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Open research questions in Spectral Theory in Mathematical Physics

81 gap statements mined from Spectral Theory in Mathematical Physics papers in our 4.5M-paper local library, which holds 635 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The study of non-self-adjoint Schrödinger operators with matrix-valued potentials is less understood. Prior work has focused on self-adjoint Schrödinger operators, leaving a gap for the non-self-adjoint case. The paper aims to fill this gap by providing a detailed analysis of resonances for the square well.

    Resonances for the One‐Dimensional Schrödinger Operator With the Matrix‐Valued Complex Square‐Well Potential · 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 need for a generalisation of the ODE/IM correspondence to the case of a classical integrable equation depending on complex coefficients or moduli.

    From classical Lax ODEs to quantum integrable theories: the moduli · 2026

  • The 'fattened' matrix A2 characteristic is a generalization of the classical A2 characteristic that has not been previously studied.

    The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform · 2026

  • 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 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 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 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 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 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 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

  • One of the key open questions is the relation between the KW (self-)dualities constructed in this work and those obtained by gauging generalized symmetries.

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

  • The properties of statistically stationary solutions to the Schrödinger Map equation and the Binormal Curvature Flow are not well understood. Prior work has established the existence of these solutions, but their properties are not well understood.

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

  • We review here two different viewpoints on the Berry-Keating operator $H_{BK}$, whose connection to the Riemann hypothesis remains an intriguing and not yet fully understood question, despite considerable attention in the recent literature.

    On the Berry-Keating Operator · 2026

  • Further study of the decay properties of solutions is needed. The authors suggest investigating the decay estimates for the eigenvectors of the discrete Schrödinger operator.

    On the existence of Nehari ground states for the nonlinear Schrödinger equation on discrete graphs · 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

  • The study of nonlinear partial differential equations on discrete structures is an active area of research. The discrete case is less well understood than the continuous case.

    On the existence of Nehari ground states for the nonlinear Schrödinger equation on discrete graphs · 2026 · DOI

  • The paper identifies that the usual methods need a safety distance to the essential spectrum.

    Quantum Systems at The Brink: Helium–Type Systems · 2026 · DOI

  • Further study of the 'fattened' matrix A2 characteristic and its applications. Investigation of the relation between the matrix A2 characteristic and other operators. Development of new techniques for studying the Hilbert transform in weighted L2 spaces.

    The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform · 2026

  • 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…

    Low-energy eigenstates in a vanishing magnetic field · 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

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Open questions raised by papers listed on this page include: (1) Future research should aim to develop a more complete understanding of the properties of NG even-denominator states (“Theory of next-generation even-denominator states”, 2026); (2) Future research may involve experimental verification of the mathematical framework (“Representations of Josephson junction on the unit circle and the derivations of…”, 2026); (3) The lack of understanding of the asymptotic behavior of the heat semigroup on manifolds with corners (“Heat Kernel Expansion and Spectral Action on the Stratified $b$-Simplex…”, 2026); (4) The properties of statistically stationary solutions to the Schrödinger Map equation and the Binormal Curvature Flow are not well understood (“On Properties of Statistically Stationary Solutions to the One-Dimensional…”, 2026); (5) Further study of the decay properties of solutions is needed (“On the existence of Nehari ground states for the nonlinear Schrödinger equation…”, 2026). 81 gap statements have been mined from Spectral Theory in Mathematical Physics papers in our 4.5M-paper local library, which holds 635 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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