PT-Symmetric Klein-Gordon Equations

• PT-symmetric Klein-Gordon equations are models where parity–time symmetry enforces spectral constraints through local gain-loss terms, imaginary velocity components, and nonlocal couplings.
• They are analyzed using perturbation methods, Birman–Schwinger reductions, and integrable system techniques to derive explicit eigenvalue shifts and stability conditions.
• Exact quantum formulations and stability criteria such as Vakhitov–Kolokolov-type conditions highlight how PT symmetry controls metastability and resonance behavior.

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 (Demirkaya et al., 2014, Borisov et al., 2015, Demirkaya et al., 2014, Jia et al., 2022, Semorádová, 2018, Giachetti et al., 2010, Zaghou et al., 2019). 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 $\varepsilon\,\gamma(x,y)\,u_t$0 and $\varepsilon\,\gamma(x,y)\,u_t$1, so the equation is invariant (Demirkaya et al., 2014). A higher-dimensional analogue replaces $\varepsilon\,\gamma(x,y)\,u_t$2 by an exponentially localized $\varepsilon\,\gamma(x,y)\,u_t$3 that is odd in one or two coordinates, again producing a PT-symmetric gain-loss defect without modifying the static kink profile (Borisov et al., 2015).

A different local realization uses a complex field $\varepsilon\,\gamma(x,y)\,u_t$4 and the perturbed equation

$\varepsilon\,\gamma(x,y)\,u_t$5

with $\varepsilon\,\gamma(x,y)\,u_t$6 real, even, and of moderate decay. Here the term $\varepsilon\,\gamma(x,y)\,u_t$7 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 (Demirkaya et al., 2014).

PT symmetry also appears in genuinely nonlocal Klein-Gordon systems. Imposing the reduction

$\varepsilon\,\gamma(x,y)\,u_t$8

on an integrable $\varepsilon\,\gamma(x,y)\,u_t$9 Klein-Gordon system yields an equation in which $iW(x)u_t$0 is coupled directly to $iW(x)u_t$1 through the mixed derivative $iW(x)u_t$2 (Jia et al., 2022). In operator-theoretic and quantum-mechanical formulations, PT symmetry is encoded by relations such as

$iW(x)u_t$3

or by the construction of closed PT-symmetric extensions of formal Klein-Gordon operators (Semorádová, 2018, Giachetti et al., 2010).

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

$iW(x)u_t$4

because the dashpot term vanishes for time-independent states. Two explicit examples are the sine-Gordon kink

$iW(x)u_t$5

and the $iW(x)u_t$6 kink

$iW(x)u_t$7

with analogous anti-kinks (Demirkaya et al., 2014).

Linearization about a kink, $iW(x)u_t$8, gives

$iW(x)u_t$9

With the modal ansatz $u(-x,-t)$0 and

$u(-x,-t)$1

one obtains the quadratic pencil

$u(-x,-t)$2

Since $u(-x,-t)$3 exponentially, the essential spectrum of $u(-x,-t)$4 is $u(-x,-t)$5, and hence

$u(-x,-t)$6

If $u(-x,-t)$7 has simple eigenvalues

$u(-x,-t)$8

then for $u(-x,-t)$9 the point spectrum consists of pairs $u_{tt}-u_{xx}+V'(u)=0,$0; the translational mode at $u_{tt}-u_{xx}+V'(u)=0,$1 has algebraic multiplicity two for the quadratic pencil (Demirkaya et al., 2014).

For small $u_{tt}-u_{xx}+V'(u)=0,$2, the discrete spectral points shift according to explicit first-order perturbation formulas. For the translational mode,

$u_{tt}-u_{xx}+V'(u)=0,$3

while for internal modes,

$u_{tt}-u_{xx}+V'(u)=0,$4

The sign of $u_{tt}-u_{xx}+V'(u)=0,$5 therefore controls whether the perturbed eigenvalue acquires positive or negative real part (Demirkaya et al., 2014).

The location of the kink center relative to the gain-loss interface is decisive. If $u_{tt}-u_{xx}+V'(u)=0,$6, then $u_{tt}-u_{xx}+V'(u)=0,$7 is even, the linearization inherits the full Hamiltonian symmetry $u_{tt}-u_{xx}+V'(u)=0,$8, and $u_{tt}-u_{xx}+V'(u)=0,$9 for all $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$0; all discrete eigenvalues remain purely imaginary to all orders $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$1, so the kink is spectrally neutral. If $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$2, the kink lies on the lossy side and the former zero mode shifts into the stable half-plane, $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$3. If $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$4, the kink lies on the gain side and $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$5, producing spectral instability. The same sign-sensitive mechanism applies to higher internal modes (Demirkaya et al., 2014).

The sine-Gordon and $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$6 examples with

$u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$7

make this dependence explicit. For the sine-Gordon kink, $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$8, giving $u_{tt}-u_{xx}+\epsilon\,\gamma(x)\,u_t+f(u)=0,$9 on the gain side and $\epsilon\ge 0$0 on the lossy side. For the $\epsilon\ge 0$1 kink, $\epsilon\ge 0$2. In reflectionless cases such as sine-Gordon and $\epsilon\ge 0$3, the continuous-spectrum edge may also generate an $\epsilon\ge 0$4 pair with real part $\epsilon\ge 0$5 (Demirkaya et al., 2014).

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

$\epsilon\ge 0$6

with a static kink depending only on $\epsilon\ge 0$7 and solving

$\epsilon\ge 0$8

The examples listed are the sine-Gordon kink,

$\epsilon\ge 0$9

and the $\gamma(x)$0 kink,

$\gamma(x)$1

The associated one-dimensional linearization operator is

$\gamma(x)$2

with essential spectrum $\gamma(x)$3 and finitely many simple eigenvalues below $\gamma(x)$4 (Borisov et al., 2015).

The PT-symmetric perturbation is introduced as a spatially localized viscous-friction term,

$\gamma(x)$5

where $\gamma(x)$6 is small and $\gamma(x)$7 is continuous, exponentially localized, and odd in one or two coordinates. Since the perturbation is proportional to $\gamma(x)$8, it does not affect the static kink itself (Borisov et al., 2015).

Setting

$\gamma(x)$9

and using $\gamma(-x)=-\gamma(x)$0 produces the non-self-adjoint eigenvalue problem

$\gamma(-x)=-\gamma(x)$1

For a simple discrete eigenvalue $\gamma(-x)=-\gamma(x)$2 of $\gamma(-x)=-\gamma(x)$3 with normalized eigenfunction $\gamma(-x)=-\gamma(x)$4, one defines

$\gamma(-x)=-\gamma(x)$5

where $\gamma(-x)=-\gamma(x)$6 solves

$\gamma(-x)=-\gamma(x)$7

The main theorem states that for each simple eigenvalue $\gamma(-x)=-\gamma(x)$8 there are exactly two small eigenvalues with

$\gamma(-x)=-\gamma(x)$9

as $\mathcal{PT}:(x,t)\to(-x,-t)$0, and if $\mathcal{PT}:(x,t)\to(-x,-t)$1 then

$\mathcal{PT}:(x,t)\to(-x,-t)$2

If $\mathcal{PT}:(x,t)\to(-x,-t)$3 but $\mathcal{PT}:(x,t)\to(-x,-t)$4, the splitting is purely along the imaginary axis,

$\mathcal{PT}:(x,t)\to(-x,-t)$5

For $\mathcal{PT}:(x,t)\to(-x,-t)$6, the emerging spectral points may be resonances rather than eigenvalues (Borisov et al., 2015).

The proof uses a non-self-adjoint Birman-Schwinger reduction, the substitution $\mathcal{PT}:(x,t)\to(-x,-t)$7, a scalar equation

$\mathcal{PT}:(x,t)\to(-x,-t)$8

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 $\mathcal{PT}:(x,t)\to(-x,-t)$9 (Borisov et al., 2015).

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

A different PT-symmetric Klein-Gordon problem considers complex standing waves of

$\varepsilon\,\gamma(x,y)\,u_t$00

with $\varepsilon\,\gamma(x,y)\,u_t$01 real and even. The standing-wave ansatz

$\varepsilon\,\gamma(x,y)\,u_t$02

reduces the equation to

$\varepsilon\,\gamma(x,y)\,u_t$03

and in one dimension to

$\varepsilon\,\gamma(x,y)\,u_t$04

Under standard hypotheses on $\varepsilon\,\gamma(x,y)\,u_t$05 and $\varepsilon\,\gamma(x,y)\,u_t$06, this admits localized pulse solutions $\varepsilon\,\gamma(x,y)\,u_t$07 (Demirkaya et al., 2014).

Linearization with

$\varepsilon\,\gamma(x,y)\,u_t$08

leads to

$\varepsilon\,\gamma(x,y)\,u_t$09

Writing $\varepsilon\,\gamma(x,y)\,u_t$10 and $\varepsilon\,\gamma(x,y)\,u_t$11, the normal-mode ansatz gives the operator pencil

$\varepsilon\,\gamma(x,y)\,u_t$12

with

$\varepsilon\,\gamma(x,y)\,u_t$13

where

$\varepsilon\,\gamma(x,y)\,u_t$14

Spectral stability means absence of any $\varepsilon\,\gamma(x,y)\,u_t$15 with $\varepsilon\,\gamma(x,y)\,u_t$16 (Demirkaya et al., 2014).

The paper derives a sharp frequency criterion: $\varepsilon\,\gamma(x,y)\,u_t$17 Equivalently, defining

$\varepsilon\,\gamma(x,y)\,u_t$18

one has

$\varepsilon\,\gamma(x,y)\,u_t$19

This is an explicit PT-symmetric analogue of the classical Vakhitov-Kolokolov condition (Demirkaya et al., 2014).

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

$\varepsilon\,\gamma(x,y)\,u_t$20

the quantity

$\varepsilon\,\gamma(x,y)\,u_t$21

changes slope at $\varepsilon\,\gamma(x,y)\,u_t$22. For $\varepsilon\,\gamma(x,y)\,u_t$23 a real pair with $\varepsilon\,\gamma(x,y)\,u_t$24 appears, whereas for $\varepsilon\,\gamma(x,y)\,u_t$25 all eigenvalues lie on the imaginary axis. The crossing of $\varepsilon\,\gamma(x,y)\,u_t$26 coincides with the bifurcation of the real eigenvalue pair (Demirkaya et al., 2014).

5. Integrable Klein-Gordon equations with PT nonlocality

The integrable nonlocal construction starts from a generic system

$\varepsilon\,\gamma(x,y)\,u_t$27

and imposes the PT reduction

$\varepsilon\,\gamma(x,y)\,u_t$28

Writing $\varepsilon\,\gamma(x,y)\,u_t$29 gives the PT-symmetric Klein-Gordon equation

$\varepsilon\,\gamma(x,y)\,u_t$30

Unlike a local Klein-Gordon equation $\varepsilon\,\gamma(x,y)\,u_t$31, this equation couples the field at $\varepsilon\,\gamma(x,y)\,u_t$32 directly to its mirror point $\varepsilon\,\gamma(x,y)\,u_t$33 (Jia et al., 2022).

Its integrability is expressed through the compatibility of the $\varepsilon\,\gamma(x,y)\,u_t$34 linear system

$\varepsilon\,\gamma(x,y)\,u_t$35

with spectral parameter $\varepsilon\,\gamma(x,y)\,u_t$36, where

$\varepsilon\,\gamma(x,y)\,u_t$37

and

$\varepsilon\,\gamma(x,y)\,u_t$38

The zero-curvature condition

$\varepsilon\,\gamma(x,y)\,u_t$39

reproduces the PT-nonlocal equation exactly (Jia et al., 2022).

Because $\varepsilon\,\gamma(x,y)\,u_t$40, any solution decomposes as

$\varepsilon\,\gamma(x,y)\,u_t$41

with

$\varepsilon\,\gamma(x,y)\,u_t$42

A one-soliton solution is given in terms of

$\varepsilon\,\gamma(x,y)\,u_t$43

$\varepsilon\,\gamma(x,y)\,u_t$44

so that

$\varepsilon\,\gamma(x,y)\,u_t$45

The center moves along $\varepsilon\,\gamma(x,y)\,u_t$46, hence $\varepsilon\,\gamma(x,y)\,u_t$47 when $\varepsilon\,\gamma(x,y)\,u_t$48; larger $\varepsilon\,\gamma(x,y)\,u_t$49 gives a narrower, faster kink. The PT-antisymmetric component $\varepsilon\,\gamma(x,y)\,u_t$50 introduces an antisymmetric phase twist and deforms the usual kink profile (Jia et al., 2022).

The model admits a Lax pair of Zakharov-Shabat type and an infinite hierarchy of conserved densities $\varepsilon\,\gamma(x,y)\,u_t$51. In the local limit $\varepsilon\,\gamma(x,y)\,u_t$52, 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 $\varepsilon\,\gamma(x,y)\,u_t$53 that generates further integrable nonlocal variants (Jia et al., 2022).

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

In the crypto-Hermitian approach, the free Klein-Gordon equation

$\varepsilon\,\gamma(x,y)\,u_t$54

is rewritten using the two-component Feshbach-Villars wavefunction

$\varepsilon\,\gamma(x,y)\,u_t$55

so that

$\varepsilon\,\gamma(x,y)\,u_t$56

With a scalar potential, one replaces $\varepsilon\,\gamma(x,y)\,u_t$57 by $\varepsilon\,\gamma(x,y)\,u_t$58. The Hamiltonian is not Hermitian in the naive inner product, but Hermiticity is restored by a positive-definite metric $\varepsilon\,\gamma(x,y)\,u_t$59 satisfying

$\varepsilon\,\gamma(x,y)\,u_t$60

A spectral construction gives

$\varepsilon\,\gamma(x,y)\,u_t$61

with positivity conditions

$\varepsilon\,\gamma(x,y)\,u_t$62

The spectrum is

$\varepsilon\,\gamma(x,y)\,u_t$63

and since $\varepsilon\,\gamma(x,y)\,u_t$64 is Hermitian and strictly positive, all $\varepsilon\,\gamma(x,y)\,u_t$65 are real. The corresponding density

$\varepsilon\,\gamma(x,y)\,u_t$66

supports a conserved current $\varepsilon\,\gamma(x,y)\,u_t$67. The PT-symmetric specialization is characterized by

$\varepsilon\,\gamma(x,y)\,u_t$68

with real spectra in the unbroken regime and complex-conjugate pairs in the broken regime (Semorádová, 2018).

A more explicitly PT-symmetric spectral problem arises for the one-dimensional Klein-Gordon oscillator. The formal operator is

$\varepsilon\,\gamma(x,y)\,u_t$69

Its closed PT-symmetric realization $\varepsilon\,\gamma(x,y)\,u_t$70 on $\varepsilon\,\gamma(x,y)\,u_t$71 is defined by subdominant behavior at $\varepsilon\,\gamma(x,y)\,u_t$72, and its adjoint satisfies

$\varepsilon\,\gamma(x,y)\,u_t$73

By results on PT-symmetric anharmonic oscillators, $\varepsilon\,\gamma(x,y)\,u_t$74 has purely discrete spectrum

$\varepsilon\,\gamma(x,y)\,u_t$75

Physical energies are determined from

$\varepsilon\,\gamma(x,y)\,u_t$76

In the nonrelativistic limit $\varepsilon\,\gamma(x,y)\,u_t$77, one recovers

$\varepsilon\,\gamma(x,y)\,u_t$78

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 $\varepsilon\,\gamma(x,y)\,u_t$79 is singled out as physically preferred. Numerically, the PT-symmetric eigenvalue $\varepsilon\,\gamma(x,y)\,u_t$80 and the complex-dilated resonance $\varepsilon\,\gamma(x,y)\,u_t$81 satisfy

$\varepsilon\,\gamma(x,y)\,u_t$82

and for $\varepsilon\,\gamma(x,y)\,u_t$83 one example is

$\varepsilon\,\gamma(x,y)\,u_t$84

The PT-symmetric levels therefore reproduce resonance positions up to the order of the width and are interpreted as metastable states (Giachetti et al., 2010).

Exact PT-symmetric bound states also arise in the $\varepsilon\,\gamma(x,y)\,u_t$85-dimensional time-independent Klein-Gordon equation with position-dependent mass $\varepsilon\,\gamma(x,y)\,u_t$86, scalar potential $\varepsilon\,\gamma(x,y)\,u_t$87, and PT-symmetric vector potential $\varepsilon\,\gamma(x,y)\,u_t$88,

$\varepsilon\,\gamma(x,y)\,u_t$89

This is rewritten as a Schrödinger-type equation with an $\varepsilon\,\gamma(x,y)\,u_t$90-dependent potential $\varepsilon\,\gamma(x,y)\,u_t$91 and auxiliary eigenvalue $\varepsilon\,\gamma(x,y)\,u_t$92; the true Klein-Gordon energies are obtained from

$\varepsilon\,\gamma(x,y)\,u_t$93

The SUSYQM factorization uses a superpotential $\varepsilon\,\gamma(x,y)\,u_t$94 and shape invariance to construct the spectrum algebraically (Zaghou et al., 2019).

For Model I,

$\varepsilon\,\gamma(x,y)\,u_t$95

with

$\varepsilon\,\gamma(x,y)\,u_t$96

the auxiliary spectrum is

$\varepsilon\,\gamma(x,y)\,u_t$97

and the Klein-Gordon energies are

$\varepsilon\,\gamma(x,y)\,u_t$98

All $\varepsilon\,\gamma(x,y)\,u_t$99 are real for any $iW(x)u_t$00, and the wavefunctions are Hermite-polynomial states

$iW(x)u_t$01

The nonrelativistic limit recovers the harmonic oscillator spectrum (Zaghou et al., 2019).

For Model II,

$iW(x)u_t$02

the effective potential is of Rosen-Morse II type, the wavefunctions reduce to Jacobi-polynomial form, and the energies are

$iW(x)u_t$03

subject to

$iW(x)u_t$04

This condition is necessary and sufficient to keep the entire spectrum real. Throughout these models, the PT-symmetric choice $iW(x)u_t$05 shifts the wavefunctions off the real axis but does not spoil normalizability under the PT inner-product (Zaghou et al., 2019).

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 $iW(x)u_t$06, $iW(x)u_t$07, $iW(x)u_t$08, and explicit parameter inequalities in exactly solvable quantum models.