former zero mode shifts into the stable half-plane, $\Re\lambda_0<0$. If $x_0>0$, the kink lies on the gain side and $\Re\lambda_0>0$, producing spectral instability. The same sign-sensitive mechanism applies to higher internal modes [1402.2942].

The sine-Gordon and $\phi^4$ examples with
\[
\gamma(x)=-x\,e^{-x^2/2}
\]
make this dependence explicit. For the sine-Gordon kink, $C_{\rm sG}(2)\approx -0.2675$, giving $\lambda_0\approx +0.2675\,\epsilon$ on the gain side and $\lambda_0\approx -0.2675\,\epsilon$ on the lossy side. For the $\phi^4$ kink, $C_{\phi^4}(2)\approx -0.2876$. In reflectionless cases such as sine-Gordon and $\phi^4$, the continuous-spectrum edge may also generate an $O(\epsilon^2)$ pair with real part $O(\epsilon^3)$ [1402.2942].

A common misconception is that a PT-symmetric perturbation necessarily destabilizes a kink. In these models, instability is instead conditional on geometric placement: centered kinks remain spectrally neutral, lossy-side kinks are stabilized, and gain-side kinks are destabilized.

## 3. Two-dimensional kink perturbations and non-self-adjoint spectral splitting

Borisov and Dmitriev studied the two-dimensional Klein-Gordon field
\[
u_{tt}-\Delta u+f(u)=0,\qquad (x,y)\in\mathbb R^2,\ t>0,
\]
with a static kink depending only on $x$ and solving
\[
-\phi''(x)+f(\phi(x))=0, \qquad \phi(x)\to\phi_\pm \text{ as } x\to\pm\infty.
\]
The examples listed are the sine-Gordon kink,
\[
f(u)=\sin u,\qquad \phi(x)=4\arctan(e^x),\qquad \Lambda_\pm=1,
\]
and the $\varphi^4$ kink,
\[
f(u)=2(u-u^3),\qquad \phi(x)=\tanh(x),\qquad \Lambda_\pm=2.
\]
The associated one-dimensional linearization operator is
\[
H_0:=-\frac{d^2}{dx^2}+f'(\phi(x)),
\]
with essential spectrum $[\Lambda_e,\infty)$ and finitely many simple eigenvalues below $\Lambda_e$ [1512.01103].

The PT-symmetric perturbation is introduced as a spatially localized viscous-friction term,
\[
u_{tt}-\Delta u+\varepsilon\,\gamma(x,y)\,u_t+f(u)=0,
\]
where $\varepsilon\ge0$ is small and $\gamma(x,y)$ is continuous, exponentially localized, and odd in one or two coordinates. Since the perturbation is proportional to $u_t$, it does not affect the static kink itself [1512.01103].

Setting
\[
u(x,y,t)=\phi(x)+\psi(x,y,t),\qquad |\psi|\ll1,
\]
and using $\psi(x,y,t)=e^{\lambda t}\Psi(x,y)$ produces the non-self-adjoint eigenvalue problem
\[
\bigl(-\Delta+f'(\phi(x))\bigr)\Psi+\varepsilon\,\lambda\,\gamma(x,y)\Psi+\lambda^2\Psi=0.
\]
For a simple discrete eigenvalue $\Lambda_*$ of $H_0$ with normalized eigenfunction $\psi_*(x)$, one defines
\[
K_1^*=\iint_{\mathbb R^2}\gamma(x,y)\,\psi_*^2(x)\,dx\,dy,
\qquad
K_2^*=\iint_{\mathbb R^2}\gamma(x,y)\,\psi_*(x)\,U_*(x,y)\,dx\,dy,
\]
where $U_*$ solves
\[
\bigl(-\Delta+f'(\phi)-\Lambda_*\bigr)U_*
=
\gamma\,\psi_*
-
\psi_*\iint\gamma\,\psi_*^2.
\]
The main theorem states that for each simple eigenvalue $\Lambda_*>0$ there are exactly two small eigenvalues with
\[
\lambda(\varepsilon)\to \pm i\,\kappa_*, \qquad \kappa_*=\sqrt{\Lambda_*},
\]
as $\varepsilon\to0$, and if $K_1^*\neq0$ then
\[
\lambda
=
\pm i\kappa_*
\;\pm\;
\frac{i\,\kappa_*\,K_1^*}{2}\,\varepsilon
+\frac{\Lambda_*\,K_2^*}{2\kappa_*}\,\varepsilon
+O(\varepsilon^2).
\]
If $K_1^*=0$ but $K_2^*\neq0$, the splitting is purely along the imaginary axis,
\[
\lambda
=
\pm i\kappa_*
\mp i\,\kappa_*\,K_2^*\,\varepsilon^2
+O(\varepsilon^3).
\]
For $\Lambda_*<0$, the emerging spectral points may be resonances rather than eigenvalues [1512.01103].

The proof uses a non-self-adjoint Birman-Schwinger reduction, the substitution $\lambda^2-\Lambda_*=k$, a scalar equation
\[
2k=\varepsilon\,F(\varepsilon,k),
\]
and Rouché’s theorem. The paper contains no explicit numerical simulations, but the asymptotic formulas predict how a localized PT-symmetric defect splits embedded one-dimensional kink eigenvalues off the imaginary axis, turning neutrally stable modes into growing or decaying modes according to the sign of $K_1^*$ [1512.01103].

## 4. Standing waves and the PT-symmetric Vakhitov-Kolokolov-type condition

A different PT-symmetric Klein-Gordon problem considers complex standing waves of
\[
u_{tt}+i\,W(x)\,u_t-\Delta u+u-f(|u|^2)u=0,
\]
with $W(x)$ real and even. The standing-wave ansatz
\[
u(x,t)=e^{i\omega t}\varphi(x),
\qquad \omega\in(-1,1),
\]
reduces the equation to
\[
-\Delta\varphi(x)+(1-\omega^2)\varphi(x)-\omega\,W(x)\varphi(x)-f(\varphi(x)^2)\varphi(x)=0,
\]
and in one dimension to
\[
-\varphi''(x)+(1-\omega^2)\varphi(x)-\omega\,W(x)\varphi(x)-f(\varphi^2)\varphi(x)=0.
\]
Under standard hypotheses on $f$ and $W$, this admits localized pulse solutions $\varphi=\varphi_\omega$ [1409.7095].

Linearization with
\[
u(x,t)=e^{i\omega t}\bigl[\varphi(x)+v(x,t)\bigr]
\]
leads to
\[
v_{tt}
+i\bigl(2\omega+W(x)\bigr)v_t
+\bigl[-\Delta+(1-\omega^2)-\omega W(x)-f(\varphi^2)\bigr]v
-2\varphi^2 f'(\varphi^2)\Re v=0.
\]
Writing $v=v_1+i v_2$ and $\mathbf V=(v_1,v_2)^T$, the normal-mode ansatz gives the operator pencil
\[
\bigl(\lambda^2\mathbb I+\lambda J+\mathcal H\bigr)\mathbf\Psi=0,
\]
with
\[
J=
\begin{pmatrix}
0 & -(2\omega+W(x))\\[6pt]
2\omega+W(x) & 0
\end{pmatrix},
\qquad
\mathcal H=
\begin{pmatrix}
L_+ & 0\\[6pt]
0 & L_-
\end{pmatrix},
\]
where
\[
L_-=-\Delta+(1-\omega^2)-\omega W(x)-f(\varphi^2),
\qquad
L_+=L_--2\varphi^2 f'(\varphi^2).
\]
Spectral stability means absence of any $\lambda$ with $\Re\lambda>0$ [1409.7095].

The paper derives a sharp frequency criterion:
\[
\frac{d}{d\omega}\!\Bigl[\int_{\mathbb R^d}(2\omega+W(x))\,\varphi_\omega^2(x)\,dx\Bigr]\le 0
\quad\Longleftrightarrow\quad
\text{spectral stability}.
\]
Equivalently, defining
\[
Q(\omega):=\int_{\mathbb R^d}(2\omega+W(x))\,\varphi_\omega^2(x)\,dx,
\]
one has
\[
Q'(\omega)\le 0
\quad\Longleftrightarrow\quad
\text{spectral stability}.
\]
This is an explicit PT-symmetric analogue of the classical Vakhitov-Kolokolov condition [1409.7095].

The numerical analysis discretizes the PDE on a uniform grid, computes standing waves by a Newton-type fixed-point solver, and then computes the spectrum of the finite-dimensional pencil. For
\[
f(\rho)=\rho,\qquad W(x)=a\,e^{-b x^2},\qquad a=0.1,\ b=1,
\]
the quantity
\[
I(\omega)=\int(2\omega+W(x))\,\varphi_\omega^2(x)\,dx
\]
changes slope at $\omega^*\approx0.705$. For $\omega<\omega^*$ a real pair with $\Re\lambda>0$ appears, whereas for $\omega>\omega^*$ all eigenvalues lie on the imaginary axis. The crossing of $I'(\omega)=0$ coincides with the bifurcation of the real eigenvalue pair [1409.7095].

## 5. Integrable Klein-Gordon equations with PT nonlocality

The integrable nonlocal construction starts from a generic system
\[
p_{xt}
=
\frac12\,[\cos(3p-3q)-\sin(3p-3q)]\,e^{-(p+q)}
+\frac12\,e^{2(p+q)},
\]
and imposes the PT reduction
\[
q(x,t)=\hat P\hat T\,p(x,t)=p(-x,-t).
\]
Writing $u(x,t)=p(x,t)$ gives the PT-symmetric Klein-Gordon equation
\[
u_{xt}(x,t)
=
\frac12\Big\{\cos\big[3u(x,t)-3u(-x,-t)\big]
-\sin\big[3u(x,t)-3u(-x,-t)\big]\Big\}
e^{-[u(x,t)+u(-x,-t)]}
+\frac12 e^{2[u(x,t)+u(-x,-t)]}.
\]
Unlike a local Klein-Gordon equation $u_{xt}=F(u)$, this equation couples the field at $(x,t)$ directly to its mirror point $(-x,-t)$ [2202.08990].

Its integrability is expressed through the compatibility of the $3\times 3$ linear system
\[
(\partial_x-M_1)\varphi=0,\qquad (\partial_t-N_1)\varphi=0,
\]
with spectral parameter $\lambda$, where
\[
M_1(x,t,\lambda)=
\begin{pmatrix}
(p+q)_x+i(p-q)_x & \lambda & 0\\
0 & -2i(p-q)_x & \lambda\\
\lambda & 0 & -(p+q)_x+i(p-q)_x
\end{pmatrix},
\]
and
\[
N_1(x,t,\lambda)
=
\frac1\lambda
\begin{pmatrix}
0 & 0 & e^{2(p+q)}\\
-\,e^{-(p+q)-3i(p-q)} & 0 & 0\\
0 & -\,e^{-(p+q)+3i(p-q)} & 0
\end{pmatrix}.
\]
The zero-curvature condition
\[
M_{1,t}-N_{1,x}+[M_1,N_1]=0
\]
reproduces the PT-nonlocal equation exactly [2202.08990].

Because $\hat P\hat T^2=1$, any solution decomposes as
\[
p=A+B,
\]
with
\[
A(x,t)=A(-x,-t),\qquad B(x,t)=-B(-x,-t).
\]
A one-soliton solution is given in terms of
\[
\eta=kx+3k^{-1}t,
\]
\[
A(\eta)=\frac14\ln\!\Big[\frac{\cosh(3\eta/2)}{4\cosh^3(\eta/2)}\Big],
\qquad
B(\eta)=\frac12\arctan\!\Big[\sqrt{3}\,\tanh(\eta/2)\Big],
\]
so that
\[
p(x,t)=A(kx+3k^{-1}t)+B(kx+3k^{-1}t).
\]
The center moves along $\eta=\text{constant}$, hence $x=-3k^{-2}t$ when $\eta=0$; larger $k$ gives a narrower, faster kink. The PT-antisymmetric component $B(\eta)$ introduces an antisymmetric phase twist and deforms the usual kink profile [2202.08990].

The model admits a Lax pair of Zakharov-Shabat type and an infinite hierarchy of conserved densities $\partial_t\rho_n=\partial_x J_n$. In the local limit $q=p$, it reduces to the well-known Tzitzéica, or Liouville/Sinh-Gordon, Klein-Gordon equation, which is Hamiltonian. The same construction also sits inside a four-element symmetry group $\mathcal G=\{1,\mathcal{PT},E_{x,t},\mathcal{PT}E_{x,t}\}$ that generates further integrable nonlocal variants [2202.08990].

## 6. Crypto-Hermitian, PT-symmetric operator, and exactly solvable quantum formulations

In the crypto-Hermitian approach, the free Klein-Gordon equation
\[
(\partial_t^2-\Delta+m^2)\psi(t,x)=0
\]
is rewritten using the two-component Feshbach-Villars wavefunction
\[
\Psi(t,x)=
\begin{pmatrix}
i\,\partial_t\psi\\
\psi
\end{pmatrix},
\]
so that
\[
i\,\partial_t\Psi(t,x)=H\Psi(t,x),\qquad
H=
\begin{pmatrix}
0 & K\\[6pt]
I & 0
\end{pmatrix},
\qquad
K=-\Delta+m^2.
\]
With a scalar potential, one replaces $K$ by $-\Delta+m^2+V(x)$. The Hamiltonian is not Hermitian in the naive inner product, but Hermiticity is restored by a positive-definite metric $\Theta=\Theta^\dagger>0$ satisfying
\[
H^\dagger\Theta=\Theta H.
\]
A spectral construction gives
\[
\Theta=
\begin{pmatrix}
\alpha & \beta K^{1/2}\\[4pt]
\beta K^{1/2} & \alpha K
\end{pmatrix},
\]
with positivity conditions
\[
\alpha>0,\qquad \alpha^2K>\beta^2.
\]
The spectrum is
\[
E_n^{(\pm)}=\pm\sqrt{\epsilon_n},\qquad K\psi_n=\epsilon_n\psi_n,
\]
and since $K$ is Hermitian and strictly positive, all $E_n^{(\pm)}$ are real. The corresponding density
\[
\rho(t,x)=\Psi^\dagger(t,x)\Theta\Psi(t,x)>0
\]
supports a conserved current $\partial_\mu J^\mu=0$. The PT-symmetric specialization is characterized by
\[
V(x)=V^*(-x),\qquad [H,PT]=0,
\]
with real spectra in the unbroken regime and complex-conjugate pairs in the broken regime [1801.09602].

A more explicitly PT-symmetric spectral problem arises for the one-dimensional Klein-Gordon oscillator. The formal operator is
\[
\widetilde K(\Omega,E)
=
-\,\frac{d^2}{dx^2}
+(1+2\Omega E)x^2
-\Omega x^4
+2i\sqrt{\Omega}\,x.
\]
Its closed PT-symmetric realization $K_+(\Omega,E)$ on $L^2(\mathbb R)$ is defined by subdominant behavior at $x\to\pm\infty$, and its adjoint satisfies
\[
K_-(\Omega,E)=P\,K_+(\Omega,E)\,P=T\,K_+(\Omega,E)\,T=K_+(\Omega,E)^*,
\qquad [K_\pm,PT]=0.
\]
By results on PT-symmetric anharmonic oscillators, $K_+(\Omega,E)$ has purely discrete spectrum
\[
0<\lambda_0<\lambda_1<\cdots\to+\infty.
\]
Physical energies are determined from
\[
E+\Omega E^2=\lambda_n(\Omega,E),
\qquad
E_n(\Omega)=\frac{1}{2\Omega}\Bigl(\sqrt{1+4\Omega\,\lambda_n(\Omega,E_n)}-1\Bigr).
\]
In the nonrelativistic limit $\Omega\to0$, one recovers
\[
E_n(0)=2n+1.
\]
In the Klein-Gordon case there are infinitely many distinct closed extensions, each corresponding to a different choice of in/out boundary conditions on the six Stokes sectors. Among these infinitely many dynamics, the PT-symmetric pair $K_\pm$ is singled out as physically preferred. Numerically, the PT-symmetric eigenvalue $E_{t0}(\Omega)$ and the complex-dilated resonance $E_{d0}(\Omega)$ satisfy
\[
E_{t0}-\Re(E_{d0})=O\bigl((\Im E_{d0})^2\bigr),
\]
and for $\Omega=0.002$ one example is
\[
E_{d0}=1.0005017620+i\,1.17\times10^{-144}.
\]
The PT-symmetric levels therefore reproduce resonance positions up to the order of the width and are interpreted as metastable states [1005.3633].

Exact PT-symmetric bound states also arise in the $(1+1)$-dimensional time-independent Klein-Gordon equation with position-dependent mass $M(x)$, scalar potential $S(x)$, and PT-symmetric vector potential $V(x)$,
\[
-\hbar^2\frac{d^2\phi}{dx^2}+[M(x)c^2+S(x)]^2\phi(x)-[E-V(x)]^2\phi(x)=0.
\]
This is rewritten as a Schrödinger-type equation with an $E$-dependent potential $V_E(x)$ and auxiliary eigenvalue $\varepsilon$; the true Klein-Gordon energies are obtained from
\[
\varepsilon_n(E)=0.
\]
The SUSYQM factorization uses a superpotential $W(x)$ and shape invariance to construct the spectrum algebraically [1911.11182].

For Model I,
\[
M(x)=\frac{\mu+\lambda x}{c},\qquad V(x)=ic\eta x,\qquad \mu,\lambda,\eta\in\mathbb R,\ \lambda>0,
\]
with
\[
A=\frac{\sqrt{\lambda^2+\eta^2}}{c},
\]
the auxiliary spectrum is
\[
\varepsilon_n(E)=\mu c^2+(2n+1)\hbar A-\frac{E^2}{A},
\]
and the Klein-Gordon energies are
\[
E_n=\pm\frac{A}{\lambda}\bigl[\mu c^2+(2n+1)\hbar A\bigr],\qquad n=0,1,2,\dots.
\]
All $E_n$ are real for any $\mu,\lambda>0,\eta$, and the wavefunctions are Hermite-polynomial states
\[
\psi_n(x)=N_n\,e^{i\pi n/2}\,e^{-z^2}H_n(z).
\]
The nonrelativistic limit recovers the harmonic oscillator spectrum [1911.11182].

For Model II,
\[
M(x)=\Bigl[\mu+\frac{\lambda}{\alpha c}\tanh(\alpha x)\Bigr]^{1/2},
\qquad
V(x)=i\,\frac{\eta}{\alpha c}\tanh(\alpha x),
\]
the effective potential is of Rosen-Morse II type, the wavefunctions reduce to Jacobi-polynomial form, and the energies are
\[
E_n
=
\pm\sqrt{
\frac{
\mu c^2+c^2B(B+\hbar\alpha)-c^2(B-n\hbar\alpha)^2
}{
1-\eta^2
}},
\]
subject to
\[
\lambda>\hbar\alpha\,\eta,
\qquad
0\le n\le n_{\max}=\Bigl\lfloor\frac{B-\eta}{\hbar\alpha}\Bigr\rfloor.
\]
This condition is necessary and sufficient to keep the entire spectrum real. Throughout these models, the PT-symmetric choice $V(x)=i\,(\text{real function})$ shifts the wavefunctions off the real axis but does not spoil normalizability under the PT inner-product [1911.11182].

Taken together, these constructions show that PT-symmetric Klein-Gordon equations are not a single model class but a family of analytically tractable non-Hermitian systems. Their spectral outcome is controlled not by PT symmetry alone, but by concrete quantities such as $\langle\gamma\psi_j,\psi_j\rangle$, $K_1^*$, $Q'(\omega)$, and explicit parameter inequalities in exactly solvable quantum models.

Source: https://www.emergentmind.com/topics/pt-symmetric-klein-gordon-equations