Open research questions in Quantum Mechanics and Non-Hermitian Physics
30 unresolved questions extracted from the limitations and future-work sections of 207 Quantum Mechanics and Non-Hermitian Physics papers in our library. Each links back to the study that raised it.
What the literature leaves open
Abstract The origin of the finite dip in the curvature power spectrum of instantaneous Slow-Roll (SR)-Ulta-Slow-Roll (USR)-SR inflation remains controversial at linear order, and its full spectral features still lack a complete asymptotic analytical description.
Evolution of linear perturbations under time-dependent hubble friction I: SR-USR-SR inflation · 2026 · DOIIn this context, an important open problem is to determine whether these geometric structures can be exploited to intrin- sically characterize quantum features such as entanglement, quantum correlations, and 14 nonclassicality.
Our goal was to solve the Hermitian Yang-Mills equations (1.1) on a holomorphic vector bundle V → X. This is a challenging nonlinear elliptic system, but we have shown that – 22 – JHEP06(2026)0930123b0123ca=00123b0123ca=10123b0123ca=20123b0123ca=3−6−5−4−3−2log|κabc| elementary differential-geometric considerations can be rather decisive on this matter. In particular, embedding the geometric constraints directly into the neural ansatz used to model the connection provides an effective means of solving this system. We have demonstrated an alternating optimisation procedure which numerically approximates the HYM connection on stable holomorphic bundles V to within (0.1%) precision. At each stage, we construct O an appropriate equivariant neural ansatz for the required geometric data via the Kodaira embedding of X into some suitable Grassmannian determined by the bundle V. Our approach applies to any holomorphic bundle that admits a HYM connection provided one may explicitly write down this embedding. We have illustrated our approach on a general class of vector bundles obtained via the monad construction. Our work has a clear phenomenological application to the study of string compactifications. We now find ourselves armed with the ability to forge a connection between the data of the compactification and quantitative physical observables for string models involving a non-Abelian gauge group, and hope this motivates further data-driven exploration of the vast string landscape. Beyond direct phenomenology, it would be of interest to apply these numerical techniques to investigate the structure of the matter field Kähler potential and superpotential couplings of heterotic theories [28, 29], in the hope of revealing vanishing patterns that may be studied analytically. In general the normalised couplings (4.4) are a complicated function of the bundle, complex structure and Kähler moduli, and studying the moduli dependence of κabc may uncover interesting structure within the combined moduli spaces. L L O O (N 2 k) ∼ The main drawback of our approach is the polynomial growth of Nk = dim H 0(X; V ⊗ k) in k. The computational time and memory requirements of the forward pass through the neural (k2n). As computing network ansatz for the non-Abelian stage (2.14) scales as the curvature two-form involves the Hessian ¯∂∂ of the endomorphism network (2.3), our algorithm is prohibitively expensive for bundles which require a large degree of twisting such k is globally generated. While reasonably performant for a given choice of moduli that V ⊗ for the bundle V → X, this precludes the application of our procedure at scale to large classes of phenomenologically interesting bundles across a range of bundle and complex structure moduli. Arguing by analogy with an expansion in Fourier modes, it is unlikely that all the k contribute equally to the hypothesis (2.14). A modest reduction, basis sections of V ⊗ linear in k, in the number of sections we must consider to attain similar performance would translate to a quadratic decrease in space complexity. Bearing this in mind, it would be fruitful k by extending the to identify and work directly with an effective subspace of sections of V ⊗ optimisation procedure to the associated Grassmannian. Lastly, we will use the methods illustrated within to derive the physically normalised Yukawa couplings for a collection of phenomenologically interesting string compactification scenarios in forthcoming work.
Hermitian Yang-Mills connections on general vector bundles: geometry and physical Yukawa couplings · 2026 · DOIA Conventions B Kodaira embedding C Global function parameterisation D Experimental details D.1 Ricci-flat representative optimisation D.2 HYM connection approximation D.3 Harmonic representative optimisation E Untwisting F Equivariance G Harmonic representative optimisation G.1 Global versus local H Visualisations…
Hermitian Yang-Mills connections on general vector bundles: geometry and physical Yukawa couplings · 2026 · DOIConclusion In this paper, we explored the transition matrix τ that describes a post-selection process transferring part of the information of a quantum state, either pure or mixed, from one side to another. In contrast to teleportation protocols, the transition matrix is not a trace-preserving map, due to post-selection. Hence, we employ entropy-based measures to quantify the amount of information transferred. We introduced the ABB entropy of the transition matrix to quantify information transfer, which could be interpreted as the relative entropy between the input maximally mixed state and its output state bridged by the transition matrix (Sec. 2.2). The ABB entropy also avoids the issues encountered in the (modified) pseudo entropy and the SVD entropy. The pseudo entropy suffers from ambiguity due to the multi-valued logarithm. Both the pseudo and modified pseudo entropies can diverge or take complex values. Although the SVD entropy circumvents these problems, it lacks a clear probabilistic interpretation based on entanglement distillation of the two quantum states that construct the transition matrix (Sec. 3). Subsequently, we demonstrated that the ABB entropy of the transition matrix does not increase under the addition of measurements and non-unitary operations, as illustrated in Fig. 3, in agreement with the behavior of conventional entanglement entropy for pure quantum states under LOCC. By contrast, the (modified) pseudo entropy and the SVD entropy do not necessarily exhibit this monotonicity. We showed that the ABB entropy of the large-copy transition matrices can be concentrated into that of the sub transition matrix with the highest probability (Sec. 3.2.1), following the notion of entanglement distillation in. That probability corresponds to the highest probability of generalized measurement on a mixed state. We also examined the probabilistic interpretation of the pseudo and SVD entropies of the transition matrix in [16, 67]. We found that a meaningful concentration interpretation for the (modified) pseudo entropy is possible only when the normalized transition matrix has a real and nonnegative spectrum, allowing them to be associated with a sub transition matrix. However, even in this case, the so-called “probability” of the concentration does not correspond to a true measurement probability in the quantum mechanical sense. For the transition matrices with negative or complex eigenvalues, it is even not possible to construct any probability-like distribution over the sub-transition matrices (Sec. 3.2.2). Similarly, for the SVD entropy, although the singular spectrum of the transition matrix is real and nonnegative, the associated distribution over sub transition matrices still does not represent a genuine probability distribution in a quantum measurement process. We computed the ensemble averages of the four entropies for transition matrices constructed from two independent Haar random states in Sec.
33 7.1 Conclusion.................................... 33 7.2 Outlook.....................................
The paper treats spherical, hyperbolic, and deSitterian hypergeometric Hamiltonians as distinct cases but does not systematically investigate whether intermediate or deformed geometries yield new families of exactly solvable operators or whether the classification is exhaustive.
The identities in Appendix C.6 relating hypergeometric function identities to elementary function decompositions (referenced for equations 6.16-6.20, 6.31-6.35, 6.42-6.43) are stated but their full derivation and scope of validity for different parameter regimes of the hypergeometric function are not detailed in the excerpt.
Section 7 outlines separation of variables for pseudo-Laplacians on pseudo-spheres but does not provide explicit computations for the general d-dimensional case beyond illustrative examples; the conditions under which hypergeometric Hamiltonians arise from arbitrary (pseudo-)Riemannian manifolds remain uncharacterized.
While Section 6 demonstrates that Laplacians with Dirichlet and Neumann boundary conditions on intervals, half-lines, and lines are special cases of hypergeometric Hamiltonians, the paper does not explore mixed or Robin boundary conditions or their corresponding Green function representations in terms of hypergeometric functions.
The paper establishes transmutation identities relating hypergeometric Hamiltonians in different coordinate systems (equation 5.82), but does not investigate whether these identities generalize to parameter ranges beyond those explicitly treated or to higher-dimensional analogues of the spherical, hyperbolic, and deSitterian cases.
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.
The paper derives closed-form expressions for the dynamical matrix coefficients φ_αβ assuming specific boundary conditions and lattice symmetries. Validation against experimental phonon dispersion data or comparison with alternative computational methods (e.g., ab initio calculations or molecular dynamics simulations) for two-body systems has not been demonstrated.
The derivation of the dynamical matrix elements D_ij for two-body central interactions is presented only for the specific coordinate system and atom arrangements referenced in the tables. Extension to non-central interactions, including tensor forces or velocity-dependent potentials, has not been addressed in the formulation of equations (17) and (18).
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 impact of exceptional points on evolution time and minimum angular separation near the EP exhibits different behaviors in different parameter regimes (monotonic decrease vs. non-monotonic behavior), but the underlying mechanisms and conditions determining these behaviors are not fully characterized.
The analysis is restricted to domains constituted by functions that are differentiable in the interval [a, b], limiting the scope of applicability to more general function spaces.
The definition of a self-adjoint operator that corresponds to the canonical radial momentum of a particle in three-dimensional motion is a quantization problem rooted in an elementary classical system but has no solution.
The numerical simulations show distinct trends in minimum distinguishable angular separation near exceptional points for different non-Hermitian systems, but a comprehensive analytical framework explaining these distinct behaviors across the parameter space is lacking.
The paper focuses on unambiguous discrimination of non-orthogonal quantum states in non-Hermitian systems, but the extension of these results to mixed states and more complex quantum information processing tasks remains unexplored.
Experimental realization of the theoretical predictions regarding finite-sized non-Hermitian systems with diverse phenomena such as cross-shaped localized modes is needed.
The findings on finite-sized systems with non-Hermitian phenomena including scale-free localized states and exceptional points may guide future experimental investigations across photonics and related platforms.
The unexpected behavior regarding bound-state formation in non-Hermitian disordered systems may prompt further investigation, especially toward understanding how single impurity insights complement scaling-based approaches for larger systems.
Most-cited papers in Quantum Mechanics and Non-Hermitian Physics
- Amoeba Formulation of Non-Bloch Band Theory in Arbitrary Dimensions · Physical Review X · 2024 · 89 citations
- Observation of dynamic non-Hermitian skin effects · Nature Communications · 2024 · 70 citations
- Non-Hermitian dynamics and non-reciprocity of optically coupled nanoparticles · Nature Physics · 2024 · 69 citations
- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric quantum mechanics · Reviews of Modern Physics · 2024 · 64 citations
- Energy-momentum Tensor of a Hydrogen Atom: Stability, \(D\)-term, and the Lamb Shift · Acta Physica Polonica B Proceedings Supplement · 2023 · 12 citations
- The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory · East European Journal of Physics · 2023 · 6 citations
- Thermodynamic Properties of the Noncommutative Dirac Oscillator with a Permanent Electric Dipole Moment · Acta Physica Polonica B · 2024 · 5 citations
- Ground states solutions for some non-autonomousSchrödinger-Bopp-Podolsky system · Electronic journal of qualitative theory of differential equations · 2022 · 3 citations
- Involving Nikiforov-Uvarov Method in Schrodinger Equation Obtaining Hartmann Potential · East European Journal of Physics · 2023 · 3 citations
- Non-Hermitian impurity problem · Communications Physics · 2026 · 2 citations
Most recent work
- 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
- Recent advances in the combination of nonlinearity and exceptional points · Applied Physics Reviews · 2026
- Fractal Waves and Caustic Signatures in a Superdeterministic Framework: Benchmarking PINNs and PI-GNNs for the Fractional Klein–Gordon Equation · Fractal and Fractional · 2026
- Exceptional deficiency of non-Hermitian systems · Nature Physics · 2026
- Fisher information and thermodynamic properties subject to harmonic oscillator with linear and inverse square potential · Physics Letters A · 2026
- Statistical Measures and Complexity of Supersymmetric Polynomials in Quantum Mechanics · Mathematics · 2026
- Non-Hermitian quantum state discrimination and information flow · Scientific Reports · 2026
Find a gap in your own Quantum Mechanics and Non-Hermitian Physics sub-topic
This page shows what the Quantum Mechanics and Non-Hermitian Physics literature already flags as unresolved. To narrow it to your specific question, run the guided finder — it searches the gap library on demand and checks candidates against 250M+ OpenAlex works.
Open the Research Gap Finder →