Open research questions in Spectral Theory in Mathematical Physics

What the literature leaves open

• The interaction of Floquet states with time-periodic potentials can modify the non-equilibrium properties of target materials. The measurement of TrARPES spectral weight is challenging. The conservation of total spectral weight upon below-gap driving is a complex phenomenon.

  Occupation Dynamics of Floquet–Volkov States and Spectral Sum Rule · 2026 · DOI

• The presence of intersecting boundary hypersurfaces introduces polyhomogeneous terms. The need to resolve the singularities of the heat kernel at t = 0 and the boundary corners.

  Heat Kernel Expansion and Spectral Action on the Stratified $b$-Simplex $\Delta^7$ · 2026 · DOI

• The zero locus of the magnetic field plays a significant role in the spectral analysis - The paper needs to reduce the spectral study to that of effective semiclassical pseudodifferential operators in dimension 1

  Low-energy eigenstates in a vanishing magnetic field · 2026 · DOI

• The unique continuation principle does not hold in the lattice setting. The need for a quantitative substitute for the unique continuation principle. The challenge of obtaining a sharp lower bound in high dimensions.

  On support cardinality for the discrete Schrödinger equation · 2026 · DOI

• It is difficult to say something about how the gap changes under perturbations. The configuration space on which the operator is defined remains fixed.

  On the Asymptotic Behavior of the Spectral Gap for Discrete Schrödinger Operators · 2026 · DOI

• The lack of understanding of nonlocal eigenvalue problems and superposition operators of mixed order. The need for a detailed analysis of disconnected domains.

  Nonlocal eigenvalue problems and superposition operators · 2026 · DOI

• The infrared divergence of the Coulomb potential. The lack of a general method for solving the eigenvalue problem.

  Novel properties of the Birman-Schwinger operator of the one-dimensional Coulomb Hamiltonian · 2026 · DOI

• The complexity of the Riemann zeta function. The lack of a clear understanding of the distribution of prime numbers.

  Proof of the Riemann Hypotesis(New version) · 2026 · DOI

• There is a gap in the existing literature on the asymptotic behaviour of the first two eigenvalues. The paper identifies the need to study the asymptotic behaviour of the first two eigenvalues.

  Asymptotic Analysis of the First Two Eigenvalues for Sturm-Liouville Problems with Two Parameters on the Jump Set with Applications · 2026 · DOI

• Future research should aim to develop a more complete understanding of the properties of NG even-denominator states. The study of these states can lead to the development of new quantum computing architectures. The paper's results can inform the design of new quantum devices.

  Theory of next-generation even-denominator states · 2026 · DOI

• The properties of even-denominator states are not well understood. The paper identifies a gap in the understanding of NG even-denominator states. The study of these states is important for understanding topological phases of matter.

  Theory of next-generation even-denominator states · 2026 · DOI

• The paper identifies a gap in the understanding of Floquet population dynamics. The work addresses the need for critical insights into the occupation dynamics of Floquet–Volkov states.

  Occupation Dynamics of Floquet–Volkov States and Spectral Sum Rule · 2026 · DOI

• The eigenvalue problem for the discrete spectrum of the one-dimensional Coulomb Hamiltonian remains unsolved. There is a need for a new method to solve the eigenvalue problem.

  Novel properties of the Birman-Schwinger operator of the one-dimensional Coulomb Hamiltonian · 2026 · DOI

• To analyze the detailed dependence on the shell parameters in the rotationally symmetric case. To explicitly calculate the scattering matrix for several concentric shells. To study the behavior of Schrödinger operators with concentric δ-shell interactions in other contexts.

  Schrödinger operators with concentric $$\delta $$–shell interactions · 2026 · DOI

• The lack of a fully three-dimensional boundary integral formulation for Schrödinger operators with concentric δ-shell interactions. The need for a detailed description of the negative spectrum of H N. The lack of an analysis of the tunneling splitting of the lowest s-wave eigenvalues for the double δ-shell operator.

  Schrödinger operators with concentric $$\delta $$–shell interactions · 2026 · DOI

• The paper suggests future research on the analysis of the numerical scheme. The paper proposes future research on the application of the method to real-world problems. The paper suggests future research on the extension of the method to other areas of physics and chemistry.

  Sparse Approximation of the Lieb Functional with Moment Constraints · 2026 · DOI

• The paper identifies a gap in the existing literature on the Lieb functional. The paper addresses the need for a new method for electronic structure calculations. The paper fills a gap in the understanding of the convergence of the MCAL functional.

  Sparse Approximation of the Lieb Functional with Moment Constraints · 2026 · DOI

• Future research may involve experimental verification of the mathematical framework. The analysis of the junction's behavior in different scenarios may be a direction for future research.

  Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns · 2026 · DOI

• The paper identifies a gap in the understanding of the Josephson junction's behavior. The authors aim to provide a novel mathematical framework to address this gap.

  Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns · 2026 · DOI

• The lack of understanding of the asymptotic behavior of the heat semigroup on manifolds with corners. The need for a novel method to construct the heat kernel on stratified spaces.

  Heat Kernel Expansion and Spectral Action on the Stratified $b$-Simplex $\Delta^7$ · 2026 · DOI

• The paper does not provide a comprehensive study of all possible quantum many-body models, - The study is limited to the derivation of ground-state energies and wavefunctions, - The paper does not discuss the experimental realization of the models.

  Taxonomy of integrable and ground-state solvable models: Jastrow wave functions on graphs and parent Hamiltonians · 2026 · DOI

• The study of quantum many-body systems is challenging due to the complexity of their interactions, - There is a need for a framework to understand the properties of these systems, - The paper aims to address this gap by deriving ground-state energies and wavefunctions for various models.

  Taxonomy of integrable and ground-state solvable models: Jastrow wave functions on graphs and parent Hamiltonians · 2026 · DOI

• The study of the applications of the bosonization framework to condensed matter physics and high energy physics. The exploration of the implications of the bosonization framework for topological phases and quantum computing. The generalization of the bosonization framework to other systems and dimensions.

  Bosonization and Kramers-Wannier dualities in general dimensions · 2026 · DOI

• The lack of a general and systematic framework for bosonization in higher dimensions. The need for a novel approach to establish new connections between fermionic and spin systems.

  Bosonization and Kramers-Wannier dualities in general dimensions · 2026 · DOI

• To further characterize the measure μ. To apply the paper's methods to other equations in physics. To use the paper's results to improve numerical simulations of the Schrödinger Map equation and the Binormal Curvature Flow.

  On Properties of Statistically Stationary Solutions to the One-Dimensional Schrödinger Map Equation. · 2026 · DOI

Most-cited papers in Spectral Theory in Mathematical Physics

• TRANSMISSION EIGENVALUES FOR MULTIPOINT SCATTERERS · Eurasian Journal of Mathematical and Computer Applications · 2021 · 3 citations
• The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well · Electronic journal of qualitative theory of differential equations · 2022 · 2 citations
• Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term · Electronic journal of qualitative theory of differential equations · 2022 · 2 citations
• Asymptotic derivation of nonclassical SPN equations from the spherical harmonic approximation · Annals of Nuclear Energy · 2026 · 1 citations
• B-Weyl Operator Pencils: Structural Characterization, Perturbation Stability, and Examples · Asian-European Journal of Mathematics · 2026 · 0 citations
• Schrödinger operators with non-integer power-law potentials and Lie-Rinehart algebras · Analysis and Mathematical Physics · 2026 · 0 citations
• Non-perturbative Isospectrality for Quasi-periodic Gasymov-Type Potentials · Communications in Mathematical Physics · 2026 · 0 citations
• Asymptotic Analysis of the First Two Eigenvalues for Sturm-Liouville Problems with Two Parameters on the Jump Set with Applications · Results in Mathematics · 2026 · 0 citations
• Insights into a new model of matrix block 3 × 3 · Asian-European Journal of Mathematics · 2026 · 0 citations
• Ground states of Fermi systems with Hardy terms and periodic potentials · Physica Scripta · 2026 · 0 citations

Most recent work

• Asymptotic derivation of nonclassical SPN equations from the spherical harmonic approximation · Annals of Nuclear Energy · 2026
• B-Weyl Operator Pencils: Structural Characterization, Perturbation Stability, and Examples · Asian-European Journal of Mathematics · 2026
• Schrödinger operators with non-integer power-law potentials and Lie-Rinehart algebras · Analysis and Mathematical Physics · 2026
• Non-perturbative Isospectrality for Quasi-periodic Gasymov-Type Potentials · Communications in Mathematical Physics · 2026
• Asymptotic Analysis of the First Two Eigenvalues for Sturm-Liouville Problems with Two Parameters on the Jump Set with Applications · Results in Mathematics · 2026
• Insights into a new model of matrix block 3 × 3 · Asian-European Journal of Mathematics · 2026
• Ground states of Fermi systems with Hardy terms and periodic potentials · Physica Scripta · 2026
• Boundary conditions and Dirac fields on AdS <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mi>n</mml:mi> </mml:msub> </mml:math> · Physical Review D · 2026
• Resonances for the One‐Dimensional Schrödinger Operator With the Matrix‐Valued Complex Square‐Well Potential · Studies in Applied Mathematics · 2026
• The Rational Solenoid Quantum Graph and the Adelic Scaling Operator HA · Zenodo (CERN European Organization for Nuclear Research) · 2026

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