Non-Hermitian Complex Potentials

• Non-Hermitian complex potentials are defined as complex-valued functions with a nonzero imaginary part that disrupts the Hermitian property of Hamiltonians.
• They induce unique spectral and localization phenomena, including mobility edges in the complex energy plane and directional-dependent scattering behavior.
• Practical applications span open quantum systems and synthetic photonics, enabling controlled gain/loss mechanisms and phenomena like coherent perfect absorption.

A non-Hermitian complex potential, in the context of quantum and wave physics, refers to a spatially varying function $V(x)$ or $V(\vec{r})$ that enters the Hamiltonian and possesses a nonzero imaginary component, thus rendering the Hamiltonian $H = T + V$ non-Hermitian: $H \neq H^\dagger$. Such potentials fundamentally modify spectral, localization, and scattering properties, and have served as paradigms for exploring physics beyond conventional Hermitian quantum mechanics, especially in the presence of gain, loss, or effective open-system couplings.

1. Defining Non-Hermitian Complex Potentials

A complex potential $V(x) = V_R(x) + i V_I(x)$, with $V_I(x) \not\equiv 0$, breaks Hermiticity. In Schrödinger-type systems, this leads to non-unitary time evolution, non-conservative norm, and potentially non-real spectra. Hamiltonians with such potentials may (in special cases) exhibit entirely real spectra if they obey certain symmetry conditions—most notably $\mathcal{PT}$ symmetry ($V^*(x) = V(-x)$), or more generally, pseudo-Hermiticity under an appropriate metric.

The physical interpretation of $V_I(x)$ is system-dependent: positive $V_I$ acts as a source (gain), negative as a sink (loss) of probability or energy. In open quantum systems, non-Hermitian complex potentials can effectively describe couplings to reservoirs or environments through Lindblad or master-equation reductions, or as practical absorbing boundaries in wave propagation and quantum transport simulations (Elenewski et al., 2015).

2. Spectral and Localization Phenomena in Complex Potentials

Non-Hermitian complex potentials fundamentally alter spectral and localization properties compared to Hermitian systems:

• Localized and Extended States: Models such as the non-Hermitian Maryland model and non-Hermitian Aubry-André-type chains admit transitions between exponentially localized and extended eigenstates, with the transition points, or "mobility edges," generically occurring in the complex energy plane rather than the real axis (Longhi, 2021, Pang et al., 31 Mar 2025, Wang et al., 2019).
• Mobility "lines" and "rings": In non-Hermitian quasiperiodic models, mobility edges may form either real-line segments or closed "mobility rings" in the complex energy plane, distinguishing extended phases (zero Lyapunov exponent) from localized phases (positive Lyapunov exponent) (Pang et al., 31 Mar 2025).
• Absence of Critical States: Critical (multifractal) states, prevalent in Hermitian quasiperiodic potentials at intermediate couplings, are typically absent in the non-Hermitian case due to the generically complex potential denominator. The phase space is sharply partitioned into only localized and extended eigenstates (Pang et al., 31 Mar 2025).

The behavior of the localization length near mobility edges also acquires non-universality: the critical exponent $\nu$ for the divergence of the localization length $\xi \sim |E-E_c|^{-\nu}$ varies with the location of $E_c$ in the complex plane, unlike the constant values found in Hermitian systems.

3. Scattering Theory, Reciprocity, and Spectral Singularities

Non-Hermitian complex potentials generate rich new scattering phenomena:

• Non-reciprocity: For generic non-Hermitian potentials, both reflectivity and transmitivity are direction-dependent: $R(-k)\ne R(k)$ and $T(-k)\ne T(k)$, except for real or $\mathcal{PT}$-symmetric potentials (Ahmed, 2011, Ahmed et al., 2017).
• $\mathcal{PT}$ Symmetry and Pseudo-unitarity: Potentials with $\mathcal{PT}$ symmetry satisfy $V(x) = V^*(-x)$, ensuring $T(-k) = T(k)$ and $R_{\rm left}(-k) = R_{\rm right}(k)$, while still generally breaking left–right reciprocity of reflection. In special parameter regimes, exact (bi-directional) reciprocity and unitarity, $R+T=1$, can be restored even in the non-Hermitian case (Ahmed, 2012).
• Spectral Singularities and Coherent Perfect Absorption: A non-Hermitian complex potential can support spectral singularities—real energies at which both the reflection and transmission amplitudes diverge. These correspond to zero-width resonances and, in certain configurations, to coherent perfect absorption (CPA) where incoming beams are completely absorbed (Ahmed, 2011, Ghatak et al., 2012, Ahmed et al., 2017). The semi-infinite models in (Ahmed et al., 2017) demonstrate that non-reciprocal scattering, spectral singularities, and CPA can be realized even for potentials with non-decaying real parts.

Phenomenon	Hermitian	Generic Complex Potential	$\mathcal{PT}$-Symmetric Potential
Reciprocity ($R(-k)=R(k)$)	Yes	No	Partial (in reflection, left/right swap)
Spectral singularities	No	Yes	Yes (fine-tuned, may coincide left/right)
Bidirectional invisibility	No	No	Yes (special parameters)

4. Real Spectra and Pseudo-Hermiticity

While non-Hermiticity generically leads to complex spectra, entirely real spectra are possible under certain structural conditions:

• $\mathcal{PT}$-Symmetry and Pseudo-Hermiticity: $\mathcal{PT}$ symmetry ensures a purely real spectrum provided the symmetry is unbroken. More generally, pseudo-Hermiticity with respect to a positive-definite metric $\eta$ ensures spectral reality. For example, double-delta complex interactions with appropriately balanced imaginary parts possess a bounded, positive-definite metric, and all observables are real and physically meaningful in the $\eta$-inner product (Mehri-Dehnavi et al., 2010).
• Isospectral Hermitian Partners: Many non-Hermitian complex Hamiltonians ($p^2 - g x^4 + a/x^2$) possess a real, isospectral Hermitian partner ($p^2 + 4g x^4 + b x$) when the parameters are related appropriately. The correspondence can hold even when the "Hermitian" partner is itself non-Hermitian, demonstrating the broader reach of pseudo-Hermiticity beyond $\mathcal{PT}$ symmetry (Nanayakkara et al., 2014).

5. Physical Realizations and Applications

Non-Hermitian complex potentials serve as fundamental elements in analytic, experimental, and computational studies across various domains:

• Open quantum systems: Imaginary (gain/loss) potentials encode Lindblad-type couplings to reservoirs, making them central to the simulation of quantum transport and nonequilibrium steady states in mesoscopic and molecular devices. Mean-field reduction of many-body master equations leads directly to real-time TDDFT with complex source/sink boundaries, facilitating the computation of steady-state currents in nanoscale systems (Elenewski et al., 2015).
• Synthetic photonic and metamaterial systems: Implementation of spatially tailored gain and loss profiles, such as $V \tan(\dots + i\epsilon)$, in photonic lattices permits direct observation of topological mobility edges, non-Hermitian delocalization transitions, Bloch oscillations with complex trajectories, and unidirectional transport phenomena (Longhi, 2021, Longhi, 2015).
• Scattering and wave control: Engineering of $\mathcal{PT}$-symmetric and complex potentials enables design of devices with direction-dependent reflection, coherent perfect absorption, and perfect transparency (invisibility) in tailored parameter regimes (Ahmed, 2012, Ahmed et al., 2017).

6. Mathematical Structures and Integrability

Complex non-Hermitian potentials expose novel exactly solvable structures and topological invariants:

• Exactly Solvable Models: The non-Hermitian Maryland and generalized Aubry-André models remain integrable via mapping to Floquet-type or transfer-matrix problems, granting explicit closed-form solutions for eigenstates, spectra, and Lyapunov exponents—even for quasi-periodic, aperiodic, and flat-band systems (Longhi, 2021, Pang et al., 31 Mar 2025).
• Topological invariants: The localization–delocalization transition and mobility-edge phases are characterized by non-Hermitian point-gap winding numbers in the complex energy plane. The distinction of extended and localized spectra is encoded in winding transitions as generalized “bulk invariants” (Longhi, 2021).
• Supersymmetry and Shape Invariance: The $\eta$-pseudo-Hermitian formalism, combined with supersymmetric factorization and shape-invariance, provides a constructive method for analytic solutions and shows that complex partner potentials can be scattering-isospectral with real spectra, even in the absence of conventional Hermiticity (Bakhshi et al., 2020).

7. Outlook and Broader Implications

The paper of non-Hermitian complex potentials underpins significant advances in fundamental theory and applied physics. The richness of physical phenomena—mobility edges, spectral singularities, non-reciprocal and invisible scattering, topological markers, and exact integrability—stems from the interplay of complex symmetry, topology, and analyticity. This has driven cross-disciplinary impacts in wave physics, quantum simulation, photonics, and open-system quantum transport. Ongoing research continues to address critical-state suppression, spectral phase transitions, and robust realization of these effects in synthetic and solid-state platforms (Longhi, 2021, Pang et al., 31 Mar 2025, Elenewski et al., 2015).