Open research questions in Quantum Mechanics and Non-Hermitian Physics
130 unresolved questions extracted from the limitations and future-work sections of 309 Quantum Mechanics and Non-Hermitian Physics papers in our library. Each links back to the study that raised it.
What the literature leaves open
Further development of the algorithm for more complex systems, - Investigation of the connection between the gauge structure and band topology, - Application of the algorithm to topological electronic and photonic devices
The conventional construction of MLWFs involves iteratively minimizing a spread functional, which can be challenging. There is a need for a non-variational constructive algorithm that can unify gauge smoothing and the eigenvalue problem of the projected position operator. The existing approaches may not always lead to the optimal solution.
The problem is ill-posed, with small changes in data resulting in significant changes in the solution. Limited spectral data must constrain an entire potential. The true transform structure may mimic the obstructive poles that the Padé filters are intended to remove.
the conclusions are limited to the reported benchmark settings, - the method may fail when the true transform structure mimics the obstructive poles that the Padé filters are intended to remove, - the relative L 2 error may increase for certain potentials
The traditional temporal coupled-mode theory is limited to resonators with zero or small material loss or gain. Non-Hermitian optical theories and experiments require a more widely applicable theory.
Temporal coupled-mode theory and topological charges for resonances in parity symmetric layered media with arbitrary loss and gain · 2026 · DOIThe proposed asymmetric plane-wave ansatz and degeneracy-resolution technique are not limited to the present model and can be generalized to other bosonic pairing systems, including those with inhomogeneous pairing or hopping terms.
Analytical diagonalization of the open-boundary bosonic Kitaev chain: An asymmetric plane-wave ansatz approach · 2026 · DOIHowever, whether skin-localized modes in non-PT-symmetric photonic systems can be remotely excited by a single real-frequency source remains unclear.
Edge states in non-Hermitian photonic Chern insulators: remote excitation with a real-frequency point source · 2026 · DOINon-Hermitian systems have recently shown new possibilities to manipulate wave scattering by exploiting loss, yet coherent perfect absorption at an exceptional point (CPA EP) remains elusive in acoustics.
Real-to-complex spectral transitions and the associated spontaneous symmetry breaking of eigenstates are central to non-Hermitian physics, yet a comprehensive and universal theory that precisely describes the underlying physical mechanisms for each individual state remains elusive.
However, precise control of thermal phase remains elusive hitherto and can hardly be realized, due to the phase oscillations.
Despite its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for this generic phenomenon even in one spatial dimension.
How to engineer the properties of the edge state such as photonic density of state by using non-Hermiticity while ensuring topological protection has not been fully studied.
Probing P T -Symmetry Breaking of Non-Hermitian Topological Photonic States via Strong Photon-Magnon Coupling · 2024 · DOIWhile the sensitivity of the probe is ultimately limited by the measurement noise, we find the non-Hermitian sensor showing superior performance under background noises that cannot be suppressed through repetitive measurements.
Additionally, current methods for constructing higher-order EPs with robustness primarily rely on passive cavities, with almost no reports on constructing robust EPs using PT-symmetric systems that encompass both gain and loss.
Future research should investigate the behavior of bound states in more complex non-Hermitian systems. The study of non-Hermitian impurity problems can be extended to other areas, such as quantum many-body systems.
The non-Hermitian impurity problem is a key area of research in non-Hermitian physics, with a lack of understanding of bound-state formation in different lattice dimensions.
The lack of understanding of how supersymmetry affects information-theoretic and complexity measures in quantum mechanics. The need for a comprehensive analysis of the Dunkl-supersymmetric harmonic oscillator using these measures. The gap in comparing the properties of supersymmetric systems with those of standard quantum harmonic oscillators.
The existing work on quantum state discrimination in non-Hermitian systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra. There is a need to explore the underlying physics of quantum state discrimination in non-Hermitian systems with complex spectra.
While the paper demonstrates unambiguous discrimination with P-pseudo-Hermitian and PT-symmetric Hamiltonians, the feasibility and practical implementation of unambiguous discrimination with generic non-Hermitian Hamiltonians requires further experimental validation.
The paper suggests further research into the quantization of confined systems. The study highlights the need for a more general solution to the problem.
The paper identifies a gap in the understanding of the quantization of confined systems. The study highlights the need for a self-adjoint Hamiltonian and the challenges in achieving this.
The formulation of D_ij(q) in equations (12)-(16) is restricted to the long-wavelength limit and specific Brillouin zone sampling. Extension to higher-order Brillouin zones and application to systems exhibiting phonon softening or structural instabilities has not been investigated for two-body central interactions.
The treatment of the coupling parameters α₁ and α₂ in equations (4)-(6) is presented for a single set of values derived from atomic coordinates. The sensitivity analysis of the dynamical matrix eigenvalues and eigenvectors to variations in α₁ and α₂, particularly for systems with competing interactions, remains unexplored.
Existing methods for constructing Floquet-Bloch states rely on canonically normalized solutions. There is a need for a constructive, basis-independent framework for constructing Floquet-Bloch states from arbitrary fundamental systems of the Hill equation.
Explicit construction of Floquet–Bloch states from arbitrary solution bases of the Hill equation · 2026 · DOIFuture research could focus on the development of new optical devices or technologies based on the time-dependent Dirac oscillator. The work could be extended to include a more comprehensive analysis of the limitations of the model. Further research could explore the applications of the Jaynes-Cummings and anti-Jaynes-Cummings models.
Most-cited papers in Quantum Mechanics and Non-Hermitian Physics
- Observation of intrinsic chiral bound states in the continuum · Nature · 2023 · 619 citations
- Strong coupling, energy splitting, and level crossings: A classical perspective · American Journal of Physics · 2010 · 417 citations
- Supersymmetry, shape invariance, and exactly solvable potentials · American Journal of Physics · 1988 · 371 citations
- Must a Hamiltonian be Hermitian? · American Journal of Physics · 2003 · 363 citations
- Wigner functions and Weyl transforms for pedestrians · American Journal of Physics · 2008 · 326 citations
- Observation of PT phase transition in a simple mechanical system · American Journal of Physics · 2013 · 307 citations
- Topological triple phase transition in non-Hermitian Floquet quasicrystals · Nature · 2022 · 205 citations
- Quantum mechanics of the 1∕x2 potential · American Journal of Physics · 2006 · 143 citations
- How Good is the Hartree-Fock Approximation · American Journal of Physics · 1968 · 123 citations
- Reflectionless eigenstates of the sech2 potential · American Journal of Physics · 2007 · 122 citations
Most recent work
- Optics in spiral dislocation spacetime: torsion as a geometric waveguide and frequency-filtering mechanism · The European Physical Journal C · 2026
- Quantum geometrical effects in non-Hermitian systems · Physical Review Research · 2026
- Non-Hermitian optical bistability with polarization quantization in Sm3+:BiPO4 · Applied Physics Letters · 2026
- Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features · Physical Review Research · 2026
- Non-Hermitian interband coupling manipulation: from unidirectional reflectionless to unilateral perfect absorption · Optics Letters · 2026
- Non-Hermitian impurity problem · Communications Physics · 2026
- •The Quantum-Resonance Lattice (QRL) Theory: Topological Derivation of the Davidenko Constant (766.490 nm) and Unified Field Mechanics. · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Hidden time-nonlocal Floquet symmetries · Quantum · 2026
- Universal topology of exceptional points in nonlinear non-Hermitian systems · Nature Communications · 2026
- Exact construction and uniqueness of the coupled-channel Green's function · Physical Review C · 2026
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