Quantum mechanics is arguably the most successful physical theory ever developed, providing a profound understanding of the microscopic world and influencing nearly every branch of physics, as well as many other scientific disciplines. The continued development of its mathematical and theoretical foundations remains essential for further advances in both fundamental science and emerging quantum technologies. Recognizing the extraordinary impact of quantum theory, the United Nations declared 2025 the International Year of Quantum Science and Technology. Despite being a centenary theory, quantum mechanics still poses numerous theoretical and mathematical problems, and remains a vibrant field of research.
The goal of this Research Topic is twofold: to stimulate the investigation of open mathematical and theoretical problems in quantum mechanics and to promote dialogue between different areas of theoretical and mathematical physics that share common tools and perspectives when applied to quantum systems. The articles in this Research Topic reflect several complementary directions of current research in quantum mechanics and illustrate the breadth of this endeavor. Broadly speaking, they address structural aspects of quantum theory through functional analysis and spectral theory, the role of point interactions and singular perturbations in quantum Hamiltonians, the analysis of relativistic quantum models, and applications of these ideas in condensed-matter systems and quantum field-theoretic dynamics. These contributions are briefly described below.
Focusing on structural aspects of quantum mechanics from the perspective of functional analysis and spectral theory, Trapani and Tschinke investigate Parseval distribution frames within the framework of rigged Hilbert spaces (RHS). The authors analyze several types of distribution frames—including Bessel, Parseval, Gelfand, and Riesz frames—and examine their associated operators and properties. Within this formalism, such frames provide a sort of generalized resolutions of the identity and lead to the construction of some kinds of closed operators on the Hilbert space underlying the RHS structure. Closely related subjects are explored in Erman and Turgut, where the completeness relation for the eigenstates of self-adjoint Hamiltonians possessing discrete spectra and modified by a -interaction in two and three dimensions is explicitly demonstrated. By constructing the Green function of the resulting singular Hamiltonian, the authors show that the completeness property remains valid even in the presence of such perturbations, thereby contributing to the relatively small set of quantum systems for which completeness relations can be rigorously established in explicit form.
Point interactions provide analytically tractable models in quantum mechanics and are generally viewed as ideal models approximating real short-range interactions. Fundamental spectral and scattering properties of quantum systems can be investigated in detail in such models. In this context, Golovaty and Hryniv examine the spectral properties of one-dimensional Schrödinger operators with -type perturbations. Studying a sequence of regularized Hamiltonians approximating these interactions, the authors show that although the operators converge in the norm-resolvent sense to a semi-bounded limit, the approximating sequence may nonetheless contain arbitrarily many negative eigenvalues diverging to minus infinity in the singular limit. This phenomenon reveals a subtle instability associated with interactions and highlights potential limitations in the use of exactly solvable models as approximations of short-range physical interactions.
Localized interactions also play a central role in the study of quantum scattering and transport. In Zolotaryuk and Zolotaryuk, the authors analyze a one-dimensional scattering model involving two spatially separated potentials, effectively representing particles crossing thin layers in higher-dimensional structures. By examining different potential profiles, the work investigates transmission properties and the possibility of tunneling for extremely low-energy particles. A closely related perspective arises in the relativistic setting studied in Bonin et al., where the one-dimensional Dirac equation with point interactions supported at two spatial points is investigated. Using a formulation in which the interaction term in the Dirac equation is given explicitly as a distribution, the authors investigate parity-symmetric interactions and characterize critical and supercritical states, bound states, confinement properties and scattering resonances for several models. Their results illustrate the rich structure that can arise in relativistic point-interaction systems even in one dimension.
Relativistic quantum models are further explored in Alrebdi et al., where the relativistic quantum behavior of spinless particles described by the Klein–Gordon oscillator interacting with a screened Kratzer potential is studied in the presence of an external magnetic field and a cosmic-string background. Using the extended Nikiforov–Uvarov method, the authors derive the corresponding energy spectrum and analyze how it depends on the quantum numbers and on the parameters associated with the topological defect and magnetic flux. A complementary relativistic oscillator model is considered in Tsutsui et al., where the Dirac oscillator is generalized to include a time-dependent frequency. Building on the well-known correspondence between the Dirac oscillator and the Jaynes–Cummings model, the authors analyse the system from a quantum-optical perspective and investigate how temporal modulations affect dynamical observables. The work reveals interesting connections between relativistic quantum mechanics and optical analog systems.
Applications of quantum-mechanical methods to condensed-matter systems are illustrated in Martínez et al., where a solvable two-band model is developed to study the electronic energy levels in disordered narrow-band semiconductor superlattices. The interaction between electrons and impurities is modeled using separable pseudopotentials, which allow closed expressions for the configurationally averaged Green function to be obtained within the coherent potential approximation, considered the best single-site scattering theory for calculating the average spectral properties of disordered systems.
Finally, quantum-mechanical spectral problems also arise in the study of topological excitations in quantum field-theoretic models. In Guilarte, the low-energy dynamics of vibrating kinks and kink–antikink configurations in the Jackiw–Rebbi model is investigated. This -dimensional quantum field theory describes bosons and fermions coupled through a Yukawa interaction. In this framework, a collective-coordinate effective model is derived and used to analyse in detail the dynamical behaviour of vibrating kinks and kink-antikink configurations.
By combining rigorous mathematical analysis with physically motivated models the works in this Research Topic demonstrate how diverse theoretical approaches can contribute to a deeper understanding of quantum systems. We hope that the articles in this Research Topic will stimulate further advances in the understanding the mathematical structure and the theoretical aspects of quantum theory and contribute to the ongoing development of quantum science and emerging quantum technologies.
Statements
Author contributions
MG: Writing – original draft, Writing – review and editing. JL: Writing – review and editing, Writing – original draft. LM: Writing – review and editing, Writing – original draft. LN: Writing – review and editing, Writing – original draft.
Funding
The author(s) declared that financial support was received for this work and/or its publication. LM thanks the NASA - MN Space Grant Consortium, and Concordia College's Office of Undergraduate Research, Scholarship, and Creative Activity for partial financial support. This research was partially supported by the Q-CAYLE project, funded by the European Union-Next Generation UE/MICIU/Plan de Recuperaci\'on, Transformaci\'on y Resiliencia/Junta de Castilla y Le\'on (PRTRC17.11), and also by projects PID2023-148409NB-I00, funded by MICIU/AEI/10. 13039/501100011033I. We also acknowledge the financial support of Castilla y Le\'on Department of Education and the FEDER Funds (CLU-2023-1-05).
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author JL declared that they were an editorial board member of Frontiers at the time of submission. This had no impact on the peer review process and the final decision.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Summary
Keywords
applications of quantum mechanics, foundations of quantum mechanics, mathematical methods in quantum mechanics, relativistic quantum models, singular Hamiltonians, spectral theory
Citation
Gadella M, Lunardi JT, Manzoni LA and Nieto LM (2026) Editorial: Recent mathematical and theoretical progress in quantum mechanics. Front. Phys. 14:1832046. doi: 10.3389/fphy.2026.1832046
Received
16 March 2026
Accepted
31 March 2026
Published
24 April 2026
Volume
14 - 2026
Edited and reviewed by
Alexandre M. Zagoskin, Loughborough University, United Kingdom
Updates
Copyright
© 2026 Gadella, Lunardi, Manzoni and Nieto.
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*Correspondence: Manuel Gadella, manuel.gadella1@gmail.com; José T. Lunardi, jttlunardi@uepg.br; Luiz A. Manzoni, manzoni@cord.edu; Luis M. Nieto, luismiguel.nieto.calzada@uva.es
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.